Subgroups — C₂×D₄×Q₈×D₅₃ #28291004

Order 13568.

11798 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×Q₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄×Dic₁₀₆	C₂	1	3	yes	maximal, complemented
4	C₂×#16811288	C₂	1	3	yes	maximal, complemented
5	C₂×Q₈×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
6	C₂×#16811286	C₂	1	6	yes	maximal, complemented
7	D₄×Q₈×D₅₃	C₂	1	32	yes	maximal, direct product factor
8	C₂×C₄⋊Q₈×D₅₃	C₂	1	3	yes	maximal, complemented
9	C₂×C₂²⋊Q₈×D₅₃	C₂	1	6	yes	maximal, complemented
10	C₂×C₄×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄×D₅₃	C₂	1	3	yes	maximal
12	C₂×D₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Dic₁₀₆	C₂²	1	6	yes	complemented
14	C₂²×Q₈×D₅₃	—	2	32	no	—
15	C₂²×Q₈×D₅₃	C₂²	1	6	yes	—
16	C₂²×Q₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×D₅₃	C₂²	1	6	yes	complemented
18	C₂²×Q₈×D₅₃	C₂²	1	2	yes	complemented
19	C₂²×Q₈×D₅₃	C₂²	1	32	yes	complemented
20	C₂²×D₄×D₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
22	Q₈×D₂₁₂	C₂²	1	16	yes	complemented
23	D₄×Dic₁₀₆	C₂²	1	48	yes	complemented
24	#16811288	C₂²	1	48	yes	complemented
25	Q₈×C₂²⋊D₅₃	C₂²	1	32	yes	complemented
26	#16811286	C₂²	1	96	yes	complemented
27	C₂×#13133217	C₂²	1	3	yes	complemented
28	C₂×#13133216	C₂²	1	6	yes	complemented
29	C₂×#13133216	C₂²	1	6	yes	complemented
30	C₂×#13133215	C₂²	1	6	yes	complemented
31	C₄⋊Q₈×D₅₃	C₂²	1	48	yes	complemented
32	C₂×#13133197	C₂²	1	6	yes	complemented
33	C₂×#13133196	C₂²	1	6	yes	complemented
34	C₂×#13133195	C₂²	1	12	yes	complemented
35	C₂×#13133194	C₂²	1	12	yes	complemented
36	C₂×#13133193	C₂²	1	6	yes	complemented
37	C₂²⋊Q₈×D₅₃	C₂²	1	96	yes	complemented
38	C₂×Q₈×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₄×Dic₁₀₆	C₂²	1	3	yes	complemented
40	C₂×#13133182	C₂²	1	3	yes	complemented
41	C₄×Q₈×D₅₃	C₂²	1	16	yes	complemented
42	C₂×C₄×D₂₁₂	C₂²	1	3	yes	—
43	C₂×D₄×Dic₅₃	C₂²	1	3	yes	—
44	C₂×#13133179	C₂²	1	3	yes	—
45	C₂×C₄×C₂²⋊D₅₃	C₂²	1	6	yes	—
46	C₂×#13133177	C₂²	1	6	yes	—
47	C₄×D₄×D₅₃	C₂²	1	48	yes	—
48	C₂×C₄⋊C₄×D₅₃	C₂²	1	3	yes	—
49	C₂×C₄⋊C₄×D₅₃	C₂²	1	3	yes	—
50	C₂×C₄⋊C₄×D₅₃	C₂²	1	6	yes	—
51	C₂×C₂²⋊C₄×D₅₃	C₂²	1	6	yes	—
52	C₂×C₄²×D₅₃	C₂²	1	3	yes	—
53	C₂³×Q₈×C₅₃	C₂²	1	2	yes	complemented
54	D₄×Q₈×C₅₃	C₂²	1	16	yes	complemented
55	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
56	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	6	yes	complemented
57	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
58	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
59	C₂⁴×D₅₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₀₆	—	2	48	no	—
61	C₂²×Dic₁₀₆	C₂³	1	12	yes	—
62	C₂²×Dic₁₀₆	C₂³	1	6	yes	—
63	C₂²×Dic₁₀₆	C₂³	1	3	yes	—
64	C₂²×Dic₁₀₆	C₂³	1	48	yes	complemented
65	C₂×Q₈×D₅₃	—	2	16	no	—
66	C₂×Q₈×D₅₃	—	2	256	no	—
67	C₂×Q₈×D₅₃	—	2	48	no	—
68	C₂×Q₈×D₅₃	—	2	48	no	—
69	C₂×Q₈×D₅₃	D₄	1	16	yes	direct product factor
70	C₂×Q₈×D₅₃	C₂³	1	48	yes	—
71	C₂×Q₈×D₅₃	C₂³	1	8	yes	—
