Subgroups — #28361012

Order 13952.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#13182004	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
5	C₂×#13182004	—	2	1	no	—
6	C₄⋊#11724784	C₄	1	1	yes	characteristic
7	#16812628	—	2	1	no	—
8	C₈×Dic₁₀₉	—	2	1	no	—
9	C₂×C₁₆×C₁₀₉	—	2	1	no	—
10	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
12	C₂×#11724784	—	2	1	no	—
13	C₄⋊Dic₁₀₉	C₈	1	1	yes	characteristic
14	#13182008	—	2	1	no	—
15	C₄×Dic₁₀₉	—	2	1	no	—
16	#13182004	—	4	1	no	—
17	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
19	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₀₉	—	2	2	no	—
21	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	M₁₆	1	1	yes	characteristic
24	C₂×Dic₁₀₉	—	2	1	no	—
25	C₂×Dic₁₀₉	—	2	1	no	—
26	Dic₂₁₈	—	2	1	no	—
27	Dic₂₁₈	C₁₆	1	2	yes	complemented
28	C₈×C₁₀₉	—	2	1	no	—
29	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
31	#11724784	—	4	1	no	—
32	C₂²×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₀₉	—	2	1	no	—
34	C₄×C₁₀₉	#103	1	1	yes	characteristic
35	C₄×C₁₀₉	#104	1	1	yes	characteristic
36	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
38	Dic₁₀₉	—	2	2	no	—
39	Dic₁₀₉	—	4	1	no	—
40	C₂×C₁₀₉	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₀₉	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₀₉	#350	1	1	yes	characteristic
43	Q₈⋊C₁₆	—	109	1	no	maximal, Sylow
44	C₁₀₉	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₈×Q₈	—	109	1	no	—
46	C₄⋊C₁₆	—	109	1	no	—
47	C₄×C₁₆	D₁₀₉	1	1	yes	characteristic
48	C₄×Q₈	—	109	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	218	1	no	—
51	C₄⋊C₈	—	109	1	no	—
52	C₄⋊C₈	—	218	1	no	—
53	C₄×C₈	—	218	1	no	—
54	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
55	C₂×Q₈	—	109	1	no	—
56	C₂×C₈	—	218	1	no	—
57	C₂×C₈	D₄₃₆	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
59	C₄⋊C₄	—	109	1	no	—
60	C₄⋊C₄	—	218	1	no	—
61	C₄²	—	218	1	no	—
62	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	436	1	no	—
65	Q₈	—	109	2	no	—
66	Q₈	—	218	1	no	—
67	C₂×C₄	—	218	1	no	—
68	C₂×C₄	—	218	1	no	—
69	C₂×C₄	#13182006	1	1	yes	characteristic
70	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
71	C₂×C₄	#13182012	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	436	1	no	—
74	C₈	#13182018	1	1	yes	characteristic
75	C₈	Dic₄₃₆	1	1	yes	characteristic
76	C₂²	#16812611	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	436	1	no	—
79	C₄	—	218	2	no	—
80	C₄	#16812624	1	1	yes	characteristic
81	C₄	#16812627	1	1	yes	characteristic
82	C₄	C₁₆×D₁₀₉	1	1	yes	characteristic
83	C₄	#16812638	1	1	yes	characteristic
84	C₂	#20294758	1	1	yes	characteristic
85	C₂	#20294805	1	1	yes	characteristic
86	C₂	#20294809	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₃₆
commutator	C₄×C₁₀₉	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₁₀₉
Fitting	C₄×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂²⋊C₈
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	C₄×C₁₆	D₁₀₉
hypocenter	C₁₀₉	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
27. Dic₂₁₈	C₁₆	semidirect
44. C₁₀₉	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₁₈	2
C₄⋊#13182004	2
C₄×C₁₆×C₁₀₉	2
Q₈⋊C₁₆	109