Subgroups — #34108970

Order 20096.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097799	C₂	1	1	yes	maximal, characteristic
2	#23757753	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757747	C₂	1	1	yes	maximal, characteristic, complemented
4	#23757386	C₂	1	2	yes	maximal, complemented
5	#23757375	C₂	1	2	yes	maximal, complemented
6	C₄×#14578943	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₂₅₆	C₂	1	2	yes	maximal, complemented
8	#23757315	C₂	1	1	yes	maximal, characteristic, complemented
9	#23757270	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19097693	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578954	C₂²	1	1	yes	characteristic
13	C₂²×D₆₂₈	C₂²	1	1	yes	characteristic
14	#19097799	C₄	1	8	yes	complemented
15	C₂×#14578943	C₂²	1	2	yes	—
16	C₂×D₁₂₅₆	C₂²	1	2	yes	—
17	C₄×Dic₃₁₄	C₂²	1	2	yes	complemented
18	C₄×D₆₂₈	C₂²	1	2	yes	—
19	C₄×D₆₂₈	C₂²	1	2	yes	—
20	C₄×D₆₂₈	C₂²	1	2	yes	complemented
21	#19097733	—	2	1	no	—
22	#19097733	—	2	1	no	—
23	#19097731	—	2	2	no	—
24	#19097730	C₂²	1	1	yes	characteristic
25	#19097729	—	2	1	no	—
26	C₂×C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₁₅₇	—	2	1	no	—
28	C₂²⋊C₄×D₁₅₇	—	2	2	no	—
29	#19097705	C₂²	1	2	yes	complemented
30	#19097703	C₂²	1	2	yes	complemented
31	#19097696	C₂²	1	1	yes	characteristic, complemented
32	#19097696	C₂²	1	1	yes	characteristic, complemented
33	#19097693	C₂²	1	2	yes	—
34	#19097693	C₂²	1	2	yes	—
35	#19097693	C₂²	1	1	yes	characteristic, complemented
36	#19097693	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
39	#117×C₁₅₇	C₂²	1	1	yes	characteristic
40	#97×C₁₅₇	C₂²	1	2	yes	complemented
41	C₄×C₈×C₁₅₇	C₂²	1	2	yes	complemented
42	C₂³×D₁₅₇	—	2	1	no	—
43	#14578954	—	2	2	no	—
44	#14578954	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₃₁₄	C₂³	1	1	yes	characteristic
46	C₂×D₆₂₈	—	2	1	no	—
47	C₂×D₆₂₈	—	2	2	no	—
48	C₂×D₆₂₈	C₂×C₄	1	4	yes	complemented
49	C₂×D₆₂₈	C₂³	1	2	yes	—
50	C₂×D₆₂₈	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₅₇	—	2	1	no	—
52	C₂²×Dic₁₅₇	—	2	1	no	—
53	C₂×C₄×D₁₅₇	—	2	1	no	—
54	C₂×C₄×D₁₅₇	—	2	2	no	—
55	C₂×C₄×D₁₅₇	—	2	2	no	—
56	C₂×C₄×D₁₅₇	—	2	2	no	—
57	#14578943	C₂×C₄	1	8	yes	complemented
58	D₁₂₅₆	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₁₅₇	—	2	1	no	—
60	C₄⋊Dic₁₅₇	C₂³	1	2	yes	—
61	C₄⋊Dic₁₅₇	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₁₅₇	C₂³	1	1	yes	characteristic
63	#14578933	—	2	2	no	—
64	C₂²⋊Dic₁₅₇	—	2	1	no	—
65	#14578931	—	2	2	no	—
66	#14578931	—	2	2	no	—
67	#14578931	—	2	2	no	—
68	C₄×Dic₁₅₇	—	2	2	no	—
69	C₄×Dic₁₅₇	—	2	1	no	—
70	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
71	M₁₆×C₁₅₇	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₁₅₇	—	2	2	no	—
73	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₁₅₇	D₄	1	2	yes	—
79	C₄²×C₁₅₇	C₂³	1	2	yes	—
80	C₂²×D₁₅₇	—	2	1	no	—
81	C₂²×D₁₅₇	—	2	2	no	—
82	C₂²×D₁₅₇	—	2	4	no	—
83	C₂²×D₁₅₇	—	8	1	no	—
84	C₂²×D₁₅₇	—	8	1	no	—
85	Dic₃₁₄	—	2	1	no	—
86	Dic₃₁₄	C₂²×C₄	1	2	yes	—
87	D₆₂₈	—	2	4	no	—
88	D₆₂₈	—	2	2	no	—
89	D₆₂₈	—	2	1	no	—
90	D₆₂₈	C₂²×C₄	1	4	yes	—
91	D₆₂₈	C₂²×C₄	1	2	yes	—
92	C₂²⋊D₁₅₇	—	2	4	no	—
93	C₂×Dic₁₅₇	—	2	1	no	—
94	C₂×Dic₁₅₇	—	2	1	no	—
