Subgroups — #34095127

Order 20032.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097571	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19097569	C₂	1	1	yes	maximal, characteristic, complemented
3	#23756554	C₂	1	4	yes	maximal, complemented
4	C₂×#19097562	C₂	1	1	yes	maximal, characteristic
5	#19097571	—	2	2	no	—
6	#19097569	—	2	2	no	—
7	C₂×#14578816	C₂²	1	2	yes	—
8	C₂×#14578816	C₂²	1	2	yes	—
9	#19097562	—	2	2	no	—
10	#19097562	—	2	2	no	—
11	C₂×D₁₂₅₂	C₄	1	2	yes	—
12	C₂×D₁₂₅₂	C₄	1	1	yes	characteristic
13	C₂×C₄×D₃₁₃	C₂²	1	2	yes	—
14	C₂×C₄×D₃₁₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₃₁₃	C₄	1	1	yes	characteristic
16	#14578816	—	2	2	no	—
17	#14578816	—	2	2	no	—
18	C₂²×D₃₁₃	C₂×C₄	1	2	yes	—
19	C₂²×D₃₁₃	C₂×C₄	1	1	yes	characteristic
20	D₁₂₅₂	—	2	4	no	—
21	D₁₂₅₂	—	2	2	no	—
22	C₂×Dic₃₁₃	C₂³	1	1	yes	characteristic
23	C₄×D₃₁₃	—	2	2	no	—
24	C₄×D₃₁₃	—	2	2	no	—
25	C₄×D₃₁₃	D₄	1	2	yes	—
26	C₄×D₃₁₃	D₄	1	2	yes	—
27	Q₈×C₃₁₃	—	2	2	no	—
28	C₂×C₄×C₃₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃₁₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₃₁₃	—	2	2	no	—
32	C₂×D₃₁₃	—	2	2	no	—
33	C₂×D₃₁₃	—	2	4	no	—
34	C₂×D₃₁₃	C₂²⋊C₄	1	2	yes	—
35	C₄×C₃₁₃	—	2	2	no	—
36	C₄×C₃₁₃	C₂²⋊C₄	1	2	yes	—
37	Dic₃₁₃	C₂×D₄	1	1	yes	characteristic
38	Dic₃₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₃₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₃₁₃	#131	1	2	yes	—
41	D₃₁₃	—	2	2	no	—
42	D₃₁₃	—	4	2	no	—
43	C₃₁₃	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	313	1	no	maximal, Sylow
45	C₂×#41	—	313	1	no	—
46	C₂×M₁₆	—	313	1	no	—
47	C₂²×C₈	—	313	1	no	—
48	C₂²⋊C₈	—	313	4	no	—
49	#41	—	626	2	no	—
50	#41	—	626	2	no	—
51	C₂×Q₈	#11929411	1	1	yes	characteristic
52	C₂×D₄	—	313	2	no	—
53	C₂×D₄	—	313	1	no	—
54	C₂²×C₄	—	313	1	no	—
55	C₂²×C₄	—	313	2	no	—
56	M₁₆	—	626	2	no	—
57	C₂×C₈	—	313	2	no	—
58	C₂×C₈	—	313	2	no	—
59	C₂×C₈	—	626	2	no	—
60	C₂³	—	313	1	no	—
61	C₂³	—	313	2	no	—
62	Q₈	—	2	2	no	—
63	D₄	—	626	4	no	—
64	D₄	—	626	2	no	—
65	C₂×C₄	—	313	1	no	—
66	C₂×C₄	—	313	2	no	—
67	C₂×C₄	—	313	2	no	—
68	C₂×C₄	—	626	2	no	—
69	C₂×C₄	—	626	2	no	—
70	C₂×C₄	C₂×#11929411	1	2	yes	—
71	C₂×C₄	C₂×#11929411	1	1	yes	characteristic
72	C₈	—	626	2	no	—
73	C₈	—	626	2	no	—
74	C₂²	—	313	2	no	—
75	C₂²	—	626	2	no	—
76	C₂²	—	626	2	no	—
77	C₂²	—	626	4	no	—
78	C₂²	C₂²×#11929411	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	313	1	no	—
81	C₄	—	313	1	no	—
82	C₄	C₂²⋊#11929411	1	2	yes	—
83	C₂	—	626	2	no	—
84	C₂	—	1252	2	no	—
85	C₂	C₂×C₂²⋊#11929411	1	1	yes	characteristic
86	C₂	#23756585	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11929411
commutator, socle	C₂²×C₃₁₃	C₂²×C₄
Fitting	C₂×Q₈×C₃₁₃	C₄
prime radical	C₂×#19097562	C₂
cosocle	C₂×Dic₃₁₃	C₂³
hypercenter	C₂×Q₈	#11929411
hypocenter	C₃₁₃	#408
LCS	G ▷ C₂²×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19097571	C₂	semidirect
2. C₂×#19097569	C₂	semidirect
3. #23756554	C₂	semidirect
43. C₃₁₃	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097571	2
C₂×#19097569	2
#23756554	2
C₂×#19097562	2
#408	313