Subgroups — #34108782

Order 20096.

384 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14578954	C₂	1	1	yes	maximal, characteristic
2	#23757333	C₂	1	2	yes	maximal, complemented
3	#23757331	C₂	1	2	yes	maximal, complemented
4	#23757330	C₂	1	1	yes	maximal, characteristic, complemented
5	#23757328	C₂	1	1	yes	maximal, characteristic, complemented
6	#23757328	C₂	1	1	yes	maximal, characteristic, complemented
7	#23757298	C₂	1	2	yes	maximal, complemented
8	#23757238	C₂	1	2	yes	maximal
9	#23757225	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#432×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14578954	C₂²	1	1	yes	characteristic
13	C₂×#14578937	C₂²	1	2	yes	—
14	C₂×C₈×D₁₅₇	C₂²	1	2	yes	—
15	C₄×Dic₃₁₄	C₄	1	2	yes	—
16	C₄×D₆₂₈	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₅₇	C₄	1	2	yes	—
18	C₄×C₂²⋊D₁₅₇	C₄	1	2	yes	—
19	#19097730	C₂²	1	1	yes	characteristic
20	#19097728	C₂²	1	1	yes	characteristic
21	#19097728	C₂²	1	1	yes	characteristic
22	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₅₇	C₂²	1	1	yes	characteristic
24	C₄⋊#11929434	C₂²	1	2	yes	—
25	#19097700	C₂²	1	2	yes	—
26	#19097700	C₂²	1	2	yes	—
27	C₂²⋊#11929434	C₂²	1	2	yes	—
28	#19097683	C₂²	1	2	yes	—
29	#19097683	C₂²	1	2	yes	—
30	#19097682	C₂²	1	1	yes	characteristic
31	#19097682	C₂²	1	1	yes	characteristic
32	#19097681	C₂²	1	2	yes	—
33	C₈×Dic₁₅₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₅₇	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₅₇	C₂²	1	2	yes	—
37	#97×C₁₅₇	C₂²	1	1	yes	characteristic
38	#97×C₁₅₇	C₂²	1	1	yes	characteristic
39	#14578954	—	2	4	no	—
40	C₂×Dic₃₁₄	C₂×C₄	1	1	yes	characteristic
41	C₂×D₆₂₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₅₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₅₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₅₇	C₂³	1	1	yes	characteristic
48	#14578937	—	2	4	no	—
49	C₂×#11929434	C₂³	1	2	yes	—
50	C₂×#11929434	C₂³	1	2	yes	—
51	C₈×D₁₅₇	—	2	4	no	—
52	C₄⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
53	#14578933	C₂×C₄	1	2	yes	—
54	#14578933	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₅₇	C₂×C₄	1	2	yes	—
56	#14578931	C₂×C₄	1	2	yes	—
57	#14578931	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₅₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₅₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
65	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
66	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
67	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
68	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
69	C₂²×D₁₅₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₅₇	C₂²×C₄	1	1	yes	characteristic
71	Dic₃₁₄	—	2	2	no	—
72	D₆₂₈	—	2	2	no	—
73	C₂²⋊D₁₅₇	—	2	2	no	—
74	C₂²⋊D₁₅₇	—	2	2	no	—
75	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₅₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Dic₁₅₇	C₂²×C₄	1	1	yes	characteristic
79	C₄×D₁₅₇	—	2	2	no	—
80	C₄×D₁₅₇	—	4	2	no	—
81	C₄×D₁₅₇	—	4	2	no	—
82	C₄×D₁₅₇	#41	1	4	yes	—
83	#11929434	—	2	2	no	—
84	#11929434	—	2	2	no	—
85	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₅₇	—	2	2	no	—
87	C₂×C₄×C₁₅₇	—	4	2	no	—
88	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₅₇	—	2	2	no	—
93	C₈×C₁₅₇	—	2	2	no	—
94	C₂²×C₁₅₇	—	2	1	no	—
95	C₂²×C₁₅₇	—	2	2	no	—
96	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₅₇	—	2	2	no	—
98	C₂×D₁₅₇	—	2	1	no	—
99	C₂×D₁₅₇	—	2	2	no	—
100	C₂×D₁₅₇	#117	1	2	yes	—
101	C₄×C₁₅₇	—	2	1	no	—
102	C₄×C₁₅₇	—	2	2	no	—
103	C₄×C₁₅₇	—	2	2	no	—
104	C₄×C₁₅₇	C₂×#41	1	2	yes	—
105	Dic₁₅₇	—	2	1	no	—
106	Dic₁₅₇	—	2	2	no	—
107	Dic₁₅₇	—	2	2	no	—
108	Dic₁₅₇	#117	1	2	yes	—
109	C₂×C₁₅₇	—	4	1	no	—
110	C₂×C₁₅₇	C₂×#117	1	1	yes	characteristic
