Subgroups — C₂×#26013694 #36375102

Order 22336.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14584102	C₂	1	2	yes	maximal
2	#26013694	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#14584102	C₄	1	8	yes	complemented
5	C₂×#14584102	C₂²	1	4	yes	—
6	C₂²×Dic₃₄₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₃₄₉	C₄	1	2	yes	—
9	C₄×Dic₃₄₉	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₄₉	C₄	1	1	yes	characteristic
11	#14584102	C₂×C₄	1	8	yes	complemented
12	C₂×Dic₃₄₉	—	2	2	no	—
13	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
14	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₃₄₉	C₂×C₄	1	4	yes	—
16	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
17	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₃₄₉	—	2	2	no	—
19	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
20	C₂²×C₃₄₉	C₄²	1	2	yes	—
21	C₂²×C₃₄₉	M₁₆	1	2	yes	—
22	C₂²×C₃₄₉	M₁₆	1	2	yes	—
23	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₄₉	—	2	2	no	—
25	Dic₃₄₉	—	2	2	no	—
26	Dic₃₄₉	C₄²	1	2	yes	—
27	Dic₃₄₉	C₂²×C₄	1	2	yes	—
28	C₂×C₃₄₉	#97	1	4	yes	complemented
29	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
30	C₂×C₃₄₉	C₂×M₁₆	1	1	yes	characteristic
31	C₂×C₃₄₉	C₂×M₁₆	1	1	yes	characteristic
32	C₃₄₉	C₂×#97	1	1	yes	characteristic, complemented, Sylow
33	C₂×#97	—	349	1	no	maximal, Sylow
34	C₂²×C₈	—	349	2	no	—
35	C₂×C₄²	—	349	1	no	—
36	#97	—	349	4	no	—
37	C₂²×C₄	—	349	1	no	—
38	C₂²×C₄	—	349	1	no	—
39	C₂²×C₄	#13063703	1	1	yes	characteristic
40	C₂×C₈	—	349	4	no	—
41	C₂×C₈	—	349	8	no	—
42	C₄²	—	349	2	no	—
43	C₄²	—	349	2	no	—
44	C₂³	C₂×#13063703	1	1	yes	characteristic
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	349	4	no	—
47	C₂×C₄	—	349	1	no	—
48	C₂×C₄	—	349	1	no	—
49	C₂×C₄	—	349	2	no	—
50	C₂×C₄	—	698	2	no	—
51	C₂×C₄	C₂×#13063703	1	2	yes	—
52	C₈	—	349	8	no	—
53	C₂²	C₄×#13063703	1	2	yes	—
54	C₂²	#19150242	1	2	yes	—
55	C₂²	#19150243	1	2	yes	—
56	C₂²	C₂²×#13063703	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	349	2	no	—
59	C₄	—	349	2	no	—
60	C₄	—	698	2	no	—
61	C₂	#26013694	1	4	yes	direct product factor
62	C₂	C₂×C₄×#13063703	1	1	yes	characteristic
63	C₂	C₂×#19150242	1	1	yes	characteristic
64	C₂	C₂×#19150243	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13063703
commutator	C₂×C₃₄₉	C₂×C₄²
Frattini	C₂²	C₂²×#13063703
Fitting	C₂²×C₄×C₃₄₉	C₄
socle, prime radical	C₂³×C₃₄₉	C₂×C₄
cosocle	C₂×Dic₃₄₉	C₂³
hypercenter	C₂²×C₄	#13063703
hypocenter	C₃₄₉	C₂×#97
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #26013694	C₂	direct (N × H)
4. C₂×#14584102	C₄	semidirect
11. #14584102	C₂×C₄	semidirect
28. C₂×C₃₄₉	#97	semidirect
32. C₃₄₉	C₂×#97	semidirect
61. C₂	#26013694	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×#14584102	2
#26013694	2
C₂×C₄×Dic₃₄₉	2
C₂×#97	349

Subgroups — C2×#26013694 #36375102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013694 #36375102