Subgroups — C₂×#26013556 #36374121

Order 22336.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150217	C₂	1	2	yes	maximal, complemented
2	#26013556	C₂	1	4	yes	maximal, direct product factor
3	C₂²×D₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	#19150217	—	2	2	no	—
5	#19150217	C₂²	1	4	yes	complemented
6	C₂×#14584095	—	2	2	no	—
7	C₂⁴×C₃₄₉	C₄	1	2	yes	—
8	C₂×D₄×C₃₄₉	—	2	2	no	—
9	C₂×D₄×C₃₄₉	—	2	2	no	—
10	C₂×D₄×C₃₄₉	C₄	1	2	yes	—
11	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
13	#14584095	—	2	4	no	—
14	C₂³×C₃₄₉	—	2	2	no	—
15	C₂³×C₃₄₉	—	2	2	no	—
16	C₂³×C₃₄₉	—	4	4	no	—
17	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₃₄₉	C₂×C₄	1	4	yes	—
19	D₄×C₃₄₉	—	2	4	no	—
20	D₄×C₃₄₉	—	2	4	no	—
21	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
23	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
24	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
25	C₂²×C₃₄₉	—	2	2	no	—
26	C₂²×C₃₄₉	—	2	4	no	—
27	C₂²×C₃₄₉	—	4	4	no	—
28	C₂²×C₃₄₉	—	4	4	no	—
29	C₂²×C₃₄₉	—	4	4	no	—
30	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₃₄₉	C₂²⋊C₄	1	2	yes	—
34	C₄×C₃₄₉	C₂×D₄	1	2	yes	—
35	C₂×C₃₄₉	—	2	1	no	—
36	C₂×C₃₄₉	—	2	1	no	—
37	C₂×C₃₄₉	—	4	4	no	—
38	C₂×C₃₄₉	#100	1	2	yes	complemented
39	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₃₄₉	C₂×#100	1	1	yes	characteristic, complemented, Sylow
41	C₂×#100	—	349	1	no	maximal, Sylow
42	C₂²×D₄	D₃₄₉	1	1	yes	characteristic
43	C₂×M₁₆	—	349	2	no	—
44	#100	—	349	4	no	—
45	C₂⁴	Dic₃₄₉	1	2	yes	—
46	C₂×D₄	—	2	2	no	—
47	C₂×D₄	—	2	2	no	—
48	C₂×D₄	Dic₃₄₉	1	2	yes	—
49	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
50	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
51	M₁₆	—	349	4	no	—
52	M₁₆	—	698	2	no	—
53	C₂×C₈	—	698	2	no	—
54	C₂³	—	2	2	no	—
55	C₂³	—	2	2	no	—
56	C₂³	—	4	4	no	—
57	C₂³	C₂×Dic₃₄₉	1	1	yes	characteristic
58	C₂³	C₂×Dic₃₄₉	1	4	yes	—
59	D₄	—	2	4	no	—
60	D₄	—	2	4	no	—
61	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
63	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
64	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
65	C₈	—	698	4	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	4	no	—
68	C₂²	—	4	4	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	4	4	no	—
71	C₂²	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
72	C₂²	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
73	C₂²	C₂²×Dic₃₄₉	1	1	yes	characteristic
74	C₄	C₂²⋊Dic₃₄₉	1	2	yes	—
75	C₄	C₂×C₂²⋊D₃₄₉	1	2	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	4	4	no	—
79	C₂	#26013556	1	2	yes	direct product factor
80	C₂	C₂×C₂²⋊Dic₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₃₄₉
commutator	C₂²×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting, prime radical	C₂²×D₄×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂²⋊C₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₂²×D₄	D₃₄₉
hypocenter	C₃₄₉	C₂×#100
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19150217	C₂	semidirect
2. #26013556	C₂	direct (N × H)
5. #19150217	C₂²	semidirect
38. C₂×C₃₄₉	#100	semidirect
40. C₃₄₉	C₂×#100	semidirect
79. C₂	#26013556	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19150217	2
#26013556	2
C₂²×D₄×C₃₄₉	2
C₂×#100	349

Subgroups — C2×#26013556 #36374121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013556 #36374121