Subgroups — #28377999

Order 13952.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13182032	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182032	C₂	1	1	yes	maximal, characteristic
3	#20295150	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182032	—	2	1	no	—
5	C₂×#13182032	—	2	1	no	—
6	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
7	#16812628	—	2	1	no	—
8	C₄×#11724784	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₀₉	—	2	1	no	—
10	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
11	#13182032	—	2	2	no	—
12	#13182032	—	4	1	no	—
13	C₂×Dic₂₁₈	C₈	1	1	yes	characteristic
14	C₂×#11724784	D₄	1	1	yes	characteristic
15	C₂×#11724784	D₄	1	1	yes	characteristic
16	C₄⋊Dic₁₀₉	C₈	1	1	yes	characteristic
17	#13182008	—	2	1	no	—
18	C₄×Dic₁₀₉	—	2	1	no	—
19	C₂×C₈×C₁₀₉	—	2	1	no	—
20	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
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	—	4	1	no	—
29	#11724784	—	2	1	no	—
30	#11724784	SD₁₆	1	1	yes	characteristic
31	#11724784	Q₁₆	1	1	yes	characteristic
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	—	109	1	no	—
48	C₄×Q₈	—	109	1	no	—
49	C₂×C₁₆	—	218	1	no	—
50	C₂×C₁₆	—	218	1	no	—
51	C₄⋊C₈	—	218	1	no	—
52	C₄⋊C₈	#89137	1	1	yes	characteristic
53	C₄×C₈	—	109	1	no	—
54	C₄×C₈	—	218	1	no	—
55	C₂×Q₈	—	109	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	109	1	no	—
58	C₂×C₈	—	109	1	no	—
59	C₄⋊C₄	—	109	1	no	—
60	C₄⋊C₄	—	218	1	no	—
61	C₄²	—	218	1	no	—
62	C₄²	C₂×#89137	1	1	yes	characteristic
63	C₁₆	—	436	1	no	—
64	C₁₆	—	218	2	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	C₂²⋊#89137	1	1	yes	characteristic
70	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
71	C₂×C₄	#13182039	1	1	yes	characteristic
72	C₈	—	4	1	no	—
73	C₈	—	109	1	no	—
74	C₈	—	109	1	no	—
75	C₈	—	218	1	no	—
76	C₂²	#16812756	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	436	1	no	—
79	C₄	—	218	2	no	—
80	C₄	#16812762	1	1	yes	characteristic
81	C₄	#16812763	1	1	yes	characteristic
82	C₄	D₁₀₉⋊C₁₆	1	1	yes	characteristic
83	C₄	#16812774	1	1	yes	characteristic
84	C₂	#20295876	1	1	yes	characteristic
85	C₂	#20295921	1	1	yes	characteristic
86	C₂	#20295925	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂×C₁₆
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₂²⋊C₈
cosocle	C₄×#11724784	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	Q₈⋊C₁₆
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
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
C₄⋊#13182032	2
C₄×#13182032	2
#20295150	2
Q₈⋊C₁₆	109