Subgroups — #28360951

Order 13952.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182004	C₂	1	1	yes	maximal, characteristic
2	C₈×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182004	C₄	1	2	yes	—
5	C₄×#11724784	C₄	1	1	yes	characteristic
6	C₄×#11724784	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₀₉	C₄	1	2	yes	—
8	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₂×#11724784	C₈	1	2	yes	—
10	C₂×#11724784	C₈	1	2	yes	—
11	#13182004	—	2	2	no	—
12	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₀₉	—	2	2	no	—
16	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₀₉	M₁₆	1	2	yes	—
20	C₈×C₁₀₉	M₁₆	1	2	yes	—
21	#11724784	—	2	2	no	—
22	#11724784	C₁₆	1	4	yes	complemented
23	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
24	C₄×C₁₀₉	#97	1	1	yes	characteristic
25	C₄×C₁₀₉	#97	1	1	yes	characteristic
26	C₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
27	C₄×C₁₀₉	M₃₂	1	2	yes	—
28	C₂×C₁₀₉	#322	1	1	yes	characteristic
29	C₂×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
30	C₂×C₁₀₉	#346	1	1	yes	characteristic
31	#1312	—	109	1	no	maximal, Sylow
32	C₁₀₉	#1312	1	1	yes	characteristic, complemented, Sylow
33	C₄×C₁₆	—	109	1	no	—
34	C₄×C₁₆	D₁₀₉	1	1	yes	characteristic
35	C₈²	—	109	1	no	—
36	C₂×C₁₆	—	109	2	no	—
37	C₂×C₁₆	Dic₁₀₉	1	2	yes	—
38	C₄×C₈	—	109	1	no	—
39	C₄×C₈	—	109	1	no	—
40	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
41	C₂×C₈	—	109	2	no	—
42	C₂×C₈	—	109	2	no	—
43	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
44	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
45	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
46	C₁₆	—	2	2	no	—
47	C₁₆	—	218	2	no	—
48	C₂×C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
49	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
50	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
51	C₈	—	109	4	no	—
52	C₈	—	218	2	no	—
53	C₈	#13182012	1	2	yes	—
54	C₈	#13182013	1	2	yes	—
55	C₂²	C₈×Dic₁₀₉	1	1	yes	characteristic
56	C₄	#16812609	1	1	yes	characteristic
57	C₄	#16812610	1	1	yes	characteristic
58	C₄	C₁₆×D₁₀₉	1	2	yes	—
59	C₄	#16812638	1	2	yes	—
60	C₂	#20294748	1	1	yes	characteristic
61	C₂	C₁₆×Dic₁₀₉	1	1	yes	characteristic
62	C₂	#20294801	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₄×C₁₆
Fitting	C₄×C₁₆×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₄×C₈
cosocle	C₄×C₈×C₁₀₉	C₂²
hypercenter	C₄×C₁₆	D₁₀₉
hypocenter	C₁₀₉	#1312
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
22. #11724784	C₁₆	semidirect
32. C₁₀₉	#1312	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182004	2
C₈×#11724784	2
C₄×C₁₆×C₁₀₉	2
#1312	109