Subgroups — C₂²×C₄×#11719947 #28290664

Order 13568.

12878 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719947	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719947	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133182	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133177	C₂	1	2	yes	maximal, complemented
8	C₂²×#13133175	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13133173	C₂	1	2	yes	maximal, complemented
10	C₂³×C₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#11719947	C₄	1	32	yes	direct product factor
14	C₂²×#11719947	C₂²	1	24	yes	—
15	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
17	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
18	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
19	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
22	C₄×#11719947	C₂²	1	256	yes	direct product factor
23	C₂×C₄×Dic₁₀₆	C₂²	1	24	yes	complemented
24	C₂×#13133182	C₂²	1	24	yes	complemented
25	C₂×D₄×Dic₅₃	C₂²	1	24	yes	complemented
26	C₂×C₄×C₂²⋊D₅₃	C₂²	1	48	yes	complemented
27	C₂×#13133177	C₂²	1	48	yes	complemented
28	C₂×#13133175	C₂²	1	24	yes	complemented
29	C₂×#13133173	C₂²	1	48	yes	complemented
30	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂²×#11719919	C₂²	1	2	yes	—
32	C₂²×#11719919	C₂²	1	2	yes	—
33	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
34	C₂²×#11719921	C₂²	1	2	yes	—
35	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
38	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
39	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
40	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
41	C₂²×C₄×Dic₅₃	C₂²	1	48	yes	complemented
42	C₂×C₄²×D₅₃	C₂²	1	24	yes	complemented
43	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
45	C₂×C₄×D₄×C₅₃	C₂²	1	24	yes	complemented
46	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
48	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
49	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
50	C₂×#11719947	C₂×C₄	1	384	yes	direct product factor
51	C₂×#11719947	C₂³	1	64	yes	—
52	C₂²×Dic₁₀₆	C₂×C₄	1	16	yes	complemented
53	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
54	C₂²×C₂²⋊D₅₃	C₂×C₄	1	32	yes	complemented
55	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
56	C₂³×Dic₅₃	—	2	16	no	—
57	C₂³×Dic₅₃	C₂×C₄	1	32	yes	complemented
58	C₂³×Dic₅₃	C₂³	1	2	yes	—
59	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
60	C₂³×Dic₅₃	C₂³	1	24	yes	—
61	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₂³×Dic₅₃	C₂³	1	2	yes	—
63	C₂³×Dic₅₃	C₂³	1	24	yes	—
64	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₂²×C₄×D₅₃	—	2	8	no	—
66	C₂²×C₄×D₅₃	—	2	8	no	—
67	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	complemented
