Subgroups — #34098017

Order 20096.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097682	C₂	1	2	yes	maximal
2	#382×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11929434	—	2	2	no	—
4	#19097682	—	2	2	no	—
5	#19097682	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₅₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
8	#95×C₁₅₇	—	2	2	no	—
9	C₂×#11929434	—	2	8	no	—
10	C₂×#11929434	—	2	4	no	—
11	C₂²×C₄×C₁₅₇	—	2	2	no	—
12	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
14	C₄⋊C₄×C₁₅₇	—	4	2	no	—
15	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
17	C₄²×C₁₅₇	Q₈	1	1	yes	characteristic
18	#11929434	—	2	8	no	—
19	C₂³×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₅₇	—	2	1	no	—
21	C₂×C₄×C₁₅₇	—	2	1	no	—
22	C₂×C₄×C₁₅₇	—	2	4	no	—
23	C₂×C₄×C₁₅₇	—	4	2	no	—
24	C₂×C₄×C₁₅₇	—	4	4	no	—
25	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
27	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₅₇	#95	1	1	yes	characteristic
30	C₂²×C₁₅₇	#100	1	1	yes	characteristic
31	C₂²×C₁₅₇	#101	1	1	yes	characteristic
32	C₂²×C₁₅₇	#104	1	4	yes	—
33	C₄×C₁₅₇	—	2	2	no	—
34	C₄×C₁₅₇	—	2	2	no	—
35	C₄×C₁₅₇	—	2	1	no	—
36	C₄×C₁₅₇	—	2	1	no	—
37	C₄×C₁₅₇	—	8	2	no	—
38	C₂×C₁₅₇	#329	1	2	yes	—
39	C₂×C₁₅₇	#330	1	2	yes	—
40	C₂×C₁₅₇	#339	1	2	yes	—
41	C₂×C₁₅₇	#343	1	1	yes	characteristic
42	C₁₅₇	#1295	1	1	yes	characteristic, complemented, Sylow
43	#1295	—	157	1	no	maximal, Sylow
44	C₂×#97	—	157	2	no	—
45	#382	D₁₅₇	1	1	yes	characteristic
46	C₂²×C₈	—	314	2	no	—
47	C₂×C₄⋊C₄	Dic₁₅₇	1	2	yes	—
48	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
49	#97	—	314	2	no	—
50	#97	—	157	4	no	—
51	#95	—	2	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
55	C₂×C₈	—	314	8	no	—
56	C₂×C₈	—	314	4	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	C₄×D₁₅₇	1	2	yes	—
59	C₄²	D₆₂₈	1	1	yes	characteristic
60	C₄²	Dic₃₁₄	1	1	yes	characteristic
61	C₂³	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	C₄×Dic₁₅₇	1	1	yes	characteristic
68	C₂×C₄	#14578931	1	2	yes	—
69	C₂×C₄	#14578933	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
71	C₈	—	314	8	no	—
72	C₂²	#19097678	1	1	yes	characteristic
73	C₂²	#19097688	1	1	yes	characteristic
74	C₂²	#19097690	1	1	yes	characteristic
75	C₂²	#19097698	1	4	yes	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	8	2	no	—
81	C₂	#23757013	1	2	yes	—
82	C₂	#23757015	1	2	yes	—
83	C₂	#23757032	1	2	yes	—
84	C₂	#23757042	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₅₇
commutator	C₂×C₄×C₁₅₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₅₇
Fitting	#382×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₅₇	C₂²
hypercenter	#382	D₁₅₇
hypocenter	C₁₅₇	#1295
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
5. #19097682	C₄	semidirect
42. C₁₅₇	#1295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097682	2
#382×C₁₅₇	2
#1295	157