Subgroups — #41002896

Order 26944.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄₂₁	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#16809890	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212278	C₂	1	2	yes	maximal
4	#27212276	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄₂₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄₂₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
9	C₂×#16809890	C₂²	1	2	yes	—
10	C₄×Dic₄₂₁	C₄	1	1	yes	characteristic
11	C₄×Dic₄₂₁	C₄	1	1	yes	characteristic
12	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
14	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄₂₁	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₄₂₁	C₂×C₄	1	1	yes	characteristic
19	C₄×D₄₂₁	—	2	2	no	—
20	C₄×D₄₂₁	—	2	2	no	—
21	C₄×D₄₂₁	—	2	1	no	—
22	C₄×D₄₂₁	—	2	1	no	—
23	#16809890	—	2	2	no	—
24	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
27	C₈×C₄₂₁	—	2	2	no	—
28	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×D₄₂₁	—	2	1	no	—
30	C₂×D₄₂₁	—	2	1	no	—
31	C₂×D₄₂₁	M₁₆	1	2	yes	—
32	C₄×C₄₂₁	—	2	1	no	—
33	C₄×C₄₂₁	M₁₆	1	2	yes	—
34	C₄×C₄₂₁	#41	1	1	yes	characteristic
35	C₄×C₄₂₁	#41	1	1	yes	characteristic
36	Dic₄₂₁	—	2	1	no	—
37	Dic₄₂₁	—	2	1	no	—
38	Dic₄₂₁	—	2	1	no	—
39	C₂×C₄₂₁	#117	1	1	yes	characteristic
40	C₂×C₄₂₁	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄₂₁	C₂×M₁₆	1	1	yes	characteristic
42	D₄₂₁	—	2	2	no	—
43	C₄₂₁	#432	1	1	yes	characteristic, complemented, Sylow
44	#432	—	421	1	no	maximal, Sylow
45	C₂×C₄²	—	421	1	no	—
46	C₄⋊C₈	—	421	1	no	—
47	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic, complemented
48	C₂²⋊C₈	—	421	2	no	—
49	#97	—	421	2	no	—
50	C₂²×C₄	—	421	1	no	—
51	C₂²×C₄	—	421	1	no	—
52	C₂²×C₄	—	421	1	no	—
53	C₂×C₈	—	421	2	no	—
54	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
55	C₄²	—	421	1	no	—
56	C₄²	—	421	1	no	—
57	C₄²	—	421	1	no	—
58	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
59	C₂³	—	421	1	no	—
60	C₂×C₄	—	842	2	no	—
61	C₂×C₄	—	842	2	no	—
62	C₂×C₄	—	842	1	no	—
63	C₂×C₄	—	842	1	no	—
64	C₂×C₄	—	421	1	no	—
65	C₂×C₄	—	421	1	no	—
66	C₂×C₄	—	421	1	no	—
67	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
68	C₂×C₄	C₄×D₄₂₁	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	842	2	no	—
72	C₂²	—	842	1	no	—
73	C₂²	—	842	1	no	—
74	C₂²	—	421	2	no	—
75	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	842	1	no	—
78	C₄	—	842	1	no	—
79	C₄	—	842	1	no	—
80	C₄	#20269754	1	2	yes	—
81	C₄	#20269770	1	1	yes	characteristic
82	C₄	#20269772	1	1	yes	characteristic
83	C₂	—	842	2	no	—
84	C₂	#27212324	1	1	yes	characteristic
85	C₂	M₁₆×D₄₂₁	1	1	yes	characteristic
86	C₂	C₂×#20269754	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₄₂₁
commutator, socle	C₂²×C₄₂₁	C₂²×C₄
Fitting	C₄⋊C₈×C₄₂₁	C₂
prime radical	C₂²×D₄₂₁	C₂×C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#432
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊#16809890	C₂	semidirect
4. #27212276	C₂	semidirect
5. C₄⋊C₈×C₄₂₁	C₂	semidirect
43. C₄₂₁	#432	semidirect
47. C₄⋊C₈	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄₂₁	2
C₄⋊#16809890	2
#27212278	2
#27212276	2
C₄⋊C₈×C₄₂₁	2
#432	421