Subgroups — #28361204

Order 13952.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295394	C₂	1	1	yes	maximal, characteristic
2	#20294825	C₂	1	1	yes	maximal, characteristic
3	#356×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812691	C₄	1	1	yes	characteristic
5	#16812623	—	2	1	no	—
6	#16812623	—	2	1	no	—
7	#16812618	—	2	1	no	—
8	#16812610	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	#101×C₁₀₉	—	2	1	no	—
11	#13182013	—	4	1	no	—
12	C₂×#11724784	—	2	1	no	—
13	C₄⋊Dic₁₀₉	—	2	1	no	—
14	#13182008	—	4	1	no	—
15	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
17	M₁₆×C₁₀₉	—	4	1	no	—
18	C₄⋊C₄×C₁₀₉	—	4	1	no	—
19	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
20	Q₈×C₁₀₉	—	4	1	no	—
21	Q₈×C₁₀₉	—	4	1	no	—
22	C₂×C₄×C₁₀₉	—	2	1	no	—
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₀₉	—	4	1	no	—
27	C₈×C₁₀₉	—	8	1	no	—
28	#11724784	—	4	1	no	—
29	#11724784	—	8	1	no	—
30	C₂²×C₁₀₉	#99	1	1	yes	characteristic
31	C₄×C₁₀₉	—	2	1	no	—
32	C₄×C₁₀₉	—	4	1	no	—
33	C₄×C₁₀₉	—	4	1	no	—
34	C₄×C₁₀₉	—	4	1	no	—
35	Dic₁₀₉	—	8	1	no	—
36	C₂×C₁₀₉	—	2	1	no	—
37	C₂×C₁₀₉	#354	1	1	yes	characteristic
38	#1413	—	109	1	no	maximal, Sylow
39	C₁₀₉	#1413	1	1	yes	characteristic, complemented, Sylow
40	#491	—	109	1	no	—
41	#356	—	109	1	no	—
42	#356	D₁₀₉	1	1	yes	characteristic
43	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
44	#125	—	109	1	no	—
45	#103	—	218	1	no	—
46	#103	—	218	1	no	—
47	#101	—	2	1	no	—
48	#101	—	218	1	no	—
49	#97	—	109	1	no	—
50	C₂×Q₈	D₄₃₆	1	1	yes	characteristic
51	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	436	1	no	—
54	C₂×C₈	—	218	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	436	1	no	—
57	C₄⋊C₄	—	218	1	no	—
58	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	436	1	no	—
65	C₂×C₄	#13182006	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	872	1	no	—
68	C₈	—	436	1	no	—
69	C₂²	#16812613	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	872	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#20294820	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20294820
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₀₉
Fitting	#356×C₁₀₉	C₂
socle	C₂×C₁₀₉	#354
prime radical	C₂²×C₁₀₉	#99
cosocle	C₄⋊Q₈×C₁₀₉	C₂²
hypercenter	#356	D₁₀₉
hypocenter	C₁₀₉	#1413
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
39. C₁₀₉	#1413	semidirect

Maximal subgroups

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

subgroup	index
#20295394	2
#20294825	2
#356×C₁₀₉	2
#1413	109