Subgroups — #36374494

Order 22336.

98 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	#26013563	C₂	1	4	yes	maximal
3	C₄×#14584095	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₉₈	—	2	1	no	—
6	C₂×#14584095	C₂²	1	2	yes	—
7	C₄⋊Dic₃₄₉	—	2	1	no	—
8	C₄⋊Dic₃₄₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₃₄₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₃₄₉	—	2	1	no	—
11	C₄⋊C₄×C₃₄₉	—	2	1	no	—
12	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
13	Dic₆₉₈	—	4	2	no	—
14	C₂×Dic₃₄₉	—	2	2	no	—
15	#14584095	—	2	2	no	—
16	Q₈×C₃₄₉	—	2	4	no	—
17	C₂×C₄×C₃₄₉	—	2	2	no	—
18	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
21	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₃₄₉	—	4	2	no	—
23	C₄×C₃₄₉	SD₁₆	1	2	yes	—
24	C₄×C₃₄₉	Q₁₆	1	2	yes	—
25	C₄×C₃₄₉	#41	1	2	yes	—
26	Dic₃₄₉	—	4	2	no	—
27	C₂×C₃₄₉	#124	1	1	yes	characteristic
28	C₂×C₃₄₉	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₃₄₉	C₂×Q₁₆	1	1	yes	characteristic
30	C₃₄₉	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	349	1	no	maximal, Sylow
32	C₄⋊Q₈	—	349	1	no	—
33	C₄⋊Q₈	D₃₄₉	1	1	yes	characteristic
34	#103	—	349	4	no	—
35	C₄×C₈	—	349	1	no	—
36	C₂×Q₈	—	698	1	no	—
37	C₂×Q₈	C₂×D₃₄₉	1	2	yes	—
38	C₂×C₈	—	349	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	349	2	no	—
42	C₄⋊C₄	—	698	1	no	—
43	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	1396	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	698	2	no	—
48	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
51	C₈	—	698	2	no	—
52	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	1396	2	no	—
55	C₄	#19150220	1	2	yes	—
56	C₄	#19150223	1	2	yes	—
57	C₄	#19150232	1	2	yes	—
58	C₂	#26013626	1	1	yes	characteristic
59	C₂	C₂×#19150220	1	1	yes	characteristic
60	C₂	C₂×#19150223	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₄₉
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₄⋊Q₈×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₃₄₉
hypocenter	C₃₄₉	#487
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
30. C₃₄₉	#487	semidirect

Maximal subgroups

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

subgroup	index
#26013639	2
#26013563	2
C₄×#14584095	2
C₄⋊Q₈×C₃₄₉	2
#487	349