Subgroups — #36375124

Order 22336.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013718	C₂	1	2	yes	maximal, complemented
2	#26013697	C₂	1	4	yes	maximal, complemented
3	C₂×#19150232	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13063703	—	2	2	no	—
5	C₂²⋊#13063703	C₂²	1	4	yes	—
6	C₄×#13063703	—	2	2	no	—
7	#19150232	—	2	2	no	—
8	#19150232	—	2	2	no	—
9	C₂×Dic₆₉₈	C₄	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₃₄₉	C₂²	1	2	yes	—
11	C₂²×Dic₃₄₉	C₄	1	2	yes	complemented
12	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₃₄₉	C₄	1	1	yes	characteristic, complemented
14	C₂×#13063703	—	2	4	no	—
15	C₂²×D₃₄₉	C₂³	1	1	yes	characteristic
16	Dic₆₉₈	—	2	2	no	—
17	C₂²⋊D₃₄₉	—	2	4	no	—
18	C₂×Dic₃₄₉	—	2	2	no	—
19	C₂×Dic₃₄₉	—	2	2	no	—
20	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₄₉	D₄	1	2	yes	—
23	C₄×D₃₄₉	D₄	1	2	yes	—
24	C₂³×C₃₄₉	C₂×C₄	1	2	yes	—
25	D₄×C₃₄₉	—	2	2	no	—
26	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₄₉	—	2	2	no	—
28	C₂²×C₃₄₉	—	4	2	no	—
29	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₃₄₉	—	4	1	no	—
31	C₂×D₃₄₉	C₂×D₄	1	2	yes	—
32	#13063703	—	4	4	no	—
33	C₄×C₃₄₉	C₂²⋊C₄	1	2	yes	—
34	Dic₃₄₉	—	2	2	no	—
35	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃₄₉	—	2	1	no	—
38	C₂×C₃₄₉	—	4	2	no	—
39	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	D₃₄₉	—	2	2	no	—
41	C₃₄₉	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	349	1	no	maximal, Sylow
43	C₂×#41	—	349	1	no	—
44	#117	—	349	2	no	—
45	#99	—	349	4	no	—
46	#41	—	698	2	no	—
47	#41	—	698	2	no	—
48	C₂×Q₈	—	349	1	no	—
49	C₂×D₄	—	349	2	no	—
50	C₂×D₄	#13063703	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	349	1	no	—
52	C₂²×C₄	—	349	2	no	—
53	C₄⋊C₄	—	698	2	no	—
54	C₂²⋊C₄	—	349	4	no	—
55	C₄²	—	698	2	no	—
56	C₂³	—	349	1	no	—
57	C₂³	C₂×#13063703	1	2	yes	—
58	Q₈	—	698	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	698	4	no	—
61	C₂×C₄	—	349	1	no	—
62	C₂×C₄	—	349	2	no	—
63	C₂×C₄	—	349	2	no	—
64	C₂×C₄	—	349	2	no	—
65	C₂×C₄	—	698	2	no	—
66	C₂×C₄	—	698	2	no	—
67	C₂×C₄	—	698	4	no	—
68	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	349	2	no	—
72	C₂²	—	1396	1	no	—
73	C₂²	C₂²×#13063703	1	1	yes	characteristic
74	C₄	—	349	1	no	—
75	C₄	—	349	1	no	—
76	C₄	—	1396	4	no	—
77	C₄	—	698	2	no	—
78	C₄	C₂²⋊#13063703	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	4	2	no	—
81	C₂	—	698	2	no	—
82	C₂	C₂×C₂²⋊#13063703	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#13063703
commutator	C₂²×C₃₄₉	C₂²×C₄
Frattini	C₂²	C₂²×#13063703
Fitting	C₂×D₄×C₃₄₉	C₄
socle	C₂×C₃₄₉	C₂×C₂²⋊C₄
prime radical	C₂×#19150232	C₂
cosocle	C₂²×D₃₄₉	C₂³
hypercenter	C₂×D₄	#13063703
hypocenter	C₃₄₉	#410
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #26013718	C₂	semidirect
2. #26013697	C₂	semidirect
9. C₂×Dic₆₉₈	C₄	semidirect
11. C₂²×Dic₃₄₉	C₄	semidirect
13. C₂×D₄×C₃₄₉	C₄	semidirect
41. C₃₄₉	#410	semidirect
50. C₂×D₄	#13063703	semidirect

Maximal subgroups

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

subgroup	index
#26013718	2
#26013697	2
C₂×#19150232	2
#410	349