Subgroups — #34098123

Order 20096.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757531	C₂	1	1	yes	maximal, characteristic
2	#23757055	C₂	1	1	yes	maximal, characteristic
3	H(3,ℤ/4)×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₂₈	C₄	1	1	yes	characteristic, complemented
5	#19097750	—	2	1	no	—
6	#19097711	—	2	1	no	—
7	#19097703	C₄	1	1	yes	characteristic, complemented
8	#19097696	—	2	1	no	—
9	C₂×M₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
10	#117×C₁₅₇	—	2	1	no	—
11	C₂×Dic₃₁₄	—	2	1	no	—
12	Dic₆₂₈	—	4	1	no	—
13	C₄⋊Dic₁₅₇	—	2	1	no	—
14	#14578933	—	4	1	no	—
15	C₂²⋊Dic₁₅₇	—	4	1	no	—
16	#14578929	—	4	1	no	—
17	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
18	M₁₆×C₁₅₇	—	4	1	no	—
19	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₁₅₇	—	4	1	no	—
22	C₂²⋊C₄×C₁₅₇	—	4	1	no	—
23	C₄²×C₁₅₇	—	2	2	no	—
24	Dic₃₁₄	—	4	1	no	—
25	Dic₃₁₄	—	4	1	no	—
26	C₂×Dic₁₅₇	—	4	1	no	—
27	C₂×Dic₁₅₇	—	4	1	no	—
28	C₂³×C₁₅₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	—	2	1	no	—
30	C₂×C₄×C₁₅₇	—	2	1	no	—
31	C₂×C₄×C₁₅₇	—	4	2	no	—
32	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
34	C₈×C₁₅₇	—	2	1	no	—
35	C₈×C₁₅₇	—	2	1	no	—
36	C₂²×C₁₅₇	—	2	1	no	—
37	C₂²×C₁₅₇	—	4	1	no	—
38	C₂²×C₁₅₇	#102	1	1	yes	characteristic
39	C₄×C₁₅₇	—	2	1	no	—
40	C₄×C₁₅₇	—	8	2	no	—
41	C₄×C₁₅₇	#99	1	1	yes	characteristic
42	C₄×C₁₅₇	#104	1	1	yes	characteristic
43	Dic₁₅₇	—	8	1	no	—
44	Dic₁₅₇	—	8	1	no	—
45	C₂×C₁₅₇	—	2	1	no	—
46	C₂×C₁₅₇	—	4	1	no	—
47	C₂×C₁₅₇	#327	1	1	yes	characteristic
48	C₁₅₇	#1340	1	1	yes	characteristic, complemented, Sylow
49	#1340	—	157	1	no	maximal, Sylow
50	#470	—	157	1	no	—
51	#349	—	157	1	no	—
52	H(3,ℤ/4)	D₁₅₇	1	1	yes	characteristic
53	C₂×Q₁₆	—	157	1	no	—
54	C₂×M₁₆	C₂×D₁₅₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	314	1	no	—
56	#117	—	2	1	no	—
57	M₃₂	—	314	1	no	—
58	#106	—	157	1	no	—
59	#103	—	314	1	no	—
60	C₂×Q₈	—	314	1	no	—
61	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
62	Q₁₆	—	628	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
65	C₂×C₈	D₆₂₈	1	1	yes	characteristic
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	314	1	no	—
68	C₄⋊C₄	—	628	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	628	1	no	—
71	C₄²	—	2	2	no	—
72	C₁₆	—	628	1	no	—
73	C₂³	#14578939	1	1	yes	characteristic
74	Q₈	—	628	1	no	—
75	Q₈	—	628	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	628	1	no	—
80	C₂×C₄	—	628	1	no	—
81	C₂×C₄	#14578931	1	1	yes	characteristic
82	C₂×C₄	#14578941	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#19097691	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	2	no	—
90	C₄	—	1256	1	no	—
91	C₄	—	1256	1	no	—
92	C₄	#19097685	1	1	yes	characteristic
93	C₄	#19097699	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#23757006	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757006
commutator	C₂×C₈×C₁₅₇	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₁₅₇
Fitting	H(3,ℤ/4)×C₁₅₇	C₂
socle	C₂×C₁₅₇	#327
prime radical	C₂³×C₁₅₇	D₈
cosocle	C₂×M₁₆×C₁₅₇	C₂²
hypercenter	H(3,ℤ/4)	D₁₅₇
hypocenter	C₁₅₇	#1340
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₆₂₈	C₄	semidirect
7. #19097703	C₄	semidirect
48. C₁₅₇	#1340	semidirect

Maximal subgroups

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

subgroup	index
#23757531	2
#23757055	2
H(3,ℤ/4)×C₁₅₇	2
#1340	157