Subgroups — C₂²×#16811248 #28290813

Order 13568.

9310 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811248	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133197	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133196	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133195	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133195	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133194	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133193	C₂	1	2	yes	maximal, complemented
8	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal
10	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₀₆	C₂²	1	2	yes	complemented
13	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
14	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
16	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
17	#16811248	C₂²	1	256	yes	direct product factor
18	C₂×#13133197	C₂²	1	48	yes	complemented
19	C₂×#13133196	C₂²	1	24	yes	complemented
20	C₂×#13133195	C₂²	1	48	yes	complemented
21	C₂×#13133195	C₂²	1	48	yes	complemented
22	C₂×#13133194	C₂²	1	48	yes	complemented
23	C₂×#13133193	C₂²	1	48	yes	complemented
24	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
25	C₂²×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
26	C₂²×#11719919	C₂²	1	2	yes	complemented
27	C₂²×#11719919	C₂²	1	2	yes	complemented
28	C₂²×#11719919	C₂²	1	2	yes	complemented
29	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
30	C₂²×#11719921	C₂²	1	24	yes	—
31	C₂²×#11719921	C₂²	1	2	yes	—
32	C₂²×#11719921	C₂²	1	2	yes	—
33	C₂²×#11719921	C₂²	1	2	yes	—
34	C₂²×#11719921	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	—
36	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
37	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
39	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
41	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
42	C₂⁴×D₅₃	—	4	4	no	—
43	C₂⁴×D₅₃	—	4	4	no	—
44	C₂⁴×D₅₃	Q₈	1	16	yes	complemented
45	C₂⁴×D₅₃	C₂³	1	2	yes	—
46	C₂⁴×D₅₃	C₂³	1	12	yes	—
47	C₂²×Dic₁₀₆	—	4	4	no	—
48	C₂²×Dic₁₀₆	—	4	8	no	—
49	C₂²×Dic₁₀₆	C₂³	1	24	yes	complemented
50	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
51	C₂³×Dic₅₃	—	4	4	no	—
52	C₂³×Dic₅₃	C₂³	1	2	yes	—
53	C₂³×Dic₅₃	C₂³	1	2	yes	—
54	C₂³×Dic₅₃	C₂³	1	2	yes	—
55	C₂³×Dic₅₃	C₂³	1	12	yes	—
56	C₂²×C₄×D₅₃	—	4	8	no	—
57	C₂²×C₄×D₅₃	—	4	8	no	—
58	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
59	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
60	#13133197	C₂³	1	128	yes	complemented
61	#13133196	C₂³	1	64	yes	complemented
62	#13133195	C₂³	1	128	yes	complemented
63	#13133195	C₂³	1	128	yes	complemented
64	#13133194	C₂³	1	128	yes	complemented
65	#13133193	C₂³	1	128	yes	complemented
66	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
67	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
68	C₂×#11719919	C₂³	1	24	yes	complemented
