Subgroups — #36374209

Order 22336.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₄₉	C₂	1	2	yes	maximal, complemented
2	C₄×D₁₃₉₆	C₂	1	1	yes	maximal, characteristic
3	#26013551	C₂	1	2	yes	maximal, complemented
4	C₄×#14584095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₃₉₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₃₄₉	C₂²	1	2	yes	—
8	C₂×#14584095	C₂²	1	1	yes	characteristic
9	C₂×#14584095	C₂²	1	1	yes	characteristic
10	C₈×D₃₄₉	—	2	4	no	—
11	C₄⋊Dic₃₄₉	C₄	1	1	yes	characteristic
12	#19150210	C₄	1	2	yes	—
13	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
14	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₂²×D₃₄₉	C₂×C₄	1	2	yes	—
16	D₁₃₉₆	C₈	1	4	yes	complemented
17	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
18	C₄×D₃₄₉	—	2	2	no	—
19	C₄×D₃₄₉	—	2	2	no	—
20	#14584095	—	2	1	no	—
21	#14584095	D₄	1	2	yes	complemented
22	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
25	C₈×C₃₄₉	—	2	2	no	—
26	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₄₉	—	2	2	no	—
28	C₂×D₃₄₉	—	2	2	no	—
29	C₂×D₃₄₉	C₂×C₈	1	4	yes	complemented
30	C₄×C₃₄₉	—	2	1	no	—
31	C₄×C₃₄₉	C₂×C₈	1	2	yes	complemented
32	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₃₄₉	#41	1	1	yes	characteristic
34	Dic₃₄₉	—	2	2	no	—
35	C₂×C₃₄₉	C₄×D₄	1	1	yes	characteristic
36	C₂×C₃₄₉	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₃₄₉	#131	1	1	yes	characteristic
38	D₃₄₉	—	2	4	no	—
39	C₃₄₉	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	349	1	no	maximal, Sylow
41	C₂²×C₈	—	349	2	no	—
42	C₄×D₄	—	349	1	no	—
43	C₄⋊C₈	D₃₄₉	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	349	2	no	—
45	C₄×C₈	—	349	1	no	—
46	C₂×D₄	—	349	1	no	—
47	C₂²×C₄	—	349	2	no	—
48	C₂×C₈	—	349	1	no	—
49	C₂×C₈	—	349	1	no	—
50	C₂×C₈	—	698	4	no	—
51	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
52	C₄⋊C₄	—	349	1	no	—
53	C₂²⋊C₄	—	349	2	no	—
54	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
55	C₂³	—	349	2	no	—
56	D₄	—	349	4	no	—
57	C₂×C₄	—	349	2	no	—
58	C₂×C₄	—	698	2	no	—
59	C₂×C₄	—	698	2	no	—
60	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
61	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	349	2	no	—
65	C₈	—	698	1	no	—
66	C₂²	—	349	4	no	—
67	C₂²	—	698	2	no	—
68	C₂²	—	698	2	no	—
69	C₂²	C₂×C₄×D₃₄₉	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	698	2	no	—
72	C₄	C₈×D₃₄₉	1	2	yes	complemented
73	C₄	D₄×D₃₄₉	1	1	yes	characteristic
74	C₄	#19150234	1	1	yes	characteristic
75	C₂	—	698	4	no	—
76	C₂	#26013601	1	1	yes	characteristic
77	C₂	C₂×C₈×D₃₄₉	1	1	yes	characteristic
78	C₂	#26013647	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₄₉
commutator	C₂×C₃₄₉	C₂²×C₈
Fitting	C₄⋊C₈×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂²×C₄
prime radical	C₂×D₁₃₉₆	C₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₄⋊C₈	D₃₄₉
hypocenter	C₃₄₉	C₈×D₄
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₃₄₉	C₂	semidirect
3. #26013551	C₂	semidirect
4. C₄×#14584095	C₂	semidirect
5. C₄⋊C₈×C₃₄₉	C₂	semidirect
16. D₁₃₉₆	C₈	semidirect
21. #14584095	D₄	semidirect
29. C₂×D₃₄₉	C₂×C₈	semidirect
31. C₄×C₃₄₉	C₂×C₈	semidirect
39. C₃₄₉	C₈×D₄	semidirect
43. C₄⋊C₈	D₃₄₉	semidirect
72. C₄	C₈×D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₄₉	2
C₄×D₁₃₉₆	2
#26013551	2
C₄×#14584095	2
C₄⋊C₈×C₃₄₉	2
C₈×D₄	349