Subgroups — #3262×C₁₅₇ #34097620

Order 20096.

346 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#542×C₁₅₇	C₂	1	2	yes	maximal, complemented
2	#477×C₁₅₇	C₂	1	2	yes	maximal, complemented
3	#474×C₁₅₇	C₂	1	2	yes	maximal, complemented
4	#468×C₁₅₇	C₂	1	2	yes	maximal, complemented
5	#467×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#466×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	#464×C₁₅₇	C₂	1	2	yes	maximal, complemented
8	#463×C₁₅₇	C₂	1	2	yes	maximal, complemented
9	#424×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₁₅₇	C₂²	1	2	yes	—
12	C₂×D₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
16	#126×C₁₅₇	—	2	2	no	—
17	#124×C₁₅₇	—	2	2	no	—
18	#124×C₁₅₇	C₂²	1	2	yes	—
19	#123×C₁₅₇	—	2	2	no	—
20	C₂²⋊Q₈×C₁₅₇	—	2	2	no	—
21	C₂²⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
22	#121×C₁₅₇	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₅₇	C₂²	1	2	yes	—
24	C₄×D₄×C₁₅₇	C₂²	1	2	yes	—
25	#117×C₁₅₇	C₂²	1	1	yes	characteristic
26	#107×C₁₅₇	C₂²	1	1	yes	characteristic
27	#107×C₁₅₇	C₂²	1	1	yes	characteristic
28	#107×C₁₅₇	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₅₇	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₅₇	C₂²	1	2	yes	—
31	#103×C₁₅₇	C₂²	1	2	yes	—
32	#103×C₁₅₇	C₂²	1	2	yes	—
33	#102×C₁₅₇	C₂²	1	2	yes	—
34	#102×C₁₅₇	C₂²	1	2	yes	—
35	C₂×Q₈×C₁₅₇	—	2	2	no	—
36	C₂×Q₈×C₁₅₇	C₂³	1	2	yes	—
37	C₂×D₄×C₁₅₇	—	2	2	no	—
38	C₂×D₄×C₁₅₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₅₇	—	2	2	no	—
40	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
41	Q₁₆×C₁₅₇	—	4	1	no	—
42	SD₁₆×C₁₅₇	—	4	2	no	—
43	D₈×C₁₅₇	—	4	1	no	—
44	M₁₆×C₁₅₇	—	4	1	no	—
45	C₂×C₈×C₁₅₇	—	4	1	no	—
46	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
48	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₅₇	—	2	2	no	—
50	C₄⋊C₄×C₁₅₇	—	2	2	no	—
51	C₄⋊C₄×C₁₅₇	—	2	2	no	—
52	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₁₅₇	D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₅₇	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
57	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
58	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
59	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
60	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
61	C₂²⋊C₄×C₁₅₇	D₄	1	2	yes	—
62	C₄²×C₁₅₇	—	2	2	no	—
63	C₄²×C₁₅₇	C₂³	1	2	yes	—
64	C₂³×C₁₅₇	—	2	2	no	—
65	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₅₇	—	2	2	no	—
67	Q₈×C₁₅₇	—	2	2	no	—
68	Q₈×C₁₅₇	—	8	2	no	—
69	D₄×C₁₅₇	—	2	2	no	—
70	D₄×C₁₅₇	—	2	2	no	—
71	D₄×C₁₅₇	—	8	2	no	—
72	C₂×C₄×C₁₅₇	—	2	1	no	—
73	C₂×C₄×C₁₅₇	—	2	1	no	—
74	C₂×C₄×C₁₅₇	—	2	2	no	—
75	C₂×C₄×C₁₅₇	—	2	2	no	—
76	C₂×C₄×C₁₅₇	—	2	2	no	—
77	C₂×C₄×C₁₅₇	—	8	2	no	—
78	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₅₇	—	2	1	no	—
83	C₈×C₁₅₇	—	2	1	no	—
84	C₈×C₁₅₇	—	2	2	no	—
85	C₂²×C₁₅₇	—	2	1	no	—
86	C₂²×C₁₅₇	—	2	1	no	—
87	C₂²×C₁₅₇	—	2	1	no	—
88	C₂²×C₁₅₇	—	4	2	no	—
89	C₂²×C₁₅₇	—	4	2	no	—
90	C₂²×C₁₅₇	—	4	2	no	—
91	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₁₅₇	—	2	1	no	—
93	C₄×C₁₅₇	—	2	2	no	—
94	C₄×C₁₅₇	—	2	2	no	—
95	C₄×C₁₅₇	—	4	2	no	—
96	C₄×C₁₅₇	—	4	2	no	—
97	C₄×C₁₅₇	—	4	2	no	—
98	C₄×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
99	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×C₁₅₇	—	4	1	no	—
101	C₂×C₁₅₇	—	8	2	no	—
102	C₂×C₁₅₇	#537	1	1	yes	characteristic
103	C₂×C₁₅₇	#576	1	1	yes	characteristic
104	C₂×C₁₅₇	#578	1	1	yes	characteristic
105	C₁₅₇	#3262	1	1	yes	characteristic, direct product factor, Sylow
106	#3262	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	#542	C₂×C₁₅₇	1	2	yes	complemented