72	C₂×Q₈×D₅₃	C₂³	1	48	yes	complemented
73	C₂×Q₈×D₅₃	C₂³	1	16	yes	complemented
74	C₂²×D₂₁₂	C₂³	1	1	yes	characteristic
75	C₂²×C₂²⋊D₅₃	C₂³	1	2	yes	—
76	C₂×D₄×D₅₃	Q₈	1	16	yes	direct product factor
77	C₂×D₄×D₅₃	C₂³	1	8	yes	—
78	C₂³×Dic₅₃	C₂³	1	6	yes	—
79	C₂²×C₄×D₅₃	—	2	48	no	—
80	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
81	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
82	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
83	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
84	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
85	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
86	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
87	#13133217	C₂³	1	24	yes	complemented
88	#13133216	C₂³	1	48	yes	complemented
89	#13133216	C₂³	1	48	yes	complemented
90	#13133215	C₂³	1	48	yes	complemented
91	#13133197	C₂³	1	48	yes	complemented
92	#13133196	C₂³	1	48	yes	complemented
93	#13133195	C₂³	1	96	yes	complemented
94	#13133194	C₂³	1	96	yes	complemented
95	#13133193	C₂³	1	48	yes	complemented
96	Q₈×Dic₅₃	C₂³	1	8	yes	complemented
97	C₄×Dic₁₀₆	C₂³	1	24	yes	complemented
98	#13133182	C₂³	1	24	yes	complemented
99	C₄×D₂₁₂	C₂³	1	24	yes	—
100	D₄×Dic₅₃	C₂³	1	24	yes	—
101	#13133179	C₂³	1	24	yes	—
102	C₄×C₂²⋊D₅₃	C₂³	1	48	yes	—
103	#13133177	C₂³	1	48	yes	—
104	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
105	C₂×C₄⋊Dic₅₃	C₂³	1	3	yes	—
106	C₂×C₄⋊Dic₅₃	C₂³	1	6	yes	—
107	C₂×#11719919	C₂³	1	6	yes	—
108	C₂×#11719919	C₂³	1	6	yes	—
109	C₂×#11719919	C₂³	1	12	yes	—
110	C₄⋊C₄×D₅₃	C₂³	1	24	yes	—
111	C₄⋊C₄×D₅₃	C₂³	1	24	yes	—
112	C₄⋊C₄×D₅₃	C₂³	1	48	yes	—
113	C₂×C₂²⋊Dic₅₃	C₂³	1	6	yes	—
114	C₂×#11719921	C₂³	1	6	yes	—
115	C₂×#11719921	C₂³	1	6	yes	—
116	C₂²⋊C₄×D₅₃	C₂³	1	48	yes	—
117	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
118	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
119	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
120	C₄²×D₅₃	C₂³	1	24	yes	—
121	C₂²×Q₈×C₅₃	—	2	16	no	—
122	C₂²×Q₈×C₅₃	C₂³	1	6	yes	—
123	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
124	C₂²×Q₈×C₅₃	C₂³	1	16	yes	complemented
125	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
126	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
127	C₄⋊Q₈×C₅₃	C₂³	1	24	yes	complemented
128	C₂²⋊Q₈×C₅₃	C₂³	1	48	yes	complemented
129	C₄×Q₈×C₅₃	C₂³	1	8	yes	complemented
130	C₄×D₄×C₅₃	C₂³	1	24	yes	—
131	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
132	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
133	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
134	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
135	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
136	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
137	C₂³×D₅₃	—	2	8	no	—
138	C₂³×D₅₃	—	2	8	no	—
139	C₂³×D₅₃	C₂×Q₈	1	16	yes	complemented
140	C₂³×D₅₃	C₂⁴	1	8	yes	—
141	C₂³×D₅₃	C₂⁴	1	2	yes	—
142	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
