Subgroups — #41003451

Order 26944.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212361	C₂	1	4	yes	maximal
2	#27212358	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212356	C₂	1	4	yes	maximal, complemented
4	#27212342	C₂	1	2	yes	maximal
5	Q₈×Dic₄₂₁	C₂	1	2	yes	maximal, complemented
6	#27212323	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
10	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
11	#20269750	C₂²	1	2	yes	—
12	#20269750	C₂²	1	4	yes	—
13	#20269750	C₂²	1	4	yes	—
14	#20269748	C₂²	1	2	yes	—
15	#20269748	C₂²	1	4	yes	—
16	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
17	C₄×Dic₄₂₁	C₂²	1	4	yes	—
18	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	—
19	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
20	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
21	C₂²×D₄₂₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₄₂₁	C₂³	1	2	yes	—
23	C₂×Dic₄₂₁	C₂³	1	4	yes	—
24	C₂×Dic₄₂₁	C₂³	1	1	yes	characteristic
25	C₄×D₄₂₁	—	2	2	no	—
26	Q₈×C₄₂₁	—	2	4	no	—
27	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
28	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
29	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
30	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
31	C₂×D₄₂₁	—	2	1	no	—
32	C₂×D₄₂₁	—	2	2	no	—
33	C₄×C₄₂₁	—	2	4	no	—
34	C₄×C₄₂₁	—	2	2	no	—
35	C₄×C₄₂₁	#41	1	2	yes	—
36	Dic₄₂₁	—	2	1	no	—
37	Dic₄₂₁	—	2	2	no	—
38	Dic₄₂₁	—	2	4	no	—
39	C₂×C₄₂₁	C₂×#41	1	1	yes	characteristic
40	C₂×C₄₂₁	2₋¹⁺⁴	1	2	yes	—
41	D₄₂₁	—	4	1	no	—
42	C₄₂₁	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	421	1	no	maximal, Sylow
44	C₄⋊Q₈	D₄₂₁	1	1	yes	characteristic, complemented
45	#126	—	421	4	no	—
46	#125	—	421	4	no	—
47	#125	—	421	1	no	—
48	C₂²⋊Q₈	—	421	2	no	—
49	C₄×Q₈	—	421	2	no	—
50	#117	—	421	1	no	—
51	C₂×Q₈	C₂×D₄₂₁	1	2	yes	—
52	C₂²×C₄	—	421	1	no	—
53	C₄⋊C₄	—	421	2	no	—
54	C₄⋊C₄	—	421	2	no	—
55	C₄⋊C₄	—	421	4	no	—
56	C₄⋊C₄	—	421	4	no	—
57	C₄⋊C₄	—	421	4	no	—
58	C₄⋊C₄	C₂×D₄₂₁	1	4	yes	—
59	C₂²⋊C₄	—	421	2	no	—
60	C₂²⋊C₄	—	421	4	no	—
61	C₄²	—	421	1	no	—
62	C₄²	—	421	4	no	—
63	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
64	C₂³	—	421	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	842	2	no	—
67	C₂×C₄	—	421	2	no	—
68	C₂×C₄	—	421	4	no	—
69	C₂×C₄	—	421	1	no	—
70	C₂×C₄	C₂²×D₄₂₁	1	4	yes	—
71	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
72	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
73	C₂²	—	842	1	no	—
74	C₂²	—	842	2	no	—
75	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	842	1	no	—
79	C₄	—	842	2	no	—
80	C₄	—	842	4	no	—
81	C₄	#20269770	1	2	yes	—
82	C₂	—	1684	1	no	—
83	C₂	C₂×#20269770	1	1	yes	characteristic
84	C₂	#27212414	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₄⋊Q₈×C₄₂₁	C₂
prime radical	C₂²×D₄₂₁	C₂³
hypercenter	C₄⋊Q₈	D₄₂₁
hypocenter	C₄₂₁	#541
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
2. #27212358	C₂	semidirect
3. #27212356	C₂	semidirect
5. Q₈×Dic₄₂₁	C₂	semidirect
7. C₄⋊Q₈×C₄₂₁	C₂	semidirect
42. C₄₂₁	#541	semidirect
44. 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
#27212361	2
#27212358	2
#27212356	2
#27212342	2
Q₈×Dic₄₂₁	2
#27212323	2
C₄⋊Q₈×C₄₂₁	2
#541	421