Subgroups — #28378037

Order 13952.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295918	C₂	1	2	yes	maximal
2	#20295394	C₂	1	1	yes	maximal, characteristic
3	#16812773	—	2	2	no	—
4	#16812691	C₄	1	1	yes	characteristic
5	#16812623	—	2	2	no	—
6	#16812610	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
8	#13182032	—	2	4	no	—
9	C₂×#11724784	D₄	1	2	yes	—
10	C₄⋊Dic₁₀₉	—	2	1	no	—
11	#13182008	—	4	1	no	—
12	C₂×Q₈×C₁₀₉	—	2	1	no	—
13	C₄⋊C₄×C₁₀₉	—	4	1	no	—
14	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
15	Q₈×C₁₀₉	—	4	2	no	—
16	C₂×C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄×C₁₀₉	—	4	1	no	—
18	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×Dic₁₀₉	—	4	1	no	—
20	#11724784	—	2	2	no	—
21	C₂²×C₁₀₉	#100	1	1	yes	characteristic
22	C₄×C₁₀₉	—	4	1	no	—
23	C₄×C₁₀₉	—	8	1	no	—
24	C₄×C₁₀₉	#104	1	2	yes	—
25	Dic₁₀₉	—	8	1	no	—
26	C₂×C₁₀₉	—	2	1	no	—
27	C₂×C₁₀₉	#329	1	1	yes	characteristic
28	#1356	—	109	1	no	maximal, Sylow
29	C₁₀₉	#1356	1	1	yes	characteristic, complemented, Sylow
30	#491	—	109	1	no	—
31	#347	—	109	2	no	—
32	C₄⋊Q₈	#89137	1	1	yes	characteristic
33	#125	—	109	1	no	—
34	M₃₂	—	218	2	no	—
35	#103	—	218	2	no	—
36	#97	—	109	1	no	—
37	C₂×Q₈	—	2	1	no	—
38	C₂×C₈	—	109	2	no	—
39	C₄⋊C₄	—	4	1	no	—
40	C₄⋊C₄	—	436	1	no	—
41	C₄⋊C₄	—	218	1	no	—
42	C₄²	C₂×#89137	1	1	yes	characteristic
43	C₁₆	—	218	4	no	—
44	Q₈	—	4	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	436	1	no	—
48	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
49	C₈	—	218	2	no	—
50	C₂²	#16812759	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	872	1	no	—
54	C₄	#16812763	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#20295881	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295881
commutator	C₄²×C₁₀₉	C₂×C₄
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂×C₁₀₉	#329
prime radical	C₂²×C₁₀₉	#100
cosocle	#16812610	C₂²
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1356
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
29. C₁₀₉	#1356	semidirect

Maximal subgroups

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

subgroup	index
#20295918	2
#20295394	2
#1356	109