143	C₂×Q₈×C₅₃	—	2	64	no	—
144	C₂×Q₈×C₅₃	—	2	24	no	—
145	C₂×Q₈×C₅₃	C₂×D₄	1	8	yes	complemented
146	C₂×Q₈×C₅₃	C₂⁴	1	24	yes	—
147	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
148	C₂×D₄×C₅₃	C₂×Q₈	1	8	yes	complemented
149	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
150	C₂²×C₄×C₅₃	—	2	24	no	—
151	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
156	C₂×C₂²⋊D₅₃	C₂×Q₈	1	16	yes	complemented
157	C₂×C₂²⋊D₅₃	C₂⁴	1	8	yes	—
158	C₂²×Dic₅₃	—	2	24	no	—
159	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
160	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
161	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
162	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
163	C₂²×Dic₅₃	C₂⁴	1	1	yes	characteristic
164	Q₈×D₅₃	—	2	64	no	—
165	Q₈×D₅₃	—	2	192	no	—
166	Q₈×D₅₃	—	2	192	no	—
167	Q₈×D₅₃	C₂×D₄	1	64	yes	direct product factor
168	D₄×D₅₃	C₂×Q₈	1	64	yes	direct product factor
169	C₂×D₂₁₂	C₂×Q₈	1	8	yes	complemented
170	C₂×D₂₁₂	C₂⁴	1	4	yes	—
171	C₂×C₄×D₅₃	—	2	24	no	—
172	C₂×C₄×D₅₃	—	2	12	no	—
173	C₂×C₄×D₅₃	—	2	24	no	—
174	C₂×C₄×D₅₃	—	2	24	no	—
175	C₂×C₄×D₅₃	—	2	192	no	—
176	C₂×C₄×D₅₃	C₂×D₄	1	24	yes	—
177	C₂×C₄×D₅₃	C₂×Q₈	1	8	yes	complemented
178	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
179	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
180	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
181	C₂×C₄×D₅₃	C₂⁴	1	24	yes	—
182	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
183	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
184	C₂×Dic₁₀₆	—	2	24	no	—
185	C₂×Dic₁₀₆	—	2	48	no	—
186	C₂×Dic₁₀₆	—	2	192	no	—
187	C₂×Dic₁₀₆	C₂×D₄	1	24	yes	complemented
188	C₂×Dic₁₀₆	C₂⁴	1	48	yes	—
189	C₂×Dic₁₀₆	C₂⁴	1	24	yes	—
190	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
191	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
192	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
193	C₄⋊C₄×C₅₃	C₂⁴	1	24	yes	—
194	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
195	C₄²×C₅₃	C₂⁴	1	12	yes	—
196	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
197	#11719921	C₂⁴	1	24	yes	—
198	#11719921	C₂⁴	1	24	yes	—
199	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
200	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
201	C₄⋊Dic₅₃	C₂⁴	1	24	yes	—
202	#11719919	C₂⁴	1	24	yes	—
203	#11719919	C₂⁴	1	24	yes	—
204	#11719919	C₂⁴	1	48	yes	—
205	C₄×Dic₅₃	C₂⁴	1	12	yes	—
206	C₄×Dic₅₃	C₂⁴	1	12	yes	—
207	C₄×Dic₅₃	C₂⁴	1	12	yes	—
208	C₂³×C₅₃	—	2	4	no	—
209	C₂³×C₅₃	—	2	4	no	—
210	C₂³×C₅₃	C₂²×Q₈	1	8	yes	complemented
211	C₂³×C₅₃	C₂⁵	1	4	yes	—
212	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
213	C₂²×D₅₃	—	2	4	no	—
214	C₂²×D₅₃	—	2	4	no	—
215	C₂²×D₅₃	—	2	64	no	—
216	C₂²×D₅₃	—	2	16	no	—
217	C₂²×D₅₃	—	2	16	no	—
218	C₂²×D₅₃	C₂²×D₄	1	4	yes	—
219	C₂²×D₅₃	C₂²×Q₈	1	4	yes	—
220	C₂²×D₅₃	C₂²×Q₈	1	32	yes	complemented
221	C₂²×D₅₃	C₂²×Q₈	1	8	yes	complemented
222	C₂²×D₅₃	2₋¹⁺⁴	1	4	yes	—
223	C₂²×D₅₃	C₂⁵	1	4	yes	—
