Subgroups — C₂×C₄×#13133263 #28290695

Order 13568.

12378 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133263	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133263	C₂	1	32	yes	maximal, direct product factor
3	C₂×#16811106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16811105	C₂	1	2	yes	maximal, complemented
5	C₂×#16811103	C₂	1	3	yes	maximal, complemented
6	C₂×#16811097	C₂	1	3	yes	maximal, complemented
7	C₂×#16811095	C₂	1	6	yes	maximal, complemented
8	C₂×C₄×#11719949	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×#11719945	C₂	1	3	yes	maximal, complemented
10	C₂²×C₄×D₂₁₂	C₂	1	3	yes	maximal, complemented
11	C₂×C₄×D₄×D₅₃	C₂	1	6	yes	maximal, complemented
12	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13133263	C₄	1	32	yes	direct product factor
14	C₂×#13133263	C₂²	1	16	yes	—
15	C₂²×#11719949	C₂²	1	2	yes	—
16	C₂²×#11719945	C₂²	1	3	yes	—
17	C₂³×D₂₁₂	C₂²	1	3	yes	—
18	C₂²×D₄×D₅₃	C₂²	1	6	yes	—
19	C₂³×C₄×D₅₃	C₂²	1	6	yes	complemented
20	#16811106	C₂²	1	16	yes	complemented
21	#16811105	C₂²	1	32	yes	complemented
22	#16811103	C₂²	1	48	yes	complemented
23	#16811097	C₂²	1	48	yes	complemented
24	#16811095	C₂²	1	96	yes	complemented
25	C₄×#11719949	C₂²	1	32	yes	complemented
26	C₄×#11719945	C₂²	1	48	yes	complemented
27	C₂×Q₈×Dic₅₃	C₂²	1	2	yes	complemented
28	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄×Dic₁₀₆	C₂²	1	3	yes	complemented
30	C₂×C₄×D₂₁₂	C₂²	1	6	yes	complemented
31	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
32	C₂×C₄×D₂₁₂	C₂²	1	6	yes	complemented
33	C₂×C₄×D₂₁₂	C₂²	1	6	yes	complemented
34	C₂×C₄×D₂₁₂	C₂²	1	3	yes	complemented
35	C₂×C₄×D₂₁₂	C₂²	1	48	yes	complemented
36	C₂×D₄×Dic₅₃	C₂²	1	6	yes	complemented
37	C₂×#13133179	C₂²	1	6	yes	complemented
38	C₂×#13133179	C₂²	1	6	yes	complemented
39	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
40	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	complemented
41	C₂×#13133177	C₂²	1	12	yes	complemented
42	C₄×D₄×D₅₃	C₂²	1	96	yes	complemented
43	C₂×#13133176	C₂²	1	3	yes	complemented
44	C₂×#13133175	C₂²	1	6	yes	complemented
45	C₂×#13133174	C₂²	1	6	yes	complemented
46	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	complemented
47	C₂×C₄⋊C₄×D₅₃	C₂²	1	6	yes	complemented
48	C₂²×#11719921	C₂²	1	6	yes	complemented
49	C₂×C₂²⋊C₄×D₅₃	C₂²	1	12	yes	complemented
50	C₂×C₄²×D₅₃	C₂²	1	6	yes	complemented
51	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
52	C₄×#41×C₅₃	C₂²	1	16	yes	complemented
53	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
54	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	complemented
56	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	complemented
57	C₂×#117×C₅₃	C₂²	1	3	yes	complemented
58	C₂²×C₄²×C₅₃	C₂²	1	3	yes	complemented
59	C₂⁴×D₅₃	C₂³	1	6	yes	—
60	#13133263	C₂×C₄	1	256	yes	direct product factor
61	C₂×#11719949	C₂×C₄	1	32	yes	complemented
62	C₂×#11719949	C₂³	1	16	yes	—
63	C₂×#11719945	C₂×C₄	1	48	yes	complemented
64	C₂×#11719945	C₂³	1	24	yes	—
65	C₂²×Dic₁₀₆	C₂³	1	1	yes	characteristic
66	C₂²×D₂₁₂	C₂×C₄	1	48	yes	complemented
67	C₂²×D₂₁₂	C₂³	1	6	yes	—
68	C₂²×D₂₁₂	C₂³	1	3	yes	—
