Subgroups — #36375042

Order 22336.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19150236	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013646	C₂	1	1	yes	maximal, characteristic
4	#19150236	—	2	1	no	—
5	#19150236	—	2	1	no	—
6	#19150217	—	2	1	no	—
7	#19150217	C₄	1	1	yes	characteristic
8	C₂×#14584095	—	2	1	no	—
9	C₂×#14584095	C₂²	1	1	yes	characteristic
10	#41×C₃₄₉	C₄	1	1	yes	characteristic
11	#14584095	—	2	1	no	—
12	#14584095	D₄	1	1	yes	characteristic
13	#14584095	D₄	1	1	yes	characteristic
14	Q₈×C₃₄₉	C₈	1	1	yes	characteristic
15	D₄×C₃₄₉	—	2	1	no	—
16	D₄×C₃₄₉	C₈	1	1	yes	characteristic
17	C₂×C₄×C₃₄₉	—	2	1	no	—
18	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃₄₉	—	2	1	no	—
20	C₂²×C₃₄₉	M₁₆	1	1	yes	characteristic
21	C₄×C₃₄₉	—	2	1	no	—
22	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₃₄₉	C₂×C₈	1	1	yes	characteristic
24	C₂×C₃₄₉	—	2	1	no	—
25	C₂×C₃₄₉	—	4	1	no	—
26	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
27	C₃₄₉	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	349	1	no	maximal, Sylow
29	#131	—	349	1	no	—
30	M₃₂	—	349	1	no	—
31	C₂×C₁₆	—	349	1	no	—
32	#41	#13063703	1	1	yes	characteristic
33	M₁₆	—	349	1	no	—
34	M₁₆	—	698	1	no	—
35	C₂×C₈	—	349	1	no	—
36	C₂×C₈	—	698	1	no	—
37	C₁₆	—	698	1	no	—
38	C₁₆	—	698	1	no	—
39	Q₈	#14584102	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	#14584102	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
44	C₈	—	349	1	no	—
45	C₈	—	349	1	no	—
46	C₈	—	698	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	#19150242	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊#13063703	1	1	yes	characteristic
51	C₄	C₂×#14584102	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊#14584102	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊#13063703
commutator	C₄×C₃₄₉	C₂×C₈
Frattini	C₂×C₄	C₂×#13063703
Fitting, prime radical	#41×C₃₄₉	C₄
socle	C₂×C₃₄₉	C₂²⋊C₈
cosocle	C₂×#14584095	C₂²
hypercenter	#41	#13063703
hypocenter	C₃₄₉	#350
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #26013713	C₂	semidirect
2. C₂×#19150236	C₂	semidirect
27. C₃₄₉	#350	semidirect

Maximal subgroups

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

subgroup	index
#26013713	2
C₂×#19150236	2
#26013646	2
#350	349