Subgroups — C₂²×#16811134 #28290749

Order 13568.

14334 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811134	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#13133200	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133196	C₂	1	2	yes	maximal, complemented
5	C₂²×#13133191	C₂	1	2	yes	maximal, complemented
6	C₂²×#13133186	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×C₂²⋊D₅₃	C₂	1	4	yes	maximal, complemented
8	C₂²×#13133176	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719945	—	2	16	no	—
11	C₂²×#11719945	C₂²	1	24	yes	complemented
12	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
14	C₂³×C₂²⋊D₅₃	C₂²	1	4	yes	—
15	C₂³×C₂²⋊D₅₃	C₂²	1	2	yes	—
16	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
17	#16811134	C₂²	1	256	yes	direct product factor
18	C₂×#13133200	C₂²	1	48	yes	complemented
19	C₂×#13133196	C₂²	1	48	yes	complemented
20	C₂×#13133191	C₂²	1	48	yes	complemented
21	C₂×#13133186	C₂²	1	48	yes	complemented
22	C₂×C₄×C₂²⋊D₅₃	C₂²	1	96	yes	complemented
23	C₂×#13133176	C₂²	1	24	yes	complemented
24	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
25	C₂²×#11719919	C₂²	1	4	yes	complemented
26	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	—
27	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
28	C₂²×#11719921	C₂²	1	4	yes	complemented
29	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	complemented
30	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
31	C₂⁴×C₄×C₅₃	—	2	16	no	—
32	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
33	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
34	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂⁴×C₄×C₅₃	C₂²	1	24	yes	complemented
37	C₂⁴×D₅₃	C₂³	1	2	yes	—
38	C₂×#11719945	—	2	192	no	—
39	C₂×#11719945	C₂³	1	64	yes	complemented
40	C₂²×Dic₁₀₆	—	2	8	no	—
41	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
42	C₂²×D₂₁₂	—	2	8	no	—
43	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
44	C₂²×C₂²⋊D₅₃	—	2	16	no	—
45	C₂²×C₂²⋊D₅₃	—	2	32	no	—
46	C₂²×C₂²⋊D₅₃	C₂³	1	48	yes	—
47	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	—
48	C₂³×Dic₅₃	C₂³	1	4	yes	—
49	C₂³×Dic₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×D₅₃	—	2	16	no	—
51	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
52	#13133200	C₂³	1	128	yes	complemented
53	#13133196	C₂³	1	128	yes	complemented
54	#13133191	C₂³	1	128	yes	complemented
55	#13133186	C₂³	1	128	yes	complemented
56	C₄×C₂²⋊D₅₃	C₂³	1	256	yes	complemented
57	#13133176	C₂³	1	64	yes	complemented
58	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
59	C₂×#11719919	C₂³	1	48	yes	complemented