69	C₂²×D₂₁₂	C₂³	1	24	yes	—
70	C₂²×C₂²⋊D₅₃	C₂³	1	6	yes	—
71	C₂×D₄×D₅₃	C₂×C₄	1	96	yes	complemented
72	C₂×D₄×D₅₃	C₂³	1	48	yes	—
73	C₂³×Dic₅₃	C₂³	1	6	yes	—
74	C₂²×C₄×D₅₃	—	4	24	no	—
75	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
76	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
77	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
78	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	48	yes	complemented
80	Q₈×Dic₅₃	C₂³	1	16	yes	complemented
81	C₄×Dic₁₀₆	C₂³	1	8	yes	complemented
82	C₄×Dic₁₀₆	C₂³	1	24	yes	complemented
83	C₄×D₂₁₂	C₂³	1	48	yes	complemented
84	C₄×D₂₁₂	C₂³	1	24	yes	complemented
85	C₄×D₂₁₂	C₂³	1	48	yes	complemented
86	C₄×D₂₁₂	C₂³	1	48	yes	complemented
87	C₄×D₂₁₂	C₂³	1	24	yes	complemented
88	D₄×Dic₅₃	C₂³	1	48	yes	complemented
89	#13133179	C₂³	1	48	yes	complemented
90	#13133179	C₂³	1	48	yes	complemented
91	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
92	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	complemented
93	#13133177	C₂³	1	96	yes	complemented
94	#13133176	C₂³	1	24	yes	complemented
95	#13133175	C₂³	1	48	yes	complemented
96	#13133174	C₂³	1	48	yes	complemented
97	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
98	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
99	C₂×C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
101	C₂×#11719919	C₂³	1	6	yes	—
102	C₂×#11719919	C₂³	1	2	yes	—
103	C₄⋊C₄×D₅₃	C₂³	1	48	yes	complemented
104	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
105	C₂×#11719921	C₂³	1	6	yes	—
106	C₂×#11719921	C₂³	1	12	yes	—
107	C₂×#11719921	C₂³	1	6	yes	—
108	C₂×#11719921	C₂³	1	48	yes	complemented
109	C₂²⋊C₄×D₅₃	C₂³	1	96	yes	complemented
110	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
111	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
112	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
113	C₄²×D₅₃	C₂³	1	48	yes	complemented
114	C₂×#41×C₅₃	C₂×C₄	1	16	yes	complemented
115	C₂×#41×C₅₃	C₂³	1	8	yes	—
116	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
117	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
118	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
119	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
120	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
121	C₄×Q₈×C₅₃	C₂³	1	8	yes	complemented
122	C₄×Q₈×C₅₃	C₂³	1	8	yes	complemented
123	C₄×D₄×C₅₃	C₂³	1	24	yes	complemented
124	C₄×D₄×C₅₃	C₂³	1	24	yes	complemented
125	#117×C₅₃	C₂³	1	24	yes	complemented
126	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
127	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
128	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
129	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
131	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
132	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
133	C₂×C₄²×C₅₃	C₂³	1	24	yes	complemented
134	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
135	C₂³×D₅₃	—	4	12	no	—
136	C₂³×D₅₃	—	4	12	no	—
137	C₂³×D₅₃	C₂²×C₄	1	48	yes	complemented
138	C₂³×D₅₃	C₂⁴	1	6	yes	—
139	C₂³×D₅₃	C₂⁴	1	24	yes	—
140	#41×C₅₃	C₂²×C₄	1	64	yes	complemented
