Subgroups — #34109228

Order 20096.

322 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757826	C₂	1	1	yes	maximal, characteristic
2	#23757403	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757402	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757402	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757401	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757401	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757400	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×Dic₃₁₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757397	C₂	1	1	yes	maximal, characteristic, complemented
10	#23757334	C₂	1	1	yes	maximal, characteristic
11	#23757331	C₂	1	1	yes	maximal, characteristic
12	#23757331	C₂	1	1	yes	maximal, characteristic
13	#23757327	C₂	1	1	yes	maximal, characteristic
14	#23757323	C₂	1	1	yes	maximal, characteristic
15	#433×C₁₅₇	C₂	1	1	yes	maximal, characteristic
16	#19097770	C₄	1	1	yes	characteristic
17	#19097770	C₄	1	1	yes	characteristic
18	#19097762	C₄	1	1	yes	characteristic
19	#19097762	C₄	1	1	yes	characteristic
20	#19097748	C₄	1	1	yes	characteristic
21	#19097748	C₄	1	1	yes	characteristic
22	#19097748	C₄	1	1	yes	characteristic
23	#19097748	C₄	1	1	yes	characteristic
24	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
26	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₁₄	C₂²	1	1	yes	characteristic
28	#19097730	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
30	C₄⋊#11929434	C₂²	1	1	yes	characteristic
31	C₄⋊#11929434	C₂²	1	1	yes	characteristic
32	#19097700	C₂²	1	1	yes	characteristic
33	#19097700	C₂²	1	1	yes	characteristic
34	#19097700	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	C₂²⋊#11929434	C₂²	1	1	yes	characteristic
41	C₂²⋊#11929434	C₂²	1	1	yes	characteristic
42	#19097682	C₂²	1	1	yes	characteristic
43	#19097681	C₂²	1	1	yes	characteristic
44	#19097681	C₂²	1	1	yes	characteristic
45	#19097681	C₂²	1	1	yes	characteristic
46	#19097681	C₂²	1	1	yes	characteristic
47	C₄×#11929434	C₂²	1	1	yes	characteristic
48	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
49	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₅₇	C₂²	1	1	yes	characteristic
52	#117×C₁₅₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₅₇	C₂²	1	1	yes	characteristic
57	#97×C₁₅₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₁₄	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₃₁₄	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₃₁₄	C₂×C₄	1	1	yes	characteristic
62	C₂×Dic₃₁₄	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	C₂×#11929434	C₂³	1	1	yes	characteristic
66	C₂×#11929434	C₂³	1	1	yes	characteristic
67	C₂×#11929434	C₂³	1	1	yes	characteristic
68	C₂×#11929434	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
75	#14578933	C₂×C₄	1	1	yes	characteristic
76	#14578933	C₂×C₄	1	1	yes	characteristic
77	#14578933	C₂×C₄	1	1	yes	characteristic
78	#14578933	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	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
86	C₄×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
