Subgroups — #28378165

Order 13952.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295931	C₂	1	2	yes	maximal
2	#20295400	C₂	1	1	yes	maximal, characteristic
3	#16812758	—	2	2	no	—
4	C₂×#13182016	—	2	2	no	—
5	#16812696	C₂²	1	1	yes	characteristic
6	C₄×#11724784	C₄	1	1	yes	characteristic, complemented
7	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#89137	—	4	2	no	—
9	C₂×D₄₃₆	—	4	1	no	—
10	C₂×D₄₃₆	D₄	1	2	yes	—
11	#13182016	—	4	4	no	—
12	C₂×#11724784	—	2	1	no	—
13	C₂×Q₈×C₁₀₉	—	2	1	no	—
14	C₄⋊C₄×C₁₀₉	—	4	1	no	—
15	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×D₁₀₉	—	2	2	no	—
17	C₂×#89137	—	8	2	no	—
18	Q₈×C₁₀₉	—	4	2	no	—
19	C₂×C₄×C₁₀₉	—	2	1	no	—
20	C₂×C₄×C₁₀₉	—	4	1	no	—
21	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	D₄₃₆	—	4	2	no	—
23	D₄₃₆	—	8	2	no	—
24	#11724784	—	2	2	no	—
25	C₂²×C₁₀₉	#99	1	1	yes	characteristic
26	C₂×D₁₀₉	—	4	2	no	—
27	C₂×D₁₀₉	—	8	2	no	—
28	#89137	—	16	2	no	—
29	C₄×C₁₀₉	—	2	2	no	—
30	C₄×C₁₀₉	—	2	1	no	—
31	C₄×C₁₀₉	—	8	1	no	—
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	#353	1	1	yes	characteristic
34	D₁₀₉	—	8	2	no	—
35	#1409	—	109	1	no	maximal, Sylow
36	C₁₀₉	#1409	1	1	yes	characteristic, complemented, Sylow
37	#492	—	109	1	no	—
38	#353	—	109	2	no	—
39	C₂×SD₁₆	—	218	2	no	—
40	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
41	#127	—	109	1	no	—
42	#99	—	218	2	no	—
43	C₄×C₈	—	109	1	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	109	2	no	—
46	C₂×D₄	—	436	1	no	—
47	SD₁₆	—	436	4	no	—
48	C₂×C₈	—	218	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	436	2	no	—
51	C₄²	C₂×#89137	1	1	yes	characteristic
52	C₂³	—	218	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	872	2	no	—
55	D₄	—	436	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	872	2	no	—
59	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
60	C₈	—	218	2	no	—
61	C₂²	—	872	2	no	—
62	C₂²	—	436	2	no	—
63	C₂²	#16812757	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	1744	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	872	2	no	—
70	C₂	#20295930	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295930
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#353
prime radical, cosocle	#16812696	C₂²
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1409
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₄×#11724784	C₄	semidirect
7. C₄⋊Q₈×C₁₀₉	C₄	semidirect
36. C₁₀₉	#1409	semidirect
40. C₄⋊Q₈	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20295931	2
#20295400	2
#1409	109