69	C₂×#11719919	C₂³	1	24	yes	complemented
70	C₂×#11719919	C₂³	1	24	yes	complemented
71	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
72	C₂×#11719921	C₂³	1	24	yes	—
73	C₂×#11719921	C₂³	1	24	yes	—
74	C₂×#11719921	C₂³	1	24	yes	—
75	C₂×#11719921	C₂³	1	24	yes	—
76	C₂×#11719921	C₂³	1	64	yes	—
77	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	—
78	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₅₃	—	4	4	no	—
80	C₂²×Q₈×C₅₃	C₂³	1	12	yes	complemented
81	C₂³×C₄×C₅₃	—	4	4	no	—
82	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
83	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
84	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
87	C₂²⋊Q₈×C₅₃	C₂³	1	64	yes	complemented
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
91	C₂⁴×C₅₃	—	2	4	no	—
92	C₂⁴×C₅₃	—	2	4	no	—
93	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
94	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
95	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂³×D₅₃	—	2	8	no	—
97	C₂³×D₅₃	—	2	8	no	—
98	C₂³×D₅₃	—	4	64	no	—
99	C₂³×D₅₃	—	4	24	no	—
100	C₂³×D₅₃	—	4	24	no	—
101	C₂³×D₅₃	C₂×Q₈	1	16	yes	—
102	C₂³×D₅₃	C₂×Q₈	1	96	yes	complemented
103	C₂³×D₅₃	C₂⁴	1	12	yes	—
104	C₂³×D₅₃	C₂⁴	1	16	yes	—
105	C₂×Q₈×C₅₃	—	4	24	no	—
106	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
107	C₂²×C₄×C₅₃	—	2	4	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	8	no	—
110	C₂²×C₄×C₅₃	—	2	8	no	—
111	C₂²×C₄×C₅₃	—	4	24	no	—
112	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
114	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
115	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
117	C₂²×Dic₅₃	—	2	8	no	—
118	C₂²×Dic₅₃	—	2	8	no	—
119	C₂²×Dic₅₃	—	2	8	no	—
120	C₂²×Dic₅₃	—	4	24	no	—
121	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
122	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
123	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
124	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
125	C₂×C₄×D₅₃	—	4	48	no	—
126	C₂×C₄×D₅₃	—	4	48	no	—
127	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
128	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
129	C₂×Dic₁₀₆	—	4	24	no	—
130	C₂×Dic₁₀₆	—	4	48	no	—
131	C₂×Dic₁₀₆	C₂⁴	1	32	yes	complemented
132	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
133	C₄⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
134	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
135	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	—
136	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
137	#11719921	C₂⁴	1	32	yes	—
138	#11719921	C₂⁴	1	32	yes	—
139	#11719921	C₂⁴	1	32	yes	—
140	#11719921	C₂⁴	1	32	yes	—
141	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
142	C₄⋊Dic₅₃	C₂⁴	1	16	yes	complemented
143	#11719919	C₂⁴	1	32	yes	complemented
144	#11719919	C₂⁴	1	32	yes	complemented
145	#11719919	C₂⁴	1	32	yes	complemented
146	C₂³×C₅₃	—	2	32	no	—
