Subgroups — C₂×#23757036 #34099850

Order 20096.

512 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₁₅₇	C₂	1	2	yes	maximal
2	#23757036	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₅₇	—	2	2	no	—
5	C₂×C₄⋊Dic₁₅₇	—	2	4	no	—
6	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
7	C₂×C₄⋊Dic₁₅₇	C₄	1	8	yes	complemented
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
9	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₅₇	—	2	4	no	—
11	C₂²×C₈×C₁₅₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₁₅₇	C₂²	1	2	yes	—
13	C₂²×Dic₁₅₇	—	2	8	no	—
14	C₂²×Dic₁₅₇	—	2	4	no	—
15	C₂²×Dic₁₅₇	—	4	4	no	—
16	C₂²×Dic₁₅₇	—	4	4	no	—
17	C₄⋊Dic₁₅₇	—	2	8	no	—
18	C₄⋊Dic₁₅₇	—	2	4	no	—
19	C₄⋊Dic₁₅₇	C₂×C₄	1	8	yes	complemented
20	C₂⁴×C₁₅₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₅₇	D₄	1	4	yes	—
24	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₅₇	—	2	8	no	—
28	C₂×C₈×C₁₅₇	—	2	4	no	—
29	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	complemented
30	C₂×Dic₁₅₇	—	2	8	no	—
31	C₂×Dic₁₅₇	—	4	16	no	—
32	C₂×Dic₁₅₇	—	4	4	no	—
33	C₂×Dic₁₅₇	—	4	4	no	—
34	C₂³×C₁₅₇	C₂²⋊C₄	1	4	yes	—
35	C₂³×C₁₅₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₁₅₇	D₈	1	2	yes	—
37	C₂³×C₁₅₇	SD₁₆	1	4	yes	—
38	C₂³×C₁₅₇	Q₁₆	1	2	yes	—
39	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₄²	1	4	yes	—
41	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₅₇	C₄⋊C₄	1	8	yes	—
44	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
49	C₈×C₁₅₇	—	2	4	no	—
50	C₂²×C₁₅₇	#95	1	4	yes	—
51	C₂²×C₁₅₇	#102	1	8	yes	—
52	C₂²×C₁₅₇	#103	1	8	yes	—
53	C₂²×C₁₅₇	#106	1	4	yes	—
54	C₂²×C₁₅₇	#107	1	4	yes	—
55	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	2	yes	—
56	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₁₅₇	C₂×SD₁₆	1	2	yes	—
59	C₂²×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₁₅₇	#95	1	4	yes	—
61	C₄×C₁₅₇	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
64	Dic₁₅₇	—	4	8	no	—
65	C₂×C₁₅₇	#340	1	8	yes	complemented
66	C₂×C₁₅₇	C₂×#95	1	1	yes	characteristic
67	C₂×C₁₅₇	C₂×#102	1	2	yes	—
68	C₂×C₁₅₇	C₂×#103	1	2	yes	—
69	C₂×C₁₅₇	C₂×#106	1	1	yes	characteristic
70	C₂×C₁₅₇	C₂×#107	1	1	yes	characteristic
71	C₁₅₇	C₂×#340	1	1	yes	characteristic, complemented, Sylow
72	C₂×#340	—	157	1	no	maximal, Sylow
73	C₂³×C₈	D₁₅₇	1	1	yes	characteristic
74	C₂²×C₄⋊C₄	—	157	2	no	—
75	#340	—	157	4	no	—
76	C₂³×C₄	—	314	2	no	—
77	C₂³×C₄	C₂×D₁₅₇	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	Dic₁₅₇	1	4	yes	complemented
80	C₂²×C₈	C₂×D₁₅₇	1	2	yes	—
81	C₂×C₄⋊C₄	—	314	4	no	—
82	C₂×C₄⋊C₄	—	314	2	no	—
83	C₂×C₄⋊C₄	—	157	8	no	—
84	C₂×C₄⋊C₄	—	157	4	no	—
