Subgroups — #36374474

Order 22336.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013618	C₂	1	1	yes	maximal, characteristic
2	#26013570	C₂	1	2	yes	maximal, complemented
3	#26013562	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#14584095	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
7	C₂×#14584095	C₂²	1	2	yes	—
8	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
9	#19150212	—	2	1	no	—
10	C₂²⋊Dic₃₄₉	—	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	Dic₆₉₈	—	2	2	no	—
16	C₂×Dic₃₄₉	—	2	1	no	—
17	C₂×Dic₃₄₉	—	2	1	no	—
18	#14584095	—	2	2	no	—
19	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₃₄₉	—	2	2	no	—
21	C₂×C₄×C₃₄₉	—	2	2	no	—
22	C₂×C₄×C₃₄₉	—	4	2	no	—
23	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
25	C₂²×C₃₄₉	—	2	1	no	—
26	C₂²×C₃₄₉	—	2	1	no	—
27	C₂²×C₃₄₉	SD₁₆	1	2	yes	—
28	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
29	C₄×C₃₄₉	—	2	1	no	—
30	C₄×C₃₄₉	—	4	2	no	—
31	C₄×C₃₄₉	#41	1	2	yes	—
32	Dic₃₄₉	—	4	1	no	—
33	Dic₃₄₉	—	4	1	no	—
34	C₂×C₃₄₉	—	2	2	no	—
35	C₂×C₃₄₉	#123	1	1	yes	characteristic
36	C₂×C₃₄₉	C₂×SD₁₆	1	1	yes	characteristic
37	C₂×C₃₄₉	#137	1	1	yes	characteristic
38	C₃₄₉	#483	1	1	yes	characteristic, complemented, Sylow
39	#483	—	349	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	349	1	no	—
41	C₂×C₄⋊C₄	D₃₄₉	1	1	yes	characteristic
42	#106	—	349	2	no	—
43	#103	—	349	2	no	—
44	C₂²⋊C₈	—	349	1	no	—
45	C₂×Q₈	—	349	1	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
48	C₂×C₈	—	349	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	349	1	no	—
51	C₄⋊C₄	—	698	1	no	—
52	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
53	C₂²⋊C₄	—	698	1	no	—
54	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
55	Q₈	—	698	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	4	2	no	—
59	C₂×C₄	—	698	1	no	—
60	C₂×C₄	—	698	1	no	—
61	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
63	C₈	—	698	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	#19150221	1	2	yes	—
67	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	2	no	—
70	C₄	—	1396	1	no	—
71	C₄	—	1396	1	no	—
72	C₄	#19150233	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#26013622	1	1	yes	characteristic
75	C₂	C₂×#19150221	1	1	yes	characteristic
76	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	C₂³×C₃₄₉	D₄
hypercenter	C₂×C₄⋊C₄	D₃₄₉
hypocenter	C₃₄₉	#483
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. #26013570	C₂	semidirect
3. #26013562	C₂	semidirect
38. C₃₄₉	#483	semidirect

Maximal subgroups

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

subgroup	index
#26013618	2
#26013570	2
#26013562	2
C₂²⋊#14584095	2
C₂×C₄⋊C₄×C₃₄₉	2
#483	349