Subgroups — #36374870

Order 22336.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013638	C₂	1	3	yes	maximal
2	#26013631	C₂	1	3	yes	maximal
3	#26013630	C₂	1	3	yes	maximal
4	Q₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₆₉₈	C₂	1	3	yes	maximal
7	C₄×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₉₈	C₂²	1	3	yes	—
9	C₄⋊Dic₃₄₉	C₂²	1	3	yes	—
10	C₄⋊Dic₃₄₉	C₂²	1	3	yes	—
11	C₄⋊Dic₃₄₉	C₂²	1	3	yes	—
12	#19150212	C₂²	1	1	yes	characteristic
13	#19150212	C₂²	1	3	yes	—
14	#19150212	C₂²	1	6	yes	—
15	C₄×Dic₃₄₉	C₂²	1	3	yes	—
16	C₄×Dic₃₄₉	C₂²	1	3	yes	—
17	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₄₉	C₂²	1	3	yes	—
19	C₄²×C₃₄₉	C₂²	1	3	yes	—
20	Dic₆₉₈	—	2	6	no	—
21	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₃₄₉	C₂³	1	3	yes	—
23	C₂×Dic₃₄₉	C₂³	1	3	yes	—
24	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
25	Q₈×C₃₄₉	Q₈	1	4	yes	complemented
26	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₃₄₉	C₂³	1	3	yes	—
28	C₂×C₄×C₃₄₉	C₂³	1	3	yes	—
29	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
30	C₄×C₃₄₉	—	2	3	no	—
31	C₄×C₃₄₉	—	2	1	no	—
32	C₄×C₃₄₉	C₂×Q₈	1	6	yes	—
33	Dic₃₄₉	—	2	3	no	—
34	Dic₃₄₉	—	2	1	no	—
35	Dic₃₄₉	—	2	3	no	—
36	Dic₃₄₉	#41	1	2	yes	—
37	C₂×C₃₄₉	C₂²×Q₈	1	1	yes	characteristic
38	C₂×C₃₄₉	C₂×#41	1	1	yes	characteristic
39	C₂×C₃₄₉	2₋¹⁺⁴	1	1	yes	characteristic
40	C₃₄₉	#557	1	1	yes	characteristic, complemented, Sylow
41	#557	—	349	1	no	maximal, Sylow
42	C₄⋊Q₈	—	349	3	no	—
43	#125	—	349	3	no	—
44	#125	—	349	3	no	—
45	C₄×Q₈	—	349	1	no	—
46	C₄×Q₈	—	349	1	no	—
47	C₄×Q₈	—	349	3	no	—
48	C₄×Q₈	D₃₄₉	1	1	yes	characteristic
49	C₂×Q₈	—	349	3	no	—
50	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
51	C₄⋊C₄	—	349	1	no	—
52	C₄⋊C₄	—	349	3	no	—
53	C₄⋊C₄	—	349	3	no	—
54	C₄⋊C₄	—	349	3	no	—
55	C₄⋊C₄	—	349	3	no	—
56	C₄⋊C₄	—	349	6	no	—
57	C₄⋊C₄	C₂×D₃₄₉	1	3	yes	—
58	C₄²	—	349	3	no	—
59	C₄²	—	349	3	no	—
60	C₄²	C₂×D₃₄₉	1	3	yes	—
61	Q₈	—	698	6	no	—
62	Q₈	Dic₆₉₈	1	4	yes	complemented
63	C₂×C₄	—	349	1	no	—
64	C₂×C₄	—	349	1	no	—
65	C₂×C₄	—	349	3	no	—
66	C₂×C₄	—	349	3	no	—
67	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
68	C₂×C₄	C₂²×D₃₄₉	1	3	yes	—
69	C₂×C₄	C₂²×D₃₄₉	1	3	yes	—
70	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	2	1	no	—
73	C₄	—	349	2	no	—
74	C₄	—	698	3	no	—
75	C₄	—	698	1	no	—
76	C₄	—	698	3	no	—
77	C₄	C₂×Dic₆₉₈	1	6	yes	—
78	C₂	C₂²×Dic₆₉₈	1	1	yes	characteristic
79	C₂	#41×D₃₄₉	1	1	yes	characteristic
80	C₂	#26013687	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₄₉
commutator, socle, prime radical, cosocle	C₂²×C₃₄₉	C₂⁴
Fitting	C₄×Q₈×C₃₄₉	C₂
hypercenter	C₄×Q₈	D₃₄₉
hypocenter	C₃₄₉	#557
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
25. Q₈×C₃₄₉	Q₈	semidirect
40. C₃₄₉	#557	semidirect
62. Q₈	Dic₆₉₈	semidirect

Maximal subgroups

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

subgroup	index
#26013638	2
#26013631	2
#26013630	2
Q₈×Dic₃₄₉	2
C₄×Dic₆₉₈	2
C₄×Q₈×C₃₄₉	2
#557	349