141	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	complemented
142	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
143	C₂×D₄×C₅₃	C₂²×C₄	1	24	yes	complemented
144	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
145	C₂²×C₄×C₅₃	—	2	12	no	—
146	C₂²×C₄×C₅₃	—	4	6	no	—
147	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	complemented
148	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
150	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
157	C₂×C₂²⋊D₅₃	C₂²×C₄	1	48	yes	complemented
158	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
159	C₂²×Dic₅₃	—	4	12	no	—
160	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
161	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
162	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
163	C₂²×Dic₅₃	C₂⁴	1	2	yes	—
164	#11719949	C₂²×C₄	1	128	yes	complemented
165	D₄×D₅₃	C₂²×C₄	1	384	yes	complemented
166	#11719945	C₂²×C₄	1	192	yes	complemented
167	C₂×D₂₁₂	C₂²×C₄	1	48	yes	complemented
168	C₂×D₂₁₂	C₂²×C₄	1	24	yes	complemented
169	C₂×D₂₁₂	C₂²×C₄	1	192	yes	complemented
170	C₂×D₂₁₂	C₂⁴	1	24	yes	—
171	C₂×D₂₁₂	C₂⁴	1	12	yes	—
172	C₂×C₄×D₅₃	—	2	24	no	—
173	C₂×C₄×D₅₃	—	4	96	no	—
174	C₂×C₄×D₅₃	—	4	24	no	—
175	C₂×C₄×D₅₃	—	4	12	no	—
176	C₂×C₄×D₅₃	C₂²×C₄	1	48	yes	complemented
177	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
178	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
179	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
180	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
181	C₂×Dic₁₀₆	C₂²×C₄	1	8	yes	complemented
182	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
184	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
185	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
186	C₄²×C₅₃	C₂⁴	1	4	yes	—
187	C₄²×C₅₃	C₂⁴	1	12	yes	—
188	C₄²×C₅₃	C₂⁴	1	12	yes	—
189	C₄²×C₅₃	C₂⁴	1	12	yes	—
190	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
191	#11719921	C₂⁴	1	24	yes	—
192	#11719921	C₂⁴	1	48	yes	—
193	#11719921	C₂⁴	1	24	yes	—
194	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
195	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
196	C₄⋊Dic₅₃	C₂⁴	1	4	yes	—
197	#11719919	C₂⁴	1	24	yes	—
198	#11719919	C₂⁴	1	8	yes	—
199	C₄×Dic₅₃	C₂⁴	1	8	yes	—
200	C₄×Dic₅₃	C₂⁴	1	24	yes	—
201	C₄×Dic₅₃	C₂⁴	1	24	yes	—
202	C₂³×C₅₃	—	2	6	no	—
203	C₂³×C₅₃	—	2	6	no	—
204	C₂³×C₅₃	C₂³×C₄	1	12	yes	—
205	C₂³×C₅₃	C₂⁵	1	6	yes	—
206	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
207	C₂²×D₅₃	—	2	12	no	—
208	C₂²×D₅₃	—	2	12	no	—
209	C₂²×D₅₃	—	4	24	no	—
210	C₂²×D₅₃	—	4	96	no	—
211	C₂²×D₅₃	—	4	24	no	—
212	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
213	C₂²×D₅₃	C₂³×C₄	1	96	yes	complemented
214	C₂²×D₅₃	C₂⁵	1	12	yes	—
215	Q₈×C₅₃	C₂³×C₄	1	16	yes	complemented
216	D₄×C₅₃	C₂³×C₄	1	48	yes	complemented
217	C₂×C₄×C₅₃	—	2	6	no	—
218	C₂×C₄×C₅₃	—	2	6	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	2	24	no	—
221	C₂×C₄×C₅₃	—	4	12	no	—
222	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
223	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
224	C₂×C₄×C₅₃	C₂³×C₄	1	48	yes	complemented
225	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	complemented
