Subgroups — #28378147

Order 13952.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295958	C₂	1	2	yes	maximal, complemented
2	C₂×#16812639	C₂	1	1	yes	maximal, characteristic
3	#16812749	—	2	2	no	—
4	#16812639	—	2	2	no	—
5	C₂×#13182004	C₄	1	1	yes	characteristic
6	C₂×#13182004	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
8	#13182004	—	2	1	no	—
9	#13182004	D₄	1	2	yes	—
10	C₂²×C₄×C₁₀₉	C₈	1	1	yes	characteristic
11	C₂×C₈×C₁₀₉	—	2	1	no	—
12	C₂×C₈×C₁₀₉	—	2	1	no	—
13	C₂×C₈×C₁₀₉	C₈	1	1	yes	characteristic
14	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₁₀₉	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄×C₁₀₉	—	2	1	no	—
18	C₂×C₄×C₁₀₉	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₀₉	—	2	1	no	—
21	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
23	C₂²×C₁₀₉	—	2	1	no	—
24	C₂²×C₁₀₉	—	4	1	no	—
25	C₂²×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
26	C₄×C₁₀₉	—	2	1	no	—
27	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
29	C₂×C₁₀₉	—	2	1	no	—
30	C₂×C₁₀₉	—	4	1	no	—
31	C₂×C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic
32	#1400	—	109	1	no	maximal, Sylow
33	C₁₀₉	#1400	1	1	yes	characteristic, complemented, Sylow
34	C₂×M₃₂	—	109	1	no	—
35	M₆₄	—	109	2	no	—
36	C₂²×C₈	#89137	1	1	yes	characteristic
37	M₃₂	—	218	2	no	—
38	C₂×C₁₆	—	109	1	no	—
39	C₂×C₁₆	—	109	1	no	—
40	C₃₂	—	218	2	no	—
41	C₂²×C₄	#11724786	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	#11724786	1	1	yes	characteristic
45	C₂×C₈	C₂×#89137	1	1	yes	characteristic
46	C₁₆	—	109	2	no	—
47	C₁₆	—	218	1	no	—
48	C₂³	#13182032	1	1	yes	characteristic
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	#13182032	1	1	yes	characteristic
52	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	C₂²⋊#89137	1	1	yes	characteristic
55	C₈	#13182038	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×#13182032	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	C₂²⋊#11724786	1	1	yes	characteristic
61	C₄	#16812773	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	C₂²⋊#13182032	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#89137
commutator	C₂²×C₁₀₉	C₂×C₁₆
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂²⋊C₁₆
prime radical	C₂³×C₁₀₉	C₁₆
cosocle	C₂×#13182004	C₂²
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	#1400
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. #20295958	C₂	semidirect
33. C₁₀₉	#1400	semidirect

Maximal subgroups

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

subgroup	index
#20295958	2
C₂×#16812639	2
#1400	109