Subgroups — #41003137

Order 26944.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₆₈₄	C₂	1	1	yes	maximal, characteristic
2	#27212299	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212297	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊#16809890	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212286	C₂	1	1	yes	maximal, characteristic
6	#27212286	C₂	1	1	yes	maximal, characteristic
7	#125×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₂₁	—	2	1	no	—
10	C₂×#16809890	C₂²	1	1	yes	characteristic
11	C₂×#16809890	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
13	#20269748	—	2	1	no	—
14	C₄⋊C₄×C₄₂₁	—	2	1	no	—
15	C₄⋊C₄×C₄₂₁	—	2	1	no	—
16	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄₂₁	—	2	1	no	—
20	D₁₆₈₄	—	2	1	no	—
21	D₁₆₈₄	Q₈	1	2	yes	—
22	C₂×Dic₄₂₁	—	2	1	no	—
23	C₄×D₄₂₁	—	4	2	no	—
24	#16809890	—	2	1	no	—
25	#16809890	—	2	1	no	—
26	C₂×C₄×C₄₂₁	—	2	1	no	—
27	C₂×C₄×C₄₂₁	—	2	1	no	—
28	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
29	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
30	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₄₂₁	—	2	2	no	—
33	C₂×D₄₂₁	—	4	1	no	—
34	C₂×D₄₂₁	—	4	1	no	—
35	C₄×C₄₂₁	—	2	1	no	—
36	C₄×C₄₂₁	—	2	1	no	—
37	C₄×C₄₂₁	—	4	1	no	—
38	C₄×C₄₂₁	—	4	1	no	—
39	C₄×C₄₂₁	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₄₂₁	#41	1	1	yes	characteristic
41	Dic₄₂₁	—	4	1	no	—
42	C₂×C₄₂₁	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₄₂₁	#135	1	1	yes	characteristic
44	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₄₂₁	—	4	2	no	—
46	C₄₂₁	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	421	1	no	maximal, Sylow
48	#125	D₄₂₁	1	1	yes	characteristic, complemented
49	C₄×D₄	—	421	1	no	—
50	#107	—	421	1	no	—
51	#106	—	421	1	no	—
52	C₄⋊C₈	—	421	1	no	—
53	#102	—	421	1	no	—
54	#102	—	421	1	no	—
55	C₂×D₄	—	421	1	no	—
56	C₂²×C₄	—	842	1	no	—
57	C₂×C₈	—	421	1	no	—
58	C₂×C₈	—	421	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	2	1	no	—
61	C₄⋊C₄	—	421	1	no	—
62	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
63	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
64	C₂²⋊C₄	—	842	1	no	—
65	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
66	C₂³	—	842	1	no	—
67	D₄	—	842	1	no	—
68	D₄	—	421	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	842	1	no	—
72	C₂×C₄	—	1684	2	no	—
73	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊D₄₂₁	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
76	C₈	—	842	1	no	—
77	C₈	—	842	1	no	—
78	C₂²	—	842	2	no	—
79	C₂²	—	1684	1	no	—
80	C₂²	—	1684	1	no	—
81	C₂²	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	1684	1	no	—
87	C₄	Q₈×D₄₂₁	1	1	yes	characteristic
88	C₄	#20269772	1	1	yes	characteristic
89	C₂	—	1684	2	no	—
90	C₂	#27212342	1	1	yes	characteristic
91	C₂	#27212386	1	1	yes	characteristic
92	C₂	#27212391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#125×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	#125	D₄₂₁
hypocenter	C₄₂₁	#478
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #27212299	C₂	semidirect
3. #27212297	C₂	semidirect
4. C₄⋊#16809890	C₂	semidirect
7. #125×C₄₂₁	C₂	semidirect
46. C₄₂₁	#478	semidirect
48. #125	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₆₈₄	2
#27212299	2
#27212297	2
C₄⋊#16809890	2
#27212286	2
#125×C₄₂₁	2
#478	421