68	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
69	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
70	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
71	C₄×Dic₁₀₆	C₂³	1	64	yes	complemented
72	#13133182	C₂³	1	64	yes	complemented
73	D₄×Dic₅₃	C₂³	1	64	yes	complemented
74	C₄×C₂²⋊D₅₃	C₂³	1	128	yes	complemented
75	#13133177	C₂³	1	128	yes	complemented
76	#13133175	C₂³	1	64	yes	complemented
77	#13133173	C₂³	1	128	yes	complemented
78	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
79	C₂×#11719919	C₂³	1	24	yes	—
80	C₂×#11719919	C₂³	1	24	yes	—
81	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
82	C₂×#11719921	C₂³	1	24	yes	—
83	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
84	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
85	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
86	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
87	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
88	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
89	C₂×C₄×Dic₅₃	C₂³	1	128	yes	complemented
90	C₄²×D₅₃	C₂³	1	64	yes	complemented
91	C₂⁵×C₅₃	C₂³	1	2	yes	—
92	C₂²×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
93	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
94	C₂³×C₄×C₅₃	—	2	16	no	—
95	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
96	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
97	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
98	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₅₃	C₂³	1	64	yes	complemented
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
103	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
104	C₂⁴×C₅₃	—	2	8	no	—
105	C₂⁴×C₅₃	—	2	8	no	—
106	C₂⁴×C₅₃	C₂²×C₄	1	16	yes	—
107	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
108	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
109	C₂³×D₅₃	—	2	4	no	—
110	C₂³×D₅₃	—	2	4	no	—
111	C₂³×D₅₃	C₂²×C₄	1	8	yes	—
112	C₂³×D₅₃	C₂⁴	1	6	yes	—
113	C₂×D₄×C₅₃	C₂²×C₄	1	96	yes	complemented
114	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
115	C₂²×C₄×C₅₃	—	2	8	no	—
116	C₂²×C₄×C₅₃	—	2	4	no	—
117	C₂²×C₄×C₅₃	—	2	96	no	—
118	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
119	C₂²×C₄×C₅₃	#41	1	8	yes	complemented
120	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
125	C₂×C₂²⋊D₅₃	C₂²×C₄	1	192	yes	complemented
126	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
127	C₂²×Dic₅₃	—	2	4	no	—
128	C₂²×Dic₅₃	—	2	8	no	—
129	C₂²×Dic₅₃	—	2	96	no	—
130	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
131	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
132	C₂²×Dic₅₃	C₂²×C₄	1	192	yes	complemented
133	C₂²×Dic₅₃	#41	1	8	yes	complemented
134	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
