Subgroups — #36375103

Order 22336.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₄₉⋊C₈	C₂	1	2	yes	maximal
2	#26013694	C₂	1	4	yes	maximal, complemented
3	C₄²×D₃₄₉	C₂	1	1	yes	maximal, characteristic
4	D₃₄₉⋊C₈	C₄	1	8	yes	complemented
5	C₂×#14584102	C₂²	1	4	yes	—
6	C₂×C₄×D₃₄₉	C₄	1	2	yes	—
7	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₃₄₉	C₄	1	1	yes	characteristic
9	C₄×Dic₃₄₉	C₂²	1	2	yes	—
10	C₄²×C₃₄₉	C₄	1	1	yes	characteristic
11	#14584102	C₂×C₄	1	8	yes	complemented
12	C₂²×D₃₄₉	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
14	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
15	C₄×D₃₄₉	—	2	4	no	—
16	C₄×D₃₄₉	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
20	C₂×D₃₄₉	C₄²	1	2	yes	—
21	C₂×D₃₄₉	M₁₆	1	4	yes	—
22	C₄×C₃₄₉	—	2	2	no	—
23	C₄×C₃₄₉	C₄²	1	2	yes	—
24	Dic₃₄₉	—	2	2	no	—
25	Dic₃₄₉	C₂²×C₄	1	2	yes	—
26	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
27	C₂×C₃₄₉	C₂×M₁₆	1	2	yes	—
28	D₃₄₉	#97	1	4	yes	complemented
29	C₃₄₉	C₂×#97	1	1	yes	characteristic, complemented, Sylow
30	C₂×#97	—	349	1	no	maximal, Sylow
31	C₂²×C₈	—	349	2	no	—
32	C₂×C₄²	—	349	1	no	—
33	#97	—	349	4	no	—
34	C₂²×C₄	—	349	1	no	—
35	C₂²×C₄	—	349	2	no	—
36	C₂×C₈	—	349	4	no	—
37	C₂×C₈	—	349	8	no	—
38	C₄²	—	349	2	no	—
39	C₄²	—	349	1	no	—
40	C₄²	#13063703	1	1	yes	characteristic
41	C₂³	—	349	1	no	—
42	C₂×C₄	—	349	1	no	—
43	C₂×C₄	—	349	4	no	—
44	C₂×C₄	—	349	2	no	—
45	C₂×C₄	—	698	4	no	—
46	C₂×C₄	C₂×#13063703	1	2	yes	—
47	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
48	C₈	—	349	8	no	—
49	C₂²	—	349	2	no	—
50	C₂²	—	349	4	no	—
51	C₂²	C₂²×#13063703	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	—	349	2	no	—
54	C₄	—	698	2	no	—
55	C₄	C₄×#13063703	1	2	yes	—
56	C₂	—	349	4	no	—
57	C₂	C₂×C₄×#13063703	1	1	yes	characteristic
58	C₂	D₃₄₉⋊M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13063703
commutator	C₂×C₃₄₉	C₂×C₄²
Frattini	C₂²	C₂²×#13063703
Fitting	C₄²×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂²×C₄
prime radical	C₂²×D₃₄₉	C₂×C₄
cosocle	C₂×Dic₃₄₉	C₂³
hypercenter	C₄²	#13063703
hypocenter	C₃₄₉	C₂×#97
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #26013694	C₂	semidirect
4. D₃₄₉⋊C₈	C₄	semidirect
11. #14584102	C₂×C₄	semidirect
28. D₃₄₉	#97	semidirect
29. C₃₄₉	C₂×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₄₉⋊C₈	2
#26013694	2
C₄²×D₃₄₉	2
C₂×#97	349