60	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
61	C₂×C₂²⋊Dic₅₃	C₂³	1	48	yes	—
62	C₂×#11719921	C₂³	1	48	yes	complemented
63	C₂×C₄×Dic₅₃	C₂³	1	24	yes	complemented
64	C₂⁵×C₅₃	—	2	8	no	—
65	C₂⁵×C₅₃	—	2	16	no	—
66	C₂⁵×C₅₃	C₂³	1	12	yes	—
67	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
68	C₂⁵×C₅₃	C₂³	1	2	yes	—
69	C₂³×C₄×C₅₃	—	2	8	no	—
70	C₂³×C₄×C₅₃	—	2	16	no	—
71	C₂³×C₄×C₅₃	—	2	16	no	—
72	C₂³×C₄×C₅₃	—	2	192	no	—
73	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
74	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
76	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
77	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
78	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
81	C₂³×C₄×C₅₃	C₂³	1	64	yes	complemented
82	C₂⁴×C₅₃	—	2	128	no	—
83	C₂⁴×C₅₃	—	2	48	no	—
84	C₂⁴×C₅₃	—	2	8	no	—
85	C₂⁴×C₅₃	—	2	8	no	—
86	C₂⁴×C₅₃	—	2	8	no	—
87	C₂⁴×C₅₃	—	2	96	no	—
88	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
89	C₂⁴×C₅₃	#41	1	16	yes	complemented
90	C₂⁴×C₅₃	C₂⁴	1	16	yes	—
91	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
92	C₂⁴×C₅₃	C₂⁴	1	12	yes	—
93	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₂³×D₅₃	—	2	8	no	—
95	C₂³×D₅₃	—	2	16	no	—
96	C₂³×D₅₃	C₂⁴	1	12	yes	—
97	C₂²×C₄×C₅₃	—	2	48	no	—
98	C₂²×C₄×C₅₃	—	2	8	no	—
99	C₂²×C₄×C₅₃	—	2	96	no	—
100	C₂²×C₄×C₅₃	—	2	96	no	—
101	C₂²×C₄×C₅₃	—	2	256	no	—
102	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
103	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
104	C₂²×C₄×C₅₃	C₂×D₄	1	96	yes	complemented
105	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
106	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	32	yes	—
108	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
109	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
112	C₂×C₂²⋊D₅₃	—	2	96	no	—
113	C₂×C₂²⋊D₅₃	—	2	192	no	—
114	C₂×C₂²⋊D₅₃	C₂⁴	1	64	yes	—
115	C₂×C₂²⋊D₅₃	C₂⁴	1	32	yes	—
116	C₂²×Dic₅₃	—	2	8	no	—
117	C₂²×Dic₅₃	—	2	16	no	—
118	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
119	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
120	#11719945	—	2	256	no	—
121	C₂×D₂₁₂	—	2	48	no	—
122	C₂×D₂₁₂	C₂⁴	1	16	yes	complemented
123	C₂×C₄×D₅₃	—	2	96	no	—
124	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
125	C₂×Dic₁₀₆	—	2	48	no	—
126	C₂×Dic₁₀₆	C₂⁴	1	16	yes	complemented
127	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	—
128	C₂²⋊Dic₅₃	C₂⁴	1	64	yes	—
129	#11719921	C₂⁴	1	64	yes	complemented
130	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
131	#11719919	C₂⁴	1	64	yes	complemented
132	C₄×Dic₅₃	C₂⁴	1	32	yes	complemented
133	C₂³×C₅₃	—	2	384	no	—
134	C₂³×C₅₃	—	2	32	no	—
