Subgroups — #28377947

Order 13952.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182032	C₂	1	2	yes	maximal
2	C₈×#11724784	C₂	1	1	yes	maximal, characteristic
3	C₂×#13182032	C₄	1	4	yes	—
4	C₄×#11724784	C₄	1	1	yes	characteristic
5	C₄×#11724784	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
7	#13182032	—	2	4	no	—
8	C₂×#11724784	C₈	1	2	yes	—
9	C₂×#11724784	C₂×C₄	1	2	yes	—
10	C₂×C₈×C₁₀₉	C₈	1	2	yes	—
11	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
13	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
15	C₈×C₁₀₉	—	2	2	no	—
16	#11724784	C₁₆	1	4	yes	complemented
17	#11724784	M₁₆	1	4	yes	—
18	C₂²×C₁₀₉	C₄×C₈	1	1	yes	characteristic
19	C₄×C₁₀₉	#97	1	2	yes	—
20	C₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
21	C₄×C₁₀₉	M₃₂	1	2	yes	—
22	C₂×C₁₀₉	#322	1	1	yes	characteristic
23	C₂×C₁₀₉	C₄×C₁₆	1	1	yes	characteristic
24	C₂×C₁₀₉	#346	1	1	yes	characteristic
25	#1312	—	109	1	no	maximal, Sylow
26	C₁₀₉	#1312	1	1	yes	characteristic, complemented, Sylow
27	C₄×C₁₆	—	109	2	no	—
28	C₈²	—	109	1	no	—
29	C₂×C₁₆	—	109	4	no	—
30	C₄×C₈	—	109	1	no	—
31	C₄×C₈	—	109	1	no	—
32	C₄×C₈	#89137	1	1	yes	characteristic
33	C₂×C₈	—	109	2	no	—
34	C₂×C₈	—	109	2	no	—
35	C₂×C₈	#11724786	1	2	yes	—
36	C₄²	C₂×#89137	1	1	yes	characteristic
37	C₁₆	—	218	4	no	—
38	C₂×C₄	C₄×#89137	1	1	yes	characteristic
39	C₂×C₄	C₂×#11724786	1	1	yes	characteristic
40	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
41	C₈	—	2	2	no	—
42	C₈	—	109	4	no	—
43	C₈	—	109	4	no	—
44	C₂²	C₄×#11724786	1	1	yes	characteristic
45	C₄	#16812753	1	2	yes	—
46	C₄	D₁₀₉⋊C₁₆	1	2	yes	—
47	C₄	#16812773	1	2	yes	—
48	C₂	#20295867	1	1	yes	characteristic
49	C₂	#20295912	1	1	yes	characteristic
50	C₂	#20295914	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×#89137
commutator	C₂×C₁₀₉	C₄×C₁₆
Fitting	C₄×C₈×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×C₈
cosocle	C₄×#11724784	C₂²
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1312
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
16. #11724784	C₁₆	semidirect
26. 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×#13182032	2
C₈×#11724784	2
#1312	109