147	C₂³×C₅₃	—	2	12	no	—
148	C₂³×C₅₃	—	2	12	no	—
149	C₂³×C₅₃	C₂²×Q₈	1	24	yes	—
150	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
151	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
152	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
153	C₂³×C₅₃	C₂⁵	1	4	yes	—
154	C₂³×C₅₃	C₂⁵	1	3	yes	—
155	C₂²×D₅₃	—	2	64	no	—
156	C₂²×D₅₃	—	2	24	no	—
157	C₂²×D₅₃	—	2	24	no	—
158	C₂²×D₅₃	—	4	192	no	—
159	C₂²×D₅₃	—	4	16	no	—
160	C₂²×D₅₃	—	4	16	no	—
161	C₂²×D₅₃	C₂²×Q₈	1	48	yes	—
162	C₂²×D₅₃	C₂²×Q₈	1	64	yes	complemented
163	C₂²×D₅₃	C₂⁵	1	8	yes	—
164	Q₈×C₅₃	—	4	16	no	—
165	C₂×C₄×C₅₃	—	2	24	no	—
166	C₂×C₄×C₅₃	—	2	12	no	—
167	C₂×C₄×C₅₃	—	2	24	no	—
168	C₂×C₄×C₅₃	—	2	12	no	—
169	C₂×C₄×C₅₃	—	4	16	no	—
170	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
171	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
172	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
173	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
174	C₂×Dic₅₃	—	2	24	no	—
175	C₂×Dic₅₃	—	2	24	no	—
176	C₂×Dic₅₃	—	2	24	no	—
177	C₂×Dic₅₃	—	4	16	no	—
178	C₂×Dic₅₃	C₂⁵	1	8	yes	—
179	C₂×Dic₅₃	C₂⁵	1	8	yes	—
180	C₂×Dic₅₃	C₂⁵	1	8	yes	—
181	C₄×D₅₃	—	4	32	no	—
182	C₄×D₅₃	—	4	32	no	—
183	Dic₁₀₆	—	4	16	no	—
184	Dic₁₀₆	—	4	32	no	—
185	C₂²×#543	—	53	1	no	maximal, Sylow
186	C₂²×C₅₃	—	2	4	no	—
187	C₂²×C₅₃	—	2	4	no	—
188	C₂²×C₅₃	—	2	48	no	—
189	C₂²×C₅₃	#543	1	16	yes	complemented
190	C₂²×C₅₃	C₂³×Q₈	1	8	yes	—
191	C₂²×C₅₃	C₂³×Q₈	1	6	yes	—
192	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
193	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
194	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
195	C₂×D₅₃	—	2	8	no	—
196	C₂×D₅₃	—	2	8	no	—
197	C₂×D₅₃	—	2	96	no	—
198	C₂×D₅₃	—	4	64	no	—
199	C₂×D₅₃	C₂³×Q₈	1	16	yes	—
200	C₄×C₅₃	—	2	8	no	—
201	C₄×C₅₃	—	2	4	no	—
202	C₄×C₅₃	—	2	8	no	—
203	C₄×C₅₃	—	2	4	no	—
204	Dic₅₃	—	2	8	no	—
205	Dic₅₃	—	2	8	no	—
206	Dic₅₃	—	2	8	no	—
207	C₂×#543	—	53	48	no	—
208	C₂²×C₂²⋊Q₈	—	53	2	no	—
209	C₂²×C₂²⋊Q₈	—	53	1	no	—
210	C₂²×C₂²⋊Q₈	—	53	2	no	—
211	C₂²×C₂²⋊Q₈	—	53	2	no	—
212	C₂²×C₂²⋊Q₈	—	53	2	no	—
213	C₂²×C₂²⋊Q₈	—	53	2	no	—
214	C₂²×C₂²⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
215	C₂³×C₂²⋊C₄	—	53	1	no	—
216	C₂³×C₂²⋊C₄	—	53	2	no	—
217	C₂×C₅₃	—	2	8	no	—
218	C₂×C₅₃	C₂×#543	1	12	yes	complemented
219	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
220	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
222	D₅₃	—	2	16	no	—
223	C₂⁶	—	53	1	no	—
224	C₂³×Q₈	—	53	1	no	—
225	C₂³×Q₈	—	53	2	no	—
226	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
227	C₂⁴×C₄	—	53	1	no	—
228	C₂⁴×C₄	—	53	2	no	—
229	C₂⁴×C₄	—	53	2	no	—
230	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
231	#543	—	53	256	no	—
232	C₂×C₂²⋊Q₈	—	53	48	no	—
233	C₂×C₂²⋊Q₈	—	53	24	no	—
234	C₂×C₂²⋊Q₈	—	53	48	no	—
235	C₂×C₂²⋊Q₈	—	53	48	no	—
236	C₂×C₂²⋊Q₈	—	53	48	no	—
237	C₂×C₂²⋊Q₈	—	53	48	no	—
238	C₂×C₂²⋊Q₈	C₂×D₅₃	1	24	yes	complemented
