Subgroups — #28378100

Order 13952.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296036	C₂	1	2	yes	maximal
2	C₂×#16812638	C₂	1	1	yes	maximal, characteristic
3	#16812778	—	2	2	no	—
4	#16812768	—	2	2	no	—
5	#16812767	C₄	1	4	yes	complemented
6	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
7	#16812638	—	2	2	no	—
8	C₂×#13182004	C₄	1	1	yes	characteristic, complemented
9	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
10	C₄⋊#89137	—	2	4	no	—
11	C₂²⋊#89137	—	4	2	no	—
12	C₄×#89137	—	4	2	no	—
13	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
14	C₂×#11724784	Q₈	1	1	yes	characteristic
15	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
16	C₈×D₁₀₉	D₄	1	2	yes	—
17	#13182004	—	2	1	no	—
18	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₀₉	—	2	1	no	—
20	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
21	C₂×#89137	—	4	4	no	—
22	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
23	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
24	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
25	C₄×D₁₀₉	SD₁₆	1	2	yes	—
26	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
27	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
28	#11724784	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₀₉	#107	1	1	yes	characteristic
30	C₂×D₁₀₉	—	4	1	no	—
31	C₂×D₁₀₉	#102	1	2	yes	—
32	#89137	—	8	4	no	—
33	C₄×C₁₀₉	#95	1	1	yes	characteristic
34	C₄×C₁₀₉	#106	1	1	yes	characteristic
35	Dic₁₀₉	#103	1	2	yes	—
36	C₂×C₁₀₉	—	2	1	no	—
37	C₂×C₁₀₉	#340	1	1	yes	characteristic
38	D₁₀₉	—	2	2	no	—
39	#1384	—	109	1	no	maximal, Sylow
40	C₁₀₉	#1384	1	1	yes	characteristic, complemented, Sylow
41	C₂×M₃₂	—	109	1	no	—
42	#427	—	109	2	no	—
43	C₂²×C₈	—	109	1	no	—
44	#117	—	218	2	no	—
45	M₃₂	—	218	2	no	—
46	C₂×C₁₆	—	109	1	no	—
47	C₂×C₁₆	#89137	1	1	yes	characteristic, complemented
48	#107	—	218	2	no	—
49	#106	—	109	4	no	—
50	C₂²×C₄	—	109	1	no	—
51	C₂×C₈	—	109	1	no	—
52	C₂×C₈	—	109	2	no	—
53	C₂×C₈	—	109	2	no	—
54	C₂×C₈	C₂×#89137	1	1	yes	characteristic
55	C₄⋊C₄	—	218	4	no	—
56	C₂²⋊C₄	—	436	2	no	—
57	C₄²	—	436	2	no	—
58	C₁₆	—	2	1	no	—
59	C₁₆	—	218	1	no	—
60	C₂³	—	109	1	no	—
61	C₂×C₄	—	109	1	no	—
62	C₂×C₄	—	109	2	no	—
63	C₂×C₄	—	109	2	no	—
64	C₂×C₄	—	436	4	no	—
65	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
66	C₈	—	109	2	no	—
67	C₈	C₄×#89137	1	1	yes	characteristic
68	C₈	C₂²⋊#89137	1	1	yes	characteristic
69	C₂²	—	109	2	no	—
70	C₂²	—	436	1	no	—
71	C₂²	#16812768	1	1	yes	characteristic
72	C₄	—	872	4	no	—
73	C₄	—	109	2	no	—
74	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
75	C₄	#16812767	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	218	2	no	—
78	C₂	#20295900	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16812767
commutator	C₈×C₁₀₉	C₄²
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#340
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1384
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
5. #16812767	C₄	semidirect
8. C₂×#13182004	C₄	semidirect
9. C₂×C₁₆×C₁₀₉	C₄	semidirect
40. C₁₀₉	#1384	semidirect
47. C₂×C₁₆	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296036	2
C₂×#16812638	2
#1384	109