Subgroups — #41003892

Order 26944.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212433	C₂	1	1	yes	maximal, characteristic
2	#27212433	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#16809897	C₂	1	1	yes	maximal, characteristic
4	#27212421	C₂	1	1	yes	maximal, characteristic
5	#27212421	C₂	1	1	yes	maximal, characteristic
6	C₄×#16809897	C₂	1	1	yes	maximal, characteristic
7	#27212331	C₂	1	1	yes	maximal, characteristic
8	C₂×#16809897	C₂²	1	1	yes	characteristic
9	C₂×#16809897	C₂²	1	1	yes	characteristic
10	C₂×#16809897	C₂²	1	1	yes	characteristic
11	C₂×#16809897	C₂²	1	1	yes	characteristic
12	C₂×Dic₈₄₂	C₄	1	1	yes	characteristic
13	#20269750	C₄	1	1	yes	characteristic
14	#20269750	C₄	1	1	yes	characteristic
15	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
16	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₂₁	C₄	1	1	yes	characteristic
19	#16809897	—	2	1	no	—
20	#16809897	—	2	1	no	—
21	#16809897	—	2	1	no	—
22	#16809897	Q₈	1	2	yes	complemented
23	Dic₈₄₂	—	2	2	no	—
24	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₄₂₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₄₂₁	—	2	1	no	—
33	C₄×C₄₂₁	—	2	1	no	—
34	C₄×C₄₂₁	—	2	1	no	—
35	Dic₄₂₁	—	2	1	no	—
36	Dic₄₂₁	—	2	1	no	—
37	Dic₄₂₁	M₁₆	1	2	yes	—
38	Dic₄₂₁	C₂×Q₈	1	1	yes	characteristic
39	Dic₄₂₁	#41	1	1	yes	characteristic
40	C₂×C₄₂₁	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₄₂₁	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄₂₁	#131	1	1	yes	characteristic
43	C₄₂₁	#446	1	1	yes	characteristic, complemented, Sylow
44	#446	—	421	1	no	maximal, Sylow
45	C₄×Q₈	—	421	1	no	—
46	C₄⋊C₈	—	421	1	no	—
47	C₄⋊C₈	—	421	1	no	—
48	C₄⋊C₈	—	421	1	no	—
49	#97	—	421	1	no	—
50	#97	—	421	1	no	—
51	C₄×C₈	—	421	1	no	—
52	C₂×Q₈	—	421	1	no	—
53	C₂×C₈	—	421	1	no	—
54	C₂×C₈	—	421	1	no	—
55	C₂×C₈	—	421	1	no	—
56	C₂×C₈	—	421	1	no	—
57	C₄⋊C₄	—	421	1	no	—
58	C₄⋊C₄	—	421	1	no	—
59	C₄⋊C₄	#13132997	1	1	yes	characteristic
60	C₄²	—	421	1	no	—
61	C₄²	—	421	1	no	—
62	C₄²	—	421	1	no	—
63	Q₈	—	842	2	no	—
64	C₂×C₄	—	421	1	no	—
65	C₂×C₄	—	421	1	no	—
66	C₂×C₄	—	421	1	no	—
67	C₂×C₄	—	421	1	no	—
68	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
69	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
70	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
71	C₈	—	842	1	no	—
72	C₈	—	842	1	no	—
73	C₈	—	842	1	no	—
74	C₈	—	421	2	no	—
75	C₂²	C₂²×#13132997	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	842	1	no	—
80	C₄	—	842	1	no	—
81	C₄	—	421	1	no	—
82	C₄	—	421	1	no	—
83	C₄	—	421	2	no	—
84	C₂	Q₈×#13132997	1	1	yes	characteristic
85	C₂	D₄₂₁⋊M₁₆	1	1	yes	characteristic
86	C₂	#27212453	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13132997
commutator, socle, prime radical	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₄⋊C₄×C₄₂₁	C₄
cosocle	C₂×Dic₄₂₁	C₂³
hypercenter	C₄⋊C₄	#13132997
hypocenter	C₄₂₁	#446
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
22. #16809897	Q₈	semidirect
43. C₄₂₁	#446	semidirect

Maximal subgroups

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

subgroup	index
#27212433	2
C₄⋊#16809897	2
#27212421	2
C₄×#16809897	2
#27212331	2
#446	421