Subgroups — #36374988

Order 22336.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14584102	C₂	1	2	yes	maximal, complemented
2	C₂×#19150210	C₂	1	1	yes	maximal, characteristic
3	C₂×#14584102	—	2	2	no	—
4	C₂³×D₃₄₉	C₄	1	1	yes	characteristic
5	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
6	#19150210	—	2	2	no	—
7	C₂²×C₄×C₃₄₉	C₄	1	1	yes	characteristic
8	#14584102	—	4	2	no	—
9	C₂²×D₃₄₉	—	2	1	no	—
10	C₂²×D₃₄₉	—	2	1	no	—
11	C₂²×D₃₄₉	—	4	1	no	—
12	C₂²×D₃₄₉	—	4	1	no	—
13	C₂²×D₃₄₉	C₈	1	2	yes	complemented
14	C₂×Dic₃₄₉	—	4	1	no	—
15	C₂×Dic₃₄₉	D₄	1	2	yes	—
16	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₄₉	—	2	1	no	—
18	C₂×C₄×C₃₄₉	—	4	1	no	—
19	C₂²×C₃₄₉	—	2	1	no	—
20	C₂²×C₃₄₉	—	2	1	no	—
21	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
23	C₂²×C₃₄₉	M₁₆	1	1	yes	characteristic
24	C₂×D₃₄₉	—	2	2	no	—
25	C₂×D₃₄₉	—	4	2	no	—
26	C₂×D₃₄₉	—	4	1	no	—
27	C₂×D₃₄₉	—	4	1	no	—
28	C₂×D₃₄₉	—	4	2	no	—
29	C₄×C₃₄₉	—	4	1	no	—
30	Dic₃₄₉	—	2	2	no	—
31	C₂×C₃₄₉	—	2	1	no	—
32	C₂×C₃₄₉	—	2	1	no	—
33	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₃₄₉	#99	1	1	yes	characteristic
35	C₂×C₃₄₉	#100	1	1	yes	characteristic
36	D₃₄₉	—	4	2	no	—
37	C₃₄₉	#323	1	1	yes	characteristic, complemented, Sylow
38	#323	—	349	1	no	maximal, Sylow
39	C₂×C₂²⋊C₄	—	349	1	no	—
40	C₂²⋊C₈	—	349	2	no	—
41	C₂⁴	—	349	1	no	—
42	C₂²×C₄	—	349	1	no	—
43	C₂²×C₄	#13063703	1	1	yes	characteristic
44	C₂×C₈	—	698	2	no	—
45	C₂²⋊C₄	—	698	2	no	—
46	C₂³	—	349	2	no	—
47	C₂³	—	1396	1	no	—
48	C₂³	—	1396	1	no	—
49	C₂³	—	698	1	no	—
50	C₂³	—	698	1	no	—
51	C₂³	C₂×#13063703	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	349	2	no	—
55	C₂×C₄	—	1396	1	no	—
56	C₈	—	1396	2	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	1396	2	no	—
60	C₂²	—	1396	2	no	—
61	C₂²	—	1396	1	no	—
62	C₂²	—	1396	1	no	—
63	C₂²	—	698	2	no	—
64	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
65	C₂²	D₃₄₉⋊C₈	1	1	yes	characteristic
66	C₂²	#19150243	1	1	yes	characteristic
67	C₄	—	4	1	no	—
68	C₄	—	698	2	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	1396	2	no	—
72	C₂	#26013696	1	1	yes	characteristic
73	C₂	#26013698	1	1	yes	characteristic
74	C₂	#26013699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13063703
commutator	C₂²×C₃₄₉	C₂×C₈
Frattini	C₂³	C₂×#13063703
Fitting	C₂²×C₄×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂²⋊C₄
prime radical	C₂³×D₃₄₉	C₄
cosocle	C₂²×Dic₃₄₉	C₂²
hypercenter	C₂²×C₄	#13063703
hypocenter	C₃₄₉	#323
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#14584102	C₂	semidirect
13. C₂²×D₃₄₉	C₈	semidirect
37. C₃₄₉	#323	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#14584102	2
C₂×#19150210	2
#323	349