226	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
227	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
228	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
229	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
230	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
231	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
232	C₂²⋊D₅₃	C₂³×C₄	1	96	yes	complemented
233	C₂×Dic₅₃	—	2	12	no	—
234	C₂×Dic₅₃	—	2	4	no	—
235	C₂×Dic₅₃	—	4	24	no	—
236	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
237	C₂×Dic₅₃	C₂⁵	1	12	yes	—
238	C₂×Dic₅₃	C₂⁵	1	4	yes	—
239	C₂×Dic₅₃	C₂⁵	1	4	yes	—
240	D₂₁₂	C₂³×C₄	1	48	yes	complemented
241	D₂₁₂	C₂³×C₄	1	96	yes	complemented
242	C₄×D₅₃	—	2	48	no	—
243	C₄×D₅₃	—	4	48	no	—
244	C₄×D₅₃	—	4	24	no	—
245	C₄×D₅₃	C₂³×C₄	1	96	yes	complemented
246	Dic₁₀₆	C₂³×C₄	1	16	yes	complemented
247	C₂×C₄×H(5,𝔽₂)	—	53	1	no	maximal, Sylow
248	C₂²×C₅₃	—	2	6	no	—
249	C₂²×C₅₃	—	2	6	no	—
250	C₂²×C₅₃	—	2	24	no	—
251	C₂²×C₅₃	C₂⁴×C₄	1	12	yes	—
252	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
253	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
254	C₂²×C₅₃	#585	1	2	yes	—
255	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
256	C₂×D₅₃	—	2	12	no	—
257	C₂×D₅₃	—	2	12	no	—
258	C₂×D₅₃	—	2	48	no	—
259	C₂×D₅₃	—	4	96	no	—
260	C₂×D₅₃	C₂⁴×C₄	1	24	yes	—
261	C₄×C₅₃	—	2	6	no	—
262	C₄×C₅₃	—	2	6	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	C₂⁴×C₄	1	4	yes	—
265	C₄×C₅₃	C₂⁴×C₄	1	12	yes	—
266	C₄×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	complemented
267	Dic₅₃	—	2	12	no	—
268	Dic₅₃	—	2	4	no	—
269	Dic₅₃	C₂⁴×C₄	1	8	yes	—
270	C₂²×H(5,𝔽₂)	—	53	1	no	—
271	C₄×H(5,𝔽₂)	—	53	32	no	—
272	C₂×#520	—	53	1	no	—
273	C₂×#520	—	53	2	no	—
274	C₂×#520	—	53	3	no	—
275	C₂×#518	—	53	6	no	—
276	C₂×#518	—	53	3	no	—
277	C₂×C₄×#41	—	53	3	no	—
278	C₂×C₄×#41	—	53	2	no	—
279	C₂×C₄×#41	D₅₃	1	1	yes	characteristic, complemented
280	C₂²×C₄×D₄	—	53	3	no	—
281	C₂²×C₄×D₄	—	53	6	no	—
282	C₂×C₅₃	—	2	12	no	—
283	C₂×C₅₃	C₄×H(5,𝔽₂)	1	4	yes	complemented
284	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
285	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₅₃	C₂×#585	1	1	yes	characteristic
287	D₅₃	—	2	24	no	—
288	C₂×H(5,𝔽₂)	—	53	32	no	—
289	C₂×H(5,𝔽₂)	—	53	16	no	—
290	C₂²×#41	—	53	2	no	—
291	C₂²×#41	—	53	3	no	—
292	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
293	C₂³×D₄	—	53	6	no	—
294	C₂³×D₄	—	53	3	no	—
295	C₂⁴×C₄	—	53	6	no	—
296	#520	—	53	16	no	—
297	#520	—	53	32	no	—
298	#520	—	53	48	no	—
299	#518	—	53	96	no	—
300	#518	—	53	48	no	—
301	C₄×#41	—	53	32	no	—
302	C₄×#41	—	53	48	no	—
303	C₄×#41	C₂×D₅₃	1	16	yes	complemented
304	C₂×C₄×Q₈	—	53	2	no	—
305	C₂×C₄×Q₈	—	53	1	no	—
306	C₂×C₄×Q₈	—	53	3	no	—
307	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
308	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
309	C₂×C₄×D₄	—	53	6	no	—
310	C₂×C₄×D₄	—	53	6	no	—
311	C₂×C₄×D₄	—	53	6	no	—
312	C₂×C₄×D₄	—	53	96	no	—
313	C₂×C₄×D₄	—	53	6	no	—
314	C₂×C₄×D₄	—	53	6	no	—
315	C₂×C₄×D₄	—	53	12	no	—
316	C₂×C₄×D₄	—	53	3	no	—
317	C₂×C₄×D₄	—	53	6	no	—
318	C₂×C₄×D₄	—	53	6	no	—