135	C₂³×C₅₃	—	2	64	no	—
136	C₂³×C₅₃	—	2	32	no	—
137	C₂³×C₅₃	—	2	24	no	—
138	C₂³×C₅₃	—	2	24	no	—
139	C₂³×C₅₃	—	2	24	no	—
140	C₂³×C₅₃	—	2	64	no	—
141	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
142	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
143	C₂³×C₅₃	C₂×#41	1	8	yes	—
144	C₂³×C₅₃	C₂×#41	1	48	yes	complemented
145	C₂³×C₅₃	C₂⁵	1	4	yes	—
146	C₂³×C₅₃	C₂⁵	1	8	yes	—
147	C₂³×C₅₃	C₂⁵	1	3	yes	—
148	C₂²×D₅₃	—	2	24	no	—
149	C₂²×D₅₃	—	2	48	no	—
150	C₂²×D₅₃	—	4	32	no	—
151	C₂²×D₅₃	C₂⁵	1	8	yes	—
152	C₂×C₄×C₅₃	—	2	32	no	—
153	C₂×C₄×C₅₃	—	2	24	no	—
154	C₂×C₄×C₅₃	—	2	64	no	—
155	C₂×C₄×C₅₃	—	2	64	no	—
156	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
158	C₂×C₄×C₅₃	C₂²×D₄	1	64	yes	complemented
159	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
160	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
161	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
162	C₂²⋊D₅₃	—	2	64	no	—
163	C₂²⋊D₅₃	—	2	128	no	—
164	C₂×Dic₅₃	—	2	24	no	—
165	C₂×Dic₅₃	—	2	48	no	—
166	C₂×Dic₅₃	C₂⁵	1	8	yes	—
167	C₂×Dic₅₃	C₂⁵	1	16	yes	—
168	D₂₁₂	—	2	32	no	—
169	C₄×D₅₃	—	2	64	no	—
170	Dic₁₀₆	—	2	32	no	—
171	C₂²×#525	—	53	1	no	maximal, Sylow
172	C₂²×C₅₃	—	2	128	no	—
173	C₂²×C₅₃	—	2	8	no	—
174	C₂²×C₅₃	—	2	8	no	—
175	C₂²×C₅₃	—	2	8	no	—
176	C₂²×C₅₃	—	2	48	no	—
177	C₂²×C₅₃	—	2	96	no	—
178	C₂²×C₅₃	#525	1	16	yes	complemented
179	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
180	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
181	C₂²×C₅₃	C₂²×#41	1	12	yes	—
182	C₂²×C₅₃	C₂²×#41	1	16	yes	complemented
183	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
184	C₂×D₅₃	—	2	8	no	—
185	C₂×D₅₃	—	2	16	no	—
186	C₂×D₅₃	—	4	48	no	—
187	C₄×C₅₃	—	2	8	no	—
188	C₄×C₅₃	C₂³×D₄	1	8	yes	—
189	C₄×C₅₃	C₂³×D₄	1	8	yes	—
190	Dic₅₃	—	2	8	no	—
191	Dic₅₃	—	2	16	no	—
192	C₂³×#41	—	53	1	no	—
193	C₂⁵×C₄	D₅₃	1	1	yes	characteristic, complemented
194	C₂×#525	—	53	48	no	—
195	C₂²×#123	—	53	2	no	—
196	C₂²×C₂²⋊Q₈	—	53	2	no	—
197	C₂²×#121	—	53	2	no	—
198	C₂²×#120	—	53	2	no	—
199	C₂²×C₄×D₄	—	53	4	no	—
200	C₂²×#117	—	53	1	no	—
201	C₂×C₅₃	—	2	8	no	—
202	C₂×C₅₃	—	2	16	no	—
203	C₂×C₅₃	C₂×#525	1	12	yes	complemented
204	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
205	C₂×C₅₃	C₂³×#41	1	2	yes	—
206	D₅₃	—	4	8	no	—
207	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
208	C₂²×#41	—	53	24	no	—
209	C₂²×#41	—	106	16	no	—
210	C₂³×Q₈	—	53	1	no	—
211	C₂³×D₄	—	53	2	no	—
212	C₂³×D₄	—	53	4	no	—
213	C₂³×D₄	—	53	1	no	—
214	C₂⁴×C₄	—	2	16	no	—
215	C₂⁴×C₄	—	53	2	no	—
216	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
217	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