224	C₂²×D₅₃	C₂⁵	1	2	yes	—
225	Q₈×C₅₃	—	2	48	no	—
226	Q₈×C₅₃	C₂²×D₄	1	16	yes	complemented
227	D₄×C₅₃	C₂²×Q₈	1	16	yes	complemented
228	C₂×C₄×C₅₃	—	2	12	no	—
229	C₂×C₄×C₅₃	—	2	6	no	—
230	C₂×C₄×C₅₃	—	2	48	no	—
231	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
232	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	complemented
233	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
234	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
235	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
236	C₂×C₄×C₅₃	C₂⁵	1	2	yes	—
237	C₂²⋊D₅₃	C₂²×Q₈	1	32	yes	complemented
238	C₂×Dic₅₃	—	2	12	no	—
239	C₂×Dic₅₃	—	2	6	no	—
240	C₂×Dic₅₃	—	2	48	no	—
241	C₂×Dic₅₃	C₂²×D₄	1	12	yes	—
242	C₂×Dic₅₃	C₂²×Q₈	1	4	yes	complemented
243	C₂×Dic₅₃	C₂⁵	1	12	yes	—
244	C₂×Dic₅₃	C₂⁵	1	6	yes	—
245	C₂×Dic₅₃	C₂⁵	1	6	yes	—
246	C₂×Dic₅₃	C₂⁵	1	2	yes	—
247	D₂₁₂	C₂²×Q₈	1	16	yes	complemented
248	C₄×D₅₃	—	2	48	no	—
249	C₄×D₅₃	—	2	24	no	—
250	C₄×D₅₃	—	2	48	no	—
251	C₄×D₅₃	—	2	48	no	—
252	C₄×D₅₃	C₂²×D₄	1	48	yes	—
253	C₄×D₅₃	C₂²×Q₈	1	16	yes	complemented
254	Dic₁₀₆	—	2	48	no	—
255	Dic₁₀₆	—	2	96	no	—
256	Dic₁₀₆	C₂²×D₄	1	48	yes	complemented
257	C₂²×D₄×Q₈	—	53	1	no	maximal, Sylow
258	C₂²×C₅₃	—	2	4	no	—
259	C₂²×C₅₃	—	2	4	no	—
260	C₂²×C₅₃	—	2	16	no	—
261	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
262	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
263	C₂²×C₅₃	C₂³×Q₈	1	8	yes	complemented
264	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
265	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
266	C₂×D₅₃	—	2	4	no	—
267	C₂×D₅₃	—	2	4	no	—
268	C₂×D₅₃	—	2	16	no	—
269	C₂×D₅₃	—	2	64	no	—
270	C₂×D₅₃	D₄×Q₈	1	16	yes	direct product factor
271	C₂×D₅₃	C₂³×D₄	1	4	yes	—
272	C₂×D₅₃	C₂³×Q₈	1	4	yes	—
273	C₂×D₅₃	C₂³×Q₈	1	8	yes	complemented
274	C₂×D₅₃	C₂×2₋¹⁺⁴	1	4	yes	—
275	C₄×C₅₃	—	2	12	no	—
276	C₄×C₅₃	—	2	6	no	—
277	C₄×C₅₃	C₂³×D₄	1	12	yes	—
278	C₄×C₅₃	C₂³×Q₈	1	4	yes	complemented
279	Dic₅₃	—	2	12	no	—
280	Dic₅₃	—	2	6	no	—
281	Dic₅₃	C₂³×D₄	1	12	yes	—
282	Dic₅₃	C₂³×Q₈	1	4	yes	complemented
283	C₂⁴×Q₈	—	53	2	no	—
284	C₂×D₄×Q₈	—	53	2	no	—
285	C₂×D₄×Q₈	—	53	32	no	—
286	C₂×D₄×Q₈	—	53	3	no	—
287	C₂×D₄×Q₈	—	53	6	no	—
288	C₂×D₄×Q₈	—	53	1	no	—
289	C₂×D₄×Q₈	—	53	3	no	—
290	C₂×D₄×Q₈	D₅₃	1	1	yes	characteristic, direct product factor
291	C₂²×C₄⋊Q₈	—	53	3	no	—
292	C₂²×C₂²⋊Q₈	—	53	6	no	—
293	C₂²×C₄×Q₈	—	53	1	no	—
294	C₂²×C₄×D₄	—	53	3	no	—
295	C₂×C₅₃	—	2	8	no	—
296	C₂×C₅₃	C₂×D₄×Q₈	1	4	yes	complemented
297	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
298	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
299	C₂×C₅₃	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
300	D₅₃	—	2	8	no	—
301	D₅₃	C₂×D₄×Q₈	1	8	yes	direct product factor
302	C₂³×Q₈	—	53	1	no	—
303	C₂³×Q₈	—	53	6	no	—
304	C₂³×Q₈	—	53	6	no	—
305	C₂³×Q₈	—	53	2	no	—
306	C₂³×Q₈	—	53	32	no	—
307	C₂³×Q₈	—	53	6	no	—
308	C₂³×Q₈	—	106	32	no	—
309	C₂³×Q₈	C₂×D₅₃	1	2	yes	complemented
310	C₂³×D₄	—	53	1	no	—
