Subgroups — #36375015

Order 22336.

110 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	C₄²×D₃₄₉	C₂	1	1	yes	maximal, characteristic
4	#19150243	C₄	1	2	yes	complemented
5	#19150242	—	2	1	no	—
6	D₃₄₉⋊C₈	—	2	1	no	—
7	C₄⋊#13063703	C₄	1	2	yes	complemented
8	C₂²⋊#13063703	—	2	1	no	—
9	C₄×#13063703	—	2	1	no	—
10	C₂×C₄×D₃₄₉	—	2	1	no	—
11	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₄₉	—	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	—	2	4	no	—
21	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₄₉	D₄	1	1	yes	characteristic
24	C₄×D₃₄₉	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	—	2	1	no	—
26	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₃₄₉	—	2	2	no	—
29	C₂×D₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₃₄₉	C₄⋊C₄	1	1	yes	characteristic
31	#13063703	—	4	2	no	—
32	C₄×C₃₄₉	—	2	2	no	—
33	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
34	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₄₉	—	2	2	no	—
36	Dic₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₄₉	—	2	1	no	—
39	C₂×C₃₄₉	#95	1	1	yes	characteristic
40	D₃₄₉	—	2	1	no	—
41	D₃₄₉	#104	1	2	yes	—
42	C₃₄₉	#339	1	1	yes	characteristic, complemented, Sylow
43	#339	—	349	1	no	maximal, Sylow
44	C₂×M₁₆	—	349	1	no	—
45	#117	—	349	1	no	—
46	C₂×C₄²	—	349	1	no	—
47	C₂²×C₄	—	349	1	no	—
48	C₂²×C₄	—	698	1	no	—
49	M₁₆	—	349	2	no	—
50	M₁₆	—	698	1	no	—
51	C₂×C₈	—	698	1	no	—
52	C₄⋊C₄	—	349	2	no	—
53	C₂²⋊C₄	—	698	1	no	—
54	C₄²	—	349	1	no	—
55	C₄²	—	698	1	no	—
56	C₄²	—	698	1	no	—
57	C₄²	#13063703	1	1	yes	characteristic, complemented
58	C₂³	—	349	1	no	—
59	C₂×C₄	—	2	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	—	349	1	no	—
65	C₂×C₄	—	698	4	no	—
66	C₂×C₄	—	698	2	no	—
67	C₂×C₄	—	698	1	no	—
68	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
69	C₈	—	698	2	no	—
70	C₂²	—	349	1	no	—
71	C₂²	—	349	1	no	—
72	C₂²	—	698	2	no	—
73	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	349	1	no	—
76	C₄	—	349	1	no	—
77	C₄	—	1396	2	no	—
78	C₄	—	698	2	no	—
79	C₄	C₄×#13063703	1	1	yes	characteristic
80	C₄	C₄⋊#13063703	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	349	2	no	—
83	C₂	—	698	1	no	—
84	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	#339
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. #19150243	C₄	semidirect
7. C₄⋊#13063703	C₄	semidirect
13. C₄×Dic₃₄₉	C₄	semidirect
14. C₄²×C₃₄₉	C₄	semidirect
42. C₃₄₉	#339	semidirect
57. 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
C₄²×D₃₄₉	2
#339	349