Subgroups — #36374355

Order 22336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₉₆	C₂	1	2	yes	maximal
2	#26013639	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic
4	#26013564	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₉₈	C₂²	1	2	yes	—
7	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
8	Dic₁₃₉₆	—	2	4	no	—
9	C₄⋊Dic₃₄₉	—	2	1	no	—
10	C₄⋊Dic₃₄₉	—	2	1	no	—
11	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
12	#19150212	—	2	1	no	—
13	C₄×Dic₃₄₉	—	2	1	no	—
14	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
15	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
16	Dic₆₉₈	—	2	1	no	—
17	Dic₆₉₈	—	2	2	no	—
18	Dic₆₉₈	—	2	2	no	—
19	Dic₆₉₈	D₄	1	2	yes	—
20	C₂×Dic₃₄₉	—	2	1	no	—
21	C₂×Dic₃₄₉	—	2	1	no	—
22	C₂×Dic₃₄₉	—	2	2	no	—
23	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
26	C₈×C₃₄₉	—	2	2	no	—
27	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₃₄₉	—	2	1	no	—
29	C₄×C₃₄₉	Q₁₆	1	2	yes	complemented
30	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₃₄₉	#41	1	1	yes	characteristic
32	Dic₃₄₉	—	2	2	no	—
33	Dic₃₄₉	—	4	2	no	—
34	Dic₃₄₉	—	4	1	no	—
35	C₂×C₃₄₉	#121	1	1	yes	characteristic
36	C₂×C₃₄₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₃₄₉	#137	1	1	yes	characteristic
38	C₃₄₉	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	349	1	no	maximal, Sylow
40	C₂×Q₁₆	—	349	2	no	—
41	C₄⋊Q₈	—	349	1	no	—
42	C₄×Q₈	—	349	1	no	—
43	C₄⋊C₈	D₃₄₉	1	1	yes	characteristic
44	#103	—	349	2	no	—
45	C₂×Q₈	—	349	1	no	—
46	C₂×Q₈	—	349	2	no	—
47	Q₁₆	—	698	4	no	—
48	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
49	C₄⋊C₄	—	349	1	no	—
50	C₄⋊C₄	—	698	1	no	—
51	C₄⋊C₄	—	698	1	no	—
52	C₄⋊C₄	—	698	1	no	—
53	C₄²	—	698	1	no	—
54	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
55	Q₈	—	349	2	no	—
56	Q₈	—	698	2	no	—
57	Q₈	—	698	2	no	—
58	Q₈	—	698	1	no	—
59	C₂×C₄	—	698	1	no	—
60	C₂×C₄	—	698	2	no	—
61	C₂×C₄	—	698	1	no	—
62	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
63	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₃₉₆	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	1396	2	no	—
69	C₄	—	1396	1	no	—
70	C₄	—	698	2	no	—
71	C₄	Dic₁₃₉₆	1	2	yes	complemented
72	C₄	D₄×D₃₄₉	1	1	yes	characteristic
73	C₄	#19150234	1	1	yes	characteristic
74	C₂	#26013614	1	1	yes	characteristic
75	C₂	C₂×Dic₁₃₉₆	1	1	yes	characteristic
76	C₂	#26013672	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₃₉₆
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₄⋊C₈×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂×D₄
hypercenter	C₄⋊C₈	D₃₄₉
hypocenter	C₃₄₉	#462
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₃₄₉	Q₁₆	semidirect
38. C₃₄₉	#462	semidirect
71. C₄	Dic₁₃₉₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₉₆	2
#26013639	2
C₄×Dic₆₉₈	2
#26013564	2
C₄⋊C₈×C₃₄₉	2
#462	349