311	C₂⁴×C₄	—	53	6	no	—
312	D₄×Q₈	—	53	32	no	—
313	D₄×Q₈	—	53	48	no	—
314	D₄×Q₈	—	53	96	no	—
315	D₄×Q₈	—	53	16	no	—
316	D₄×Q₈	—	53	48	no	—
317	D₄×Q₈	C₂×D₅₃	1	16	yes	direct product factor
318	C₂×C₄⋊Q₈	—	53	6	no	—
319	C₂×C₄⋊Q₈	—	53	48	no	—
320	C₂×C₄⋊Q₈	—	53	6	no	—
321	C₂×C₄⋊Q₈	—	53	6	no	—
322	C₂×C₄⋊Q₈	—	53	3	no	—
323	C₂×C₄⋊Q₈	C₂×D₅₃	1	3	yes	complemented
324	C₂×C₂²⋊Q₈	—	53	6	no	—
325	C₂×C₂²⋊Q₈	—	53	6	no	—
326	C₂×C₂²⋊Q₈	—	53	96	no	—
327	C₂×C₂²⋊Q₈	—	53	12	no	—
328	C₂×C₂²⋊Q₈	—	53	6	no	—
329	C₂×C₂²⋊Q₈	—	53	12	no	—
330	C₂×C₂²⋊Q₈	C₂×D₅₃	1	6	yes	complemented
331	C₂×C₄×Q₈	—	53	1	no	—
332	C₂×C₄×Q₈	—	53	16	no	—
333	C₂×C₄×Q₈	—	53	3	no	—
334	C₂×C₄×Q₈	—	53	3	no	—
335	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
336	C₂×C₄×D₄	—	53	3	no	—
337	C₂×C₄×D₄	—	53	6	no	—
338	C₂×C₄×D₄	—	53	48	no	—
339	C₂×C₄×D₄	—	53	3	no	—
340	C₂×C₄×D₄	—	53	6	no	—
341	C₂×C₄×D₄	—	53	3	no	—
342	C₂×C₄×D₄	C₂×D₅₃	1	3	yes	—
343	C₂²×C₄⋊C₄	—	53	3	no	—
344	C₂²×C₄⋊C₄	—	53	3	no	—
345	C₂²×C₄⋊C₄	—	53	6	no	—
346	C₂²×C₂²⋊C₄	—	53	6	no	—
347	C₂²×C₄²	—	53	3	no	—
348	C₅₃	C₂²×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
349	C₂⁵	—	53	2	no	—
350	C₂²×Q₈	—	2	16	no	—
351	C₂²×Q₈	—	53	16	no	—
352	C₂²×Q₈	—	53	8	no	—
353	C₂²×Q₈	—	53	48	no	—
354	C₂²×Q₈	—	53	48	no	—
355	C₂²×Q₈	—	53	16	no	—
356	C₂²×Q₈	—	53	3	no	—
357	C₂²×Q₈	—	53	12	no	—
358	C₂²×Q₈	—	53	6	no	—
359	C₂²×Q₈	—	53	48	no	—
360	C₂²×Q₈	—	106	48	no	—
361	C₂²×Q₈	—	106	48	no	—
362	C₂²×Q₈	—	106	256	no	—
363	C₂²×Q₈	—	106	48	no	—
364	C₂²×Q₈	—	106	16	no	—
365	C₂²×Q₈	C₂²×D₅₃	1	6	yes	—
366	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
367	C₂²×Q₈	C₂²×D₅₃	1	16	yes	complemented
368	C₂²×D₄	—	53	2	no	—
369	C₂²×D₄	—	53	16	no	—
370	C₂²×D₄	—	53	8	no	—
371	C₂²×D₄	—	53	1	no	—
372	C₂²×D₄	C₂²×D₅₃	1	1	yes	characteristic
373	C₂³×C₄	—	53	1	no	—
374	C₂³×C₄	—	53	3	no	—
375	C₂³×C₄	—	53	6	no	—
376	C₂³×C₄	—	53	3	no	—
377	C₂³×C₄	—	53	6	no	—
378	C₂³×C₄	—	53	6	no	—
379	C₂³×C₄	—	53	48	no	—
380	C₂³×C₄	—	53	6	no	—
381	C₂³×C₄	—	106	48	no	—
382	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
383	C₄⋊Q₈	—	53	48	no	—
384	C₄⋊Q₈	—	53	48	no	—
385	C₄⋊Q₈	—	53	48	no	—
386	C₄⋊Q₈	—	53	24	no	—
387	C₄⋊Q₈	C₂²×D₅₃	1	24	yes	complemented
388	C₂²⋊Q₈	—	53	48	no	—
389	C₂²⋊Q₈	—	53	48	no	—
390	C₂²⋊Q₈	—	53	96	no	—
391	C₂²⋊Q₈	—	53	48	no	—
392	C₂²⋊Q₈	—	53	96	no	—
393	C₂²⋊Q₈	C₂²×D₅₃	1	48	yes	complemented
394	C₄×Q₈	—	53	8	no	—
395	C₄×Q₈	—	53	24	no	—
396	C₄×Q₈	—	53	24	no	—
397	C₄×Q₈	C₂²×D₅₃	1	8	yes	complemented
398	C₄×D₄	—	53	24	no	—
399	C₄×D₄	—	53	48	no	—
400	C₄×D₄	—	53	24	no	—
401	C₄×D₄	—	53	48	no	—
402	C₄×D₄	—	53	24	no	—
403	C₄×D₄	C₂²×D₅₃	1	24	yes	—
404	C₂×C₄⋊C₄	—	53	6	no	—
405	C₂×C₄⋊C₄	—	53	3	no	—
406	C₂×C₄⋊C₄	—	53	6	no	—
407	C₂×C₄⋊C₄	—	53	3	no	—
408	C₂×C₄⋊C₄	—	53	12	no	—
409	C₂×C₄⋊C₄	—	53	6	no	—
410	C₂×C₄⋊C₄	—	53	24	no	—
411	C₂×C₄⋊C₄	—	53	24	no	—
412	C₂×C₄⋊C₄	—	53	48	no	—
413	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
414	C₂×C₄⋊C₄	C₂²×D₅₃	1	3	yes	—
