Subgroups — C₂²×C₂²⋊Q₈×D₅₃ #28290731

Order 13568.

14078 conjugacy classes of subgroups.

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

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

Complements

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

N	H ≅ G/N	type
1. C₂³×Q₈×D₅₃	C₂	semidirect
3. C₂²×#13133197	C₂	semidirect
4. C₂²×#13133196	C₂	semidirect
5. C₂²×#13133195	C₂	semidirect
6. C₂²×#13133194	C₂	semidirect
7. C₂²×#13133193	C₂	semidirect
8. C₂×C₂²⋊Q₈×D₅₃	C₂	direct (N × H)
9. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
10. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
12. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
14. C₂³×Dic₁₀₆	C₂²	semidirect
15. C₂³×Dic₁₀₆	C₂²	semidirect
17. C₂²×Q₈×D₅₃	C₂²	semidirect
27. C₂×#13133197	C₂²	semidirect
28. C₂×#13133196	C₂²	semidirect
29. C₂×#13133195	C₂²	semidirect
30. C₂×#13133194	C₂²	semidirect
31. C₂×#13133193	C₂²	semidirect
32. C₂²⋊Q₈×D₅₃	C₂²	direct (N × H)
33. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
34. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
35. C₂²×#11719919	C₂²	semidirect
36. C₂²×#11719919	C₂²	semidirect
37. C₂²×#11719919	C₂²	semidirect
38. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
39. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
44. C₂³×Q₈×C₅₃	C₂²	semidirect
46. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
52. C₂⁴×D₅₃	Q₈	semidirect
58. C₂²×Dic₁₀₆	C₂³	semidirect
59. C₂²×Dic₁₀₆	C₂³	semidirect
61. C₂×Q₈×D₅₃	C₂³	semidirect
82. #13133197	C₂³	semidirect
83. #13133196	C₂³	semidirect
84. #13133195	C₂³	semidirect
85. #13133194	C₂³	semidirect
86. #13133193	C₂³	semidirect
87. C₂×C₄⋊Dic₅₃	C₂³	semidirect
88. C₂×C₄⋊Dic₅₃	C₂³	semidirect
89. C₂×#11719919	C₂³	semidirect
90. C₂×#11719919	C₂³	semidirect
91. C₂×#11719919	C₂³	semidirect
92. C₄⋊C₄×D₅₃	C₂³	semidirect
93. C₄⋊C₄×D₅₃	C₂³	semidirect
100. C₂²×Q₈×C₅₃	C₂³	semidirect
107. C₂²⋊Q₈×C₅₃	C₂³	semidirect
108. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
109. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
113. C₂⁴×C₅₃	C₂×Q₈	semidirect
123. C₂³×D₅₃	C₂×Q₈	semidirect
124. C₂³×D₅₃	C₂×Q₈	semidirect
130. C₂×Q₈×C₅₃	C₂⁴	semidirect
167. C₂×Dic₁₀₆	C₂⁴	semidirect
168. C₂×Dic₁₀₆	C₂⁴	semidirect
169. C₄⋊C₄×C₅₃	C₂⁴	semidirect
170. C₄⋊C₄×C₅₃	C₂⁴	semidirect
175. C₄⋊Dic₅₃	C₂⁴	semidirect
176. C₄⋊Dic₅₃	C₂⁴	semidirect
177. #11719919	C₂⁴	semidirect
178. #11719919	C₂⁴	semidirect
179. #11719919	C₂⁴	semidirect
185. C₂³×C₅₃	C₂²×Q₈	semidirect
195. C₂²×D₅₃	C₂²⋊Q₈	direct (N × H)
198. C₂²×D₅₃	C₂²×Q₈	semidirect
199. C₂²×D₅₃	C₂²×Q₈	semidirect
234. C₂²×C₅₃	C₂×C₂²⋊Q₈	semidirect
237. C₂²×C₅₃	C₂³×Q₈	semidirect
244. C₂×D₅₃	C₂×C₂²⋊Q₈	direct (N × H)
247. C₂×D₅₃	C₂³×Q₈	semidirect
265. C₂²×C₂²⋊Q₈	D₅₃	direct (N × H)
270. C₂×C₅₃	C₂²×C₂²⋊Q₈	semidirect
275. D₅₃	C₂²×C₂²⋊Q₈	direct (N × H)
281. C₂³×Q₈	C₂×D₅₃	semidirect
298. C₂×C₂²⋊Q₈	C₂×D₅₃	direct (N × H)
306. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
307. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
313. C₅₃	C₂³×C₂²⋊Q₈	semidirect
328. C₂²×Q₈	C₂²×D₅₃	semidirect
360. C₂²⋊Q₈	C₂²×D₅₃	direct (N × H)
368. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
369. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
390. C₂⁴	Q₈×D₅₃	semidirect
399. C₂×Q₈	C₂³×D₅₃	semidirect
438. C₄⋊C₄	C₂³×D₅₃	semidirect
439. C₄⋊C₄	C₂³×D₅₃	semidirect
463. C₂³	C₂×Q₈×D₅₃	semidirect
506. C₂²	C₂²⋊Q₈×D₅₃	direct (N × H)
509. C₂²	C₂²×Q₈×D₅₃	semidirect
523. C₂	C₂×C₂²⋊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
C₂⁴×C₄×D₅₃	2
C₂²×#13133197	2
C₂²×#13133196	2
C₂²×#13133195	2
C₂²×#13133194	2
C₂²×#13133193	2
C₂×C₂²⋊Q₈×D₅₃	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂³×C₂²⋊Q₈	53

Subgroups — C22×C22⋊Q8×D53 #28290731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊Q₈×D₅₃ #28290731