135	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
136	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
137	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
138	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
139	C₂²×Dic₅₃	C₂⁴	1	32	yes	—
140	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
141	#11719947	C₂²×C₄	1	512	yes	direct product factor
142	C₂×C₄×D₅₃	—	2	48	no	—
143	C₂×C₄×D₅₃	—	2	48	no	—
144	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	complemented
145	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
146	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
147	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
148	C₂×Dic₁₀₆	C₂²×C₄	1	96	yes	complemented
149	C₂×Dic₁₀₆	C₂⁴	1	16	yes	—
150	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	—
151	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
152	C₄²×C₅₃	C₂⁴	1	16	yes	—
153	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
154	#11719921	C₂⁴	1	32	yes	—
155	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
156	#11719919	C₂⁴	1	32	yes	—
157	#11719919	C₂⁴	1	32	yes	—
158	C₄×Dic₅₃	C₂⁴	1	16	yes	—
159	C₄×Dic₅₃	C₂⁴	1	16	yes	—
160	C₄×Dic₅₃	C₂⁴	1	16	yes	—
161	C₄×Dic₅₃	C₂⁴	1	32	yes	—
162	C₄×Dic₅₃	C₂⁴	1	32	yes	—
163	C₄×Dic₅₃	C₂⁴	1	32	yes	—
164	C₂³×C₅₃	—	2	64	no	—
165	C₂³×C₅₃	—	2	24	no	—
166	C₂³×C₅₃	—	2	24	no	—
167	C₂³×C₅₃	C₂³×C₄	1	48	yes	—
168	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
169	C₂³×C₅₃	C₂×#41	1	4	yes	—
170	C₂³×C₅₃	C₂×#41	1	4	yes	—
171	C₂³×C₅₃	C₂⁵	1	8	yes	—
172	C₂³×C₅₃	C₂⁵	1	3	yes	—
173	C₂²×D₅₃	—	2	32	no	—
174	C₂²×D₅₃	—	2	12	no	—
175	C₂²×D₅₃	—	2	12	no	—
176	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
177	C₂²×D₅₃	C₂⁵	1	4	yes	—
178	D₄×C₅₃	C₂³×C₄	1	64	yes	complemented
179	C₂×C₄×C₅₃	—	2	12	no	—
180	C₂×C₄×C₅₃	—	2	24	no	—
181	C₂×C₄×C₅₃	—	2	64	no	—
182	C₂×C₄×C₅₃	C₂³×C₄	1	24	yes	—
183	C₂×C₄×C₅₃	C₂×#41	1	24	yes	complemented
184	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
185	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
186	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
187	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
188	C₂²⋊D₅₃	C₂³×C₄	1	128	yes	complemented
189	C₂×Dic₅₃	—	2	12	no	—
190	C₂×Dic₅₃	—	2	24	no	—
191	C₂×Dic₅₃	—	2	64	no	—
192	C₂×Dic₅₃	C₂³×C₄	1	48	yes	—
193	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
194	C₂×Dic₅₃	C₂³×C₄	1	128	yes	complemented
195	C₂×Dic₅₃	C₂×#41	1	24	yes	complemented
196	C₂×Dic₅₃	C₂⁵	1	4	yes	—
197	C₂×Dic₅₃	C₂⁵	1	8	yes	—
198	C₂×Dic₅₃	C₂⁵	1	8	yes	—
199	C₂×Dic₅₃	C₂⁵	1	4	yes	—
200	C₂×Dic₅₃	C₂⁵	1	4	yes	—
201	C₄×D₅₃	—	2	32	no	—
202	C₄×D₅₃	—	2	32	no	—
203	C₄×D₅₃	C₂³×C₄	1	64	yes	complemented
204	Dic₁₀₆	C₂³×C₄	1	64	yes	complemented
