Subgroups — #27186161

Order 12928.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13108077	C₂	1	2	yes	maximal
2	C₈×#11717579	C₂	1	1	yes	maximal, characteristic
3	C₂×#13108077	—	2	2	no	—
4	C₄×#11717579	C₄	1	1	yes	characteristic
5	C₄×#11717579	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₀₁	C₄	1	1	yes	characteristic
7	#13108077	—	4	2	no	—
8	C₂×#11717579	—	2	1	no	—
9	C₂×#11717579	D₄	1	1	yes	characteristic
10	C₂×#11717579	Q₈	1	1	yes	characteristic
11	C₂×C₈×C₁₀₁	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₀₁	C₈	1	1	yes	characteristic
13	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₀₁	M₁₆	1	1	yes	characteristic
17	C₈×C₁₀₁	—	2	1	no	—
18	C₈×C₁₀₁	C₁₆	1	2	yes	complemented
19	#11717579	—	2	2	no	—
20	#11717579	—	2	1	no	—
21	#11717579	D₈	1	1	yes	characteristic
22	#11717579	Q₁₆	1	1	yes	characteristic
23	C₂²×C₁₀₁	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₀₁	—	2	1	no	—
25	C₄×C₁₀₁	#107	1	1	yes	characteristic
26	C₄×C₁₀₁	#108	1	1	yes	characteristic
27	C₄×C₁₀₁	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₁₀₁	M₃₂	1	1	yes	characteristic
29	C₂×C₁₀₁	#335	1	1	yes	characteristic
30	C₂×C₁₀₁	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₁₀₁	#364	1	1	yes	characteristic
32	#1370	—	101	1	no	maximal, Sylow
33	C₁₀₁	#1370	1	1	yes	characteristic, complemented, Sylow
34	C₄⋊C₁₆	—	101	2	no	—
35	C₈²	—	101	1	no	—
36	C₂×C₁₆	—	202	2	no	—
37	C₄×C₈	—	101	1	no	—
38	C₄×C₈	—	101	1	no	—
39	C₄×C₈	#87938	1	1	yes	characteristic
40	C₂×C₈	—	101	1	no	—
41	C₂×C₈	—	101	1	no	—
42	C₂×C₈	—	202	1	no	—
43	C₂×C₈	#11717581	1	1	yes	characteristic
44	C₂×C₈	#11717581	1	1	yes	characteristic
45	C₄²	C₂×#87938	1	1	yes	characteristic
46	C₁₆	—	404	2	no	—
47	C₂×C₄	C₄⋊#87938	1	1	yes	characteristic
48	C₂×C₄	C₂×#11717581	1	1	yes	characteristic
49	C₂×C₄	#13108084	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	—	101	1	no	—
52	C₈	—	101	1	no	—
53	C₈	—	202	2	no	—
54	C₈	—	202	1	no	—
55	C₈	#13108077	1	2	yes	complemented
56	C₂²	C₄⋊#11717581	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#15692324	1	1	yes	characteristic
59	C₄	#15692326	1	1	yes	characteristic
60	C₄	C₂×#13108077	1	1	yes	characteristic
61	C₄	#15692329	1	1	yes	characteristic
62	C₂	#20266040	1	1	yes	characteristic
63	C₂	C₄⋊#13108077	1	1	yes	characteristic
64	C₂	#20266099	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#87938
commutator	C₄×C₁₀₁	C₂×C₁₆
Frattini	C₄²	C₂×#87938
Fitting	C₄×C₈×C₁₀₁	C₄
socle, prime radical	C₂²×C₁₀₁	C₄⋊C₈
cosocle	C₄×#11717579	C₂²
hypercenter	C₄×C₈	#87938
hypocenter	C₁₀₁	#1370
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
18. C₈×C₁₀₁	C₁₆	semidirect
33. C₁₀₁	#1370	semidirect
55. C₈	#13108077	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13108077	2
C₈×#11717579	2
#1370	101