Subgroups — #28377992

Order 13952.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295915	C₂	1	1	yes	maximal, characteristic
2	#20295887	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182032	—	2	1	no	—
5	C₄⋊#11724786	—	2	1	no	—
6	Q₈×Dic₁₀₉	—	2	1	no	—
7	#16812633	C₄	1	1	yes	characteristic
8	#16812623	—	2	1	no	—
9	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#11724786	—	4	1	no	—
12	#13182032	—	2	2	no	—
13	C₂×#11724784	D₄	1	1	yes	characteristic
14	C₄⋊Dic₁₀₉	—	2	1	no	—
15	C₄⋊Dic₁₀₉	—	4	1	no	—
16	C₄×Dic₁₀₉	—	4	1	no	—
17	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₀₉	—	2	1	no	—
19	Q₁₆×C₁₀₉	—	2	1	no	—
20	Q₁₆×C₁₀₉	C₈	1	2	yes	complemented
21	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	Q₈×C₁₀₉	—	2	2	no	—
23	Q₈×C₁₀₉	—	4	1	no	—
24	C₂×C₄×C₁₀₉	—	4	1	no	—
25	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₀₉	—	4	1	no	—
27	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
28	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
29	#11724786	—	8	1	no	—
30	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
32	#11724784	—	2	1	no	—
33	C₂²×C₁₀₉	#102	1	1	yes	characteristic
34	C₄×C₁₀₉	—	4	2	no	—
35	C₄×C₁₀₉	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₀₉	#104	1	1	yes	characteristic
37	Dic₁₀₉	—	2	1	no	—
38	Dic₁₀₉	—	2	1	no	—
39	Dic₁₀₉	—	8	1	no	—
40	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
41	C₂×C₁₀₉	#360	1	1	yes	characteristic
42	C₂×C₁₀₉	#362	1	1	yes	characteristic
43	#1334	—	109	1	no	maximal, Sylow
44	C₁₀₉	#1334	1	1	yes	characteristic, complemented, Sylow
45	C₄×Q₁₆	—	109	1	no	—
46	#346	—	109	1	no	—
47	#334	—	109	1	no	—
48	C₂×Q₁₆	#89137	1	1	yes	characteristic
49	C₄×Q₈	—	218	1	no	—
50	C₂×C₁₆	—	218	1	no	—
51	#107	—	109	1	no	—
52	C₄⋊C₈	—	218	1	no	—
53	#103	—	218	1	no	—
54	C₄×C₈	—	109	1	no	—
55	C₂×Q₈	—	2	1	no	—
56	Q₁₆	—	2	1	no	—
57	Q₁₆	#11724786	1	2	yes	complemented
58	C₂×C₈	—	109	1	no	—
59	C₂×C₈	—	436	1	no	—
60	C₂×C₈	C₂×#89137	1	1	yes	characteristic
61	C₄⋊C₄	—	436	1	no	—
62	C₄⋊C₄	—	218	1	no	—
63	C₄²	—	109	1	no	—
64	C₄²	—	436	1	no	—
65	C₁₆	—	218	2	no	—
66	Q₈	—	2	2	no	—
67	Q₈	—	4	1	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	109	1	no	—
70	C₂×C₄	—	109	1	no	—
71	C₂×C₄	—	436	1	no	—
72	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
73	C₈	—	872	1	no	—
74	C₈	—	218	1	no	—
75	C₈	C₂×#11724786	1	1	yes	characteristic
76	C₈	#13182038	1	1	yes	characteristic
77	C₂²	#16812761	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	872	1	no	—
80	C₄	—	218	1	no	—
81	C₄	—	218	1	no	—
82	C₄	C₂²⋊#11724786	1	1	yes	characteristic
83	C₄	#16812763	1	1	yes	characteristic
84	C₂	D₄⋊#11724786	1	1	yes	characteristic
85	C₂	#20295943	1	1	yes	characteristic
86	C₂	#20295945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812761
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×Q₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	#102
cosocle	C₈×Dic₁₀₉	C₂²
hypercenter	C₂×Q₁₆	#89137
hypocenter	C₁₀₉	#1334
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
20. Q₁₆×C₁₀₉	C₈	semidirect
44. C₁₀₉	#1334	semidirect
57. Q₁₆	#11724786	semidirect

Maximal subgroups

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

subgroup	index
#20295915	2
#20295887	2
Q₁₆×Dic₁₀₉	2
#1334	109