Subgroups — #34109256

Order 20096.

398 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14578953	C₂	1	1	yes	maximal, characteristic
2	#23757402	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757401	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757347	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757345	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757342	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#23757336	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757328	C₂	1	1	yes	maximal, characteristic, complemented
10	#23757327	C₂	1	1	yes	maximal, characteristic, complemented
11	#23757323	C₂	1	1	yes	maximal, characteristic, complemented
12	#23757296	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#19097700	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄⋊C₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14578953	C₄	1	1	yes	characteristic
17	C₂×#14578938	C₂²	1	1	yes	characteristic
18	C₂²×#11929434	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₅₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
22	#19097736	C₄	1	1	yes	characteristic
23	D₄×Dic₁₅₇	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₅₇	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₅₇	C₂²	1	1	yes	characteristic
26	#19097731	C₄	1	1	yes	characteristic
27	#19097731	C₄	1	1	yes	characteristic
28	#19097729	C₄	1	1	yes	characteristic
29	#19097727	C₄	1	1	yes	characteristic
30	#19097727	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
33	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
34	C₄⋊#11929434	C₂²	1	1	yes	characteristic
35	#19097700	C₂²	1	1	yes	characteristic
36	#19097700	C₂²	1	1	yes	characteristic
37	#19097700	C₂²	1	1	yes	characteristic
38	#19097700	C₂²	1	1	yes	characteristic
39	#19097700	C₂²	1	1	yes	characteristic
40	#19097700	C₂²	1	1	yes	characteristic
41	#19097700	C₂²	1	1	yes	characteristic
42	#19097700	C₂²	1	1	yes	characteristic
43	C₂²⋊#11929434	C₂²	1	1	yes	characteristic
44	C₂²⋊#11929434	C₂²	1	1	yes	characteristic
45	#19097683	C₂²	1	1	yes	characteristic
46	#19097683	C₂²	1	1	yes	characteristic
47	#19097682	C₂²	1	1	yes	characteristic
48	#19097681	C₂²	1	1	yes	characteristic
49	#19097681	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
58	#97×C₁₅₇	C₂²	1	1	yes	characteristic
59	#14578953	—	2	4	no	—
60	C₂×Dic₃₁₄	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₅₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
70	#14578938	—	4	1	no	—
71	C₂×#11929434	—	4	1	no	—
72	C₂×#11929434	C₂³	1	1	yes	characteristic
73	C₂×#11929434	C₂³	1	1	yes	characteristic
74	C₂×#11929434	C₂³	1	1	yes	characteristic
75	C₂×#11929434	C₂³	1	1	yes	characteristic
76	C₈×D₁₅₇	—	4	1	no	—
77	C₈×D₁₅₇	—	4	1	no	—
78	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
79	#14578933	C₂×C₄	1	1	yes	characteristic
80	#14578933	C₂×C₄	1	1	yes	characteristic
81	#14578933	C₂×C₄	1	1	yes	characteristic
82	#14578933	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
85	#14578931	C₂×C₄	1	1	yes	characteristic
86	#14578931	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₅₇	—	4	1	no	—
100	C₂×C₈×C₁₅₇	—	4	1	no	—
101	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
108	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
109	C₂²×D₁₅₇	C₂²×C₄	1	1	yes	characteristic
110	Dic₃₁₄	—	2	2	no	—
111	C₂²⋊D₁₅₇	—	2	2	no	—
112	C₂²⋊D₁₅₇	M₁₆	1	4	yes	complemented