239	C₂²×C₄⋊C₄	—	53	2	no	—
240	C₂²×C₄⋊C₄	—	53	2	no	—
241	C₂²×C₄⋊C₄	—	53	2	no	—
242	C₂²×C₄⋊C₄	—	53	1	no	—
243	C₂²×C₄⋊C₄	—	53	2	no	—
244	C₂²×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
245	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
246	C₂²×C₂²⋊C₄	—	53	2	no	—
247	C₂²×C₂²⋊C₄	—	53	2	no	—
248	C₂²×C₂²⋊C₄	—	53	2	no	—
249	C₂²×C₂²⋊C₄	—	53	2	no	—
250	C₂²×C₂²⋊C₄	—	53	2	no	—
251	C₂²×C₂²⋊C₄	—	53	24	no	—
252	C₂²×C₂²⋊C₄	—	53	48	no	—
253	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
254	C₅₃	C₂²×#543	1	1	yes	characteristic, complemented, Sylow
255	C₂⁵	—	53	2	no	—
256	C₂⁵	—	53	16	no	—
257	C₂⁵	—	53	12	no	—
258	C₂⁵	—	212	4	no	—
259	C₂⁵	—	212	4	no	—
260	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
261	C₂²×Q₈	—	4	4	no	—
262	C₂²×Q₈	—	53	12	no	—
263	C₂²×Q₈	—	53	24	no	—
264	C₂²×Q₈	—	212	4	no	—
265	C₂²×Q₈	—	212	8	no	—
266	C₂²×Q₈	C₂²×D₅₃	1	12	yes	complemented
267	C₂³×C₄	—	4	4	no	—
268	C₂³×C₄	—	53	2	no	—
269	C₂³×C₄	—	53	2	no	—
270	C₂³×C₄	—	53	2	no	—
271	C₂³×C₄	—	53	12	no	—
272	C₂³×C₄	—	53	24	no	—
273	C₂³×C₄	—	53	24	no	—
274	C₂³×C₄	—	212	8	no	—
275	C₂³×C₄	—	212	8	no	—
276	C₂³×C₄	—	212	4	no	—
277	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
278	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
279	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
280	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
281	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
282	C₂²⋊Q₈	—	53	128	no	—
283	C₂²⋊Q₈	—	53	64	no	—
284	C₂²⋊Q₈	—	53	128	no	—
285	C₂²⋊Q₈	—	53	128	no	—
286	C₂²⋊Q₈	—	53	128	no	—
287	C₂²⋊Q₈	—	53	128	no	—
288	C₂²⋊Q₈	C₂²×D₅₃	1	64	yes	complemented
289	C₂×C₄⋊C₄	—	53	24	no	—
290	C₂×C₄⋊C₄	—	53	24	no	—
291	C₂×C₄⋊C₄	—	53	24	no	—
292	C₂×C₄⋊C₄	—	53	12	no	—
293	C₂×C₄⋊C₄	—	53	24	no	—
294	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
295	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
296	C₂×C₂²⋊C₄	—	53	24	no	—
297	C₂×C₂²⋊C₄	—	53	24	no	—
298	C₂×C₂²⋊C₄	—	53	24	no	—
299	C₂×C₂²⋊C₄	—	53	24	no	—
300	C₂×C₂²⋊C₄	—	53	24	no	—
301	C₂×C₂²⋊C₄	—	53	64	no	—
302	C₂×C₂²⋊C₄	—	53	128	no	—
303	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
304	C₂⁴	—	2	4	no	—
305	C₂⁴	—	2	4	no	—
306	C₂⁴	—	53	16	no	—
307	C₂⁴	—	53	12	no	—
308	C₂⁴	—	53	96	no	—
309	C₂⁴	—	53	16	no	—
310	C₂⁴	—	106	8	no	—
311	C₂⁴	—	106	8	no	—
312	C₂⁴	—	212	24	no	—
313	C₂⁴	—	212	24	no	—
314	C₂⁴	—	212	64	no	—
315	C₂⁴	Q₈×D₅₃	1	8	yes	—
316	C₂⁴	C₂³×D₅₃	1	6	yes	—
317	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
318	C₂×Q₈	—	4	24	no	—
319	C₂×Q₈	—	53	16	no	—
320	C₂×Q₈	—	53	32	no	—
321	C₂×Q₈	—	212	24	no	—
322	C₂×Q₈	—	212	48	no	—
323	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	2	4	no	—
326	C₂²×C₄	—	2	8	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	4	24	no	—
329	C₂²×C₄	—	53	12	no	—
330	C₂²×C₄	—	53	12	no	—
331	C₂²×C₄	—	53	12	no	—
332	C₂²×C₄	—	53	16	no	—
333	C₂²×C₄	—	53	32	no	—
