Subgroups — #28377988

Order 13952.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295912	C₂	1	1	yes	maximal, characteristic
2	#20295889	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	—	8	1	no	—
39	Dic₁₀₉	SD₃₂	1	1	yes	characteristic
40	Dic₁₀₉	Q₃₂	1	1	yes	characteristic
41	C₂×C₁₀₉	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₀₉	#358	1	1	yes	characteristic
43	C₂×C₁₀₉	#359	1	1	yes	characteristic
44	#1332	—	109	1	no	maximal, Sylow
45	C₁₀₉	#1332	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	109	1	no	—
47	C₄×C₁₆	—	109	1	no	—
48	#335	—	109	1	no	—
49	C₂×Q₁₆	#89137	1	1	yes	characteristic
50	C₄×Q₈	—	218	1	no	—
51	C₂×C₁₆	—	218	1	no	—
52	#107	—	109	1	no	—
53	C₄⋊C₈	—	218	1	no	—
54	#103	—	218	1	no	—
55	C₄×C₈	—	109	1	no	—
56	C₂×Q₈	—	2	1	no	—
57	Q₁₆	—	2	1	no	—
58	Q₁₆	#11724786	1	2	yes	complemented
59	C₂×C₈	—	109	1	no	—
60	C₂×C₈	—	436	1	no	—
61	C₂×C₈	C₂×#89137	1	1	yes	characteristic
62	C₄⋊C₄	—	436	1	no	—
63	C₄⋊C₄	—	218	1	no	—
64	C₄²	—	109	1	no	—
65	C₄²	—	436	1	no	—
66	C₁₆	—	218	2	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	109	1	no	—
71	C₂×C₄	—	109	1	no	—
72	C₂×C₄	—	436	1	no	—
73	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
74	C₈	—	872	1	no	—
75	C₈	—	218	1	no	—
76	C₈	C₂×#11724786	1	1	yes	characteristic
77	C₈	#13182038	1	1	yes	characteristic
78	C₂²	#16812761	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	—	872	1	no	—
81	C₄	—	109	1	no	—
82	C₄	—	109	1	no	—
83	C₄	—	218	1	no	—
84	C₄	C₂²⋊#11724786	1	1	yes	characteristic
85	C₄	#16812763	1	1	yes	characteristic
86	C₂	D₄⋊#11724786	1	1	yes	characteristic
87	C₂	#20295939	1	1	yes	characteristic
88	C₂	#20295941	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	#1332
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
45. C₁₀₉	#1332	semidirect
58. Q₁₆	#11724786	semidirect

Maximal subgroups

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

subgroup	index
#20295912	2
#20295889	2
Q₁₆×Dic₁₀₉	2
#1332	109