Subgroups — #36375039

Order 22336.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150236	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
3	#19150236	—	2	2	no	—
4	C₂×C₄×D₃₄₉	C₄	1	1	yes	characteristic
5	C₂×#14584095	C₂²	1	1	yes	characteristic
6	C₈×D₃₄₉	—	2	2	no	—
7	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic
8	C₂²×D₃₄₉	C₈	1	1	yes	characteristic
9	C₂×Dic₃₄₉	C₈	1	1	yes	characteristic
10	C₄×D₃₄₉	—	2	1	no	—
11	C₄×D₃₄₉	—	2	1	no	—
12	#14584095	D₄	1	2	yes	—
13	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
14	C₈×C₃₄₉	—	2	1	no	—
15	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
16	C₂×D₃₄₉	—	2	1	no	—
17	C₂×D₃₄₉	—	2	1	no	—
18	C₂×D₃₄₉	C₁₆	1	2	yes	complemented
19	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
21	Dic₃₄₉	—	2	1	no	—
22	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
23	C₂×C₃₄₉	C₂×C₁₆	1	1	yes	characteristic
24	C₂×C₃₄₉	M₃₂	1	1	yes	characteristic
25	D₃₄₉	—	2	2	no	—
26	C₃₄₉	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂²⋊C₁₆	—	349	1	no	maximal, Sylow
28	C₂²×C₈	—	349	1	no	—
29	C₂×C₁₆	—	349	2	no	—
30	C₂²×C₄	—	349	1	no	—
31	C₂×C₈	—	349	1	no	—
32	C₂×C₈	—	698	2	no	—
33	C₂×C₈	#13063703	1	1	yes	characteristic
34	C₁₆	—	698	2	no	—
35	C₂³	—	349	1	no	—
36	C₂×C₄	—	349	1	no	—
37	C₂×C₄	—	698	1	no	—
38	C₂×C₄	—	698	1	no	—
39	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	—	349	2	no	—
42	C₂²	—	349	2	no	—
43	C₂²	—	698	1	no	—
44	C₂²	—	698	1	no	—
45	C₂²	D₃₄₉⋊C₈	1	1	yes	characteristic
46	C₄	—	698	1	no	—
47	C₄	C₂²⋊#13063703	1	1	yes	characteristic
48	C₄	#19150243	1	1	yes	characteristic
49	C₂	—	698	2	no	—
50	C₂	#26013696	1	1	yes	characteristic
51	C₂	D₃₄₉⋊C₁₆	1	1	yes	characteristic
52	C₂	#26013714	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#13063703
commutator	C₂×C₃₄₉	C₂×C₁₆
Fitting	C₂×C₈×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂×C₈
prime radical	C₂²×D₃₄₉	C₈
cosocle	C₂×#14584095	C₂²
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	C₂²⋊C₁₆
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19150236	C₂	semidirect
18. C₂×D₃₄₉	C₁₆	semidirect
26. C₃₄₉	C₂²⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150236	2
C₂×C₈×D₃₄₉	2
C₂²⋊C₁₆	349