Subgroups — #28378149

Order 13952.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295958	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812749	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295443	C₂	1	1	yes	maximal, characteristic
4	#16812749	—	2	1	no	—
5	#16812749	—	2	1	no	—
6	#16812639	—	2	1	no	—
7	#16812639	C₄	1	1	yes	characteristic
8	C₂×#13182004	—	2	1	no	—
9	C₂×#13182004	C₂²	1	1	yes	characteristic
10	#131×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182004	—	2	1	no	—
12	#13182004	D₄	1	1	yes	characteristic
13	#13182004	D₄	1	1	yes	characteristic
14	#41×C₁₀₉	C₈	1	1	yes	characteristic
15	M₁₆×C₁₀₉	—	2	1	no	—
16	M₁₆×C₁₀₉	C₈	1	1	yes	characteristic
17	C₂×C₈×C₁₀₉	—	2	1	no	—
18	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₁₀₉	C₁₆	1	1	yes	characteristic
20	D₄×C₁₀₉	—	2	1	no	—
21	D₄×C₁₀₉	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
24	C₈×C₁₀₉	—	2	1	no	—
25	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₀₉	—	2	1	no	—
28	C₂²×C₁₀₉	M₃₂	1	1	yes	characteristic
29	C₄×C₁₀₉	—	2	1	no	—
30	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	—	4	1	no	—
34	C₂×C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic
35	#1401	—	109	1	no	maximal, Sylow
36	C₁₀₉	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	109	1	no	—
38	M₆₄	—	109	1	no	—
39	C₂×C₃₂	—	109	1	no	—
40	#131	#89137	1	1	yes	characteristic
41	M₃₂	—	109	1	no	—
42	M₃₂	—	218	1	no	—
43	C₂×C₁₆	—	109	1	no	—
44	C₂×C₁₆	—	218	1	no	—
45	C₃₂	—	218	1	no	—
46	C₃₂	—	218	1	no	—
47	#41	#11724786	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	#11724786	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×#89137	1	1	yes	characteristic
52	C₁₆	—	109	1	no	—
53	C₁₆	—	109	1	no	—
54	C₁₆	—	218	1	no	—
55	Q₈	#13182032	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	#13182032	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊#89137	1	1	yes	characteristic
62	C₈	#13182038	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	#16812773	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊#11724786	1	1	yes	characteristic
67	C₄	C₂×#13182032	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊#13182032	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₁₆
Frattini	C₂×C₈	C₂×#89137
Fitting	#131×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂²⋊C₁₆
prime radical	#41×C₁₀₉	C₈
cosocle	C₂×#13182004	C₂²
hypercenter	#131	#89137
hypocenter	C₁₀₉	#1401
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20295958	C₂	semidirect
2. C₂×#16812749	C₂	semidirect
36. C₁₀₉	#1401	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×#16812749	2
#20295443	2
#1401	109