95	C₂×Dic₁₅₇	—	2	1	no	—
96	C₂×Dic₁₅₇	—	2	2	no	—
97	C₂×Dic₁₅₇	—	8	1	no	—
98	C₄×D₁₅₇	—	2	4	no	—
99	C₄×D₁₅₇	—	8	2	no	—
100	C₄×D₁₅₇	—	8	2	no	—
101	C₄×D₁₅₇	—	8	2	no	—
102	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₅₇	#41	1	2	yes	—
108	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
109	C₈×C₁₅₇	—	2	1	no	—
110	C₈×C₁₅₇	—	2	1	no	—
111	C₈×C₁₅₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₁₅₇	—	2	1	no	—
113	C₂²×C₁₅₇	—	2	1	no	—
114	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
115	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₅₇	—	2	2	no	—
117	C₂×D₁₅₇	—	2	4	no	—
118	C₂×D₁₅₇	—	4	1	no	—
119	C₂×D₁₅₇	—	4	2	no	—
120	C₂×D₁₅₇	—	4	2	no	—
121	C₂×D₁₅₇	—	4	1	no	—
122	C₂×D₁₅₇	—	8	4	no	—
123	C₄×C₁₅₇	—	2	2	no	—
124	C₄×C₁₅₇	—	2	2	no	—
125	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
126	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
128	Dic₁₅₇	—	2	2	no	—
129	Dic₁₅₇	—	4	1	no	—
130	Dic₁₅₇	—	4	2	no	—
131	Dic₁₅₇	—	4	1	no	—
132	C₂×C₁₅₇	—	2	2	no	—
133	C₂×C₁₅₇	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₁₅₇	#576	1	1	yes	characteristic
135	C₂×C₁₅₇	#577	1	1	yes	characteristic
136	D₁₅₇	—	4	2	no	—
137	D₁₅₇	—	4	4	no	—
138	C₁₅₇	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	157	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	—	157	1	no	—
141	#520	—	157	1	no	—
142	#518	—	157	1	no	—
143	#442	—	157	2	no	—
144	#440	—	157	2	no	—
145	C₄×SD₁₆	—	157	2	no	—
146	C₄×D₈	—	157	2	no	—
147	#428	—	157	1	no	—
148	#416	—	157	1	no	—
149	C₂×#102	—	157	1	no	—
150	#405	D₁₅₇	1	1	yes	characteristic, complemented
151	C₂×#41	—	157	1	no	—
152	C₂²×D₄	—	157	1	no	—
153	AGL(1,ℤ/8)	—	157	8	no	—
154	C₂×SD₁₆	—	157	2	no	—
155	C₂×D₈	—	157	2	no	—
156	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
158	C₄×Q₈	—	157	2	no	—
159	C₄×D₄	—	314	1	no	—
160	C₄×D₄	—	314	1	no	—
161	C₄×D₄	—	314	2	no	—
162	C₄×D₄	—	157	2	no	—
163	C₄×D₄	—	157	2	no	—
164	C₄×D₄	—	157	2	no	—
165	#117	—	314	1	no	—
166	#117	—	157	1	no	—
167	#117	C₂×D₁₅₇	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	314	1	no	—
169	C₂×C₄⋊C₄	—	157	1	no	—
170	C₂×C₂²⋊C₄	—	314	2	no	—
171	#107	—	157	2	no	—
172	#106	—	157	2	no	—
173	#103	—	157	1	no	—
174	#103	—	157	1	no	—
175	#102	—	157	2	no	—
176	#102	—	157	2	no	—
177	#102	—	157	1	no	—
178	#102	—	157	1	no	—
179	#97	C₂×D₁₅₇	1	2	yes	complemented
180	C₄×C₈	C₂×D₁₅₇	1	2	yes	complemented
181	C₂⁴	—	314	1	no	—
182	#41	—	314	2	no	—
183	#41	—	157	4	no	—
184	C₂×Q₈	—	157	1	no	—
185	C₂×D₄	—	314	1	no	—
186	C₂×D₄	—	314	1	no	—
187	C₂×D₄	—	314	2	no	—
188	C₂×D₄	—	157	4	no	—
189	C₂×D₄	—	157	1	no	—
190	C₂×D₄	—	157	2	no	—
191	C₂²×C₄	—	314	1	no	—
192	C₂²×C₄	—	314	1	no	—
193	C₂²×C₄	—	314	2	no	—
194	C₂²×C₄	—	314	2	no	—
195	C₂²×C₄	—	314	2	no	—
196	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
197	SD₁₆	—	157	8	no	—
198	D₈	—	157	8	no	—
199	M₁₆	C₄×D₁₅₇	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
204	C₄⋊C₄	—	314	1	no	—
205	C₄⋊C₄	—	314	2	no	—
206	C₄⋊C₄	—	157	2	no	—
207	C₄⋊C₄	—	157	1	no	—
208	C₄⋊C₄	—	157	1	no	—
209	C₄⋊C₄	D₆₂₈	1	1	yes	characteristic
210	C₄⋊C₄	D₆₂₈	1	1	yes	characteristic
211	C₂²⋊C₄	—	314	1	no	—