85	C₂⁴	D₆₂₈	1	1	yes	characteristic
86	C₂²×C₄	—	314	4	no	—
87	C₂²×C₄	—	314	8	no	—
88	C₂²×C₄	—	628	4	no	—
89	C₂²×C₄	—	628	4	no	—
90	C₂²×C₄	C₄×D₁₅₇	1	4	yes	—
91	C₂²×C₄	C₂×Dic₁₅₇	1	2	yes	—
92	C₂²×C₄	C₂²⋊D₁₅₇	1	4	yes	—
93	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
94	C₂²×C₄	Dic₃₁₄	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	8	no	—
98	C₂×C₈	C₂×Dic₁₅₇	1	4	yes	complemented
99	C₄⋊C₄	—	314	4	no	—
100	C₄⋊C₄	—	314	8	no	—
101	C₄⋊C₄	—	157	8	no	—
102	C₂³	#14578931	1	4	yes	—
103	C₂³	C₄⋊Dic₁₅₇	1	2	yes	—
104	C₂³	D₁₂₅₆	1	2	yes	—
105	C₂³	#14578943	1	4	yes	—
106	C₂³	Dic₆₂₈	1	2	yes	—
107	C₂³	C₂×D₆₂₈	1	1	yes	characteristic
108	C₂×C₄	—	314	8	no	—
109	C₂×C₄	—	628	16	no	—
110	C₂×C₄	—	628	4	no	—
111	C₂×C₄	—	628	4	no	—
112	C₂×C₄	C₄×Dic₁₅₇	1	4	yes	—
113	C₂×C₄	#14578931	1	4	yes	—
114	C₂×C₄	C₂²⋊Dic₁₅₇	1	4	yes	—
115	C₂×C₄	#14578933	1	8	yes	—
116	C₂×C₄	C₄⋊Dic₁₅₇	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
118	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
120	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	#19097678	1	4	yes	—
123	C₂²	#19097693	1	8	yes	—
124	C₂²	#19097696	1	8	yes	—
125	C₂²	#19097703	1	4	yes	—
126	C₂²	#19097705	1	4	yes	—
127	C₂²	C₂×#14578931	1	2	yes	—
128	C₂²	C₂×C₄⋊Dic₁₅₇	1	1	yes	characteristic
129	C₂²	C₂×D₁₂₅₆	1	1	yes	characteristic
130	C₂²	C₂×#14578943	1	2	yes	—
131	C₂²	C₂×Dic₆₂₈	1	1	yes	characteristic
132	C₄	—	628	8	no	—
133	C₄	#19097678	1	4	yes	—
134	C₄	C₂×C₄×Dic₁₅₇	1	1	yes	characteristic
135	C₄	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
136	C₄	C₂×#14578933	1	2	yes	—
137	C₂	#23757036	1	8	yes	direct product factor
138	C₂	C₂×#19097678	1	1	yes	characteristic
139	C₂	C₂×#19097693	1	2	yes	—
140	C₂	C₂×#19097696	1	2	yes	—
141	C₂	C₂×#19097703	1	1	yes	characteristic
142	C₂	C₂×#19097705	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₆₂₈
commutator	C₄×C₁₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂³×C₈×C₁₅₇	C₂
socle, prime radical	C₂⁴×C₁₅₇	D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂³×C₈	D₁₅₇
hypocenter	C₁₅₇	C₂×#340
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #23757036	C₂	direct (N × H)
7. C₂×C₄⋊Dic₁₅₇	C₄	semidirect
11. C₂²×C₈×C₁₅₇	C₄	semidirect
19. C₄⋊Dic₁₅₇	C₂×C₄	semidirect
29. C₂×C₈×C₁₅₇	C₂×C₄	semidirect
65. C₂×C₁₅₇	#340	semidirect
71. C₁₅₇	C₂×#340	semidirect
79. C₂²×C₈	Dic₁₅₇	semidirect
98. C₂×C₈	C₂×Dic₁₅₇	semidirect
137. C₂	#23757036	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₁₅₇	2
#23757036	2
C₂³×C₈×C₁₅₇	2
C₂×#340	157

Subgroups — C2×#23757036 #34099850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757036 #34099850