415	C₂×C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
416	C₂×C₂²⋊C₄	—	53	6	no	—
417	C₂×C₂²⋊C₄	—	53	6	no	—
418	C₂×C₂²⋊C₄	—	53	6	no	—
419	C₂×C₂²⋊C₄	—	53	48	no	—
420	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
421	C₂×C₄²	—	53	3	no	—
422	C₂×C₄²	—	53	3	no	—
423	C₂×C₄²	—	53	3	no	—
424	C₂×C₄²	—	53	24	no	—
425	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
426	C₂⁴	—	53	1	no	—
427	C₂⁴	—	53	16	no	—
428	C₂⁴	—	53	8	no	—
429	C₂⁴	—	53	2	no	—
430	C₂⁴	—	106	8	no	—
431	C₂⁴	—	106	8	no	—
432	C₂⁴	C₂³×D₅₃	1	2	yes	—
433	C₂×Q₈	—	2	64	no	—
434	C₂×Q₈	—	2	24	no	—
435	C₂×Q₈	—	53	64	no	—
436	C₂×Q₈	—	53	24	no	—
437	C₂×Q₈	—	53	12	no	—
438	C₂×Q₈	—	53	48	no	—
439	C₂×Q₈	—	53	24	no	—
440	C₂×Q₈	—	106	48	no	—
441	C₂×Q₈	—	106	192	no	—
442	C₂×Q₈	—	106	192	no	—
443	C₂×Q₈	—	106	192	no	—
444	C₂×Q₈	—	106	64	no	—
445	C₂×Q₈	—	106	24	no	—
446	C₂×Q₈	D₄×D₅₃	1	8	yes	direct product factor
447	C₂×Q₈	C₂³×D₅₃	1	24	yes	—
448	C₂×Q₈	C₂³×D₅₃	1	4	yes	—
449	C₂×D₄	—	53	16	no	—
450	C₂×D₄	—	53	8	no	—
451	C₂×D₄	—	53	64	no	—
452	C₂×D₄	—	53	8	no	—
453	C₂×D₄	—	53	4	no	—
454	C₂×D₄	Q₈×D₅₃	1	8	yes	direct product factor
455	C₂×D₄	C₂³×D₅₃	1	4	yes	—
456	C₂²×C₄	—	2	24	no	—
457	C₂²×C₄	—	53	1	no	—
458	C₂²×C₄	—	53	3	no	—
459	C₂²×C₄	—	53	24	no	—
460	C₂²×C₄	—	53	8	no	—
461	C₂²×C₄	—	53	4	no	—
462	C₂²×C₄	—	53	12	no	—
463	C₂²×C₄	—	53	24	no	—
464	C₂²×C₄	—	53	3	no	—
465	C₂²×C₄	—	53	6	no	—
466	C₂²×C₄	—	53	12	no	—
467	C₂²×C₄	—	53	24	no	—
468	C₂²×C₄	—	53	24	no	—
469	C₂²×C₄	—	53	24	no	—
470	C₂²×C₄	—	106	24	no	—
471	C₂²×C₄	—	106	192	no	—
472	C₂²×C₄	—	106	12	no	—
473	C₂²×C₄	—	106	24	no	—
474	C₂²×C₄	—	106	24	no	—
475	C₂²×C₄	—	106	24	no	—
476	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
477	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
478	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
479	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
480	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
481	C₄⋊C₄	—	53	24	no	—
482	C₄⋊C₄	—	53	12	no	—
483	C₄⋊C₄	—	53	24	no	—
484	C₄⋊C₄	—	53	12	no	—
485	C₄⋊C₄	—	53	48	no	—
486	C₄⋊C₄	—	53	24	no	—
487	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
488	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
489	C₄⋊C₄	C₂³×D₅₃	1	24	yes	—
490	C₂²⋊C₄	—	53	24	no	—
491	C₂²⋊C₄	—	53	24	no	—
492	C₂²⋊C₄	—	53	24	no	—
493	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
494	C₄²	—	53	12	no	—
495	C₄²	—	53	12	no	—
496	C₄²	—	53	12	no	—
497	C₄²	C₂³×D₅₃	1	12	yes	—
498	C₂³	—	2	4	no	—
499	C₂³	—	2	4	no	—
500	C₂³	—	53	4	no	—
501	C₂³	—	53	4	no	—
502	C₂³	—	53	4	no	—
503	C₂³	—	53	2	no	—
504	C₂³	—	53	8	no	—
505	C₂³	—	53	32	no	—
506	C₂³	—	53	4	no	—
507	C₂³	—	106	16	no	—
508	C₂³	—	106	64	no	—
509	C₂³	—	106	16	no	—
510	C₂³	—	106	4	no	—
511	C₂³	—	106	4	no	—
512	C₂³	C₂×Q₈×D₅₃	1	8	yes	complemented
513	C₂³	C₂⁴×D₅₃	1	4	yes	—
514	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
515	Q₈	—	2	48	no	—
516	Q₈	—	53	48	no	—
517	Q₈	—	106	96	no	—
