Subgroups — C₂×#23756417 #34094138

Order 20032.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14578809	C₂	1	1	yes	maximal, characteristic, complemented
3	#23756417	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₄×C₃₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₁₃	—	2	1	no	—
6	C₂×D₁₂₅₂	—	2	1	no	—
7	C₂×D₁₂₅₂	—	2	2	no	—
8	C₂×D₁₂₅₂	C₄	1	4	yes	complemented
9	C₂×D₁₂₅₂	C₂²	1	2	yes	—
10	C₂×#14578809	—	2	2	no	—
11	C₂×#14578809	C₂²	1	2	yes	complemented
12	C₂²×C₄×C₃₁₃	—	2	1	no	—
13	C₂²×C₄×C₃₁₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₁₃	—	2	1	no	—
15	C₄⋊C₄×C₃₁₃	C₂²	1	2	yes	complemented
16	C₂²×D₃₁₃	—	2	4	no	—
17	C₂²×D₃₁₃	—	2	2	no	—
18	C₂²×D₃₁₃	—	4	2	no	—
19	C₂²×D₃₁₃	—	4	2	no	—
20	D₁₂₅₂	—	2	4	no	—
21	D₁₂₅₂	—	2	2	no	—
22	D₁₂₅₂	C₂×C₄	1	4	yes	complemented
23	#14578809	—	2	2	no	—
24	C₂³×C₃₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₁₃	—	2	2	no	—
26	C₂×C₄×C₃₁₃	—	4	2	no	—
27	C₂×C₄×C₃₁₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₃₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁₃	D₄	1	2	yes	—
30	C₂×C₄×C₃₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₃₁₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₃₁₃	D₈	1	2	yes	—
33	C₂²×C₃₁₃	SD₁₆	1	2	yes	—
34	C₂²×C₃₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×D₃₁₃	—	2	4	no	—
36	C₂×D₃₁₃	—	4	8	no	—
37	C₂×D₃₁₃	—	4	2	no	—
38	C₂×D₃₁₃	—	4	2	no	—
39	C₄×C₃₁₃	—	4	2	no	—
40	C₄×C₃₁₃	C₂²⋊C₄	1	2	yes	—
41	C₄×C₃₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₃₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃₁₃	#102	1	4	yes	complemented
44	C₂×C₃₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃₁₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₃₁₃	C₂×SD₁₆	1	1	yes	characteristic
47	D₃₁₃	—	4	4	no	—
48	C₃₁₃	C₂×#102	1	1	yes	characteristic, complemented, Sylow
49	C₂×#102	—	313	1	no	maximal, Sylow
50	C₂²×D₄	—	313	1	no	—
51	C₂²×C₈	—	313	1	no	—
52	C₂×C₄⋊C₄	D₃₁₃	1	1	yes	characteristic, complemented
53	#102	—	313	4	no	—
54	C₂⁴	—	626	1	no	—
55	C₂×D₄	—	313	4	no	—
56	C₂×D₄	—	313	2	no	—
57	C₂×D₄	—	626	1	no	—
58	C₂×D₄	—	626	2	no	—
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂×D₃₁₃	1	1	yes	characteristic
61	C₂×C₈	—	313	2	no	—
62	C₂×C₈	—	626	2	no	—
63	C₄⋊C₄	—	2	1	no	—
64	C₄⋊C₄	C₂×D₃₁₃	1	2	yes	complemented
65	C₂³	—	626	4	no	—
66	C₂³	—	626	2	no	—
67	C₂³	—	1252	2	no	—
68	C₂³	—	1252	2	no	—
69	C₂³	C₂²⋊D₃₁₃	1	1	yes	characteristic
70	D₄	—	313	4	no	—
71	D₄	—	626	4	no	—
72	D₄	—	626	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	C₄×D₃₁₃	1	2	yes	—
76	C₂×C₄	C₂²⋊D₃₁₃	1	1	yes	characteristic
77	C₂×C₄	D₁₂₅₂	1	2	yes	—
78	C₂×C₄	C₂²×D₃₁₃	1	1	yes	characteristic
79	C₈	—	626	2	no	—
80	C₂²	—	626	4	no	—
81	C₂²	—	1252	8	no	—
82	C₂²	—	1252	2	no	—
83	C₂²	—	1252	2	no	—
84	C₂²	#19097538	1	2	yes	—
85	C₂²	#19097546	1	2	yes	—
86	C₂²	#19097548	1	2	yes	—
87	C₂²	C₂×C₂²⋊D₃₁₃	1	1	yes	characteristic
88	C₄	—	4	2	no	—
89	C₄	#19097538	1	2	yes	—
90	C₄	C₂×C₄×D₃₁₃	1	1	yes	characteristic
91	C₄	C₂×D₁₂₅₂	1	1	yes	characteristic
92	C₂	—	1252	4	no	—
93	C₂	#23756417	1	4	yes	direct product factor
94	C₂	C₂×#19097538	1	1	yes	characteristic
95	C₂	C₂×#19097546	1	1	yes	characteristic
96	C₂	C₂×#19097548	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₁₃
commutator	C₄×C₃₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₃
Fitting	C₂×C₄⋊C₄×C₃₁₃	C₂
socle	C₂³×C₃₁₃	D₄
prime radical	C₂²×D₁₂₅₂	C₂
cosocle	C₂×C₄×C₃₁₃	C₂³
hypercenter	C₂×C₄⋊C₄	D₃₁₃
hypocenter	C₃₁₃	C₂×#102
LCS	G ▷ C₄×C₃₁₃ ▷ C₂×C₃₁₃ ▷ C₃₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#14578809	C₂	semidirect
3. #23756417	C₂	direct (N × H)
4. C₂×C₄⋊C₄×C₃₁₃	C₂	semidirect
8. C₂×D₁₂₅₂	C₄	semidirect
11. C₂×#14578809	C₂²	semidirect
15. C₄⋊C₄×C₃₁₃	C₂²	semidirect
22. D₁₂₅₂	C₂×C₄	semidirect
43. C₂×C₃₁₃	#102	semidirect
48. C₃₁₃	C₂×#102	semidirect
52. C₂×C₄⋊C₄	D₃₁₃	semidirect
64. C₄⋊C₄	C₂×D₃₁₃	semidirect
93. C₂	#23756417	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₅₂	2
C₂²×#14578809	2
#23756417	2
C₂×C₄⋊C₄×C₃₁₃	2
C₂×#102	313

Subgroups — C2×#23756417 #34094138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23756417 #34094138