218	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
219	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
220	C₂⁴×C₄	C₂×D₅₃	1	24	yes	complemented
221	#525	—	53	256	no	—
222	C₂×#123	—	53	48	no	—
223	C₂×C₂²⋊Q₈	—	53	48	no	—
224	C₂×#121	—	53	48	no	—
225	C₂×#120	—	53	48	no	—
226	C₂×C₄×D₄	—	53	96	no	—
227	C₂×#117	—	53	24	no	—
228	C₂²×C₄⋊C₄	—	53	4	no	—
229	C₂²×C₄⋊C₄	—	53	2	no	—
230	C₂²×C₂²⋊C₄	—	53	2	no	—
231	C₂²×C₂²⋊C₄	—	53	4	no	—
232	C₂²×C₂²⋊C₄	—	53	4	no	—
233	C₂²×C₄²	—	53	2	no	—
234	C₅₃	C₂²×#525	1	1	yes	characteristic, complemented, Sylow
235	C₂⁵	—	2	16	no	—
236	C₂⁵	—	2	8	no	—
237	C₂⁵	—	53	2	no	—
238	C₂⁵	C₂²×D₅₃	1	12	yes	—
239	C₂⁵	C₂²×D₅₃	1	2	yes	—
240	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×#41	—	53	64	no	—
242	C₂×#41	—	106	192	no	—
243	C₂²×Q₈	—	53	12	no	—
244	C₂²×Q₈	—	106	8	no	—
245	C₂²×D₄	—	53	48	no	—
246	C₂²×D₄	—	53	24	no	—
247	C₂²×D₄	—	53	12	no	—
248	C₂²×D₄	—	106	32	no	—
249	C₂²×D₄	—	106	16	no	—
250	C₂²×D₄	—	106	8	no	—
251	C₂³×C₄	—	2	8	no	—
252	C₂³×C₄	—	2	16	no	—
253	C₂³×C₄	—	2	16	no	—
254	C₂³×C₄	—	2	192	no	—
255	C₂³×C₄	—	53	4	no	—
256	C₂³×C₄	—	53	2	no	—
257	C₂³×C₄	—	53	24	no	—
258	C₂³×C₄	—	106	16	no	—
259	C₂³×C₄	C₂²⋊D₅₃	1	16	yes	complemented
260	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
261	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
262	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
263	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
264	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
265	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
266	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
267	C₂³×C₄	C₂²×D₅₃	1	64	yes	complemented
268	#123	—	53	128	no	—
269	C₂²⋊Q₈	—	53	128	no	—
270	#121	—	53	128	no	—
271	#120	—	53	128	no	—
272	C₄×D₄	—	53	256	no	—
273	#117	—	53	64	no	—
274	C₂×C₄⋊C₄	—	53	24	no	—
275	C₂×C₄⋊C₄	—	53	48	no	—
276	C₂×C₂²⋊C₄	—	53	24	no	—
277	C₂×C₂²⋊C₄	—	53	48	no	—
278	C₂×C₂²⋊C₄	—	53	48	no	—
279	C₂×C₄²	—	53	24	no	—
280	C₂⁴	—	2	128	no	—
281	C₂⁴	—	2	48	no	—
282	C₂⁴	—	2	8	no	—
283	C₂⁴	—	2	8	no	—
284	C₂⁴	—	2	8	no	—
285	C₂⁴	—	2	96	no	—
286	C₂⁴	—	53	12	no	—
287	C₂⁴	—	106	8	no	—
288	C₂⁴	—	106	16	no	—
289	C₂⁴	#11719945	1	16	yes	complemented
290	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
291	C₂⁴	C₂³×D₅₃	1	16	yes	—
292	C₂⁴	C₂³×D₅₃	1	6	yes	—
293	C₂⁴	C₂³×D₅₃	1	12	yes	—
294	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
295	#41	—	106	256	no	—
296	C₂×Q₈	—	53	16	no	—
297	C₂×Q₈	—	106	48	no	—
298	C₂×D₄	—	53	64	no	—
299	C₂×D₄	—	53	32	no	—
300	C₂×D₄	—	53	16	no	—
301	C₂×D₄	—	106	96	no	—
302	C₂×D₄	—	106	192	no	—
303	C₂×D₄	—	106	48	no	—
304	C₂²×C₄	—	2	48	no	—