518	Q₈	—	106	48	no	—
519	Q₈	C₂×D₄×D₅₃	1	16	yes	direct product factor
520	D₄	—	53	32	no	—
521	D₄	—	53	16	no	—
522	D₄	C₂×Q₈×D₅₃	1	16	yes	direct product factor
523	C₂×C₄	—	2	12	no	—
524	C₂×C₄	—	2	6	no	—
525	C₂×C₄	—	2	48	no	—
526	C₂×C₄	—	53	12	no	—
527	C₂×C₄	—	53	4	no	—
528	C₂×C₄	—	53	2	no	—
529	C₂×C₄	—	53	6	no	—
530	C₂×C₄	—	53	48	no	—
531	C₂×C₄	—	53	16	no	—
532	C₂×C₄	—	53	6	no	—
533	C₂×C₄	—	53	12	no	—
534	C₂×C₄	—	106	48	no	—
535	C₂×C₄	—	106	24	no	—
536	C₂×C₄	—	106	48	no	—
537	C₂×C₄	—	106	48	no	—
538	C₂×C₄	—	106	12	no	—
539	C₂×C₄	—	106	6	no	—
540	C₂×C₄	—	106	48	no	—
541	C₂×C₄	C₂×D₄×D₅₃	1	12	yes	—
542	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	complemented
543	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
544	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
545	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
546	C₂×C₄	C₂⁴×D₅₃	1	2	yes	—
547	C₂²	—	2	16	no	—
548	C₂²	—	2	4	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	53	16	no	—
551	C₂²	—	53	4	no	—
552	C₂²	—	53	4	no	—
553	C₂²	—	53	4	no	—
554	C₂²	—	53	8	no	—
555	C₂²	—	106	64	no	—
556	C₂²	—	106	16	no	—
557	C₂²	—	106	4	no	—
558	C₂²	—	106	4	no	—
559	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
560	C₂²	C₂²×Q₈×D₅₃	1	2	yes	—
561	C₂²	C₂²×Q₈×D₅₃	1	8	yes	complemented
562	C₂²	2₋¹⁺⁴×D₅₃	1	2	yes	—
563	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
564	C₄	—	2	12	no	—
565	C₄	—	2	6	no	—
566	C₄	—	53	12	no	—
567	C₄	—	53	4	no	—
568	C₄	—	106	12	no	—
569	C₄	—	106	6	no	—
570	C₄	C₂²×D₄×D₅₃	1	12	yes	—
571	C₄	C₂²×Q₈×D₅₃	1	4	yes	complemented
572	C₂	—	2	8	no	—
573	C₂	—	53	8	no	—
574	C₂	—	106	8	no	—
575	C₂	D₄×Q₈×D₅₃	1	4	yes	direct product factor
576	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
577	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
578	C₂	C₂×2₋¹⁺⁴×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×D₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×D₄×D₅₃	C₂²
hypercenter	C₂×D₄×Q₈	D₅₃
hypocenter	C₅₃	C₂²×D₄×Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₅₃	C₂	semidirect
2. C₂×Q₈×D₂₁₂	C₂	semidirect
3. C₂×D₄×Dic₁₀₆	C₂	semidirect
4. C₂×#16811288	C₂	semidirect
5. C₂×Q₈×C₂²⋊D₅₃	C₂	semidirect
6. C₂×#16811286	C₂	semidirect
7. D₄×Q₈×D₅₃	C₂	direct (N × H)
8. C₂×C₄⋊Q₈×D₅₃	C₂	semidirect
9. C₂×C₂²⋊Q₈×D₅₃	C₂	semidirect
10. C₂×C₄×Q₈×D₅₃	C₂	semidirect
12. C₂×D₄×Q₈×C₅₃	C₂	semidirect
13. C₂³×Dic₁₀₆	C₂²	semidirect
17. C₂²×Q₈×D₅₃	C₂²	semidirect
18. C₂²×Q₈×D₅₃	C₂²	semidirect
19. C₂²×Q₈×D₅₃	C₂²	semidirect
22. Q₈×D₂₁₂	C₂²	semidirect
23. D₄×Dic₁₀₆	C₂²	semidirect
24. #16811288	C₂²	semidirect
25. Q₈×C₂²⋊D₅₃	C₂²	semidirect
26. #16811286	C₂²	semidirect
27. C₂×#13133217	C₂²	semidirect
28. C₂×#13133216	C₂²	semidirect
29. C₂×#13133216	C₂²	semidirect
30. C₂×#13133215	C₂²	semidirect
31. C₄⋊Q₈×D₅₃	C₂²	semidirect
32. C₂×#13133197	C₂²	semidirect
33. C₂×#13133196	C₂²	semidirect
34. C₂×#13133195	C₂²	semidirect
35. C₂×#13133194	C₂²	semidirect
