Subgroups — #34100307

Order 20096.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097706	C₂	1	4	yes	maximal
2	#23757297	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11929434	C₂	1	1	yes	maximal, characteristic
4	#423×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	C₂²	1	2	yes	—
6	C₂²×#11929434	C₂²	1	2	yes	—
7	#19097706	—	2	8	no	—
8	C₄⋊#11929434	—	2	2	no	—
9	C₄×#11929434	C₄	1	2	yes	—
10	C₄×#11929434	C₄	1	2	yes	—
11	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₇	—	2	2	no	—
14	#14578938	—	2	4	no	—
15	C₂×#11929434	—	2	2	no	—
16	C₂×#11929434	C₂×C₄	1	4	yes	—
17	C₂×#11929434	D₄	1	4	yes	—
18	C₂×#11929434	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
21	M₁₆×C₁₅₇	—	2	4	no	—
22	C₂×C₈×C₁₅₇	—	2	2	no	—
23	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₅₇	D₄	1	2	yes	—
26	#11929434	—	4	2	no	—
27	#11929434	C₄⋊C₄	1	8	yes	—
28	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	—	2	4	no	—
30	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
36	C₈×C₁₅₇	—	4	2	no	—
37	C₂²×C₁₅₇	—	2	2	no	—
38	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
40	C₄×C₁₅₇	—	2	2	no	—
41	C₄×C₁₅₇	#108	1	4	yes	complemented
42	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
43	C₄×C₁₅₇	#127	1	1	yes	characteristic
44	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₅₇	—	2	2	no	—
46	C₂×C₁₅₇	#384	1	1	yes	characteristic
47	C₂×C₁₅₇	C₂×#108	1	2	yes	—
48	C₁₅₇	#1848	1	1	yes	characteristic, complemented, Sylow
49	#1848	—	157	1	no	maximal, Sylow
50	C₂×#108	—	157	4	no	—
51	#423	—	157	1	no	—
52	#423	D₁₅₇	1	1	yes	characteristic
53	C₂×C₄×C₈	—	157	1	no	—
54	C₂×M₁₆	—	157	2	no	—
55	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
56	C₂²×C₈	—	157	2	no	—
57	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
58	#108	—	314	8	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	314	2	no	—
61	C₄×C₈	—	157	2	no	—
62	C₄×C₈	—	157	2	no	—
63	C₂²×C₄	Dic₃₁₄	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	314	4	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	314	2	no	—
69	C₂×C₈	—	157	4	no	—
70	C₂×C₈	—	157	4	no	—
71	C₂×C₈	—	157	4	no	—
72	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
73	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
74	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
75	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	#14578933	1	2	yes	—
78	C₂×C₄	#14578933	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
80	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
81	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
82	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
83	C₈	—	4	2	no	—
84	C₈	—	628	2	no	—
85	C₈	—	157	8	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂×#14578933	1	1	yes	characteristic
88	C₂²	#19097770	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	#19097706	1	4	yes	complemented
91	C₄	C₄⋊C₄×D₁₅₇	1	2	yes	—
92	C₄	#19097767	1	1	yes	characteristic
93	C₄	#19097769	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	#23757142	1	1	yes	characteristic
96	C₂	C₂×#19097706	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#423×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#423	D₁₅₇
hypocenter	C₁₅₇	#1848
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
41. C₄×C₁₅₇	#108	semidirect
48. C₁₅₇	#1848	semidirect
90. C₄	#19097706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097706	2
#23757297	2
C₂×C₄×#11929434	2
#423×C₁₅₇	2
#1848	157