111	C₂×C₁₅₇	#568	1	2	yes	—
112	D₁₅₇	—	2	2	no	—
113	D₁₅₇	—	4	1	no	—
114	C₁₅₇	#2923	1	1	yes	characteristic, complemented, Sylow
115	#2923	—	157	1	no	maximal, Sylow
116	C₄×#41	—	157	1	no	—
117	#433	—	157	2	no	—
118	#433	—	157	2	no	—
119	#432	—	157	1	no	—
120	#432	—	157	1	no	—
121	#432	—	157	1	no	—
122	#432	D₁₅₇	1	1	yes	characteristic, complemented
123	#424	—	157	2	no	—
124	#408	—	157	2	no	—
125	C₄×M₁₆	—	157	1	no	—
126	C₂×#97	—	157	1	no	—
127	C₂×#41	—	157	1	no	—
128	C₂×M₁₆	—	157	2	no	—
129	C₂²×C₈	—	157	2	no	—
130	C₄×Q₈	—	157	2	no	—
131	C₄×D₄	—	157	2	no	—
132	C₄×D₄	—	157	2	no	—
133	C₄×D₄	—	157	2	no	—
134	#117	—	157	1	no	—
135	#117	—	157	1	no	—
136	#117	—	157	1	no	—
137	C₂×C₄²	—	157	1	no	—
138	C₂×C₄²	—	157	1	no	—
139	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
140	C₄⋊C₈	—	157	2	no	—
141	C₄⋊C₈	—	157	2	no	—
142	C₄⋊C₈	—	157	2	no	—
143	C₄⋊C₈	C₂×D₁₅₇	1	2	yes	—
144	C₂²⋊C₈	—	157	2	no	—
145	C₂²⋊C₈	—	157	2	no	—
146	C₂²⋊C₈	—	157	2	no	—
147	C₂²⋊C₈	C₂×D₁₅₇	1	2	yes	—
148	#97	—	157	2	no	—
149	#97	—	157	1	no	—
150	#97	—	157	1	no	—
151	#97	C₂×D₁₅₇	1	1	yes	characteristic
152	#97	C₂×D₁₅₇	1	1	yes	characteristic
153	C₄×C₈	—	157	2	no	—
154	#41	—	314	4	no	—
155	C₂×Q₈	—	157	1	no	—
156	C₂×D₄	—	157	1	no	—
157	C₂×D₄	—	157	1	no	—
158	C₂×D₄	—	157	1	no	—
159	C₂²×C₄	—	157	1	no	—
160	C₂²×C₄	—	157	2	no	—
161	C₂²×C₄	—	157	2	no	—
162	C₂²×C₄	—	157	1	no	—
163	C₂²×C₄	C₄×D₁₅₇	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
165	M₁₆	—	314	4	no	—
166	C₂×C₈	—	314	4	no	—
167	C₂×C₈	—	157	2	no	—
168	C₂×C₈	—	157	2	no	—
169	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
170	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
171	C₄⋊C₄	—	157	2	no	—
172	C₄⋊C₄	—	157	2	no	—
173	C₄⋊C₄	—	157	2	no	—
174	C₂²⋊C₄	—	157	2	no	—
175	C₂²⋊C₄	—	157	2	no	—
176	C₂²⋊C₄	—	157	2	no	—
177	C₄²	—	157	2	no	—
178	C₄²	—	157	2	no	—
179	C₄²	—	157	1	no	—
180	C₄²	—	157	1	no	—
181	C₄²	C₄×D₁₅₇	1	2	yes	—
182	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
183	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
184	C₂³	—	157	1	no	—
185	C₂³	—	157	1	no	—
186	C₂³	C₂×C₄×D₁₅₇	1	1	yes	characteristic
187	Q₈	—	314	2	no	—
188	D₄	—	314	2	no	—
189	D₄	—	314	2	no	—
190	D₄	—	314	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	314	2	no	—
194	C₂×C₄	—	628	2	no	—
195	C₂×C₄	—	628	2	no	—
196	C₂×C₄	—	157	2	no	—
197	C₂×C₄	—	157	2	no	—
198	C₂×C₄	—	157	1	no	—
199	C₂×C₄	—	157	1	no	—
200	C₂×C₄	—	157	4	no	—
201	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
204	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	314	2	no	—
208	C₈	—	314	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	314	2	no	—
212	C₂²	—	314	1	no	—
213	C₂²	—	314	2	no	—
214	C₂²	—	157	2	no	—
215	C₂²	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	314	1	no	—
220	C₄	—	314	2	no	—
221	C₄	—	314	2	no	—
222	C₄	—	157	2	no	—
223	C₄	#41×D₁₅₇	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	314	2	no	—
226	C₂	—	628	1	no	—
227	C₂	#117×D₁₅₇	1	1	yes	characteristic
228	C₂	#23758039	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂³×C₄
Fitting	#432×C₁₅₇	C₂
prime radical	C₂×#14578954	C₂²
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	#432	D₁₅₇
hypocenter	C₁₅₇	#2923
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #23757333	C₂	semidirect
3. #23757331	C₂	semidirect
4. #23757330	C₂	semidirect
5. #23757328	C₂	semidirect
6. #23757328	C₂	semidirect
7. #23757298	C₂	semidirect
9. #23757225	C₂	semidirect
10. #97×D₁₅₇	C₂	semidirect
11. #432×C₁₅₇	C₂	semidirect
114. C₁₅₇	#2923	semidirect
122. #432	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14578954	2
#23757333	2
#23757331	2
#23757330	2
#23757328	2
#23757298	2
#23757238	2
#23757225	2
#97×D₁₅₇	2
#432×C₁₅₇	2
#2923	157