Subgroups — C₂×#26013604 #36374608

Order 22336.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic
2	#26013604	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19150212	C₂	1	2	yes	maximal
4	C₂×C₄×Dic₃₄₉	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₆₉₈	C₄	1	8	yes	complemented
8	C₂×Dic₆₉₈	C₂²	1	4	yes	—
9	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
11	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
12	#19150212	C₂²	1	8	yes	—
13	C₄×Dic₃₄₉	C₂²	1	8	yes	—
14	C₄×Dic₃₄₉	C₂²	1	4	yes	—
15	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₄₉	C₂²	1	4	yes	—
18	Dic₆₉₈	C₂×C₄	1	16	yes	complemented
19	C₂×Dic₃₄₉	—	2	2	no	—
20	C₂×Dic₃₄₉	C₂×C₄	1	8	yes	—
21	C₂×Dic₃₄₉	Q₈	1	4	yes	complemented
22	C₂×Dic₃₄₉	C₂³	1	2	yes	—
23	C₂×Dic₃₄₉	C₂³	1	4	yes	—
24	C₂×Dic₃₄₉	C₂³	1	2	yes	—
25	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₃₄₉	—	2	4	no	—
27	C₂×C₄×C₃₄₉	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
29	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
30	C₂²×C₃₄₉	C₂²×C₄	1	2	yes	—
31	C₂²×C₃₄₉	C₂×Q₈	1	2	yes	—
32	C₂²×C₃₄₉	#41	1	2	yes	—
33	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
34	C₄×C₃₄₉	—	2	4	no	—
35	C₄×C₃₄₉	C₂²×C₄	1	4	yes	—
36	Dic₃₄₉	—	2	2	no	—
37	Dic₃₄₉	C₂²×C₄	1	8	yes	—
38	Dic₃₄₉	C₂×Q₈	1	4	yes	complemented
39	C₂×C₃₄₉	C₄×Q₈	1	4	yes	complemented
40	C₂×C₃₄₉	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₃₄₉	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₃₄₉	C₂×#41	1	1	yes	characteristic
43	C₃₄₉	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄×Q₈	—	349	1	no	maximal, Sylow
45	C₂²×Q₈	—	349	1	no	—
46	C₄×Q₈	—	349	8	no	—
47	C₂×C₄⋊C₄	—	349	2	no	—
48	C₂×C₄⋊C₄	D₃₄₉	1	1	yes	characteristic
49	C₂×C₄²	—	349	2	no	—
50	C₂×C₄²	—	349	1	no	—
51	C₂×Q₈	—	349	8	no	—
52	C₂×Q₈	—	349	4	no	—
53	C₂²×C₄	—	349	1	no	—
54	C₂²×C₄	—	349	2	no	—
55	C₂²×C₄	—	349	1	no	—
56	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
57	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
58	C₄⋊C₄	—	349	8	no	—
59	C₄⋊C₄	C₂×D₃₄₉	1	4	yes	—
60	C₄²	—	349	8	no	—
61	C₄²	—	349	4	no	—
62	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
63	Q₈	—	349	16	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	349	8	no	—
66	C₂×C₄	—	349	4	no	—
67	C₂×C₄	—	349	2	no	—
68	C₂×C₄	—	349	4	no	—
69	C₂×C₄	—	349	2	no	—
70	C₂×C₄	—	698	2	no	—
71	C₂×C₄	C₄×D₃₄₉	1	4	yes	—
72	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
73	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
74	C₂²	C₂×C₄×D₃₄₉	1	2	yes	—
75	C₂²	Q₈×D₃₄₉	1	2	yes	—
76	C₂²	#19150232	1	2	yes	—
77	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	349	8	no	—
80	C₄	—	349	4	no	—
81	C₄	—	698	2	no	—
82	C₄	C₂×C₄×D₃₄₉	1	4	yes	—
83	C₂	#26013604	1	4	yes	direct product factor
84	C₂	C₂²×C₄×D₃₄₉	1	1	yes	characteristic
85	C₂	C₂×Q₈×D₃₄₉	1	1	yes	characteristic
86	C₂	C₂×#19150232	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator	C₂×C₃₄₉	C₂³×C₄
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₄⋊C₄×C₃₄₉	C₂
socle, prime radical	C₂³×C₃₄₉	C₂³
cosocle	C₂²×C₃₄₉	C₂⁴
hypercenter	C₂×C₄⋊C₄	D₃₄₉
hypocenter	C₃₄₉	C₂×C₄×Q₈
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #26013604	C₂	direct (N × H)
7. C₂×Dic₆₉₈	C₄	semidirect
18. Dic₆₉₈	C₂×C₄	semidirect
21. C₂×Dic₃₄₉	Q₈	semidirect
38. Dic₃₄₉	C₂×Q₈	semidirect
39. C₂×C₃₄₉	C₄×Q₈	semidirect
43. C₃₄₉	C₂×C₄×Q₈	semidirect
83. C₂	#26013604	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×Dic₆₉₈	2
#26013604	2
C₂×#19150212	2
C₂×C₄×Dic₃₄₉	2
C₂×C₄⋊C₄×C₃₄₉	2
C₂×C₄×Q₈	349

Subgroups — C2×#26013604 #36374608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013604 #36374608