Subgroups — #36373955

Order 22336.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013661	C₂	1	1	yes	maximal, characteristic
2	#26013571	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19150233	—	2	1	no	—
5	Dic₁₃₉₆	C₄	1	1	yes	characteristic, complemented
6	#19150222	—	2	1	no	—
7	D₂₇₉₂	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₃₄₉	—	2	1	no	—
9	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₄₉	—	2	1	no	—
11	Dic₆₉₈	—	2	1	no	—
12	D₁₃₉₆	—	2	1	no	—
13	C₂²⋊D₃₄₉	—	4	1	no	—
14	C₂×Dic₃₄₉	—	4	1	no	—
15	C₄×D₃₄₉	—	4	1	no	—
16	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
17	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₃₄₉	D₄	1	1	yes	characteristic
19	C₂²×C₃₄₉	D₈	1	1	yes	characteristic
20	C₂×D₃₄₉	—	4	1	no	—
21	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₄₉	SD₁₆	1	1	yes	characteristic
23	Dic₃₄₉	—	4	1	no	—
24	Dic₃₄₉	—	8	1	no	—
25	C₂×C₃₄₉	—	2	1	no	—
26	C₂×C₃₄₉	#102	1	1	yes	characteristic
27	D₃₄₉	—	8	1	no	—
28	C₃₄₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	349	1	no	maximal, Sylow
30	#135	—	349	1	no	—
31	M₃₂	D₃₄₉	1	1	yes	characteristic, complemented
32	#106	—	349	1	no	—
33	#41	—	698	1	no	—
34	Q₁₆	—	349	1	no	—
35	SD₁₆	—	698	1	no	—
36	D₈	—	349	1	no	—
37	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
38	C₄⋊C₄	—	698	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	698	1	no	—
41	D₄	—	1396	1	no	—
42	D₄	—	698	1	no	—
43	C₂×C₄	—	1396	1	no	—
44	C₂×C₄	—	1396	1	no	—
45	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
46	C₈	C₄×D₃₄₉	1	1	yes	characteristic
47	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
48	C₂²	—	1396	1	no	—
49	C₂²	D₂₇₉₂	1	1	yes	characteristic
50	C₄	—	1396	1	no	—
51	C₄	—	2792	1	no	—
52	C₄	#19150210	1	1	yes	characteristic
53	C₄	#19150222	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	2792	1	no	—
56	C₂	#26013561	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013561
commutator	C₈×C₃₄₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₄₉
Fitting	M₃₂×C₃₄₉	C₂
socle	C₂×C₃₄₉	#102
prime radical	#26013661	C₂
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	M₃₂	D₃₄₉
hypocenter	C₃₄₉	#360
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #26013571	C₂	semidirect
3. M₃₂×C₃₄₉	C₂	semidirect
5. Dic₁₃₉₆	C₄	semidirect
7. D₂₇₉₂	C₄	semidirect
28. C₃₄₉	#360	semidirect
31. M₃₂	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013661	2
#26013571	2
M₃₂×C₃₄₉	2
#360	349