305	C₂²×C₄	—	2	8	no	—
306	C₂²×C₄	—	2	96	no	—
307	C₂²×C₄	—	2	96	no	—
308	C₂²×C₄	—	2	256	no	—
309	C₂²×C₄	—	53	24	no	—
310	C₂²×C₄	—	53	12	no	—
311	C₂²×C₄	—	53	32	no	—
312	C₂²×C₄	—	106	8	no	—
313	C₂²×C₄	—	106	16	no	—
314	C₂²×C₄	—	106	96	no	—
315	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	—
316	C₂²×C₄	C₂×C₂²⋊D₅₃	1	8	yes	—
317	C₂²×C₄	C₂×C₂²⋊D₅₃	1	96	yes	complemented
318	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
319	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
320	C₂²×C₄	C₂³×D₅₃	1	32	yes	—
321	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
322	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
323	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
324	C₂²×C₄	C₂³×D₅₃	1	16	yes	—
325	C₄⋊C₄	—	53	32	no	—
326	C₄⋊C₄	—	53	64	no	—
327	C₂²⋊C₄	—	53	32	no	—
328	C₂²⋊C₄	—	53	64	no	—
329	C₂²⋊C₄	—	53	64	no	—
330	C₄²	—	53	32	no	—
331	C₂³	—	2	384	no	—
332	C₂³	—	2	64	no	—
333	C₂³	—	2	32	no	—
334	C₂³	—	2	32	no	—
335	C₂³	—	2	24	no	—
336	C₂³	—	2	24	no	—
337	C₂³	—	2	24	no	—
338	C₂³	—	2	64	no	—
339	C₂³	—	53	8	no	—
340	C₂³	—	106	24	no	—
341	C₂³	—	106	48	no	—
342	C₂³	—	212	32	no	—
343	C₂³	C₂²×C₂²⋊D₅₃	1	24	yes	—
344	C₂³	C₂²×C₂²⋊D₅₃	1	4	yes	—
345	C₂³	C₂×#11719945	1	8	yes	—
346	C₂³	C₂×#11719945	1	48	yes	complemented
347	C₂³	C₂⁴×D₅₃	1	4	yes	—
348	C₂³	C₂⁴×D₅₃	1	8	yes	—
349	C₂³	C₂⁴×D₅₃	1	3	yes	—
350	Q₈	—	106	32	no	—
351	D₄	—	106	64	no	—
352	D₄	—	106	128	no	—
353	D₄	—	106	32	no	—
354	C₂×C₄	—	2	32	no	—
355	C₂×C₄	—	2	24	no	—
356	C₂×C₄	—	2	64	no	—
357	C₂×C₄	—	2	64	no	—
358	C₂×C₄	—	53	16	no	—
359	C₂×C₄	—	53	8	no	—
360	C₂×C₄	—	106	24	no	—
361	C₂×C₄	—	106	48	no	—
362	C₂×C₄	—	106	64	no	—
363	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
364	C₂×C₄	C₂²×C₂²⋊D₅₃	1	24	yes	—
365	C₂×C₄	C₂²×C₂²⋊D₅₃	1	64	yes	complemented
366	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
367	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
368	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
369	C₂²	—	2	128	no	—
370	C₂²	—	2	48	no	—
371	C₂²	—	2	96	no	—
372	C₂²	—	2	8	no	—
373	C₂²	—	2	8	no	—
374	C₂²	—	2	8	no	—
375	C₂²	—	106	8	no	—
376	C₂²	—	106	16	no	—
377	C₂²	—	212	48	no	—
378	C₂²	#16811134	1	16	yes	direct product factor
379	C₂²	C₂³×C₂²⋊D₅₃	1	8	yes	—
380	C₂²	C₂³×C₂²⋊D₅₃	1	6	yes	—
381	C₂²	C₂²×#11719945	1	12	yes	—
382	C₂²	C₂²×#11719945	1	16	yes	complemented
383	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
384	C₄	—	2	8	no	—
385	C₄	—	106	8	no	—
386	C₄	—	106	16	no	—
387	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
388	C₄	C₂³×C₂²⋊D₅₃	1	8	yes	—
389	C₂	—	2	8	no	—
390	C₂	—	2	16	no	—
391	C₂	—	212	8	no	—
392	C₂	C₂×#16811134	1	12	yes	direct product factor