88	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
89	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
98	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
99	C₂²⋊C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
100	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
101	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
102	Dic₃₁₄	—	2	2	no	—
103	Dic₃₁₄	—	2	2	no	—
104	Dic₃₁₄	—	4	1	no	—
105	Dic₃₁₄	—	4	1	no	—
106	C₂×Dic₁₅₇	—	2	2	no	—
107	C₂×Dic₁₅₇	—	4	1	no	—
108	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
116	#11929434	—	2	1	no	—
117	#11929434	—	2	1	no	—
118	#11929434	—	2	1	no	—
119	#11929434	—	2	1	no	—
120	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₅₇	—	2	1	no	—
122	C₂×C₄×C₁₅₇	—	2	1	no	—
123	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
129	C₈×C₁₅₇	—	2	1	no	—
130	C₈×C₁₅₇	—	2	1	no	—
131	C₈×C₁₅₇	—	2	1	no	—
132	C₈×C₁₅₇	—	2	1	no	—
133	C₂²×C₁₅₇	—	2	1	no	—
134	C₂²×C₁₅₇	—	2	1	no	—
135	C₂²×C₁₅₇	—	2	1	no	—
136	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
137	C₄×C₁₅₇	—	2	1	no	—
138	C₄×C₁₅₇	—	2	1	no	—
139	C₄×C₁₅₇	—	2	1	no	—
140	C₄×C₁₅₇	—	2	1	no	—
141	C₄×C₁₅₇	—	2	1	no	—
142	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
143	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₁₅₇	—	2	1	no	—
145	Dic₁₅₇	—	2	1	no	—
146	Dic₁₅₇	—	2	1	no	—
147	Dic₁₅₇	—	2	1	no	—
148	Dic₁₅₇	—	2	1	no	—
149	Dic₁₅₇	—	2	1	no	—
150	Dic₁₅₇	—	2	1	no	—
151	Dic₁₅₇	#131	1	2	yes	—
152	C₂×C₁₅₇	—	4	1	no	—
153	C₂×C₁₅₇	#519	1	1	yes	characteristic
154	C₂×C₁₅₇	C₂×#131	1	1	yes	characteristic
155	C₂×C₁₅₇	#568	1	1	yes	characteristic
156	C₁₅₇	#2985	1	1	yes	characteristic, complemented, Sylow
157	#2985	—	157	1	no	maximal, Sylow
158	#533	—	157	1	no	—
159	#446	—	157	1	no	—
160	#446	—	157	1	no	—
161	#446	—	157	1	no	—
162	#446	—	157	1	no	—
163	#446	—	157	1	no	—
164	#446	—	157	1	no	—
165	C₈×Q₈	—	157	1	no	—
166	C₈×Q₈	—	157	1	no	—
167	#433	—	157	1	no	—
168	#433	—	157	1	no	—
169	#433	—	157	1	no	—
170	#433	D₁₅₇	1	1	yes	characteristic
171	#432	—	157	1	no	—
172	#431	—	157	1	no	—
173	C₄⋊Q₈	—	157	1	no	—
174	C₄⋊Q₈	—	157	1	no	—
175	#125	—	157	1	no	—
176	#125	—	157	1	no	—
177	C₂²⋊Q₈	—	157	1	no	—
178	C₂²⋊Q₈	—	157	1	no	—
179	C₂²⋊Q₈	—	157	1	no	—
180	C₂²⋊Q₈	—	157	1	no	—
181	C₄×Q₈	—	157	1	no	—
182	C₄×Q₈	—	157	1	no	—
183	C₄×Q₈	—	157	1	no	—
184	C₄×Q₈	—	157	1	no	—
185	#117	—	157	1	no	—
186	#117	C₂×D₁₅₇	1	1	yes	characteristic
187	C₂×C₄²	—	157	1	no	—
188	C₄⋊C₈	—	157	1	no	—
189	C₄⋊C₈	—	157	1	no	—
190	C₄⋊C₈	—	157	1	no	—
191	C₄⋊C₈	—	157	1	no	—
192	C₄⋊C₈	—	157	1	no	—
193	C₄⋊C₈	—	157	1	no	—
194	C₄⋊C₈	—	157	1	no	—
195	C₄⋊C₈	—	157	1	no	—
196	C₄⋊C₈	—	157	1	no	—
197	C₄⋊C₈	—	157	1	no	—
198	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
200	C₂²⋊C₈	—	157	1	no	—
201	C₂²⋊C₈	—	157	1	no	—
202	C₂²⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
203	C₂²⋊C₈	C₂×D₁₅₇	1	1	yes	characteristic
204	#97	—	157	1	no	—
205	#97	—	157	1	no	—
206	#97	—	157	1	no	—
207	#97	—	157	1	no	—
208	#97	—	157	1	no	—
209	#97	C₂×D₁₅₇	1	1	yes	characteristic
210	C₄×C₈	—	157	1	no	—
211	C₄×C₈	—	157	1	no	—
212	C₄×C₈	—	157	1	no	—
213	C₄×C₈	—	157	1	no	—
