Subgroups — #36374110

Order 22336.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150233	C₂	1	1	yes	maximal, characteristic
2	C₂×#19150217	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013551	C₂	1	4	yes	maximal, complemented
4	C₂²×C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#19150233	—	2	2	no	—
6	#19150233	—	2	2	no	—
7	C₂×Dic₆₉₈	C₄	1	1	yes	characteristic
8	C₂×D₁₃₉₆	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊D₃₄₉	C₄	1	2	yes	—
10	C₂×C₄×D₃₄₉	C₂²	1	2	yes	—
11	#19150217	—	2	2	no	—
12	C₂×#14584095	C₂²	1	2	yes	—
13	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₄₉	—	2	2	no	—
15	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
16	C₂²×D₃₄₉	C₂×C₄	1	2	yes	—
17	Dic₆₉₈	—	2	2	no	—
18	D₁₃₉₆	—	2	2	no	—
19	C₂²⋊D₃₄₉	—	2	4	no	—
20	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
21	C₄×D₃₄₉	—	2	2	no	—
22	C₄×D₃₄₉	—	2	2	no	—
23	#14584095	—	2	2	no	—
24	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
27	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
28	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
29	C₈×C₃₄₉	—	2	2	no	—
30	C₂²×C₃₄₉	—	2	2	no	—
31	C₂²×C₃₄₉	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×D₃₄₉	—	2	2	no	—
34	C₂×D₃₄₉	—	2	4	no	—
35	C₄×C₃₄₉	C₂²⋊C₄	1	2	yes	—
36	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
37	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
38	Dic₃₄₉	—	2	2	no	—
39	C₂×C₃₄₉	—	2	2	no	—
40	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₃₄₉	#131	1	2	yes	—
42	D₃₄₉	—	4	2	no	—
43	C₃₄₉	#408	1	1	yes	characteristic, complemented, Sylow
44	#408	—	349	1	no	maximal, Sylow
45	C₂×#41	—	349	1	no	—
46	C₂×M₁₆	—	349	1	no	—
47	C₂²×C₈	D₃₄₉	1	1	yes	characteristic, complemented
48	C₂²⋊C₈	—	349	4	no	—
49	#41	—	698	2	no	—
50	#41	—	698	2	no	—
51	C₂×Q₈	—	349	1	no	—
52	C₂×D₄	—	349	2	no	—
53	C₂×D₄	—	349	1	no	—
54	C₂²×C₄	—	349	2	no	—
55	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
56	M₁₆	—	698	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	349	2	no	—
59	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
60	C₂³	—	349	2	no	—
61	C₂³	C₄×D₃₄₉	1	1	yes	characteristic
62	Q₈	—	698	2	no	—
63	D₄	—	698	4	no	—
64	D₄	—	698	2	no	—
65	C₂×C₄	—	349	2	no	—
66	C₂×C₄	—	698	2	no	—
67	C₂×C₄	—	698	2	no	—
68	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
70	C₂×C₄	D₁₃₉₆	1	2	yes	—
71	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	698	2	no	—
74	C₂²	—	2	2	no	—
75	C₂²	—	698	2	no	—
76	C₂²	—	698	4	no	—
77	C₂²	#19150210	1	2	yes	—
78	C₂²	C₂×C₄×D₃₄₉	1	1	yes	characteristic
79	C₄	—	698	2	no	—
80	C₄	#19150210	1	2	yes	—
81	C₄	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
82	C₄	C₂×D₁₃₉₆	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	1396	2	no	—
85	C₂	C₂×#19150210	1	1	yes	characteristic
86	C₂	#26013645	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₄₉
commutator, socle	C₂²×C₃₄₉	C₂²×C₄
Fitting	C₂²×C₈×C₃₄₉	C₂
prime radical	C₂×#19150233	C₂
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₂²×C₈	D₃₄₉
hypocenter	C₃₄₉	#408
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19150217	C₂	semidirect
3. #26013551	C₂	semidirect
4. C₂²×C₈×C₃₄₉	C₂	semidirect
43. C₃₄₉	#408	semidirect
47. C₂²×C₈	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150233	2
C₂×#19150217	2
#26013551	2
C₂²×C₈×C₃₄₉	2
#408	349