205	C₂²×C₄×#41	—	53	1	no	maximal, Sylow
206	C₂²×C₅₃	—	2	8	no	—
207	C₂²×C₅₃	—	2	8	no	—
208	C₂²×C₅₃	—	2	96	no	—
209	C₂²×C₅₃	C₄×#41	1	16	yes	complemented
210	C₂²×C₅₃	C₂⁴×C₄	1	16	yes	—
211	C₂²×C₅₃	C₂⁴×C₄	1	6	yes	—
212	C₂²×C₅₃	C₂²×#41	1	6	yes	—
213	C₂²×C₅₃	C₂²×#41	1	6	yes	—
214	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
215	C₂×D₅₃	—	2	4	no	—
216	C₂×D₅₃	—	2	4	no	—
217	C₂×D₅₃	—	2	48	no	—
218	C₂×D₅₃	C₂⁴×C₄	1	8	yes	—
219	C₄×C₅₃	—	2	4	no	—
220	C₄×C₅₃	—	2	8	no	—
221	C₄×C₅₃	C₂⁴×C₄	1	8	yes	—
222	C₄×C₅₃	C₂²×#41	1	8	yes	complemented
223	Dic₅₃	—	2	4	no	—
224	Dic₅₃	—	2	8	no	—
225	Dic₅₃	C₂⁴×C₄	1	16	yes	—
226	Dic₅₃	C₂⁴×C₄	1	8	yes	—
227	Dic₅₃	C₂²×#41	1	8	yes	complemented
228	C₂³×#41	—	53	1	no	—
229	C₂×C₄×#41	—	53	48	no	—
230	C₂²×C₄×Q₈	—	53	1	no	—
231	C₂²×C₄×Q₈	—	53	1	no	—
232	C₂²×C₄×D₄	—	53	1	no	—
233	C₂²×C₄×D₄	—	53	2	no	—
234	C₂²×C₄×D₄	—	53	2	no	—
235	C₂²×C₄×D₄	D₅₃	1	1	yes	characteristic, complemented
236	C₂²×#117	—	53	1	no	—
237	C₂²×#117	—	53	2	no	—
238	C₂³×C₄²	—	53	1	no	—
239	C₂³×C₄²	—	53	2	no	—
240	C₂×C₅₃	—	2	16	no	—
241	C₂×C₅₃	C₂×C₄×#41	1	12	yes	complemented
242	C₂×C₅₃	C₂⁵×C₄	1	1	yes	characteristic
243	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
244	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
245	D₅₃	—	2	8	no	—
246	C₂²×#41	—	53	32	no	—
247	C₂²×#41	—	53	24	no	—
248	C₂³×Q₈	—	53	1	no	—
249	C₂³×D₄	—	53	2	no	—
250	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
251	C₂⁴×C₄	—	53	2	no	—
252	C₂⁴×C₄	—	53	1	no	—
253	C₂⁴×C₄	—	53	2	no	—
254	C₂⁴×C₄	—	53	1	no	—
255	C₂⁴×C₄	—	53	1	no	—
256	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
257	C₄×#41	—	53	256	no	—
258	C₂×C₄×Q₈	—	53	24	no	—
259	C₂×C₄×Q₈	—	53	24	no	—
260	C₂×C₄×D₄	—	53	24	no	—
261	C₂×C₄×D₄	—	53	48	no	—
262	C₂×C₄×D₄	—	53	48	no	—
263	C₂×C₄×D₄	C₂×D₅₃	1	24	yes	complemented
264	C₂×#117	—	53	24	no	—
265	C₂×#117	—	53	48	no	—
266	C₂²×C₄⋊C₄	—	53	2	no	—
267	C₂²×C₄⋊C₄	—	53	2	no	—
268	C₂²×C₄⋊C₄	—	53	1	no	—
269	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
270	C₂²×C₂²⋊C₄	—	53	2	no	—
271	C₂²×C₂²⋊C₄	—	53	2	no	—
272	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
273	C₂²×C₄²	—	53	2	no	—
274	C₂²×C₄²	—	53	1	no	—
275	C₂²×C₄²	—	53	2	no	—
276	C₂²×C₄²	—	53	1	no	—
277	C₂²×C₄²	—	53	48	no	—
278	C₂²×C₄²	—	53	1	no	—
279	C₂²×C₄²	—	53	2	no	—
280	C₂²×C₄²	—	53	24	no	—
281	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
282	C₅₃	C₂²×C₄×#41	1	1	yes	characteristic, complemented, Sylow
283	C₂⁵	—	53	1	no	—
284	C₂⁵	C₂²×D₅₃	1	2	yes	—
285	C₂×#41	—	53	384	no	—
286	C₂×#41	—	53	64	no	—
287	C₂²×Q₈	—	53	16	no	—
288	C₂²×Q₈	—	53	12	no	—
289	C₂²×D₄	—	53	32	no	—
290	C₂²×D₄	—	53	24	no	—
291	C₂²×D₄	C₄×D₅₃	1	16	yes	complemented
292	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
