Subgroups — #36373944

Order 22336.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₇₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19150208	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₉₆	—	2	1	no	—
5	D₂₇₉₂	—	2	1	no	—
6	D₂₇₉₂	C₄	1	2	yes	complemented
7	#19150208	—	2	1	no	—
8	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₃₄₉	—	2	1	no	—
10	C₂²×D₃₄₉	—	4	1	no	—
11	D₁₃₉₆	—	2	2	no	—
12	D₁₃₉₆	—	4	1	no	—
13	C₂×C₄×C₃₄₉	—	4	1	no	—
14	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
15	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
16	C₈×C₃₄₉	D₄	1	1	yes	characteristic
17	C₂²×C₃₄₉	D₈	1	1	yes	characteristic
18	C₂×D₃₄₉	—	4	2	no	—
19	C₂×D₃₄₉	—	8	1	no	—
20	C₂×D₃₄₉	—	8	1	no	—
21	C₄×C₃₄₉	—	8	1	no	—
22	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₃₄₉	SD₁₆	1	1	yes	characteristic
24	C₂×C₃₄₉	#102	1	1	yes	characteristic
25	C₂×C₃₄₉	D₁₆	1	1	yes	characteristic
26	C₂×C₃₄₉	SD₃₂	1	1	yes	characteristic
27	D₃₄₉	—	8	2	no	—
28	C₃₄₉	#357	1	1	yes	characteristic, complemented, Sylow
29	#357	—	349	1	no	maximal, Sylow
30	C₂×D₈	—	349	1	no	—
31	C₂×C₁₆	—	349	1	no	—
32	#107	D₃₄₉	1	1	yes	characteristic, complemented
33	C₂×D₄	—	698	1	no	—
34	D₈	—	349	2	no	—
35	D₈	—	698	1	no	—
36	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
37	C₄⋊C₄	—	2	1	no	—
38	C₁₆	—	698	1	no	—
39	C₂³	—	1396	1	no	—
40	D₄	—	1396	1	no	—
41	D₄	—	698	2	no	—
42	C₂×C₄	—	4	1	no	—
43	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
44	C₈	C₄×D₃₄₉	1	1	yes	characteristic
45	C₈	D₁₃₉₆	1	1	yes	characteristic
46	C₂²	—	1396	2	no	—
47	C₂²	—	2792	1	no	—
48	C₂²	—	2792	1	no	—
49	C₂²	#19150218	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	#19150210	1	1	yes	characteristic
52	C₄	#19150220	1	1	yes	characteristic
53	C₂	—	2792	2	no	—
54	C₂	#26013559	1	1	yes	characteristic
55	C₂	#26013580	1	1	yes	characteristic
56	C₂	#26013582	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19150218
commutator	C₈×C₃₄₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₄₉
Fitting	#107×C₃₄₉	C₂
socle	C₂²×C₃₄₉	D₈
prime radical	C₂×D₂₇₉₂	C₂
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	#107	D₃₄₉
hypocenter	C₃₄₉	#357
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
2. C₂×#19150208	C₂	semidirect
3. #107×C₃₄₉	C₂	semidirect
6. D₂₇₉₂	C₄	semidirect
28. C₃₄₉	#357	semidirect
32. #107	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₇₉₂	2
C₂×#19150208	2
#107×C₃₄₉	2
#357	349