Subgroups — #36374701

Order 22336.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013639	C₂	1	2	yes	maximal, complemented
2	#26013631	C₂	1	2	yes	maximal, complemented
3	#26013618	C₂	1	4	yes	maximal
4	C₄×Dic₆₉₈	C₂	1	4	yes	maximal, complemented
5	#26013598	C₂	1	2	yes	maximal
6	C₂×C₄²×C₃₄₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₆₉₈	C₂²	1	4	yes	—
8	C₄⋊Dic₃₄₉	C₂²	1	4	yes	—
9	C₄⋊Dic₃₄₉	C₂²	1	4	yes	—
10	#19150212	C₂²	1	8	yes	—
11	C₂²⋊Dic₃₄₉	C₂²	1	4	yes	—
12	C₄×Dic₃₄₉	C₂²	1	4	yes	—
13	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₄²×C₃₄₉	C₂²	1	2	yes	—
16	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
17	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
18	Dic₆₉₈	—	2	8	no	—
19	C₂×Dic₃₄₉	C₂³	1	8	yes	—
20	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₃₄₉	—	2	4	no	—
22	C₂×C₄×C₃₄₉	Q₈	1	4	yes	complemented
23	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
24	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
26	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
27	C₂²×C₃₄₉	—	2	2	no	—
28	C₂²×C₃₄₉	C₂×Q₈	1	2	yes	—
29	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
30	C₄×C₃₄₉	—	2	2	no	—
31	C₄×C₃₄₉	C₂×Q₈	1	2	yes	—
32	C₄×C₃₄₉	C₂×Q₈	1	2	yes	—
33	C₄×C₃₄₉	#41	1	4	yes	—
34	Dic₃₄₉	—	2	8	no	—
35	C₂×C₃₄₉	—	2	2	no	—
36	C₂×C₃₄₉	C₂²×Q₈	1	1	yes	characteristic
37	C₂×C₃₄₉	C₂×#41	1	2	yes	—
38	C₃₄₉	#533	1	1	yes	characteristic, complemented, Sylow
39	#533	—	349	1	no	maximal, Sylow
40	C₄⋊Q₈	—	349	2	no	—
41	#125	—	349	2	no	—
42	C₂²⋊Q₈	—	349	4	no	—
43	C₄×Q₈	—	349	4	no	—
44	#117	—	349	2	no	—
45	C₂×C₄²	D₃₄₉	1	1	yes	characteristic
46	C₂×Q₈	—	349	4	no	—
47	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
48	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
49	C₄⋊C₄	—	349	4	no	—
50	C₄⋊C₄	—	349	8	no	—
51	C₄⋊C₄	—	349	4	no	—
52	C₂²⋊C₄	—	349	4	no	—
53	C₄²	—	349	4	no	—
54	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
55	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
56	C₄²	C₂×D₃₄₉	1	2	yes	—
57	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
58	Q₈	—	698	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	349	8	no	—
61	C₂×C₄	Dic₆₉₈	1	4	yes	complemented
62	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
63	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
65	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Dic₆₉₈	1	2	yes	—
68	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	698	8	no	—
71	C₄	C₂×Dic₆₉₈	1	2	yes	—
72	C₄	C₂×Dic₆₉₈	1	2	yes	—
73	C₄	#19150233	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₆₉₈	1	1	yes	characteristic
76	C₂	C₂×#19150233	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₄₉
commutator, socle, cosocle	C₂²×C₃₄₉	C₂⁴
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₄²×C₃₄₉	C₂
prime radical	C₂³×C₃₄₉	C₂³
hypercenter	C₂×C₄²	D₃₄₉
hypocenter	C₃₄₉	#533
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #26013639	C₂	semidirect
2. #26013631	C₂	semidirect
4. C₄×Dic₆₉₈	C₂	semidirect
22. C₂×C₄×C₃₄₉	Q₈	semidirect
38. C₃₄₉	#533	semidirect
61. C₂×C₄	Dic₆₉₈	semidirect

Maximal subgroups

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

subgroup	index
#26013639	2
#26013631	2
#26013618	2
C₄×Dic₆₉₈	2
#26013598	2
C₂×C₄²×C₃₄₉	2
#533	349