293	C₂³×C₄	—	2	16	no	—
294	C₂³×C₄	—	53	32	no	—
295	C₂³×C₄	—	53	2	no	—
296	C₂³×C₄	—	53	24	no	—
297	C₂³×C₄	—	53	1	no	—
298	C₂³×C₄	—	53	1	no	—
299	C₂³×C₄	—	53	16	no	—
300	C₂³×C₄	—	53	12	no	—
301	C₂³×C₄	—	53	2	no	—
302	C₂³×C₄	—	53	1	no	—
303	C₂³×C₄	—	53	24	no	—
304	C₂³×C₄	—	53	12	no	—
305	C₂³×C₄	—	53	12	no	—
306	C₂³×C₄	—	106	8	no	—
307	C₂³×C₄	—	106	16	no	—
308	C₂³×C₄	—	106	8	no	—
309	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
310	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
311	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
312	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
313	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
314	C₄×Q₈	—	53	64	no	—
315	C₄×Q₈	—	53	64	no	—
316	C₄×D₄	—	53	64	no	—
317	C₄×D₄	—	53	128	no	—
318	C₄×D₄	—	53	128	no	—
319	C₄×D₄	C₂²×D₅₃	1	64	yes	complemented
320	#117	—	53	64	no	—
321	#117	—	53	128	no	—
322	C₂×C₄⋊C₄	—	53	24	no	—
323	C₂×C₄⋊C₄	—	53	24	no	—
324	C₂×C₄⋊C₄	—	53	12	no	—
325	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
326	C₂×C₂²⋊C₄	—	53	24	no	—
327	C₂×C₂²⋊C₄	—	53	24	no	—
328	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
329	C₂×C₄²	—	53	24	no	—
330	C₂×C₄²	—	53	12	no	—
331	C₂×C₄²	—	53	24	no	—
332	C₂×C₄²	—	53	12	no	—
333	C₂×C₄²	—	53	24	no	—
334	C₂×C₄²	—	53	128	no	—
335	C₂×C₄²	—	53	12	no	—
336	C₂×C₄²	—	53	64	no	—
337	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
338	C₂⁴	—	2	8	no	—
339	C₂⁴	—	2	8	no	—
340	C₂⁴	—	53	8	no	—
341	C₂⁴	—	53	6	no	—
342	C₂⁴	—	106	4	no	—
343	C₂⁴	—	106	4	no	—
344	C₂⁴	C₂×C₄×D₅₃	1	16	yes	—
345	C₂⁴	C₂³×D₅₃	1	12	yes	—
346	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
347	#41	—	53	512	no	—
348	C₂×Q₈	—	53	96	no	—
349	C₂×Q₈	—	53	16	no	—
350	C₂×D₄	—	53	192	no	—
351	C₂×D₄	—	53	32	no	—
352	C₂×D₄	C₂×C₄×D₅₃	1	96	yes	complemented
353	C₂×D₄	C₂³×D₅₃	1	16	yes	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	8	no	—
356	C₂²×C₄	—	2	96	no	—
357	C₂²×C₄	—	53	8	no	—
358	C₂²×C₄	—	53	16	no	—
359	C₂²×C₄	—	53	192	no	—
360	C₂²×C₄	—	53	8	no	—
361	C₂²×C₄	—	53	6	no	—
362	C₂²×C₄	—	53	6	no	—
363	C₂²×C₄	—	53	12	no	—
364	C₂²×C₄	—	53	32	no	—
365	C₂²×C₄	—	53	96	no	—
366	C₂²×C₄	—	53	16	no	—
367	C₂²×C₄	—	53	12	no	—
368	C₂²×C₄	—	53	6	no	—
369	C₂²×C₄	—	53	32	no	—
370	C₂²×C₄	—	53	16	no	—
371	C₂²×C₄	—	53	16	no	—
372	C₂²×C₄	—	106	48	no	—
373	C₂²×C₄	—	106	96	no	—
374	C₂²×C₄	—	106	8	no	—
375	C₂²×C₄	—	106	4	no	—
376	C₂²×C₄	—	106	48	no	—
377	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
378	C₂²×C₄	#11719947	1	8	yes	direct product factor
379	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
380	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
381	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