393	C₂	C₂⁴×C₂²⋊D₅₃	1	1	yes	characteristic
394	C₂	C₂³×#11719945	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂²×#525
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11719945	C₂	semidirect
2. C₂×#16811134	C₂	direct (N × H)
3. C₂²×#13133200	C₂	semidirect
4. C₂²×#13133196	C₂	semidirect
5. C₂²×#13133191	C₂	semidirect
6. C₂²×#13133186	C₂	semidirect
7. C₂²×C₄×C₂²⋊D₅₃	C₂	semidirect
8. C₂²×#13133176	C₂	semidirect
9. C₂⁵×C₄×C₅₃	C₂	semidirect
11. C₂²×#11719945	C₂²	semidirect
12. C₂³×Dic₁₀₆	C₂²	semidirect
13. C₂³×D₂₁₂	C₂²	semidirect
16. C₂³×C₄×D₅₃	C₂²	semidirect
17. #16811134	C₂²	direct (N × H)
18. C₂×#13133200	C₂²	semidirect
19. C₂×#13133196	C₂²	semidirect
20. C₂×#13133191	C₂²	semidirect
21. C₂×#13133186	C₂²	semidirect
22. C₂×C₄×C₂²⋊D₅₃	C₂²	semidirect
23. C₂×#13133176	C₂²	semidirect
24. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
25. C₂²×#11719919	C₂²	semidirect
28. C₂²×#11719921	C₂²	semidirect
29. C₂²×C₄×Dic₅₃	C₂²	semidirect
36. C₂⁴×C₄×C₅₃	C₂²	semidirect
39. C₂×#11719945	C₂³	semidirect
41. C₂²×Dic₁₀₆	C₂³	semidirect
43. C₂²×D₂₁₂	C₂³	semidirect
51. C₂²×C₄×D₅₃	C₂³	semidirect
52. #13133200	C₂³	semidirect
53. #13133196	C₂³	semidirect
54. #13133191	C₂³	semidirect
55. #13133186	C₂³	semidirect
56. C₄×C₂²⋊D₅₃	C₂³	semidirect
57. #13133176	C₂³	semidirect
58. C₂×C₄⋊Dic₅₃	C₂³	semidirect
59. C₂×#11719919	C₂³	semidirect
62. C₂×#11719921	C₂³	semidirect
63. C₂×C₄×Dic₅₃	C₂³	semidirect
73. C₂³×C₄×C₅₃	D₄	semidirect
81. C₂³×C₄×C₅₃	C₂³	semidirect
89. C₂⁴×C₅₃	#41	semidirect
104. C₂²×C₄×C₅₃	C₂×D₄	semidirect
122. C₂×D₂₁₂	C₂⁴	semidirect
124. C₂×C₄×D₅₃	C₂⁴	semidirect
126. C₂×Dic₁₀₆	C₂⁴	semidirect
129. #11719921	C₂⁴	semidirect
130. C₄⋊Dic₅₃	C₂⁴	semidirect
131. #11719919	C₂⁴	semidirect
132. C₄×Dic₅₃	C₂⁴	semidirect
144. C₂³×C₅₃	C₂×#41	semidirect
158. C₂×C₄×C₅₃	C₂²×D₄	semidirect
178. C₂²×C₅₃	#525	semidirect
182. C₂²×C₅₃	C₂²×#41	semidirect
193. C₂⁵×C₄	D₅₃	semidirect
203. C₂×C₅₃	C₂×#525	semidirect
220. C₂⁴×C₄	C₂×D₅₃	semidirect
234. C₅₃	C₂²×#525	semidirect
259. C₂³×C₄	C₂²⋊D₅₃	semidirect
267. C₂³×C₄	C₂²×D₅₃	semidirect
289. C₂⁴	#11719945	semidirect
317. C₂²×C₄	C₂×C₂²⋊D₅₃	semidirect
346. C₂³	C₂×#11719945	semidirect
365. C₂×C₄	C₂²×C₂²⋊D₅₃	semidirect
378. C₂²	#16811134	direct (N × H)
382. C₂²	C₂²×#11719945	semidirect
392. C₂	C₂×#16811134	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×#11719945	2
C₂×#16811134	2
C₂²×#13133200	2
C₂²×#13133196	2
C₂²×#13133191	2
C₂²×#13133186	2
C₂²×C₄×C₂²⋊D₅₃	2
C₂²×#13133176	2
C₂⁵×C₄×C₅₃	2
C₂²×#525	53

Subgroups — C22×#16811134 #28290749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811134 #28290749