Subgroups — #36375085

Order 22336.

506 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#13063703	C₂	1	6	yes	maximal, complemented
2	C₂⁴×D₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13063703	C₂²	1	4	yes	complemented
4	C₂²⋊#13063703	—	2	12	no	—
5	C₂³×D₃₄₉	—	2	12	no	—
6	C₂³×D₃₄₉	C₄	1	3	yes	complemented
7	C₂³×D₃₄₉	C₂²	1	3	yes	—
8	C₂⁴×C₃₄₉	C₄	1	1	yes	characteristic, complemented
9	C₂×#13063703	—	2	12	no	—
10	C₂²×D₃₄₉	—	2	32	no	—
11	C₂²×D₃₄₉	—	2	6	no	—
12	C₂²×D₃₄₉	—	2	12	no	—
13	C₂²×D₃₄₉	—	2	12	no	—
14	C₂²×D₃₄₉	C₂×C₄	1	3	yes	—
15	C₂²×D₃₄₉	D₄	1	12	yes	—
16	C₂²×D₃₄₉	C₂³	1	1	yes	characteristic
17	C₂³×C₃₄₉	—	2	6	no	—
18	C₂³×C₃₄₉	C₂×C₄	1	3	yes	—
19	C₂²×C₃₄₉	—	2	8	no	—
20	C₂²×C₃₄₉	—	2	6	no	—
21	C₂²×C₃₄₉	C₂²⋊C₄	1	6	yes	complemented
22	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
23	C₂×D₃₄₉	—	2	24	no	—
24	C₂×D₃₄₉	—	2	6	no	—
25	C₂×D₃₄₉	—	2	24	no	—
26	C₂×D₃₄₉	C₂²⋊C₄	1	6	yes	complemented
27	C₂×D₃₄₉	C₂×D₄	1	6	yes	—
28	#13063703	—	4	4	no	—
29	C₂×C₃₄₉	—	2	6	no	—
30	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	3	yes	—
31	D₃₄₉	—	2	6	no	—
32	D₃₄₉	#120	1	4	yes	—
33	C₃₄₉	#379	1	1	yes	characteristic, complemented, Sylow
34	#379	—	349	1	no	maximal, Sylow
35	C₂⁵	—	349	1	no	—
36	C₂×C₂²⋊C₄	—	349	6	no	—
37	C₂⁴	—	349	3	no	—
38	C₂⁴	—	349	3	no	—
39	C₂⁴	—	698	12	no	—
40	C₂⁴	#13063703	1	1	yes	characteristic, complemented
41	C₂²×C₄	—	349	4	no	—
42	C₂²⋊C₄	—	698	12	no	—
43	C₂³	—	2	6	no	—
44	C₂³	—	349	1	no	—
45	C₂³	—	349	12	no	—
46	C₂³	—	349	3	no	—
47	C₂³	—	698	12	no	—
48	C₂³	—	698	12	no	—
49	C₂³	—	698	32	no	—
50	C₂³	—	698	6	no	—
51	C₂³	C₂×#13063703	1	3	yes	—
52	C₂×C₄	—	698	12	no	—
53	C₂²	—	2	8	no	—
54	C₂²	—	2	6	no	—
55	C₂²	—	349	6	no	—
56	C₂²	—	349	6	no	—
57	C₂²	—	698	24	no	—
58	C₂²	—	698	24	no	—
59	C₂²	—	698	6	no	—
60	C₂²	C₂²⋊#13063703	1	6	yes	complemented
61	C₂²	C₂²×#13063703	1	1	yes	characteristic
62	C₄	—	1396	4	no	—
63	C₂	—	2	6	no	—
64	C₂	—	349	4	no	—
65	C₂	—	698	6	no	—
66	C₂	C₂×C₂²⋊#13063703	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13063703
commutator, socle	C₂²×C₃₄₉	C₂²×C₄
Fitting	C₂⁴×C₃₄₉	C₄
prime radical	C₂⁴×D₃₄₉	C₂
cosocle	C₂²×D₃₄₉	C₂³
hypercenter	C₂⁴	#13063703
hypocenter	C₃₄₉	#379
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊#13063703	C₂	semidirect
3. C₂²×#13063703	C₂²	semidirect
6. C₂³×D₃₄₉	C₄	semidirect
8. C₂⁴×C₃₄₉	C₄	semidirect
21. C₂²×C₃₄₉	C₂²⋊C₄	semidirect
26. C₂×D₃₄₉	C₂²⋊C₄	semidirect
33. C₃₄₉	#379	semidirect
40. C₂⁴	#13063703	semidirect
60. C₂²	C₂²⋊#13063703	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#13063703	2
C₂⁴×D₃₄₉	2
#379	349