382	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
383	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
384	C₄⋊C₄	—	53	32	no	—
385	C₄⋊C₄	—	53	32	no	—
386	C₄⋊C₄	—	53	16	no	—
387	C₄⋊C₄	C₂³×D₅₃	1	16	yes	—
388	C₂²⋊C₄	—	53	32	no	—
389	C₂²⋊C₄	—	53	32	no	—
390	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
391	C₄²	—	53	16	no	—
392	C₄²	—	53	32	no	—
393	C₄²	—	53	16	no	—
394	C₄²	—	53	32	no	—
395	C₄²	—	53	16	no	—
396	C₄²	—	53	32	no	—
397	C₄²	C₂³×D₅₃	1	16	yes	—
398	C₂³	—	2	64	no	—
399	C₂³	—	2	24	no	—
400	C₂³	—	2	24	no	—
401	C₂³	—	53	24	no	—
402	C₂³	—	53	4	no	—
403	C₂³	—	106	32	no	—
404	C₂³	—	106	12	no	—
405	C₂³	—	106	12	no	—
406	C₂³	C₂²×C₄×D₅₃	1	48	yes	—
407	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
408	C₂³	#41×D₅₃	1	4	yes	—
409	C₂³	C₂×#11719947	1	4	yes	—
410	C₂³	C₂⁴×D₅₃	1	8	yes	—
411	C₂³	C₂⁴×D₅₃	1	3	yes	—
412	Q₈	—	53	64	no	—
413	D₄	—	53	128	no	—
414	D₄	C₂²×C₄×D₅₃	1	64	yes	complemented
415	C₂×C₄	—	2	12	no	—
416	C₂×C₄	—	2	24	no	—
417	C₂×C₄	—	2	64	no	—
418	C₂×C₄	—	53	24	no	—
419	C₂×C₄	—	53	48	no	—
420	C₂×C₄	—	53	128	no	—
421	C₂×C₄	—	53	24	no	—
422	C₂×C₄	—	53	4	no	—
423	C₂×C₄	—	53	4	no	—
424	C₂×C₄	—	53	8	no	—
425	C₂×C₄	—	53	64	no	—
426	C₂×C₄	—	53	8	no	—
427	C₂×C₄	—	53	4	no	—
428	C₂×C₄	—	106	32	no	—
429	C₂×C₄	—	106	64	no	—
430	C₂×C₄	—	106	24	no	—
431	C₂×C₄	—	106	12	no	—
432	C₂×C₄	—	106	32	no	—
433	C₂×C₄	C₂²×C₄×D₅₃	1	24	yes	—
434	C₂×C₄	C₂×#11719947	1	24	yes	direct product factor
435	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
436	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
437	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
438	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
439	C₂²	—	2	96	no	—
440	C₂²	—	2	8	no	—
441	C₂²	—	2	8	no	—
442	C₂²	—	53	8	no	—
443	C₂²	—	106	48	no	—
444	C₂²	—	106	4	no	—
445	C₂²	—	106	4	no	—
446	C₂²	C₄×#11719947	1	16	yes	direct product factor
447	C₂²	C₂³×C₄×D₅₃	1	16	yes	—
448	C₂²	C₂³×C₄×D₅₃	1	6	yes	—
449	C₂²	C₂×#41×D₅₃	1	6	yes	—
450	C₂²	C₂²×#11719947	1	6	yes	—
451	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
452	C₄	—	2	4	no	—
453	C₄	—	2	8	no	—
454	C₄	—	53	8	no	—
455	C₄	—	53	16	no	—
456	C₄	—	53	8	no	—
457	C₄	—	106	8	no	—
458	C₄	—	106	4	no	—
459	C₄	C₂³×C₄×D₅₃	1	8	yes	—
460	C₄	C₂²×#11719947	1	8	yes	direct product factor
461	C₂	—	2	16	no	—
462	C₂	—	106	8	no	—
463	C₂	C₂×C₄×#11719947	1	12	yes	direct product factor
464	C₂	C₂⁴×C₄×D₅₃	1	1	yes	characteristic
465	C₂	C₂²×#41×D₅₃	1	1	yes	characteristic
466	C₂	C₂³×#11719947	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×C₄×D₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×#11719947	C₂
cosocle	C₂²×C₅₃	C₂⁶
hypercenter	C₂²×C₄×D₄	D₅₃
hypocenter	C₅₃	C₂²×C₄×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#11719947	C₂	direct (N × H)
