Subgroups — #41003512

Order 26944.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269772	C₂	1	2	yes	maximal, complemented
2	#27212363	C₂	1	3	yes	maximal, complemented
3	#27212341	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₆₈₄	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269772	—	2	8	no	—
7	C₂×D₁₆₈₄	C₂²	1	3	yes	—
8	C₂×D₁₆₈₄	C₂²	1	6	yes	—
9	C₂×D₁₆₈₄	C₂²	1	6	yes	—
10	C₂×C₄×D₄₂₁	C₂²	1	6	yes	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269748	C₂²	1	6	yes	—
13	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₄₂₁	C₂²	1	3	yes	—
15	C₄²×C₄₂₁	C₂²	1	3	yes	—
16	C₂²×D₄₂₁	C₂³	1	6	yes	—
17	D₁₆₈₄	—	2	12	no	—
18	D₁₆₈₄	—	2	6	no	—
19	D₁₆₈₄	—	4	6	no	—
20	C₂×Dic₄₂₁	C₂³	1	2	yes	—
21	C₄×D₄₂₁	—	2	12	no	—
22	Q₈×C₄₂₁	D₄	1	4	yes	complemented
23	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
24	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
25	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
27	C₂×D₄₂₁	—	2	6	no	—
28	C₂×D₄₂₁	—	2	6	no	—
29	C₂×D₄₂₁	—	2	6	no	—
30	C₄×C₄₂₁	—	2	3	no	—
31	C₄×C₄₂₁	C₂×D₄	1	6	yes	—
32	C₄×C₄₂₁	#41	1	2	yes	complemented
33	Dic₄₂₁	—	2	2	no	—
34	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₄₂₁	C₂×#41	1	1	yes	characteristic
36	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
37	D₄₂₁	—	4	6	no	—
38	C₄₂₁	#550	1	1	yes	characteristic, complemented, Sylow
39	#550	—	421	1	no	maximal, Sylow
40	C₂×#41	—	421	2	no	—
41	#127	—	421	3	no	—
42	#121	—	421	6	no	—
43	C₄×Q₈	D₄₂₁	1	1	yes	characteristic, complemented
44	C₄×D₄	—	421	3	no	—
45	#41	—	842	8	no	—
46	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
47	C₂×D₄	—	421	3	no	—
48	C₂×D₄	—	421	6	no	—
49	C₂×D₄	—	421	6	no	—
50	C₂²×C₄	—	421	6	no	—
51	C₄⋊C₄	—	421	1	no	—
52	C₄⋊C₄	C₂×D₄₂₁	1	3	yes	—
53	C₂²⋊C₄	—	421	6	no	—
54	C₄²	C₂×D₄₂₁	1	3	yes	—
55	C₂³	—	421	6	no	—
56	Q₈	D₁₆₈₄	1	4	yes	complemented
57	D₄	—	842	12	no	—
58	D₄	—	842	6	no	—
59	D₄	—	1684	6	no	—
60	C₂×C₄	—	842	12	no	—
61	C₂×C₄	—	421	2	no	—
62	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
63	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
64	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
65	C₂²	—	842	6	no	—
66	C₂²	—	842	6	no	—
67	C₂²	—	842	6	no	—
68	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	842	2	no	—
71	C₄	C₂×D₁₆₈₄	1	6	yes	—
72	C₄	#20269772	1	2	yes	complemented
73	C₂	—	1684	6	no	—
74	C₂	C₂²×D₁₆₈₄	1	1	yes	characteristic
75	C₂	C₂×#20269772	1	1	yes	characteristic
76	C₂	#27212412	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	#550
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. C₂×#20269772	C₂	semidirect
2. #27212363	C₂	semidirect
3. #27212341	C₂	semidirect
4. C₄×D₁₆₈₄	C₂	semidirect
5. C₄×Q₈×C₄₂₁	C₂	semidirect
22. Q₈×C₄₂₁	D₄	semidirect
32. C₄×C₄₂₁	#41	semidirect
38. C₄₂₁	#550	semidirect
43. C₄×Q₈	D₄₂₁	semidirect
56. Q₈	D₁₆₈₄	semidirect
72. C₄	#20269772	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269772	2
#27212363	2
#27212341	2
C₄×D₁₆₈₄	2
C₄×Q₈×C₄₂₁	2
#550	421