Subgroups — #36374411

Order 22336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013665	C₂	1	2	yes	maximal, complemented
2	#26013645	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013575	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013556	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150233	C₂²	1	1	yes	characteristic
7	#19150221	—	2	2	no	—
8	#19150218	—	2	2	no	—
9	#19150217	C₂²	1	1	yes	characteristic
10	#19150217	C₂²	1	2	yes	—
11	#19150216	—	2	1	no	—
12	C₈×D₃₄₉	—	2	1	no	—
13	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
14	D₈×C₃₄₉	—	2	1	no	—
15	D₈×C₃₄₉	—	2	1	no	—
16	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
17	Dic₆₉₈	D₄	1	1	yes	characteristic
18	D₁₃₉₆	D₄	1	1	yes	characteristic
19	C₂²⋊D₃₄₉	—	2	1	no	—
20	C₄×D₃₄₉	—	2	1	no	—
21	#14584095	—	2	1	no	—
22	#14584095	—	2	2	no	—
23	C₂³×C₃₄₉	—	2	2	no	—
24	D₄×C₃₄₉	—	2	2	no	—
25	D₄×C₃₄₉	—	2	2	no	—
26	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
27	C₈×C₃₄₉	D₄	1	1	yes	characteristic
28	C₈×C₃₄₉	D₄	1	1	yes	characteristic
29	C₂²×C₃₄₉	—	4	2	no	—
30	C₂²×C₃₄₉	—	8	2	no	—
31	C₂²×C₃₄₉	#41	1	1	yes	characteristic
32	C₂×D₃₄₉	—	2	1	no	—
33	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
35	Dic₃₄₉	—	2	1	no	—
36	C₂×C₃₄₉	—	2	1	no	—
37	C₂×C₃₄₉	—	8	2	no	—
38	C₂×C₃₄₉	#121	1	1	yes	characteristic
39	D₃₄₉	—	4	1	no	—
40	C₃₄₉	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	349	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	349	2	no	—
43	C₂×D₈	D₃₄₉	1	1	yes	characteristic, complemented
44	#131	—	349	1	no	—
45	#108	—	349	1	no	—
46	#100	—	349	2	no	—
47	#41	—	349	1	no	—
48	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
49	SD₁₆	—	698	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	698	2	no	—
53	M₁₆	—	349	1	no	—
54	M₁₆	—	349	2	no	—
55	M₁₆	—	698	1	no	—
56	C₂×C₈	—	698	1	no	—
57	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	349	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	349	1	no	—
63	D₄	—	698	1	no	—
64	C₂×C₄	—	698	1	no	—
65	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
66	C₈	—	698	2	no	—
67	C₈	—	698	1	no	—
68	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
69	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	698	1	no	—
73	C₂²	#19150232	1	1	yes	characteristic
74	C₄	—	698	1	no	—
75	C₄	D₄×D₃₄₉	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	1396	1	no	—
80	C₂	#26013610	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013610
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₂×D₈×C₃₄₉	C₂
socle	C₂×C₃₄₉	#121
hypercenter	C₂×D₈	D₃₄₉
hypocenter	C₃₄₉	#472
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #26013665	C₂	semidirect
2. #26013645	C₂	semidirect
3. #26013575	C₂	semidirect
4. #26013556	C₂	semidirect
5. C₂×D₈×C₃₄₉	C₂	semidirect
40. C₃₄₉	#472	semidirect
43. C₂×D₈	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013665	2
#26013645	2
#26013575	2
#26013556	2
C₂×D₈×C₃₄₉	2
#472	349