3. C₂²×C₄×Dic₁₀₆	C₂	semidirect
4. C₂²×#13133182	C₂	semidirect
5. C₂²×D₄×Dic₅₃	C₂	semidirect
6. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
7. C₂²×#13133177	C₂	semidirect
8. C₂²×#13133175	C₂	semidirect
9. C₂²×#13133173	C₂	semidirect
10. C₂³×C₄×Dic₅₃	C₂	semidirect
11. C₂²×C₄²×D₅₃	C₂	semidirect
12. C₂²×C₄×D₄×C₅₃	C₂	semidirect
13. C₂²×#11719947	C₄	direct (N × H)
22. C₄×#11719947	C₂²	direct (N × H)
23. C₂×C₄×Dic₁₀₆	C₂²	semidirect
24. C₂×#13133182	C₂²	semidirect
25. C₂×D₄×Dic₅₃	C₂²	semidirect
26. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
27. C₂×#13133177	C₂²	semidirect
28. C₂×#13133175	C₂²	semidirect
29. C₂×#13133173	C₂²	semidirect
41. C₂²×C₄×Dic₅₃	C₂²	semidirect
42. C₂×C₄²×D₅₃	C₂²	semidirect
45. C₂×C₄×D₄×C₅₃	C₂²	semidirect
50. C₂×#11719947	C₂×C₄	direct (N × H)
52. C₂²×Dic₁₀₆	C₂×C₄	semidirect
54. C₂²×C₂²⋊D₅₃	C₂×C₄	semidirect
57. C₂³×Dic₅₃	C₂×C₄	semidirect
67. C₂²×C₄×D₅₃	C₂×C₄	semidirect
71. C₄×Dic₁₀₆	C₂³	semidirect
72. #13133182	C₂³	semidirect
73. D₄×Dic₅₃	C₂³	semidirect
74. C₄×C₂²⋊D₅₃	C₂³	semidirect
75. #13133177	C₂³	semidirect
76. #13133175	C₂³	semidirect
77. #13133173	C₂³	semidirect
89. C₂×C₄×Dic₅₃	C₂³	semidirect
90. C₄²×D₅₃	C₂³	semidirect
92. C₂²×D₄×C₅₃	C₂×C₄	semidirect
100. C₄×D₄×C₅₃	C₂³	semidirect
113. C₂×D₄×C₅₃	C₂²×C₄	semidirect
119. C₂²×C₄×C₅₃	#41	semidirect
125. C₂×C₂²⋊D₅₃	C₂²×C₄	semidirect
132. C₂²×Dic₅₃	C₂²×C₄	semidirect
133. C₂²×Dic₅₃	#41	semidirect
141. #11719947	C₂²×C₄	direct (N × H)
144. C₂×C₄×D₅₃	C₂²×C₄	semidirect
148. C₂×Dic₁₀₆	C₂²×C₄	semidirect
178. D₄×C₅₃	C₂³×C₄	semidirect
183. C₂×C₄×C₅₃	C₂×#41	semidirect
188. C₂²⋊D₅₃	C₂³×C₄	semidirect
194. C₂×Dic₅₃	C₂³×C₄	semidirect
195. C₂×Dic₅₃	C₂×#41	semidirect
203. C₄×D₅₃	C₂³×C₄	semidirect
204. Dic₁₀₆	C₂³×C₄	semidirect
209. C₂²×C₅₃	C₄×#41	semidirect
222. C₄×C₅₃	C₂²×#41	semidirect
227. Dic₅₃	C₂²×#41	semidirect
235. C₂²×C₄×D₄	D₅₃	semidirect
241. C₂×C₅₃	C₂×C₄×#41	semidirect
263. C₂×C₄×D₄	C₂×D₅₃	semidirect
282. C₅₃	C₂²×C₄×#41	semidirect
291. C₂²×D₄	C₄×D₅₃	semidirect
319. C₄×D₄	C₂²×D₅₃	semidirect
352. C₂×D₄	C₂×C₄×D₅₃	semidirect
378. C₂²×C₄	#11719947	direct (N × H)
414. D₄	C₂²×C₄×D₅₃	semidirect
434. C₂×C₄	C₂×#11719947	direct (N × H)
446. C₂²	C₄×#11719947	direct (N × H)
460. C₄	C₂²×#11719947	direct (N × H)
463. C₂	C₂×C₄×#11719947	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719947	2
C₂×C₄×#11719947	2
C₂²×C₄×Dic₁₀₆	2
C₂²×#13133182	2
C₂²×D₄×Dic₅₃	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#13133177	2
C₂²×#13133175	2
C₂²×#13133173	2
C₂³×C₄×Dic₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂²×C₄×#41	53

Subgroups — C22×C4×#11719947 #28290664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11719947 #28290664