Subgroups — C₂×#26013568 #36374164

Order 22336.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14584095	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	#26013568	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₄₉	C₄	1	1	yes	characteristic
6	C₂²×Dic₃₄₉	C₄	1	1	yes	characteristic
7	C₂×#14584095	—	2	2	no	—
8	C₂×#14584095	C₂²	1	2	yes	—
9	C₄×Dic₃₄₉	C₄	1	2	yes	—
10	C₄×Dic₃₄₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₄₉	—	2	2	no	—
13	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
14	C₂×Dic₃₄₉	—	2	2	no	—
15	C₂×Dic₃₄₉	C₈	1	4	yes	complemented
16	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
17	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
18	#14584095	—	2	2	no	—
19	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
22	C₂×C₄×C₃₄₉	Q₈	1	2	yes	—
23	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
24	C₈×C₃₄₉	—	2	2	no	—
25	C₂²×C₃₄₉	C₄⋊C₄	1	2	yes	—
26	C₂²×C₃₄₉	C₂×C₈	1	2	yes	—
27	C₂²×C₃₄₉	M₁₆	1	2	yes	—
28	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₄₉	C₄⋊C₄	1	2	yes	—
30	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
32	Dic₃₄₉	—	2	2	no	—
33	Dic₃₄₉	C₂×C₈	1	4	yes	complemented
34	C₂×C₃₄₉	C₄⋊C₈	1	4	yes	complemented
35	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₃₄₉	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₃₄₉	C₂×M₁₆	1	1	yes	characteristic
38	C₃₄₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₄⋊C₈	—	349	1	no	maximal, Sylow
40	C₂²×C₈	—	349	1	no	—
41	C₂²×C₈	D₃₄₉	1	1	yes	characteristic
42	C₂×C₄²	—	349	1	no	—
43	C₄⋊C₈	—	349	4	no	—
44	C₂²×C₄	—	349	1	no	—
45	C₂²×C₄	—	349	1	no	—
46	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	349	2	no	—
49	C₂×C₈	—	698	2	no	—
50	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
51	C₄²	—	349	2	no	—
52	C₄²	—	349	2	no	—
53	C₂³	C₄×D₃₄₉	1	1	yes	characteristic
54	C₂×C₄	—	349	4	no	—
55	C₂×C₄	—	349	2	no	—
56	C₂×C₄	—	349	2	no	—
57	C₂×C₄	—	698	2	no	—
58	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
60	C₂×C₄	Dic₆₉₈	1	2	yes	—
61	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	698	2	no	—
64	C₂²	#19150212	1	2	yes	—
65	C₂²	C₈×D₃₄₉	1	2	yes	—
66	C₂²	#19150216	1	2	yes	—
67	C₂²	C₂×C₄×D₃₄₉	1	1	yes	characteristic
68	C₄	—	349	4	no	—
69	C₄	—	698	2	no	—
70	C₄	#19150212	1	2	yes	—
71	C₄	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
72	C₄	C₂×Dic₆₉₈	1	1	yes	characteristic
73	C₂	#26013568	1	4	yes	direct product factor
74	C₂	C₂×#19150212	1	1	yes	characteristic
75	C₂	C₂×C₈×D₃₄₉	1	1	yes	characteristic
76	C₂	C₂×#19150216	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. #26013568	C₂	direct (N × H)
15. C₂×Dic₃₄₉	C₈	semidirect
33. Dic₃₄₉	C₂×C₈	semidirect
34. C₂×C₃₄₉	C₄⋊C₈	semidirect
38. C₃₄₉	C₂×C₄⋊C₈	semidirect
73. C₂	#26013568	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14584095	2
C₂×C₄×Dic₃₄₉	2
#26013568	2
C₂²×C₈×C₃₄₉	2
C₂×C₄⋊C₈	349

Subgroups — C2×#26013568 #36374164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013568 #36374164