Subgroups — #36375011

Order 22336.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₄₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#26013718	C₂	1	1	yes	maximal, characteristic
3	#26013596	C₂	1	1	yes	maximal, characteristic
4	#19150243	—	2	1	no	—
5	#19150242	—	2	1	no	—
6	D₃₄₉⋊C₈	C₄	1	2	yes	complemented
7	C₄⋊#13063703	—	2	1	no	—
8	C₂²⋊#13063703	—	2	1	no	—
9	C₄×#13063703	C₄	1	2	yes	complemented
10	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
11	#19150212	—	2	1	no	—
12	#19150210	—	2	1	no	—
13	C₄×Dic₃₄₉	C₄	1	1	yes	characteristic, complemented
14	C₄²×C₃₄₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#13063703	—	2	2	no	—
16	#14584102	—	2	2	no	—
17	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
18	C₂×Dic₃₄₉	—	2	1	no	—
19	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
20	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
21	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₄₉	D₄	1	1	yes	characteristic
23	C₄×D₃₄₉	Q₈	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	—	2	1	no	—
25	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₃₄₉	—	4	1	no	—
28	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₃₄₉	C₄⋊C₄	1	1	yes	characteristic
30	#13063703	—	4	2	no	—
31	C₄×C₃₄₉	—	4	1	no	—
32	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
33	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
34	Dic₃₄₉	—	4	1	no	—
35	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₃₄₉	—	2	1	no	—
38	C₂×C₃₄₉	#95	1	1	yes	characteristic
39	D₃₄₉	—	2	1	no	—
40	D₃₄₉	—	2	1	no	—
41	C₃₄₉	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
42	H(3,ℤ/4)	—	349	1	no	maximal, Sylow
43	C₂×M₁₆	—	349	1	no	—
44	#117	—	349	1	no	—
45	#117	—	349	1	no	—
46	C₂²×C₄	—	349	1	no	—
47	M₁₆	—	698	1	no	—
48	M₁₆	—	698	1	no	—
49	C₂×C₈	—	349	2	no	—
50	C₄⋊C₄	—	698	1	no	—
51	C₄⋊C₄	—	698	1	no	—
52	C₂²⋊C₄	—	698	1	no	—
53	C₂²⋊C₄	—	698	1	no	—
54	C₄²	—	349	2	no	—
55	C₄²	—	349	1	no	—
56	C₄²	#13063703	1	1	yes	characteristic, complemented
57	C₂³	—	349	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	349	1	no	—
60	C₂×C₄	—	349	1	no	—
61	C₂×C₄	—	349	1	no	—
62	C₂×C₄	—	349	1	no	—
63	C₂×C₄	—	349	1	no	—
64	C₂×C₄	—	698	2	no	—
65	C₂×C₄	—	698	1	no	—
66	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
67	C₈	—	698	2	no	—
68	C₂²	—	349	1	no	—
69	C₂²	—	349	1	no	—
70	C₂²	—	1396	1	no	—
71	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	349	1	no	—
74	C₄	—	349	1	no	—
75	C₄	—	1396	1	no	—
76	C₄	—	1396	2	no	—
77	C₄	C₄×#13063703	1	1	yes	characteristic
78	C₄	C₄⋊#13063703	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	698	1	no	—
81	C₂	—	698	1	no	—
82	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#13063703
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂×C₄	C₂×#13063703
Fitting	C₄²×C₃₄₉	C₄
socle	C₂×C₃₄₉	#95
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×D₃₄₉	C₂²
hypercenter	C₄²	#13063703
hypocenter	C₃₄₉	H(3,ℤ/4)
LCS	G ▷ C₄×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. D₃₄₉⋊C₈	C₄	semidirect
9. C₄×#13063703	C₄	semidirect
13. C₄×Dic₃₄₉	C₄	semidirect
14. C₄²×C₃₄₉	C₄	semidirect
41. C₃₄₉	H(3,ℤ/4)	semidirect
56. C₄²	#13063703	semidirect

Maximal subgroups

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

subgroup	index
D₃₄₉⋊M₁₆	2
#26013718	2
#26013596	2
H(3,ℤ/4)	349