212	C₂²⋊C₄	—	314	2	no	—
213	C₂²⋊C₄	—	314	2	no	—
214	C₂²⋊C₄	—	314	2	no	—
215	C₂²⋊C₄	D₆₂₈	1	2	yes	—
216	C₄²	—	314	2	no	—
217	C₄²	—	314	1	no	—
218	C₄²	C₂²×D₁₅₇	1	2	yes	—
219	C₂³	—	314	1	no	—
220	C₂³	—	314	2	no	—
221	C₂³	—	314	4	no	—
222	C₂³	—	1256	1	no	—
223	C₂³	—	1256	1	no	—
224	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
225	Q₈	—	314	1	no	—
226	Q₈	—	157	2	no	—
227	D₄	—	314	4	no	—
228	D₄	—	314	4	no	—
229	D₄	—	314	2	no	—
230	D₄	—	314	1	no	—
231	D₄	—	157	2	no	—
232	D₄	—	157	4	no	—
233	C₂×C₄	—	314	1	no	—
234	C₂×C₄	—	314	4	no	—
235	C₂×C₄	—	314	1	no	—
236	C₂×C₄	—	314	1	no	—
237	C₂×C₄	—	314	2	no	—
238	C₂×C₄	—	1256	2	no	—
239	C₂×C₄	—	1256	2	no	—
240	C₂×C₄	—	1256	2	no	—
241	C₂×C₄	—	1256	1	no	—
242	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
243	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
244	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
245	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
246	C₂×C₄	#14578954	1	2	yes	—
247	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×C₄×D₁₅₇	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	314	2	no	—
254	C₂²	—	314	4	no	—
255	C₂²	—	628	2	no	—
256	C₂²	—	628	1	no	—
257	C₂²	—	628	1	no	—
258	C₂²	—	628	2	no	—
259	C₂²	—	1256	4	no	—
260	C₂²	C₄×D₆₂₈	1	2	yes	—
261	C₂²	C₂²×D₆₂₈	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	314	2	no	—
265	C₄	—	628	1	no	—
266	C₄	—	628	1	no	—
267	C₄	—	628	2	no	—
268	C₄	C₄×D₆₂₈	1	2	yes	—
269	C₄	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
270	C₄	C₂×#14578954	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	628	4	no	—
273	C₂	—	628	2	no	—
274	C₂	C₂×C₄×D₆₂₈	1	1	yes	characteristic
275	C₂	#23758078	1	1	yes	characteristic
276	C₂	#23758085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₂₈
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	#405×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×D₄
prime radical	C₂×#19097799	C₂
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#405	D₁₅₇
hypocenter	C₁₅₇	#2946
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #23757753	C₂	semidirect
3. #23757747	C₂	semidirect
4. #23757386	C₂	semidirect
5. #23757375	C₂	semidirect
6. C₄×#14578943	C₂	semidirect
7. C₄×D₁₂₅₆	C₂	semidirect
8. #23757315	C₂	semidirect
9. #23757270	C₂	semidirect
10. C₂×#19097693	C₂	semidirect
11. #405×C₁₅₇	C₂	semidirect
14. #19097799	C₄	semidirect
17. C₄×Dic₃₁₄	C₂²	semidirect
20. C₄×D₆₂₈	C₂²	semidirect
29. #19097705	C₂²	semidirect
30. #19097703	C₂²	semidirect
31. #19097696	C₂²	semidirect
32. #19097696	C₂²	semidirect
35. #19097693	C₂²	semidirect
36. #19097693	C₂²	semidirect
40. #97×C₁₅₇	C₂²	semidirect
41. C₄×C₈×C₁₅₇	C₂²	semidirect
44. #14578954	C₂×C₄	semidirect
48. C₂×D₆₂₈	C₂×C₄	semidirect
57. #14578943	C₂×C₄	semidirect
58. D₁₂₅₆	C₂×C₄	semidirect
71. M₁₆×C₁₅₇	C₂×C₄	semidirect
138. C₁₅₇	#2946	semidirect
150. #405	D₁₅₇	semidirect
179. #97	C₂×D₁₅₇	semidirect
180. C₄×C₈	C₂×D₁₅₇	semidirect
199. M₁₆	C₄×D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097799	2
#23757753	2
#23757747	2
#23757386	2
#23757375	2
C₄×#14578943	2
C₄×D₁₂₅₆	2
#23757315	2
#23757270	2
C₂×#19097693	2
#405×C₁₅₇	2
#2946	157