Subgroups — #36374457

Order 22336.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013613	C₂	1	1	yes	maximal, characteristic
2	#26013572	C₂	1	2	yes	maximal, complemented
3	#26013559	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#14584095	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₃₄₉	—	2	1	no	—
8	C₂×#14584095	C₂²	1	2	yes	—
9	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
10	#19150210	—	2	1	no	—
11	C₂²×C₄×C₃₄₉	—	2	1	no	—
12	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₄₉	—	2	1	no	—
14	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
15	C₂²×D₃₄₉	—	2	1	no	—
16	D₁₃₉₆	—	2	2	no	—
17	C₂²⋊D₃₄₉	—	4	2	no	—
18	C₂×Dic₃₄₉	—	2	1	no	—
19	#14584095	—	2	2	no	—
20	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₃₄₉	—	2	2	no	—
22	C₂×C₄×C₃₄₉	—	2	2	no	—
23	C₂×C₄×C₃₄₉	—	4	2	no	—
24	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
26	C₂²×C₃₄₉	—	2	1	no	—
27	C₂²×C₃₄₉	—	2	1	no	—
28	C₂²×C₃₄₉	D₈	1	2	yes	—
29	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
30	C₂×D₃₄₉	—	4	1	no	—
31	C₂×D₃₄₉	—	4	1	no	—
32	C₂×D₃₄₉	—	4	1	no	—
33	C₄×C₃₄₉	—	2	1	no	—
34	C₄×C₃₄₉	—	4	2	no	—
35	C₄×C₃₄₉	#41	1	2	yes	—
36	Dic₃₄₉	—	4	1	no	—
37	C₂×C₃₄₉	—	2	2	no	—
38	C₂×C₃₄₉	#123	1	1	yes	characteristic
39	C₂×C₃₄₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₃₄₉	#137	1	1	yes	characteristic
41	D₃₄₉	—	8	1	no	—
42	C₃₄₉	#480	1	1	yes	characteristic, complemented, Sylow
43	#480	—	349	1	no	maximal, Sylow
44	#121	—	349	1	no	—
45	C₂×C₄⋊C₄	D₃₄₉	1	1	yes	characteristic, complemented
46	#107	—	349	2	no	—
47	#102	—	349	2	no	—
48	C₂²⋊C₈	—	349	1	no	—
49	C₂×D₄	—	349	1	no	—
50	C₂×D₄	—	698	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
53	C₂×C₈	—	349	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	349	1	no	—
56	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
57	C₂²⋊C₄	—	698	1	no	—
58	C₂³	—	698	1	no	—
59	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
60	D₄	—	1396	2	no	—
61	D₄	—	698	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	698	1	no	—
66	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
68	C₈	—	698	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	1396	1	no	—
72	C₂²	—	1396	1	no	—
73	C₂²	—	1396	1	no	—
74	C₂²	#19150218	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	1396	1	no	—
79	C₄	#19150233	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	2792	1	no	—
82	C₂	#26013622	1	1	yes	characteristic
83	C₂	C₂×#19150218	1	1	yes	characteristic
84	C₂	#26013670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₄₉
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₂×C₄⋊C₄×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	#26013613	C₂
hypercenter	C₂×C₄⋊C₄	D₃₄₉
hypocenter	C₃₄₉	#480
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #26013572	C₂	semidirect
3. #26013559	C₂	semidirect
4. C₂²⋊#14584095	C₂	semidirect
5. C₂×C₄⋊C₄×C₃₄₉	C₂	semidirect
42. C₃₄₉	#480	semidirect
45. C₂×C₄⋊C₄	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013613	2
#26013572	2
#26013559	2
C₂²⋊#14584095	2
C₂×C₄⋊C₄×C₃₄₉	2
#480	349