Subgroups — #34115082

Order 20096.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758169	C₂	1	2	yes	maximal
2	#23757297	C₂	1	1	yes	maximal, characteristic
3	#19097845	—	2	2	no	—
4	C₂×#14578938	C₄	1	1	yes	characteristic
5	C₂×#14578938	C₂²	1	1	yes	characteristic
6	C₄⋊#11929434	—	2	2	no	—
7	C₂×C₄²×C₁₅₇	C₄	1	1	yes	characteristic
8	#14578957	—	4	2	no	—
9	#14578938	—	4	1	no	—
10	#14578938	—	4	1	no	—
11	C₂×#11929434	—	2	1	no	—
12	C₂×#11929434	D₄	1	2	yes	—
13	C₂²×C₄×C₁₅₇	—	2	1	no	—
14	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
15	C₄²×C₁₅₇	—	2	1	no	—
16	C₄²×C₁₅₇	C₈	1	1	yes	characteristic
17	C₄²×C₁₅₇	C₈	1	1	yes	characteristic
18	#11929434	—	2	2	no	—
19	#11929434	—	4	1	no	—
20	C₂³×C₁₅₇	M₁₆	1	1	yes	characteristic
21	C₂×C₄×C₁₅₇	—	2	1	no	—
22	C₂×C₄×C₁₅₇	—	2	1	no	—
23	C₂×C₄×C₁₅₇	—	2	1	no	—
24	C₂×C₄×C₁₅₇	—	2	1	no	—
25	C₂×C₄×C₁₅₇	—	4	2	no	—
26	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₅₇	—	2	1	no	—
29	C₂²×C₁₅₇	—	4	1	no	—
30	C₂²×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₅₇	—	2	2	no	—
32	C₄×C₁₅₇	—	2	1	no	—
33	C₄×C₁₅₇	—	4	1	no	—
34	C₄×C₁₅₇	#99	1	1	yes	characteristic
35	C₄×C₁₅₇	#101	1	1	yes	characteristic
36	C₂×C₁₅₇	—	2	1	no	—
37	C₂×C₁₅₇	—	4	1	no	—
38	C₂×C₁₅₇	#324	1	1	yes	characteristic
39	C₁₅₇	#1327	1	1	yes	characteristic, complemented, Sylow
40	#1327	—	157	1	no	maximal, Sylow
41	#423	—	157	1	no	—
42	#349	—	157	2	no	—
43	C₂×M₁₆	—	157	1	no	—
44	C₂×M₁₆	—	157	1	no	—
45	C₂×C₄²	#10597845	1	1	yes	characteristic
46	M₃₂	—	314	2	no	—
47	C₄⋊C₈	—	314	2	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	C₂×#10597845	1	1	yes	characteristic
50	M₁₆	—	628	1	no	—
51	M₁₆	—	628	1	no	—
52	C₂×C₈	—	314	1	no	—
53	C₂×C₈	—	157	2	no	—
54	C₄²	—	2	1	no	—
55	C₄²	#11929441	1	1	yes	characteristic
56	C₄²	#11929441	1	1	yes	characteristic
57	C₁₆	—	628	2	no	—
58	C₂³	#14578963	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
65	C₂×C₄	C₂×#11929441	1	1	yes	characteristic
66	C₈	—	314	2	no	—
67	C₈	—	628	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂²⋊#11929441	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	2	1	no	—
73	C₄	—	4	1	no	—
74	C₄	#19097830	1	1	yes	characteristic
75	C₄	#19097832	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#23758118	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19097830
commutator	C₂×C₄×C₁₅₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#10597845
Fitting	C₂×C₄²×C₁₅₇	C₄
socle	C₂×C₁₅₇	#324
prime radical	C₂³×C₁₅₇	M₁₆
cosocle	C₂×#14578938	C₂²
hypercenter	C₂×C₄²	#10597845
hypocenter	C₁₅₇	#1327
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
39. C₁₅₇	#1327	semidirect

Maximal subgroups

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

subgroup	index
#23758169	2
#23757297	2
#1327	157