334	C₂²×C₄	—	53	32	no	—
335	C₂²×C₄	—	106	8	no	—
336	C₂²×C₄	—	106	8	no	—
337	C₂²×C₄	—	106	8	no	—
338	C₂²×C₄	—	212	48	no	—
339	C₂²×C₄	—	212	48	no	—
340	C₂²×C₄	—	212	24	no	—
341	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
342	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
343	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
344	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
345	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
346	C₄⋊C₄	—	53	32	no	—
347	C₄⋊C₄	—	53	32	no	—
348	C₄⋊C₄	—	53	32	no	—
349	C₄⋊C₄	—	53	16	no	—
350	C₄⋊C₄	—	53	32	no	—
351	C₄⋊C₄	C₂³×D₅₃	1	16	yes	complemented
352	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
353	C₂²⋊C₄	—	53	32	no	—
354	C₂²⋊C₄	—	53	32	no	—
355	C₂²⋊C₄	—	53	32	no	—
356	C₂²⋊C₄	—	53	32	no	—
357	C₂²⋊C₄	—	53	32	no	—
358	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	—
359	C₂³	—	2	12	no	—
360	C₂³	—	2	12	no	—
361	C₂³	—	2	32	no	—
362	C₂³	—	53	48	no	—
363	C₂³	—	53	8	no	—
364	C₂³	—	53	64	no	—
365	C₂³	—	106	24	no	—
366	C₂³	—	106	24	no	—
367	C₂³	—	106	64	no	—
368	C₂³	—	212	192	no	—
369	C₂³	—	212	16	no	—
370	C₂³	—	212	16	no	—
371	C₂³	C₂×Q₈×D₅₃	1	24	yes	—
372	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
373	C₂³	#13133262	1	4	yes	—
374	C₂³	#13133263	1	4	yes	—
375	C₂³	C₂⁴×D₅₃	1	4	yes	—
376	C₂³	C₂⁴×D₅₃	1	3	yes	—
377	Q₈	—	4	16	no	—
378	Q₈	—	212	16	no	—
379	Q₈	—	212	32	no	—
380	C₂×C₄	—	2	12	no	—
381	C₂×C₄	—	2	12	no	—
382	C₂×C₄	—	2	24	no	—
383	C₂×C₄	—	2	24	no	—
384	C₂×C₄	—	4	16	no	—
385	C₂×C₄	—	53	8	no	—
386	C₂×C₄	—	53	8	no	—
387	C₂×C₄	—	53	8	no	—
388	C₂×C₄	—	106	24	no	—
389	C₂×C₄	—	106	24	no	—
390	C₂×C₄	—	106	24	no	—
391	C₂×C₄	—	212	32	no	—
392	C₂×C₄	—	212	32	no	—
393	C₂×C₄	—	212	16	no	—
394	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
395	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
396	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
397	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
398	C₂²	—	2	48	no	—
399	C₂²	—	2	4	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	53	16	no	—
402	C₂²	—	106	96	no	—
403	C₂²	—	106	8	no	—
404	C₂²	—	106	8	no	—
405	C₂²	—	212	64	no	—
406	C₂²	#16811248	1	16	yes	direct product factor
407	C₂²	C₂²×Q₈×D₅₃	1	8	yes	—
408	C₂²	C₂²×Q₈×D₅₃	1	6	yes	—
409	C₂²	C₂×#13133262	1	6	yes	—
410	C₂²	C₂×#13133263	1	6	yes	—
411	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
412	C₄	—	2	4	no	—
413	C₄	—	2	4	no	—
414	C₄	—	2	8	no	—
415	C₄	—	2	8	no	—
416	C₄	—	106	8	no	—
417	C₄	—	106	8	no	—
418	C₄	—	106	8	no	—
419	C₂	—	2	8	no	—
420	C₂	—	106	16	no	—
421	C₂	C₂×#16811248	1	12	yes	direct product factor
422	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
423	C₂	C₂²×#13133262	1	1	yes	characteristic
424	C₂	C₂²×#13133263	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	C₂²×#543
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. C₂×#16811248	C₂	direct (N × H)
