Subgroups — #36373878

Order 22336.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14584095	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013546	C₂	1	1	yes	maximal, characteristic, complemented
3	#121×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₄₉	—	2	1	no	—
5	C₂×#14584095	—	2	1	no	—
6	C₂×D₄×C₃₄₉	—	2	1	no	—
7	C₂×D₄×C₃₄₉	C₄	1	1	yes	characteristic, complemented
8	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₃₄₉	C₄	1	1	yes	characteristic, complemented
10	C₂²⋊C₄×C₃₄₉	—	2	1	no	—
11	C₂×Dic₃₄₉	—	2	2	no	—
12	C₂×Dic₃₄₉	—	4	1	no	—
13	C₂×Dic₃₄₉	—	4	1	no	—
14	#14584095	—	4	1	no	—
15	C₂³×C₃₄₉	—	2	1	no	—
16	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
17	D₄×C₃₄₉	—	4	1	no	—
18	D₄×C₃₄₉	—	4	1	no	—
19	D₄×C₃₄₉	—	4	1	no	—
20	C₂×C₄×C₃₄₉	—	2	1	no	—
21	C₂×C₄×C₃₄₉	—	4	1	no	—
22	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
24	C₂²×C₃₄₉	—	2	1	no	—
25	C₂²×C₃₄₉	—	2	1	no	—
26	C₂²×C₃₄₉	—	4	1	no	—
27	C₂²×C₃₄₉	—	4	1	no	—
28	C₂²×C₃₄₉	—	4	1	no	—
29	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₃₄₉	D₈	1	1	yes	characteristic
31	C₂²×C₃₄₉	SD₁₆	1	1	yes	characteristic
32	C₄×C₃₄₉	—	2	1	no	—
33	C₄×C₃₄₉	—	2	1	no	—
34	C₄×C₃₄₉	—	4	1	no	—
35	Dic₃₄₉	—	4	2	no	—
36	C₂×C₃₄₉	—	2	1	no	—
37	C₂×C₃₄₉	—	2	1	no	—
38	C₂×C₃₄₉	—	8	1	no	—
39	C₂×C₃₄₉	#99	1	1	yes	characteristic
40	C₂×C₃₄₉	#102	1	1	yes	characteristic
41	C₂×C₃₄₉	#104	1	1	yes	characteristic
42	C₃₄₉	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	349	1	no	maximal, Sylow
44	#121	D₃₄₉	1	1	yes	characteristic
45	C₂²⋊C₈	—	349	1	no	—
46	#95	—	349	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	Dic₃₄₉	1	1	yes	characteristic, complemented
49	C₂²×C₄	—	698	1	no	—
50	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
51	C₂×C₈	—	698	1	no	—
52	C₄⋊C₄	Dic₃₄₉	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	2	1	no	—
54	C₂³	—	2	1	no	—
55	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
56	D₄	—	4	1	no	—
57	D₄	—	4	1	no	—
58	D₄	—	4	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	1396	1	no	—
62	C₂×C₄	—	1396	1	no	—
63	C₂×C₄	—	698	2	no	—
64	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
66	C₈	—	1396	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
73	C₂²	#19150218	1	1	yes	characteristic
74	C₂²	#19150221	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	1396	2	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#26013554	1	1	yes	characteristic
83	C₂	#26013560	1	1	yes	characteristic
84	C₂	#26013566	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₃₄₉
commutator	C₂×C₄×C₃₄₉	C₂×C₄
Frattini	C₂²×C₄	C₂×D₃₄₉
Fitting, prime radical	#121×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂²⋊C₄
cosocle	C₂²×C₄×C₃₄₉	C₂²
hypercenter	#121	D₃₄₉
hypocenter	C₃₄₉	#327
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#14584095	C₂	semidirect
2. #26013546	C₂	semidirect
7. C₂×D₄×C₃₄₉	C₄	semidirect
9. C₄⋊C₄×C₃₄₉	C₄	semidirect
42. C₃₄₉	#327	semidirect
48. C₂×D₄	Dic₃₄₉	semidirect
52. C₄⋊C₄	Dic₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#14584095	2
#26013546	2
#121×C₃₄₉	2
#327	349