Subgroups — #41003048

Order 26944.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269761	C₂	1	1	yes	maximal, characteristic
2	#27212364	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
4	#27212295	C₂	1	1	yes	maximal, characteristic
5	#27212291	C₂	1	1	yes	maximal, characteristic
6	#27212290	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₁₆×C₄₂₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
9	#20269761	—	2	2	no	—
10	C₂×#16809890	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₂₁	—	2	1	no	—
12	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
13	#20269750	—	2	1	no	—
14	#20269750	—	2	1	no	—
15	C₄×Dic₄₂₁	—	2	1	no	—
16	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
19	Q₁₆×C₄₂₁	—	2	2	no	—
20	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
21	Dic₈₄₂	—	2	1	no	—
22	Dic₈₄₂	—	2	1	no	—
23	C₂×Dic₄₂₁	—	2	1	no	—
24	C₂×Dic₄₂₁	—	2	1	no	—
25	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
26	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
27	#16809890	—	2	1	no	—
28	Q₈×C₄₂₁	—	2	1	no	—
29	Q₈×C₄₂₁	—	2	1	no	—
30	Q₈×C₄₂₁	—	2	1	no	—
31	Q₈×C₄₂₁	D₄	1	2	yes	—
32	C₂×C₄×C₄₂₁	—	2	1	no	—
33	C₂×C₄×C₄₂₁	—	2	1	no	—
34	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
35	C₈×C₄₂₁	—	2	1	no	—
36	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
37	C₄×C₄₂₁	—	2	2	no	—
38	C₄×C₄₂₁	—	4	1	no	—
39	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
40	C₄×C₄₂₁	#41	1	1	yes	characteristic
41	Dic₄₂₁	—	2	1	no	—
42	Dic₄₂₁	—	4	1	no	—
43	Dic₄₂₁	—	4	1	no	—
44	Dic₄₂₁	Q₁₆	1	2	yes	complemented
45	C₂×C₄₂₁	#121	1	1	yes	characteristic
46	C₂×C₄₂₁	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×C₄₂₁	#137	1	1	yes	characteristic
48	C₄₂₁	#462	1	1	yes	characteristic, complemented, Sylow
49	#462	—	421	1	no	maximal, Sylow
50	C₂×Q₁₆	—	421	1	no	—
51	C₂×Q₁₆	D₄₂₁	1	1	yes	characteristic
52	C₄⋊Q₈	—	421	1	no	—
53	C₄×Q₈	—	421	1	no	—
54	C₄⋊C₈	—	421	1	no	—
55	#103	—	421	1	no	—
56	#103	—	421	1	no	—
57	C₂×Q₈	—	421	1	no	—
58	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
59	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
60	Q₁₆	—	2	2	no	—
61	Q₁₆	—	842	2	no	—
62	C₂×C₈	—	421	1	no	—
63	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
64	C₄⋊C₄	—	842	1	no	—
65	C₄⋊C₄	—	842	1	no	—
66	C₄⋊C₄	—	842	1	no	—
67	C₄⋊C₄	—	421	1	no	—
68	C₄²	—	842	1	no	—
69	C₄²	—	421	1	no	—
70	Q₈	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	—	2	1	no	—
73	Q₈	—	842	1	no	—
74	Q₈	—	842	1	no	—
75	Q₈	C₂²⋊D₄₂₁	1	2	yes	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	842	1	no	—
79	C₂×C₄	—	842	1	no	—
80	C₂×C₄	—	421	1	no	—
81	C₂×C₄	—	421	1	no	—
82	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	842	1	no	—
85	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	1	no	—
88	C₄	—	842	1	no	—
89	C₄	—	1684	1	no	—
90	C₄	—	1684	1	no	—
91	C₄	—	421	2	no	—
92	C₄	C₂×C₂²⋊D₄₂₁	1	1	yes	characteristic
93	C₄	#20269770	1	1	yes	characteristic
94	C₂	#27212338	1	1	yes	characteristic
95	C₂	Q₁₆×D₄₂₁	1	1	yes	characteristic
96	C₂	#27212398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂×Q₁₆×C₄₂₁	C₂
socle, prime radical	C₂²×C₄₂₁	C₂×D₄
hypercenter	C₂×Q₁₆	D₄₂₁
hypocenter	C₄₂₁	#462
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
44. Dic₄₂₁	Q₁₆	semidirect
48. C₄₂₁	#462	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269761	2
#27212364	2
Q₈×Dic₄₂₁	2
#27212295	2
#27212291	2
#27212290	2
C₂×Q₁₆×C₄₂₁	2
#462	421