2. C₂²×#13133197	C₂	semidirect
3. C₂²×#13133196	C₂	semidirect
4. C₂²×#13133195	C₂	semidirect
5. C₂²×#13133195	C₂	semidirect
6. C₂²×#13133194	C₂	semidirect
7. C₂²×#13133193	C₂	semidirect
10. C₂²×C₂²⋊Q₈×C₅₃	C₂	semidirect
12. C₂³×Dic₁₀₆	C₂²	semidirect
13. C₂³×Dic₁₀₆	C₂²	semidirect
17. #16811248	C₂²	direct (N × H)
18. C₂×#13133197	C₂²	semidirect
19. C₂×#13133196	C₂²	semidirect
20. C₂×#13133195	C₂²	semidirect
21. C₂×#13133195	C₂²	semidirect
22. C₂×#13133194	C₂²	semidirect
23. C₂×#13133193	C₂²	semidirect
24. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
25. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
26. C₂²×#11719919	C₂²	semidirect
27. C₂²×#11719919	C₂²	semidirect
28. C₂²×#11719919	C₂²	semidirect
36. C₂³×Q₈×C₅₃	C₂²	semidirect
38. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
39. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
40. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
44. C₂⁴×D₅₃	Q₈	semidirect
49. C₂²×Dic₁₀₆	C₂³	semidirect
50. C₂²×Dic₁₀₆	C₂³	semidirect
60. #13133197	C₂³	semidirect
61. #13133196	C₂³	semidirect
62. #13133195	C₂³	semidirect
63. #13133195	C₂³	semidirect
64. #13133194	C₂³	semidirect
65. #13133193	C₂³	semidirect
66. C₂×C₄⋊Dic₅₃	C₂³	semidirect
67. C₂×C₄⋊Dic₅₃	C₂³	semidirect
68. C₂×#11719919	C₂³	semidirect
69. C₂×#11719919	C₂³	semidirect
70. C₂×#11719919	C₂³	semidirect
80. C₂²×Q₈×C₅₃	C₂³	semidirect
87. C₂²⋊Q₈×C₅₃	C₂³	semidirect
88. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
89. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
102. C₂³×D₅₃	C₂×Q₈	semidirect
106. C₂×Q₈×C₅₃	C₂⁴	semidirect
131. C₂×Dic₁₀₆	C₂⁴	semidirect
132. C₂×Dic₁₀₆	C₂⁴	semidirect
133. C₄⋊C₄×C₅₃	C₂⁴	semidirect
134. C₄⋊C₄×C₅₃	C₂⁴	semidirect
141. C₄⋊Dic₅₃	C₂⁴	semidirect
142. C₄⋊Dic₅₃	C₂⁴	semidirect
143. #11719919	C₂⁴	semidirect
144. #11719919	C₂⁴	semidirect
145. #11719919	C₂⁴	semidirect
162. C₂²×D₅₃	C₂²×Q₈	semidirect
189. C₂²×C₅₃	#543	semidirect
214. C₂²×C₂²⋊Q₈	D₅₃	semidirect
218. C₂×C₅₃	C₂×#543	semidirect
226. C₂³×Q₈	C₂×D₅₃	semidirect
238. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
244. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
245. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
254. C₅₃	C₂²×#543	semidirect
266. C₂²×Q₈	C₂²×D₅₃	semidirect
288. C₂²⋊Q₈	C₂²×D₅₃	semidirect
294. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
295. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
323. C₂×Q₈	C₂³×D₅₃	semidirect
351. C₄⋊C₄	C₂³×D₅₃	semidirect
352. C₄⋊C₄	C₂³×D₅₃	semidirect
406. C₂²	#16811248	direct (N × H)
421. C₂	C₂×#16811248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811248	2
C₂²×#13133197	2
C₂²×#13133196	2
C₂²×#13133195	2
C₂²×#13133194	2
C₂²×#13133193	2
C₂³×#11719921	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₂²×#543	53

Subgroups — C22×#16811248 #28290813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811248 #28290813