319	C₂×C₄×D₄	—	53	6	no	—
320	C₂×C₄×D₄	—	53	3	no	—
321	C₂×C₄×D₄	—	53	48	no	—
322	C₂×C₄×D₄	C₂×D₅₃	1	3	yes	complemented
323	C₂×C₄×D₄	C₂×D₅₃	1	3	yes	complemented
324	C₂×#117	—	53	3	no	—
325	C₂×#117	—	53	6	no	—
326	C₂×#117	—	53	6	no	—
327	C₂×#117	C₂×D₅₃	1	3	yes	complemented
328	C₂²×C₄⋊C₄	—	53	3	no	—
329	C₂²×C₄⋊C₄	—	53	6	no	—
330	C₂²×C₂²⋊C₄	—	53	6	no	—
331	C₂²×C₂²⋊C₄	—	53	12	no	—
332	C₂²×C₄²	—	53	6	no	—
333	C₂²×C₄²	C₂×D₅₃	1	3	yes	complemented
334	C₅₃	C₂×C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
335	C₂⁵	—	53	6	no	—
336	H(5,𝔽₂)	—	53	256	no	—
337	C₂×#41	—	53	32	no	—
338	C₂×#41	—	53	16	no	—
339	C₂×#41	—	53	48	no	—
340	C₂×#41	—	53	24	no	—
341	C₂×#41	C₄×D₅₃	1	16	yes	complemented
342	C₂×#41	C₂²×D₅₃	1	8	yes	—
343	C₂²×Q₈	—	53	1	no	—
344	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
345	C₂²×D₄	—	53	6	no	—
346	C₂²×D₄	—	53	96	no	—
347	C₂²×D₄	—	53	48	no	—
348	C₂²×D₄	—	53	48	no	—
349	C₂²×D₄	—	53	6	no	—
350	C₂²×D₄	—	53	3	no	—
351	C₂²×D₄	—	53	24	no	—
352	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
353	C₂³×C₄	—	53	6	no	—
354	C₂³×C₄	—	53	6	no	—
355	C₂³×C₄	—	53	6	no	—
356	C₂³×C₄	—	53	6	no	—
357	C₂³×C₄	—	53	12	no	—
358	C₂³×C₄	—	53	48	no	—
359	C₂³×C₄	—	212	24	no	—
360	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
361	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
362	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
363	C₄×Q₈	—	53	16	no	—
364	C₄×Q₈	—	53	8	no	—
365	C₄×Q₈	—	53	24	no	—
366	C₄×Q₈	C₂²×D₅₃	1	8	yes	complemented
367	C₄×Q₈	C₂²×D₅₃	1	8	yes	complemented
368	C₄×D₄	—	53	48	no	—
369	C₄×D₄	—	53	48	no	—
370	C₄×D₄	—	53	48	no	—
371	C₄×D₄	—	53	48	no	—
372	C₄×D₄	—	53	48	no	—
373	C₄×D₄	—	53	96	no	—
374	C₄×D₄	—	53	48	no	—
375	C₄×D₄	—	53	24	no	—
376	C₄×D₄	—	53	48	no	—
377	C₄×D₄	—	53	48	no	—
378	C₄×D₄	—	53	24	no	—
379	C₄×D₄	C₂²×D₅₃	1	24	yes	complemented
380	C₄×D₄	C₂²×D₅₃	1	24	yes	complemented
381	#117	—	53	24	no	—
382	#117	—	53	48	no	—
383	#117	—	53	48	no	—
384	#117	C₂²×D₅₃	1	24	yes	complemented
385	C₂×C₄⋊C₄	—	53	6	no	—
386	C₂×C₄⋊C₄	—	53	6	no	—
387	C₂×C₄⋊C₄	—	53	3	no	—
388	C₂×C₄⋊C₄	—	53	24	no	—
389	C₂×C₄⋊C₄	—	53	2	no	—
390	C₂×C₄⋊C₄	—	53	1	no	—
391	C₂×C₄⋊C₄	—	53	48	no	—
392	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
393	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
394	C₂×C₂²⋊C₄	—	53	6	no	—
395	C₂×C₂²⋊C₄	—	53	6	no	—
396	C₂×C₂²⋊C₄	—	53	12	no	—
397	C₂×C₂²⋊C₄	—	53	6	no	—
398	C₂×C₂²⋊C₄	—	53	48	no	—
399	C₂×C₂²⋊C₄	—	53	96	no	—
400	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
401	C₂×C₄²	—	53	6	no	—
402	C₂×C₄²	—	53	6	no	—
403	C₂×C₄²	—	53	2	no	—
404	C₂×C₄²	—	53	48	no	—
405	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
406	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
407	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
408	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
409	C₂×C₄²	C₂²×D₅₃	1	24	yes	complemented
410	C₂⁴	—	53	6	no	—
411	C₂⁴	—	53	48	no	—
412	C₂⁴	—	53	24	no	—
413	C₂⁴	—	212	12	no	—
414	C₂⁴	—	212	12	no	—
415	C₂⁴	C₂³×D₅₃	1	3	yes	—
416	#41	—	53	128	no	—
417	#41	—	53	192	no	—
418	#41	C₂×C₄×D₅₃	1	64	yes	complemented