36. C₂×#13133193	C₂²	semidirect
37. C₂²⋊Q₈×D₅₃	C₂²	semidirect
38. C₂×Q₈×Dic₅₃	C₂²	semidirect
39. C₂×C₄×Dic₁₀₆	C₂²	semidirect
40. C₂×#13133182	C₂²	semidirect
41. C₄×Q₈×D₅₃	C₂²	semidirect
53. C₂³×Q₈×C₅₃	C₂²	semidirect
54. D₄×Q₈×C₅₃	C₂²	semidirect
55. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
56. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
57. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
64. C₂²×Dic₁₀₆	C₂³	semidirect
69. C₂×Q₈×D₅₃	D₄	direct (N × H)
72. C₂×Q₈×D₅₃	C₂³	semidirect
73. C₂×Q₈×D₅₃	C₂³	semidirect
76. C₂×D₄×D₅₃	Q₈	direct (N × H)
87. #13133217	C₂³	semidirect
88. #13133216	C₂³	semidirect
89. #13133216	C₂³	semidirect
90. #13133215	C₂³	semidirect
91. #13133197	C₂³	semidirect
92. #13133196	C₂³	semidirect
93. #13133195	C₂³	semidirect
94. #13133194	C₂³	semidirect
95. #13133193	C₂³	semidirect
96. Q₈×Dic₅₃	C₂³	semidirect
97. C₄×Dic₁₀₆	C₂³	semidirect
98. #13133182	C₂³	semidirect
124. C₂²×Q₈×C₅₃	C₂³	semidirect
127. C₄⋊Q₈×C₅₃	C₂³	semidirect
128. C₂²⋊Q₈×C₅₃	C₂³	semidirect
129. C₄×Q₈×C₅₃	C₂³	semidirect
139. C₂³×D₅₃	C₂×Q₈	semidirect
145. C₂×Q₈×C₅₃	C₂×D₄	semidirect
148. C₂×D₄×C₅₃	C₂×Q₈	semidirect
156. C₂×C₂²⋊D₅₃	C₂×Q₈	semidirect
167. Q₈×D₅₃	C₂×D₄	direct (N × H)
168. D₄×D₅₃	C₂×Q₈	direct (N × H)
169. C₂×D₂₁₂	C₂×Q₈	semidirect
177. C₂×C₄×D₅₃	C₂×Q₈	semidirect
187. C₂×Dic₁₀₆	C₂×D₄	semidirect
210. C₂³×C₅₃	C₂²×Q₈	semidirect
220. C₂²×D₅₃	C₂²×Q₈	semidirect
221. C₂²×D₅₃	C₂²×Q₈	semidirect
226. Q₈×C₅₃	C₂²×D₄	semidirect
227. D₄×C₅₃	C₂²×Q₈	semidirect
232. C₂×C₄×C₅₃	C₂²×Q₈	semidirect
237. C₂²⋊D₅₃	C₂²×Q₈	semidirect
242. C₂×Dic₅₃	C₂²×Q₈	semidirect
247. D₂₁₂	C₂²×Q₈	semidirect
253. C₄×D₅₃	C₂²×Q₈	semidirect
256. Dic₁₀₆	C₂²×D₄	semidirect
263. C₂²×C₅₃	C₂³×Q₈	semidirect
270. C₂×D₅₃	D₄×Q₈	direct (N × H)
273. C₂×D₅₃	C₂³×Q₈	semidirect
278. C₄×C₅₃	C₂³×Q₈	semidirect
282. Dic₅₃	C₂³×Q₈	semidirect
290. C₂×D₄×Q₈	D₅₃	direct (N × H)
296. C₂×C₅₃	C₂×D₄×Q₈	semidirect
301. D₅₃	C₂×D₄×Q₈	direct (N × H)
309. C₂³×Q₈	C₂×D₅₃	semidirect
317. D₄×Q₈	C₂×D₅₃	direct (N × H)
323. C₂×C₄⋊Q₈	C₂×D₅₃	semidirect
330. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
335. C₂×C₄×Q₈	C₂×D₅₃	semidirect
348. C₅₃	C₂²×D₄×Q₈	semidirect
367. C₂²×Q₈	C₂²×D₅₃	semidirect
387. C₄⋊Q₈	C₂²×D₅₃	semidirect
393. C₂²⋊Q₈	C₂²×D₅₃	semidirect
397. C₄×Q₈	C₂²×D₅₃	semidirect
446. C₂×Q₈	D₄×D₅₃	direct (N × H)
454. C₂×D₄	Q₈×D₅₃	direct (N × H)
512. C₂³	C₂×Q₈×D₅₃	semidirect
519. Q₈	C₂×D₄×D₅₃	direct (N × H)
522. D₄	C₂×Q₈×D₅₃	direct (N × H)
542. C₂×C₄	C₂×Q₈×D₅₃	semidirect
561. C₂²	C₂²×Q₈×D₅₃	semidirect
571. C₄	C₂²×Q₈×D₅₃	semidirect
575. C₂	D₄×Q₈×D₅₃	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×Q₈×D₅₃	2
C₂×Q₈×D₂₁₂	2
C₂×D₄×Dic₁₀₆	2
C₂×#16811288	2
C₂×Q₈×C₂²⋊D₅₃	2
C₂×#16811286	2
D₄×Q₈×D₅₃	2
C₂×C₄⋊Q₈×D₅₃	2
C₂×C₂²⋊Q₈×D₅₃	2
C₂×C₄×Q₈×D₅₃	2
C₂×C₄×D₄×D₅₃	2
C₂×D₄×Q₈×C₅₃	2
C₂²×D₄×Q₈	53

Subgroups — C2×D4×Q8×D53 #28291004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×Q₈×D₅₃ #28291004