113	C₂×Dic₁₅₇	—	2	2	no	—
114	C₂×Dic₁₅₇	—	2	2	no	—
115	C₂×Dic₁₅₇	—	2	2	no	—
116	C₂×Dic₁₅₇	—	2	2	no	—
117	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×D₁₅₇	—	2	2	no	—
125	C₄×D₁₅₇	—	2	1	no	—
126	C₄×D₁₅₇	—	2	1	no	—
127	C₄×D₁₅₇	—	2	2	no	—
128	#11929434	—	2	1	no	—
129	#11929434	—	2	1	no	—
130	#11929434	—	2	1	no	—
131	#11929434	—	2	1	no	—
132	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
133	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
134	D₄×C₁₅₇	—	2	2	no	—
135	C₂×C₄×C₁₅₇	—	2	1	no	—
136	C₂×C₄×C₁₅₇	—	2	1	no	—
137	C₂×C₄×C₁₅₇	—	2	1	no	—
138	C₂×C₄×C₁₅₇	—	2	1	no	—
139	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
144	C₈×C₁₅₇	—	2	1	no	—
145	C₈×C₁₅₇	—	2	1	no	—
146	C₈×C₁₅₇	—	2	1	no	—
147	C₈×C₁₅₇	—	2	1	no	—
148	C₂²×C₁₅₇	—	2	1	no	—
149	C₂²×C₁₅₇	—	2	1	no	—
150	C₂²×C₁₅₇	—	2	1	no	—
151	C₂²×C₁₅₇	—	2	1	no	—
152	C₂²×C₁₅₇	—	2	1	no	—
153	C₂²×C₁₅₇	C₂×M₁₆	1	2	yes	—
154	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₁₅₇	—	2	1	no	—
156	C₂×D₁₅₇	—	2	1	no	—
157	C₂×D₁₅₇	C₂×M₁₆	1	2	yes	—
158	C₄×C₁₅₇	—	2	1	no	—
159	C₄×C₁₅₇	—	2	1	no	—
160	C₄×C₁₅₇	—	2	1	no	—
161	C₄×C₁₅₇	—	2	1	no	—
162	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
163	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
164	Dic₁₅₇	—	2	1	no	—
165	Dic₁₅₇	—	2	1	no	—
166	Dic₁₅₇	—	2	1	no	—
167	Dic₁₅₇	—	2	1	no	—
168	Dic₁₅₇	—	2	1	no	—
169	Dic₁₅₇	C₂×M₁₆	1	2	yes	—
170	Dic₁₅₇	#131	1	2	yes	—
171	C₂×C₁₅₇	—	2	2	no	—
172	C₂×C₁₅₇	—	4	1	no	—
173	C₂×C₁₅₇	#520	1	1	yes	characteristic
174	C₂×C₁₅₇	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₁₅₇	C₂×#131	1	1	yes	characteristic
176	D₁₅₇	—	2	2	no	—
177	C₁₅₇	#2990	1	1	yes	characteristic, complemented, Sylow
178	#2990	—	157	1	no	maximal, Sylow
179	C₄×#41	—	157	1	no	—
180	#446	—	157	1	no	—
181	#446	—	157	1	no	—
182	#435	—	157	1	no	—
183	#435	—	157	1	no	—
184	#435	—	157	1	no	—
185	#435	D₁₅₇	1	1	yes	characteristic, complemented
186	C₈×D₄	—	157	1	no	—
187	C₈×D₄	—	157	1	no	—
188	#432	—	157	1	no	—
189	#432	—	157	1	no	—
190	#431	—	157	1	no	—
191	#423	—	157	1	no	—
192	C₂×C₄⋊C₈	—	157	1	no	—
193	C₂×C₄⋊C₈	—	157	1	no	—
194	C₂×#41	—	157	1	no	—
195	C₂×M₁₆	—	157	1	no	—
196	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
197	C₂²×C₈	—	157	1	no	—
198	C₂²×C₈	—	157	1	no	—
199	C₂²×C₈	—	157	1	no	—
200	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
201	C₄×Q₈	—	157	1	no	—
202	C₄×Q₈	—	157	1	no	—
203	C₄×D₄	—	157	1	no	—
204	C₄×D₄	—	157	1	no	—
205	C₄×D₄	—	157	1	no	—
206	C₄×D₄	—	157	1	no	—
207	C₄×D₄	—	157	1	no	—
208	C₄×D₄	C₂×D₁₅₇	1	1	yes	characteristic
209	#117	—	157	1	no	—
210	#117	—	157	1	no	—
211	#117	—	157	1	no	—
212	C₂×C₄²	—	157	1	no	—
213	C₂×C₄²	—	157	1	no	—
214	C₂×C₄²	—	157	1	no	—
215	C₄⋊C₈	—	157	1	no	—
216	C₄⋊C₈	—	157	1	no	—
217	C₄⋊C₈	—	157	1	no	—
218	C₄⋊C₈	—	157	1	no	—
219	C₄⋊C₈	—	157	1	no	—
220	C₄⋊C₈	—	157	1	no	—
221	C₄⋊C₈	—	157	1	no	—
222	C₄⋊C₈	—	157	1	no	—
223	C₄⋊C₈	—	157	1	no	—
224	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
225	C₂²⋊C₈	—	157	1	no	—
226	C₂²⋊C₈	—	157	1	no	—
227	C₂²⋊C₈	—	157	1	no	—
228	C₂²⋊C₈	—	157	1	no	—
229	C₂²⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
230	C₂²⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
231	#97	—	157	1	no	—
232	#97	—	157	1	no	—
233	#97	—	157	1	no	—
234	#97	C₂×D₁₅₇	1	1	yes	characteristic
235	C₄×C₈	—	157	1	no	—
236	C₄×C₈	—	157	1	no	—
237	#41	—	314	4	no	—
238	C₂×Q₈	—	157	1	no	—