419	C₂×Q₈	—	53	8	no	—
420	C₂×Q₈	—	53	4	no	—
421	C₂×Q₈	C₂×C₄×D₅₃	1	8	yes	complemented
422	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
423	C₂×D₄	—	53	48	no	—
424	C₂×D₄	—	53	24	no	—
425	C₂×D₄	—	53	384	no	—
426	C₂×D₄	—	53	48	no	—
427	C₂×D₄	—	53	24	no	—
428	C₂×D₄	—	53	192	no	—
429	C₂×D₄	—	53	24	no	—
430	C₂×D₄	—	53	12	no	—
431	C₂×D₄	C₂×C₄×D₅₃	1	24	yes	complemented
432	C₂×D₄	C₂³×D₅₃	1	12	yes	—
433	C₂²×C₄	—	2	12	no	—
434	C₂²×C₄	—	4	6	no	—
435	C₂²×C₄	—	53	2	no	—
436	C₂²×C₄	—	53	48	no	—
437	C₂²×C₄	—	53	24	no	—
438	C₂²×C₄	—	53	2	no	—
439	C₂²×C₄	—	53	6	no	—
440	C₂²×C₄	—	53	24	no	—
441	C₂²×C₄	—	53	24	no	—
442	C₂²×C₄	—	53	24	no	—
443	C₂²×C₄	—	53	48	no	—
444	C₂²×C₄	—	106	24	no	—
445	C₂²×C₄	—	212	96	no	—
446	C₂²×C₄	—	212	12	no	—
447	C₂²×C₄	—	212	24	no	—
448	C₂²×C₄	—	212	12	no	—
449	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	complemented
450	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
451	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
453	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
454	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
455	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
456	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
457	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
458	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
459	C₄⋊C₄	—	53	24	no	—
460	C₄⋊C₄	—	53	24	no	—
461	C₄⋊C₄	—	53	12	no	—
462	C₄⋊C₄	—	53	8	no	—
463	C₄⋊C₄	—	53	4	no	—
464	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
465	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
466	C₂²⋊C₄	—	53	24	no	—
467	C₂²⋊C₄	—	53	24	no	—
468	C₂²⋊C₄	—	53	48	no	—
469	C₂²⋊C₄	—	53	24	no	—
470	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
471	C₄²	—	53	8	no	—
472	C₄²	—	53	24	no	—
473	C₄²	—	53	24	no	—
474	C₄²	C₂³×D₅₃	1	12	yes	—
475	C₄²	C₂³×D₅₃	1	4	yes	—
476	C₄²	C₂³×D₅₃	1	12	yes	—
477	C₄²	C₂³×D₅₃	1	12	yes	—
478	C₂³	—	2	6	no	—
479	C₂³	—	2	6	no	—
480	C₂³	—	53	24	no	—
481	C₂³	—	53	12	no	—
482	C₂³	—	53	96	no	—
483	C₂³	—	106	12	no	—
484	C₂³	—	106	12	no	—
485	C₂³	—	212	24	no	—
486	C₂³	—	212	96	no	—
487	C₂³	—	212	24	no	—
488	C₂³	C₂²×C₄×D₅₃	1	12	yes	—
489	C₂³	C₂⁴×D₅₃	1	6	yes	—
490	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
491	Q₈	—	53	16	no	—
492	Q₈	C₂²×C₄×D₅₃	1	16	yes	complemented
493	D₄	—	53	96	no	—
494	D₄	—	53	96	no	—
495	D₄	—	53	48	no	—
496	D₄	C₂²×C₄×D₅₃	1	48	yes	complemented
497	C₂×C₄	—	2	6	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	6	no	—
500	C₂×C₄	—	2	24	no	—
501	C₂×C₄	—	4	12	no	—
502	C₂×C₄	—	53	8	no	—
503	C₂×C₄	—	53	4	no	—
504	C₂×C₄	—	53	96	no	—
505	C₂×C₄	—	53	4	no	—
506	C₂×C₄	—	53	12	no	—
507	C₂×C₄	—	106	12	no	—
508	C₂×C₄	—	106	4	no	—
509	C₂×C₄	—	106	48	no	—
510	C₂×C₄	—	212	24	no	—
511	C₂×C₄	—	212	48	no	—
512	C₂×C₄	—	212	24	no	—
513	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
514	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
515	C₂×C₄	C₂²×C₄×D₅₃	1	48	yes	complemented
516	C₂×C₄	#13133263	1	4	yes	direct product factor
517	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
518	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
