Subgroups — #41003692

Order 26944.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212423	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×#13132997)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212341	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13132997	—	2	1	no	—
5	C₂×#16809897	—	2	1	no	—
6	C₂×D₁₆₈₄	—	2	1	no	—
7	C₂×D₁₆₈₄	C₄	1	1	yes	characteristic, complemented
8	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
9	#20269748	—	2	1	no	—
10	C₄⋊C₄×C₄₂₁	C₄	1	1	yes	characteristic, complemented
11	C₂×#13132997	—	2	2	no	—
12	C₂×#13132997	—	4	1	no	—
13	C₂×#13132997	—	4	1	no	—
14	#16809897	—	4	1	no	—
15	C₂²×D₄₂₁	—	2	1	no	—
16	C₂²×D₄₂₁	D₄	1	1	yes	characteristic
17	D₁₆₈₄	—	4	1	no	—
18	D₁₆₈₄	—	4	1	no	—
19	D₁₆₈₄	—	4	1	no	—
20	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
21	C₄×D₄₂₁	—	4	1	no	—
22	C₂×C₄×C₄₂₁	—	2	1	no	—
23	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₄₂₁	—	2	1	no	—
26	C₂×D₄₂₁	—	2	1	no	—
27	C₂×D₄₂₁	—	4	1	no	—
28	C₂×D₄₂₁	—	4	1	no	—
29	C₂×D₄₂₁	—	4	1	no	—
30	C₂×D₄₂₁	D₈	1	1	yes	characteristic
31	C₂×D₄₂₁	SD₁₆	1	1	yes	characteristic
32	#13132997	—	4	2	no	—
33	C₄×C₄₂₁	—	2	1	no	—
34	C₄×C₄₂₁	—	4	1	no	—
35	Dic₄₂₁	—	2	1	no	—
36	C₂×C₄₂₁	#99	1	1	yes	characteristic
37	C₂×C₄₂₁	#102	1	1	yes	characteristic
38	C₂×C₄₂₁	#104	1	1	yes	characteristic
39	D₄₂₁	—	2	1	no	—
40	D₄₂₁	—	2	1	no	—
41	D₄₂₁	—	8	1	no	—
42	C₄₂₁	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	421	1	no	maximal, Sylow
44	#121	—	421	1	no	—
45	C₂²⋊C₈	—	421	1	no	—
46	#95	—	421	1	no	—
47	C₂×D₄	—	842	1	no	—
48	C₂×D₄	—	421	1	no	—
49	C₂²×C₄	—	842	1	no	—
50	C₂²×C₄	—	421	1	no	—
51	C₂×C₈	—	842	1	no	—
52	C₄⋊C₄	#13132997	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	842	1	no	—
54	C₂³	—	842	1	no	—
55	C₂³	—	421	1	no	—
56	D₄	—	1684	1	no	—
57	D₄	—	1684	1	no	—
58	D₄	—	1684	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	842	2	no	—
61	C₂×C₄	—	1684	1	no	—
62	C₂×C₄	—	1684	1	no	—
63	C₂×C₄	—	1684	1	no	—
64	C₂×C₄	—	421	1	no	—
65	C₂×C₄	C₂×#13132997	1	1	yes	characteristic
66	C₈	—	1684	1	no	—
67	C₂²	—	842	1	no	—
68	C₂²	—	842	1	no	—
69	C₂²	—	1684	1	no	—
70	C₂²	—	1684	1	no	—
71	C₂²	—	1684	1	no	—
72	C₂²	—	421	1	no	—
73	C₂²	—	421	1	no	—
74	C₂²	C₂²⋊#13132997	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	842	1	no	—
78	C₄	—	1684	2	no	—
79	C₂	—	3368	1	no	—
80	C₂	—	842	1	no	—
81	C₂	—	842	1	no	—
82	C₂	#27212425	1	1	yes	characteristic
83	C₂	#27212428	1	1	yes	characteristic
84	C₂	#27212431	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13132997
commutator	C₂×C₄×C₄₂₁	C₂×C₄
Frattini	C₂×C₄	C₂×#13132997
Fitting	C₄⋊C₄×C₄₂₁	C₄
socle	C₂²×C₄₂₁	C₂²⋊C₄
prime radical	#27212341	C₂
cosocle	C₂×C₄×D₄₂₁	C₂²
hypercenter	C₄⋊C₄	#13132997
hypocenter	C₄₂₁	#327
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #27212423	C₂	semidirect
2. (C₂×#13132997)⋊C₄	C₂	semidirect
7. C₂×D₁₆₈₄	C₄	semidirect
10. C₄⋊C₄×C₄₂₁	C₄	semidirect
42. C₄₂₁	#327	semidirect
52. C₄⋊C₄	#13132997	semidirect

Maximal subgroups

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

subgroup	index
#27212423	2
(C₂×#13132997)⋊C₄	2
#27212341	2
#327	421