214	C₄×C₈	—	157	1	no	—
215	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
216	C₂×Q₈	—	157	1	no	—
217	C₂×Q₈	—	157	1	no	—
218	C₂×Q₈	—	157	1	no	—
219	C₂×Q₈	—	157	1	no	—
220	C₂²×C₄	—	157	1	no	—
221	C₂²×C₄	—	157	1	no	—
222	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
223	C₂×C₈	—	157	1	no	—
224	C₂×C₈	—	157	1	no	—
225	C₂×C₈	—	157	1	no	—
226	C₂×C₈	—	157	1	no	—
227	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
231	C₄⋊C₄	—	157	1	no	—
232	C₄⋊C₄	—	157	1	no	—
233	C₄⋊C₄	—	157	1	no	—
234	C₄⋊C₄	—	157	1	no	—
235	C₄⋊C₄	—	157	1	no	—
236	C₄⋊C₄	—	157	1	no	—
237	C₄⋊C₄	—	157	1	no	—
238	C₄⋊C₄	—	157	1	no	—
239	C₄⋊C₄	—	157	1	no	—
240	C₄⋊C₄	—	157	1	no	—
241	C₄⋊C₄	—	157	1	no	—
242	C₄⋊C₄	—	157	1	no	—
243	C₄⋊C₄	—	157	1	no	—
244	C₄⋊C₄	—	157	1	no	—
245	C₄⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
246	C₄⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
247	C₂²⋊C₄	—	157	1	no	—
248	C₂²⋊C₄	—	157	1	no	—
249	C₂²⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
250	C₂²⋊C₄	C₄×D₁₅₇	1	1	yes	characteristic
251	C₄²	—	157	1	no	—
252	C₄²	—	157	1	no	—
253	C₄²	—	157	1	no	—
254	C₄²	—	157	1	no	—
255	C₄²	—	157	1	no	—
256	C₄²	—	157	1	no	—
257	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
258	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
259	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
260	Q₈	—	314	2	no	—
261	Q₈	—	314	2	no	—
262	Q₈	—	628	1	no	—
263	Q₈	—	628	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	314	2	no	—
267	C₂×C₄	—	628	1	no	—
268	C₂×C₄	—	157	1	no	—
269	C₂×C₄	—	157	1	no	—
270	C₂×C₄	—	157	1	no	—
271	C₂×C₄	—	157	1	no	—
272	C₂×C₄	—	157	1	no	—
273	C₂×C₄	—	157	1	no	—
274	C₂×C₄	—	157	1	no	—
275	C₂×C₄	—	157	1	no	—
276	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
278	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
281	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	314	1	no	—
287	C₈	—	314	1	no	—
288	C₈	—	314	1	no	—
289	C₈	—	314	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	314	1	no	—
300	C₄	—	314	1	no	—
301	C₄	—	314	1	no	—
302	C₄	—	314	1	no	—
303	C₄	—	314	1	no	—
304	C₄	—	314	1	no	—
305	C₄	—	314	1	no	—
306	C₄	—	157	2	no	—
307	C₄	#19097818	1	1	yes	characteristic
308	C₄	#19097818	1	1	yes	characteristic
309	C₂	—	4	1	no	—
310	C₂	#23757750	1	1	yes	characteristic
311	C₂	#131×D₁₅₇	1	1	yes	characteristic
312	C₂	#23758038	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	#433×C₁₅₇	C₂
prime radical	C₂³×C₁₅₇	C₂²×C₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#433	D₁₅₇
hypocenter	C₁₅₇	#2985
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #23757403	C₂	semidirect
3. #23757402	C₂	semidirect
4. #23757402	C₂	semidirect
5. #23757401	C₂	semidirect
6. #23757401	C₂	semidirect
7. #23757400	C₂	semidirect
8. C₈×Dic₃₁₄	C₂	semidirect
9. #23757397	C₂	semidirect
156. C₁₅₇	#2985	semidirect

Maximal subgroups

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

subgroup	index
#23757826	2
#23757403	2
#23757402	2
#23757401	2
#23757400	2
C₈×Dic₃₁₄	2
#23757397	2
#23757334	2
#23757331	2
#23757327	2
#23757323	2
#433×C₁₅₇	2
#2985	157