Subgroups — #41003556

Order 26944.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212355	C₂	1	4	yes	maximal, complemented
2	#27212341	C₂	1	4	yes	maximal, complemented
3	C₄×D₁₆₈₄	C₂	1	2	yes	maximal, complemented
4	#27212324	C₂	1	4	yes	maximal, complemented
5	C₄⋊Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	C₂²	1	2	yes	—
7	C₂×D₁₆₈₄	C₂²	1	4	yes	—
8	C₂×C₄×D₄₂₁	C₂²	1	4	yes	—
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	#20269748	C₂²	1	8	yes	—
11	#20269748	C₂²	1	4	yes	—
12	C₄×Dic₄₂₁	C₂²	1	4	yes	—
13	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	—
14	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
15	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄₂₁	C₂³	1	4	yes	—
17	D₁₆₈₄	—	2	4	no	—
18	D₁₆₈₄	—	4	4	no	—
19	C₂×Dic₄₂₁	C₂³	1	4	yes	—
20	C₄×D₄₂₁	—	2	8	no	—
21	Q₈×C₄₂₁	—	4	2	no	—
22	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
25	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₂₁	—	2	4	no	—
27	C₂×D₄₂₁	—	2	4	no	—
28	C₂×D₄₂₁	—	2	4	no	—
29	C₄×C₄₂₁	—	2	1	no	—
30	C₄×C₄₂₁	—	2	4	no	—
31	C₄×C₄₂₁	#41	1	4	yes	—
32	Dic₄₂₁	—	2	4	no	—
33	C₂×C₄₂₁	C₂×#41	1	2	yes	—
34	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
35	D₄₂₁	—	4	4	no	—
36	C₄₂₁	#555	1	1	yes	characteristic, complemented, Sylow
37	#555	—	421	1	no	maximal, Sylow
38	C₄⋊Q₈	D₄₂₁	1	1	yes	characteristic, complemented
39	#124	—	421	4	no	—
40	#121	—	421	4	no	—
41	C₄×D₄	—	421	2	no	—
42	#117	—	421	4	no	—
43	C₂×Q₈	C₂×D₄₂₁	1	2	yes	—
44	C₂×D₄	—	421	2	no	—
45	C₂×D₄	—	421	4	no	—
46	C₂²×C₄	—	421	4	no	—
47	C₄⋊C₄	—	421	2	no	—
48	C₄⋊C₄	C₂×D₄₂₁	1	4	yes	—
49	C₂²⋊C₄	—	421	8	no	—
50	C₂²⋊C₄	—	421	4	no	—
51	C₄²	—	421	4	no	—
52	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
53	C₂³	—	421	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	842	4	no	—
56	D₄	—	1684	4	no	—
57	C₂×C₄	—	842	8	no	—
58	C₂×C₄	—	421	4	no	—
59	C₂×C₄	C₂²×D₄₂₁	1	4	yes	—
60	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
62	C₂²	—	842	4	no	—
63	C₂²	—	842	4	no	—
64	C₂²	—	842	4	no	—
65	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	842	4	no	—
69	C₄	#20269772	1	4	yes	—
70	C₂	—	1684	4	no	—
71	C₂	C₂×#20269772	1	2	yes	—
72	C₂	#27212411	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₄⋊Q₈×C₄₂₁	C₂
hypercenter	C₄⋊Q₈	D₄₂₁
hypocenter	C₄₂₁	#555
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #27212355	C₂	semidirect
2. #27212341	C₂	semidirect
3. C₄×D₁₆₈₄	C₂	semidirect
4. #27212324	C₂	semidirect
5. C₄⋊Q₈×C₄₂₁	C₂	semidirect
36. C₄₂₁	#555	semidirect
38. C₄⋊Q₈	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212355	2
#27212341	2
C₄×D₁₆₈₄	2
#27212324	2
C₄⋊Q₈×C₄₂₁	2
#555	421