519	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
520	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
521	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
522	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
523	C₂²	—	2	24	no	—
524	C₂²	—	2	6	no	—
525	C₂²	—	2	6	no	—
526	C₂²	—	53	24	no	—
527	C₂²	—	106	48	no	—
528	C₂²	—	106	12	no	—
529	C₂²	—	106	12	no	—
530	C₂²	—	212	96	no	—
531	C₂²	C₂³×C₄×D₅₃	1	12	yes	—
532	C₂²	C₂³×C₄×D₅₃	1	2	yes	—
533	C₂²	C₂×#13133263	1	2	yes	—
534	C₂²	#16811459	1	2	yes	—
535	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
536	C₄	—	2	6	no	—
537	C₄	—	2	2	no	—
538	C₄	—	2	6	no	—
539	C₄	—	53	8	no	—
540	C₄	—	106	12	no	—
541	C₄	—	106	4	no	—
542	C₄	C₂³×C₄×D₅₃	1	12	yes	—
543	C₄	C₂³×C₄×D₅₃	1	4	yes	—
544	C₄	C₂×#13133263	1	4	yes	direct product factor
545	C₂	—	2	12	no	—
546	C₂	—	106	24	no	—
547	C₂	C₄×#13133263	1	4	yes	direct product factor
548	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
549	C₂	C₂²×#13133263	1	1	yes	characteristic
550	C₂	C₂×#16811459	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₄×#41×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#13133263	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂×C₄×#41	D₅₃
hypocenter	C₅₃	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#13133263	C₂	direct (N × H)
3. C₂×#16811106	C₂	semidirect
4. C₂×#16811105	C₂	semidirect
5. C₂×#16811103	C₂	semidirect
6. C₂×#16811097	C₂	semidirect
7. C₂×#16811095	C₂	semidirect
8. C₂×C₄×#11719949	C₂	semidirect
9. C₂×C₄×#11719945	C₂	semidirect
10. C₂²×C₄×D₂₁₂	C₂	semidirect
11. C₂×C₄×D₄×D₅₃	C₂	semidirect
12. C₂×C₄×#41×C₅₃	C₂	semidirect
13. C₂×#13133263	C₄	direct (N × H)
19. C₂³×C₄×D₅₃	C₂²	semidirect
20. #16811106	C₂²	semidirect
21. #16811105	C₂²	semidirect
22. #16811103	C₂²	semidirect
23. #16811097	C₂²	semidirect
24. #16811095	C₂²	semidirect
25. C₄×#11719949	C₂²	semidirect
26. C₄×#11719945	C₂²	semidirect
27. C₂×Q₈×Dic₅₃	C₂²	semidirect
28. C₂×C₄×Dic₁₀₆	C₂²	semidirect
29. C₂×C₄×Dic₁₀₆	C₂²	semidirect
30. C₂×C₄×D₂₁₂	C₂²	semidirect
31. C₂×C₄×D₂₁₂	C₂²	semidirect
32. C₂×C₄×D₂₁₂	C₂²	semidirect
33. C₂×C₄×D₂₁₂	C₂²	semidirect
34. C₂×C₄×D₂₁₂	C₂²	semidirect
35. C₂×C₄×D₂₁₂	C₂²	semidirect
36. C₂×D₄×Dic₅₃	C₂²	semidirect
37. C₂×#13133179	C₂²	semidirect
38. C₂×#13133179	C₂²	semidirect
39. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
40. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
41. C₂×#13133177	C₂²	semidirect
42. C₄×D₄×D₅₃	C₂²	semidirect
43. C₂×#13133176	C₂²	semidirect
44. C₂×#13133175	C₂²	semidirect
45. C₂×#13133174	C₂²	semidirect
46. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
47. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
48. C₂²×#11719921	C₂²	semidirect
49. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
50. C₂×C₄²×D₅₃	C₂²	semidirect
52. C₄×#41×C₅₃	C₂²	semidirect
53. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
54. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
55. C₂×C₄×D₄×C₅₃	C₂²	semidirect
56. C₂×C₄×D₄×C₅₃	C₂²	semidirect
57. C₂×#117×C₅₃	C₂²	semidirect
58. C₂²×C₄²×C₅₃	C₂²	semidirect
60. #13133263	C₂×C₄	direct (N × H)
