Subgroups — #36374872

Order 22336.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013639	C₂	1	1	yes	maximal, characteristic
2	#26013637	C₂	1	2	yes	maximal
3	#26013631	C₂	1	2	yes	maximal
4	#26013629	C₂	1	4	yes	maximal
5	C₄×Dic₆₉₈	C₂	1	2	yes	maximal
6	C₄×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic
7	#26013604	C₂	1	2	yes	maximal
8	C₄×Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₆₉₈	C₂²	1	2	yes	—
10	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₄₉	C₂²	1	2	yes	—
13	C₄⋊Dic₃₄₉	C₂²	1	2	yes	—
14	#19150212	C₂²	1	4	yes	—
15	#19150212	C₂²	1	2	yes	—
16	#19150212	C₂²	1	4	yes	—
17	#19150212	C₂²	1	4	yes	—
18	C₄×Dic₃₄₉	C₂²	1	4	yes	—
19	C₄×Dic₃₄₉	C₂²	1	2	yes	—
20	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₃₄₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
23	C₄²×C₃₄₉	C₂²	1	2	yes	—
24	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
25	Dic₆₉₈	—	2	4	no	—
26	Dic₆₉₈	Q₈	1	4	yes	complemented
27	C₂×Dic₃₄₉	C₂³	1	2	yes	—
28	C₂×Dic₃₄₉	C₂³	1	4	yes	—
29	C₂×Dic₃₄₉	C₂³	1	2	yes	—
30	Q₈×C₃₄₉	—	2	2	no	—
31	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
32	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
34	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
36	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
37	C₄×C₃₄₉	—	2	2	no	—
38	C₄×C₃₄₉	—	2	2	no	—
39	C₄×C₃₄₉	—	2	1	no	—
40	C₄×C₃₄₉	C₂×Q₈	1	2	yes	—
41	C₄×C₃₄₉	#41	1	2	yes	—
42	Dic₃₄₉	—	2	4	no	—
43	Dic₃₄₉	—	2	2	no	—
44	Dic₃₄₉	C₂×Q₈	1	4	yes	—
45	C₂×C₃₄₉	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₃₄₉	C₂×#41	1	1	yes	characteristic
47	C₂×C₃₄₉	2₋¹⁺⁴	1	1	yes	characteristic
48	C₃₄₉	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	349	1	no	maximal, Sylow
50	C₄⋊Q₈	—	349	2	no	—
51	C₄⋊Q₈	—	349	1	no	—
52	#125	—	349	4	no	—
53	#125	—	349	2	no	—
54	C₄×Q₈	—	349	2	no	—
55	C₄×Q₈	—	349	2	no	—
56	C₄×Q₈	—	349	1	no	—
57	C₄×Q₈	D₃₄₉	1	1	yes	characteristic
58	C₂×Q₈	—	349	1	no	—
59	C₂×Q₈	—	349	2	no	—
60	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
61	C₄⋊C₄	—	349	2	no	—
62	C₄⋊C₄	—	349	4	no	—
63	C₄⋊C₄	—	349	1	no	—
64	C₄⋊C₄	—	349	4	no	—
65	C₄⋊C₄	—	349	2	no	—
66	C₄⋊C₄	—	349	4	no	—
67	C₄⋊C₄	—	349	2	no	—
68	C₄⋊C₄	C₂×D₃₄₉	1	1	yes	characteristic
69	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
70	C₄²	—	349	4	no	—
71	C₄²	—	349	2	no	—
72	C₄²	C₂×D₃₄₉	1	2	yes	—
73	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
74	Q₈	—	2	2	no	—
75	Q₈	—	349	4	no	—
76	Q₈	—	698	4	no	—
77	C₂×C₄	—	349	2	no	—
78	C₂×C₄	—	349	2	no	—
79	C₂×C₄	—	349	4	no	—
80	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
81	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
83	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
85	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	349	4	no	—
90	C₄	—	698	4	no	—
91	C₄	—	698	2	no	—
92	C₄	Q₈×D₃₄₉	1	2	yes	—
93	C₄	#19150233	1	2	yes	—
94	C₂	C₂×Q₈×D₃₄₉	1	1	yes	characteristic
95	C₂	C₂×#19150233	1	1	yes	characteristic
96	C₂	#26013686	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
26. Dic₆₉₈	Q₈	semidirect
48. C₃₄₉	#557	semidirect

Maximal subgroups

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

subgroup	index
#26013639	2
#26013637	2
#26013631	2
#26013629	2
C₄×Dic₆₉₈	2
#26013604	2
C₄×Q₈×C₃₄₉	2
#557	349