Subgroups — #28378054

Order 13952.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13182032	C₂	1	2	yes	maximal
2	C₈×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182032	—	2	2	no	—
4	C₄×#11724784	C₄	1	1	yes	characteristic
5	C₄×#11724784	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
7	#13182032	—	4	2	no	—
8	C₂×#11724784	—	2	1	no	—
9	C₂×#11724784	D₄	1	1	yes	characteristic
10	C₂×#11724784	Q₈	1	1	yes	characteristic
11	C₂×C₈×C₁₀₉	C₈	1	1	yes	characteristic
12	C₂×C₈×C₁₀₉	C₈	1	1	yes	characteristic
13	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
17	C₈×C₁₀₉	—	2	1	no	—
18	C₈×C₁₀₉	C₁₆	1	2	yes	complemented
19	#11724784	—	2	2	no	—
20	#11724784	—	2	1	no	—
21	#11724784	SD₁₆	1	2	yes	—
22	C₂²×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₀₉	—	2	1	no	—
24	C₄×C₁₀₉	#106	1	1	yes	characteristic
25	C₄×C₁₀₉	#108	1	1	yes	characteristic
26	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
28	C₂×C₁₀₉	#334	1	1	yes	characteristic
29	C₂×C₁₀₉	C₄⋊C₁₆	1	1	yes	characteristic
30	C₂×C₁₀₉	#364	1	1	yes	characteristic
31	#1367	—	109	1	no	maximal, Sylow
32	C₁₀₉	#1367	1	1	yes	characteristic, complemented, Sylow
33	C₄⋊C₁₆	—	109	2	no	—
34	C₈²	—	109	1	no	—
35	C₂×C₁₆	—	218	2	no	—
36	C₄×C₈	—	109	1	no	—
37	C₄×C₈	—	109	1	no	—
38	C₄×C₈	#89137	1	1	yes	characteristic
39	C₂×C₈	—	109	1	no	—
40	C₂×C₈	—	109	1	no	—
41	C₂×C₈	—	218	1	no	—
42	C₂×C₈	#11724786	1	1	yes	characteristic
43	C₂×C₈	#11724786	1	1	yes	characteristic
44	C₄²	C₂×#89137	1	1	yes	characteristic
45	C₁₆	—	436	2	no	—
46	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
47	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
48	C₂×C₄	#13182039	1	1	yes	characteristic
49	C₈	—	2	1	no	—
50	C₈	—	109	2	no	—
51	C₈	—	218	2	no	—
52	C₈	—	218	1	no	—
53	C₈	#13182032	1	2	yes	complemented
54	C₂²	C₄⋊#11724786	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#16812767	1	1	yes	characteristic
57	C₄	#16812769	1	1	yes	characteristic
58	C₄	C₂×#13182032	1	1	yes	characteristic
59	C₄	#16812773	1	1	yes	characteristic
60	C₂	#20295887	1	1	yes	characteristic
61	C₂	C₄⋊#13182032	1	1	yes	characteristic
62	C₂	#20295948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#89137
commutator	C₄×C₁₀₉	C₂×C₁₆
Frattini	C₄²	C₂×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄⋊C₈
cosocle	C₄×#11724784	C₂²
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1367
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
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×C₁₀₉	C₁₆	semidirect
32. C₁₀₉	#1367	semidirect
53. C₈	#13182032	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13182032	2
C₈×#11724784	2
#1367	109