108	#477	C₂×C₁₅₇	1	2	yes	complemented
109	#474	C₂×C₁₅₇	1	2	yes	complemented
110	#468	C₂×C₁₅₇	1	2	yes	complemented
111	#467	C₂×C₁₅₇	1	1	yes	characteristic, complemented
112	#466	C₂×C₁₅₇	1	1	yes	characteristic
113	#464	C₂×C₁₅₇	1	2	yes	complemented
114	#463	C₂×C₁₅₇	1	2	yes	complemented
115	#424	C₂×C₁₅₇	1	1	yes	characteristic, complemented
116	C₂×Q₁₆	C₂²×C₁₅₇	1	1	yes	characteristic
117	C₂×SD₁₆	C₂²×C₁₅₇	1	2	yes	—
118	C₂×D₈	C₂²×C₁₅₇	1	1	yes	characteristic
119	C₂×M₁₆	C₂²×C₁₅₇	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₁₅₇	1	1	yes	characteristic
121	C₄⋊Q₈	C₂²×C₁₅₇	1	2	yes	—
122	#126	—	2	2	no	—
123	#124	—	2	2	no	—
124	#124	C₂²×C₁₅₇	1	2	yes	—
125	#123	—	2	2	no	—
126	C₂²⋊Q₈	—	2	2	no	—
127	C₂²⋊Q₈	C₂²×C₁₅₇	1	2	yes	—
128	#121	C₂²×C₁₅₇	1	2	yes	—
129	C₄×Q₈	C₂²×C₁₅₇	1	2	yes	—
130	C₄×D₄	C₂²×C₁₅₇	1	2	yes	—
131	#117	C₂²×C₁₅₇	1	1	yes	characteristic
132	#107	C₂²×C₁₅₇	1	1	yes	characteristic
133	#107	C₂²×C₁₅₇	1	1	yes	characteristic
134	#107	C₂²×C₁₅₇	1	2	yes	—
135	C₄⋊C₈	C₂²×C₁₅₇	1	2	yes	—
136	C₄⋊C₈	C₂²×C₁₅₇	1	2	yes	—
137	#103	C₂²×C₁₅₇	1	2	yes	—
138	#103	C₂²×C₁₅₇	1	2	yes	—
139	#102	C₂²×C₁₅₇	1	2	yes	—
140	#102	C₂²×C₁₅₇	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	C₂³×C₁₅₇	1	2	yes	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂³×C₁₅₇	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂³×C₁₅₇	1	1	yes	characteristic
147	Q₁₆	—	4	1	no	—
148	SD₁₆	—	4	2	no	—
149	D₈	—	4	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂³×C₁₅₇	1	1	yes	characteristic
153	C₂×C₈	C₂³×C₁₅₇	1	2	yes	—
154	C₂×C₈	C₂³×C₁₅₇	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	D₄×C₁₅₇	1	1	yes	characteristic
159	C₄⋊C₄	D₄×C₁₅₇	1	1	yes	characteristic
160	C₄⋊C₄	C₂³×C₁₅₇	1	2	yes	—
161	C₄⋊C₄	C₂³×C₁₅₇	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	D₄×C₁₅₇	1	2	yes	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂³×C₁₅₇	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂×D₄×C₁₅₇	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	8	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₅₇	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₅₇	1	1	yes	characteristic
187	C₂×C₄	C₂⁴×C₁₅₇	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂²×D₄×C₁₅₇	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	H(5,𝔽₂)×C₁₅₇	1	1	yes	characteristic
205	C₄	2₋¹⁺⁴×C₁₅₇	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	8	2	no	—
208	C₂	#537×C₁₅₇	1	1	yes	characteristic
209	C₂	#576×C₁₅₇	1	1	yes	characteristic
210	C₂	#578×C₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₅₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₇
prime radical	#466×C₁₅₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ #117×C₁₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #542×C₁₅₇	C₂	semidirect
2. #477×C₁₅₇	C₂	semidirect
3. #474×C₁₅₇	C₂	semidirect
4. #468×C₁₅₇	C₂	semidirect
5. #467×C₁₅₇	C₂	semidirect
7. #464×C₁₅₇	C₂	semidirect
8. #463×C₁₅₇	C₂	semidirect
9. #424×C₁₅₇	C₂	semidirect
105. C₁₅₇	#3262	direct (N × H)
106. #3262	C₁₅₇	direct (N × H)
107. #542	C₂×C₁₅₇	semidirect
108. #477	C₂×C₁₅₇	semidirect
109. #474	C₂×C₁₅₇	semidirect
110. #468	C₂×C₁₅₇	semidirect
111. #467	C₂×C₁₅₇	semidirect
113. #464	C₂×C₁₅₇	semidirect
114. #463	C₂×C₁₅₇	semidirect
115. #424	C₂×C₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#542×C₁₅₇	2
#477×C₁₅₇	2
#474×C₁₅₇	2
#468×C₁₅₇	2
#467×C₁₅₇	2
#466×C₁₅₇	2
#464×C₁₅₇	2
#463×C₁₅₇	2
#424×C₁₅₇	2
#3262	157

Subgroups — #3262×C157 #34097620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3262×C₁₅₇ #34097620