Subgroups — #28378167

Order 13952.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295937	C₂	1	2	yes	maximal, complemented
2	#20295398	C₂	1	1	yes	maximal, characteristic
3	#16812760	—	2	2	no	—
4	C₂×#13182017	—	2	2	no	—
5	#16812699	C₂²	1	1	yes	characteristic
6	C₄×#11724784	C₄	1	1	yes	characteristic
7	#127×C₁₀₉	C₄	1	1	yes	characteristic
8	#13182038	—	4	2	no	—
9	C₂×Dic₂₁₈	D₄	1	2	yes	—
10	#13182017	—	4	4	no	—
11	C₂×#11724784	—	2	1	no	—
12	C₄⋊Dic₁₀₉	—	4	1	no	—
13	C₂×D₄×C₁₀₉	—	2	1	no	—
14	C₂×D₄×C₁₀₉	—	4	1	no	—
15	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₁₀₉	—	4	1	no	—
17	D₄×C₁₀₉	—	4	2	no	—
18	D₄×C₁₀₉	—	8	2	no	—
19	C₂×C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×Dic₁₀₉	—	2	2	no	—
22	Dic₂₁₈	—	4	2	no	—
23	#11724786	—	8	2	no	—
24	#11724784	—	2	2	no	—
25	C₂²×C₁₀₉	—	8	2	no	—
26	C₂²×C₁₀₉	—	8	1	no	—
27	C₂²×C₁₀₉	#99	1	1	yes	characteristic
28	C₄×C₁₀₉	—	2	2	no	—
29	C₄×C₁₀₉	—	2	1	no	—
30	Dic₁₀₉	—	4	2	no	—
31	C₂×C₁₀₉	—	2	1	no	—
32	C₂×C₁₀₉	—	16	1	no	—
33	C₂×C₁₀₉	#353	1	1	yes	characteristic
34	#1410	—	109	1	no	maximal, Sylow
35	C₁₀₉	#1410	1	1	yes	characteristic, complemented, Sylow
36	#492	—	109	1	no	—
37	#356	—	109	2	no	—
38	C₂×SD₁₆	—	218	2	no	—
39	C₄⋊Q₈	—	109	1	no	—
40	#127	#89137	1	1	yes	characteristic
41	#101	—	218	2	no	—
42	C₄×C₈	—	109	1	no	—
43	C₂×Q₈	—	109	2	no	—
44	C₂×D₄	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	SD₁₆	—	436	4	no	—
47	M₁₆	—	436	2	no	—
48	C₂×C₈	—	218	1	no	—
49	C₄⋊C₄	—	436	1	no	—
50	C₄²	C₂×#89137	1	1	yes	characteristic
51	C₂³	—	4	1	no	—
52	Q₈	—	436	2	no	—
53	D₄	—	4	2	no	—
54	D₄	—	8	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	218	2	no	—
57	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
58	C₈	—	872	2	no	—
59	C₈	—	218	2	no	—
60	C₂²	—	8	2	no	—
61	C₂²	—	8	1	no	—
62	C₂²	#16812757	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	436	2	no	—
66	C₂	—	2	1	no	—
67	C₂	—	16	1	no	—
68	C₂	#20295930	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295930
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting, prime radical	#127×C₁₀₉	C₄
socle	C₂×C₁₀₉	#353
cosocle	#16812699	C₂²
hypercenter	#127	#89137
hypocenter	C₁₀₉	#1410
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #20295937	C₂	semidirect
35. C₁₀₉	#1410	semidirect

Maximal subgroups

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

subgroup	index
#20295937	2
#20295398	2
#1410	109