61. C₂×#11719949	C₂×C₄	semidirect
63. C₂×#11719945	C₂×C₄	semidirect
66. C₂²×D₂₁₂	C₂×C₄	semidirect
71. C₂×D₄×D₅₃	C₂×C₄	semidirect
79. C₂²×C₄×D₅₃	C₂³	semidirect
80. Q₈×Dic₅₃	C₂³	semidirect
81. C₄×Dic₁₀₆	C₂³	semidirect
82. C₄×Dic₁₀₆	C₂³	semidirect
83. C₄×D₂₁₂	C₂³	semidirect
84. C₄×D₂₁₂	C₂³	semidirect
85. C₄×D₂₁₂	C₂³	semidirect
86. C₄×D₂₁₂	C₂³	semidirect
87. C₄×D₂₁₂	C₂³	semidirect
88. D₄×Dic₅₃	C₂³	semidirect
89. #13133179	C₂³	semidirect
90. #13133179	C₂³	semidirect
91. C₄×C₂²⋊D₅₃	C₂³	semidirect
92. C₄×C₂²⋊D₅₃	C₂³	semidirect
93. #13133177	C₂³	semidirect
94. #13133176	C₂³	semidirect
95. #13133175	C₂³	semidirect
96. #13133174	C₂³	semidirect
100. C₂×C₄⋊Dic₅₃	C₂³	semidirect
103. C₄⋊C₄×D₅₃	C₂³	semidirect
108. C₂×#11719921	C₂³	semidirect
109. C₂²⋊C₄×D₅₃	C₂³	semidirect
113. C₄²×D₅₃	C₂³	semidirect
114. C₂×#41×C₅₃	C₂×C₄	semidirect
121. C₄×Q₈×C₅₃	C₂³	semidirect
122. C₄×Q₈×C₅₃	C₂³	semidirect
123. C₄×D₄×C₅₃	C₂³	semidirect
124. C₄×D₄×C₅₃	C₂³	semidirect
125. #117×C₅₃	C₂³	semidirect
133. C₂×C₄²×C₅₃	C₂³	semidirect
137. C₂³×D₅₃	C₂²×C₄	semidirect
140. #41×C₅₃	C₂²×C₄	semidirect
141. C₂×Q₈×C₅₃	C₂²×C₄	semidirect
143. C₂×D₄×C₅₃	C₂²×C₄	semidirect
147. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
157. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
164. #11719949	C₂²×C₄	semidirect
165. D₄×D₅₃	C₂²×C₄	semidirect
166. #11719945	C₂²×C₄	semidirect
167. C₂×D₂₁₂	C₂²×C₄	semidirect
168. C₂×D₂₁₂	C₂²×C₄	semidirect
169. C₂×D₂₁₂	C₂²×C₄	semidirect
176. C₂×C₄×D₅₃	C₂²×C₄	semidirect
181. C₂×Dic₁₀₆	C₂²×C₄	semidirect
213. C₂²×D₅₃	C₂³×C₄	semidirect
215. Q₈×C₅₃	C₂³×C₄	semidirect
216. D₄×C₅₃	C₂³×C₄	semidirect
224. C₂×C₄×C₅₃	C₂³×C₄	semidirect
225. C₂×C₄×C₅₃	H(5,𝔽₂)	semidirect
232. C₂²⋊D₅₃	C₂³×C₄	semidirect
240. D₂₁₂	C₂³×C₄	semidirect
241. D₂₁₂	C₂³×C₄	semidirect
245. C₄×D₅₃	C₂³×C₄	semidirect
246. Dic₁₀₆	C₂³×C₄	semidirect
266. C₄×C₅₃	C₂×H(5,𝔽₂)	semidirect
279. C₂×C₄×#41	D₅₃	semidirect
283. C₂×C₅₃	C₄×H(5,𝔽₂)	semidirect
303. C₄×#41	C₂×D₅₃	semidirect
307. C₂×C₄×Q₈	C₂×D₅₃	semidirect
308. C₂×C₄×Q₈	C₂×D₅₃	semidirect
322. C₂×C₄×D₄	C₂×D₅₃	semidirect
323. C₂×C₄×D₄	C₂×D₅₃	semidirect
327. C₂×#117	C₂×D₅₃	semidirect
333. C₂²×C₄²	C₂×D₅₃	semidirect
334. C₅₃	C₂×C₄×H(5,𝔽₂)	semidirect
341. C₂×#41	C₄×D₅₃	semidirect
366. C₄×Q₈	C₂²×D₅₃	semidirect
367. C₄×Q₈	C₂²×D₅₃	semidirect
379. C₄×D₄	C₂²×D₅₃	semidirect
380. C₄×D₄	C₂²×D₅₃	semidirect
384. #117	C₂²×D₅₃	semidirect
409. C₂×C₄²	C₂²×D₅₃	semidirect
418. #41	C₂×C₄×D₅₃	semidirect
421. C₂×Q₈	C₂×C₄×D₅₃	semidirect
431. C₂×D₄	C₂×C₄×D₅₃	semidirect
449. C₂²×C₄	C₂×C₄×D₅₃	semidirect
492. Q₈	C₂²×C₄×D₅₃	semidirect
496. D₄	C₂²×C₄×D₅₃	semidirect
515. C₂×C₄	C₂²×C₄×D₅₃	semidirect
516. C₂×C₄	#13133263	direct (N × H)
544. C₄	C₂×#13133263	direct (N × H)
547. C₂	C₄×#13133263	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13133263	2
C₄×#13133263	2
C₂×#16811106	2
C₂×#16811105	2
C₂×#16811103	2
C₂×#16811097	2
C₂×#16811095	2
C₂×C₄×#11719949	2
C₂×C₄×#11719945	2
C₂²×C₄×D₂₁₂	2
C₂×C₄×D₄×D₅₃	2
C₂×C₄×#41×C₅₃	2
C₂×C₄×H(5,𝔽₂)	53

Subgroups — C2×C4×#13133263 #28290695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133263 #28290695