Subgroups — #36374506

Order 22336.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013627	C₂	1	1	yes	maximal, characteristic
2	#26013562	C₂	1	2	yes	maximal, complemented
3	#26013559	C₂	1	2	yes	maximal
4	#26013550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
7	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic
8	C₂×#14584095	C₂²	1	2	yes	—
9	#19150210	—	2	1	no	—
10	#19150210	—	2	1	no	—
11	C₂×Q₈×C₃₄₉	—	2	1	no	—
12	C₄⋊C₄×C₃₄₉	—	2	1	no	—
13	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
14	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₂²×D₃₄₉	—	2	1	no	—
16	Dic₆₉₈	—	2	2	no	—
17	D₁₃₉₆	—	2	2	no	—
18	C₂×Dic₃₄₉	—	2	1	no	—
19	#14584095	—	2	2	no	—
20	Q₈×C₃₄₉	—	4	2	no	—
21	C₂×C₄×C₃₄₉	—	2	2	no	—
22	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
25	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₃₄₉	—	4	1	no	—
27	C₂×D₃₄₉	—	4	1	no	—
28	C₂×D₃₄₉	—	4	1	no	—
29	C₄×C₃₄₉	—	2	1	no	—
30	C₄×C₃₄₉	—	2	1	no	—
31	C₄×C₃₄₉	—	4	2	no	—
32	C₄×C₃₄₉	#41	1	2	yes	—
33	Dic₃₄₉	—	4	1	no	—
34	C₂×C₃₄₉	#124	1	1	yes	characteristic
35	C₂×C₃₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₃₄₉	#137	1	1	yes	characteristic
37	D₃₄₉	—	8	1	no	—
38	C₃₄₉	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	349	1	no	maximal, Sylow
40	C₄⋊Q₈	D₃₄₉	1	1	yes	characteristic, complemented
41	#124	—	349	1	no	—
42	#103	—	349	2	no	—
43	#102	—	349	2	no	—
44	#97	—	349	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	349	1	no	—
47	C₂×D₄	—	349	1	no	—
48	C₂×C₈	—	349	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
51	C₂²⋊C₄	—	698	1	no	—
52	C₂²⋊C₄	—	698	1	no	—
53	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
54	C₂³	—	698	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	698	2	no	—
57	D₄	—	698	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	698	1	no	—
60	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
63	C₈	—	698	2	no	—
64	C₂²	—	1396	1	no	—
65	C₂²	—	1396	1	no	—
66	C₂²	—	1396	1	no	—
67	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	1396	1	no	—
72	C₄	#19150234	1	2	yes	—
73	C₂	—	2792	1	no	—
74	C₂	#26013628	1	1	yes	characteristic
75	C₂	#26013665	1	1	yes	characteristic
76	C₂	#26013670	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	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂×D₁₃₉₆	C₂²
hypercenter	C₄⋊Q₈	D₃₄₉
hypocenter	C₃₄₉	#489
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
2. #26013562	C₂	semidirect
4. #26013550	C₂	semidirect
5. C₄⋊Q₈×C₃₄₉	C₂	semidirect
38. C₃₄₉	#489	semidirect
40. C₄⋊Q₈	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013627	2
#26013562	2
#26013559	2
#26013550	2
C₄⋊Q₈×C₃₄₉	2
#489	349