Subgroups — #36375107

Order 22336.

130 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	#26013693	C₂	1	2	yes	maximal, complemented
4	C₈×#13063703	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
6	#19150243	C₄	1	2	yes	complemented
7	#19150242	C₄	1	2	yes	complemented
8	D₃₄₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊#13063703	C₄	1	2	yes	—
10	C₂²⋊#13063703	C₄	1	2	yes	—
11	C₄×#13063703	C₂²	1	2	yes	—
12	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
13	C₂×#14584095	C₄	1	1	yes	characteristic, complemented
14	C₈×D₃₄₉	C₄	1	1	yes	characteristic, complemented
15	C₈×D₃₄₉	C₄	1	1	yes	characteristic, complemented
16	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
17	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic, complemented
19	C₂×#13063703	C₂×C₄	1	4	yes	—
20	#14584102	C₂×C₄	1	4	yes	—
21	C₂²×D₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₄×D₃₄₉	C₂³	1	1	yes	characteristic
27	#14584095	C₂×C₄	1	1	yes	characteristic
28	#14584095	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
30	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
31	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃₄₉	C₄²	1	1	yes	characteristic
33	C₂×D₃₄₉	—	2	2	no	—
34	C₂×D₃₄₉	C₄²	1	1	yes	characteristic
35	C₂×D₃₄₉	C₂²×C₄	1	1	yes	characteristic
36	#13063703	—	2	4	no	—
37	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
38	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
39	Dic₃₄₉	C₄²	1	1	yes	characteristic
40	Dic₃₄₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₃₄₉	—	2	1	no	—
42	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
43	D₃₄₉	—	2	1	no	—
44	D₃₄₉	#131	1	2	yes	—
45	C₃₄₉	#405	1	1	yes	characteristic, complemented, Sylow
46	#405	—	349	1	no	maximal, Sylow
47	C₂×M₁₆	—	349	1	no	—
48	C₂²×C₈	—	349	1	no	—
49	#117	—	349	1	no	—
50	#97	—	349	2	no	—
51	C₄×C₈	—	349	2	no	—
52	C₂²×C₄	—	349	1	no	—
53	M₁₆	—	349	2	no	—
54	M₁₆	—	349	2	no	—
55	C₂×C₈	—	349	2	no	—
56	C₂×C₈	—	349	1	no	—
57	C₂×C₈	—	349	1	no	—
58	C₂×C₈	—	349	1	no	—
59	C₂×C₈	—	349	1	no	—
60	C₂×C₈	—	349	1	no	—
61	C₂×C₈	#13063703	1	1	yes	characteristic, complemented
62	C₄⋊C₄	—	349	2	no	—
63	C₂²⋊C₄	—	349	2	no	—
64	C₄²	—	349	2	no	—
65	C₂³	—	349	1	no	—
66	C₂×C₄	—	349	1	no	—
67	C₂×C₄	—	349	1	no	—
68	C₂×C₄	—	349	1	no	—
69	C₂×C₄	—	349	1	no	—
70	C₂×C₄	—	349	1	no	—
71	C₂×C₄	—	349	4	no	—
72	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
73	C₈	—	349	4	no	—
74	C₈	—	349	1	no	—
75	C₈	—	349	1	no	—
76	C₈	C₂×#13063703	1	1	yes	characteristic
77	C₈	C₂×#13063703	1	1	yes	characteristic
78	C₂²	—	349	1	no	—
79	C₂²	—	349	1	no	—
80	C₂²	—	698	2	no	—
81	C₂²	C₄×#13063703	1	1	yes	characteristic
82	C₄	—	349	1	no	—
83	C₄	—	349	1	no	—
84	C₄	—	698	4	no	—
85	C₄	C₄×#13063703	1	1	yes	characteristic
86	C₄	C₂²×#13063703	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	349	2	no	—
89	C₂	—	698	1	no	—
90	C₂	C₂×C₄×#13063703	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×#13063703
commutator, socle	C₂×C₃₄₉	C₂×C₄²
Frattini	C₄	C₂²×#13063703
Fitting	C₂×C₈×C₃₄₉	C₄
prime radical	C₂²×D₃₄₉	C₂×C₄
cosocle	C₄×D₃₄₉	C₂³
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	#405
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. #26013693	C₂	semidirect
4. C₈×#13063703	C₂	semidirect
6. #19150243	C₄	semidirect
7. #19150242	C₄	semidirect
13. C₂×#14584095	C₄	semidirect
14. C₈×D₃₄₉	C₄	semidirect
15. C₈×D₃₄₉	C₄	semidirect
18. C₂×C₈×C₃₄₉	C₄	semidirect
45. C₃₄₉	#405	semidirect
61. C₂×C₈	#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
#26013693	2
C₈×#13063703	2
C₂×C₈×D₃₄₉	2
#405	349