239	C₂×D₄	—	157	1	no	—
240	C₂×D₄	—	157	1	no	—
241	C₂×D₄	C₄×D₁₅₇	1	1	yes	characteristic
242	C₂²×C₄	—	157	1	no	—
243	C₂²×C₄	—	157	1	no	—
244	C₂²×C₄	—	157	1	no	—
245	C₂²×C₄	—	157	1	no	—
246	C₂²×C₄	—	157	1	no	—
247	C₂²×C₄	—	157	1	no	—
248	C₂²×C₄	—	157	1	no	—
249	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	628	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	628	1	no	—
255	C₂×C₈	—	628	1	no	—
256	C₂×C₈	—	628	1	no	—
257	C₂×C₈	—	157	1	no	—
258	C₂×C₈	—	157	1	no	—
259	C₂×C₈	—	157	1	no	—
260	C₂×C₈	—	157	1	no	—
261	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
265	C₄⋊C₄	—	157	1	no	—
266	C₄⋊C₄	—	157	1	no	—
267	C₄⋊C₄	—	157	1	no	—
268	C₄⋊C₄	—	157	1	no	—
269	C₄⋊C₄	—	157	1	no	—
270	C₄⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
271	C₂²⋊C₄	—	157	1	no	—
272	C₂²⋊C₄	—	157	1	no	—
273	C₂²⋊C₄	—	157	1	no	—
274	C₂²⋊C₄	—	157	1	no	—
275	C₂²⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
277	C₄²	—	157	1	no	—
278	C₄²	—	157	1	no	—
279	C₄²	—	157	1	no	—
280	C₄²	—	157	1	no	—
281	C₄²	—	157	1	no	—
282	C₄²	—	157	1	no	—
283	C₄²	—	157	1	no	—
284	C₄²	—	157	1	no	—
285	C₄²	—	157	1	no	—
286	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
287	C₂³	—	157	1	no	—
288	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
290	Q₈	—	314	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	314	2	no	—
293	D₄	—	157	4	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	314	2	no	—
299	C₂×C₄	—	314	2	no	—
300	C₂×C₄	—	314	2	no	—
301	C₂×C₄	—	314	2	no	—
302	C₂×C₄	—	314	2	no	—
303	C₂×C₄	—	314	2	no	—
304	C₂×C₄	—	314	1	no	—
305	C₂×C₄	—	314	1	no	—
306	C₂×C₄	—	157	1	no	—
307	C₂×C₄	—	157	1	no	—
308	C₂×C₄	—	157	1	no	—
309	C₂×C₄	—	157	1	no	—
310	C₂×C₄	—	157	1	no	—
311	C₂×C₄	—	157	1	no	—
312	C₂×C₄	—	157	1	no	—
313	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	—	314	1	no	—
323	C₈	—	314	1	no	—
324	C₈	—	314	1	no	—
325	C₈	—	314	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	314	1	no	—
332	C₂²	—	314	1	no	—
333	C₂²	—	157	2	no	—
334	C₂²	M₁₆×D₁₅₇	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	314	1	no	—
341	C₄	—	314	1	no	—
342	C₄	—	314	1	no	—
343	C₄	—	314	1	no	—
344	C₄	—	314	1	no	—
345	C₄	—	157	2	no	—
346	C₄	—	157	2	no	—
347	C₄	#19097816	1	1	yes	characteristic
348	C₄	#19097818	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	314	2	no	—
352	C₂	#23757752	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₁₅₇	1	1	yes	characteristic
354	C₂	#131×D₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	#435×C₁₅₇	C₂
prime radical	C₂×#14578953	C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#435	D₁₅₇
hypocenter	C₁₅₇	#2990
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #23757402	C₂	semidirect
3. #23757401	C₂	semidirect
4. #23757347	C₂	semidirect
5. #23757345	C₂	semidirect
6. #23757342	C₂	semidirect
7. C₈×C₂²⋊D₁₅₇	C₂	semidirect
8. #23757336	C₂	semidirect
9. #23757328	C₂	semidirect
10. #23757327	C₂	semidirect
11. #23757323	C₂	semidirect
12. #23757296	C₂	semidirect
13. C₂×#19097700	C₂	semidirect
14. C₄⋊C₈×D₁₅₇	C₂	semidirect
15. #435×C₁₅₇	C₂	semidirect
112. C₂²⋊D₁₅₇	M₁₆	semidirect
177. C₁₅₇	#2990	semidirect
185. #435	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14578953	2
#23757402	2
#23757401	2
#23757347	2
#23757345	2
#23757342	2
C₈×C₂²⋊D₁₅₇	2
#23757336	2
#23757328	2
#23757327	2
#23757323	2
#23757296	2
C₂×#19097700	2
C₄⋊C₈×D₁₅₇	2
#435×C₁₅₇	2
#2990	157