Subgroups — #36373906

Order 22336.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14584095	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₄₉	—	2	1	no	—
5	C₂×#14584095	—	2	1	no	—
6	C₂×#14584095	—	2	1	no	—
7	C₂×#14584095	C₄	1	2	yes	complemented
8	C₄⋊Dic₃₄₉	—	2	1	no	—
9	C₄⋊Dic₃₄₉	C₄	1	2	yes	complemented
10	C₂²×C₄×C₃₄₉	—	2	1	no	—
11	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₄₉	—	2	1	no	—
13	C₄⋊C₄×C₃₄₉	C₄	1	2	yes	complemented
14	C₂×Dic₃₄₉	—	2	2	no	—
15	C₂×Dic₃₄₉	—	4	1	no	—
16	C₂×Dic₃₄₉	—	4	1	no	—
17	#14584095	—	2	2	no	—
18	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₃₄₉	—	2	2	no	—
20	C₂×C₄×C₃₄₉	—	4	1	no	—
21	C₂×C₄×C₃₄₉	—	4	1	no	—
22	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₄₉	Q₈	1	1	yes	characteristic
28	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₃₄₉	D₈	1	1	yes	characteristic
32	C₂²×C₃₄₉	SD₁₆	1	1	yes	characteristic
33	C₂²×C₃₄₉	SD₁₆	1	1	yes	characteristic
34	C₂²×C₃₄₉	Q₁₆	1	1	yes	characteristic
35	C₄×C₃₄₉	—	4	2	no	—
36	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
37	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
40	Dic₃₄₉	—	4	2	no	—
41	C₂×C₃₄₉	#95	1	1	yes	characteristic
42	C₂×C₃₄₉	#102	1	1	yes	characteristic
43	C₂×C₃₄₉	#102	1	1	yes	characteristic
44	C₂×C₃₄₉	#103	1	1	yes	characteristic
45	C₂×C₃₄₉	#103	1	1	yes	characteristic
46	C₂×C₃₄₉	#106	1	1	yes	characteristic
47	C₂×C₃₄₉	#107	1	1	yes	characteristic
48	C₃₄₉	#340	1	1	yes	characteristic, complemented, Sylow
49	#340	—	349	1	no	maximal, Sylow
50	C₂²×C₈	—	349	1	no	—
51	C₂×C₄⋊C₄	—	349	1	no	—
52	C₂×C₄⋊C₄	D₃₄₉	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	698	1	no	—
55	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
56	C₂×C₈	—	349	2	no	—
57	C₂×C₈	—	698	1	no	—
58	C₂×C₈	—	698	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	349	2	no	—
61	C₄⋊C₄	—	698	1	no	—
62	C₄⋊C₄	Dic₃₄₉	1	2	yes	complemented
63	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	1396	1	no	—
68	C₂×C₄	—	1396	1	no	—
69	C₂×C₄	—	698	2	no	—
70	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
71	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
72	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
74	C₂×C₄	D₁₃₉₆	1	1	yes	characteristic
75	C₂×C₄	Dic₆₉₈	1	1	yes	characteristic
76	C₈	—	698	2	no	—
77	C₂²	#19150210	1	1	yes	characteristic
78	C₂²	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
79	C₂²	#19150212	1	1	yes	characteristic
80	C₂²	#19150218	1	1	yes	characteristic
81	C₂²	#19150220	1	1	yes	characteristic
82	C₂²	#19150221	1	1	yes	characteristic
83	C₂²	#19150223	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	1396	2	no	—
86	C₄	C₄×Dic₃₄₉	1	1	yes	characteristic
87	C₄	#19150210	1	1	yes	characteristic
88	C₄	#19150212	1	1	yes	characteristic
89	C₄	C₄⋊Dic₃₄₉	1	1	yes	characteristic
90	C₂	#26013546	1	1	yes	characteristic
91	C₂	#26013559	1	1	yes	characteristic
92	C₂	#26013560	1	1	yes	characteristic
93	C₂	#26013562	1	1	yes	characteristic
94	C₂	#26013563	1	1	yes	characteristic
95	C₂	#26013570	1	1	yes	characteristic
96	C₂	#26013572	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₄₉
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂²×C₄	C₂×D₃₄₉
Fitting	C₂×C₄⋊C₄×C₃₄₉	C₂
socle, prime radical	C₂³×C₃₄₉	D₄
cosocle	C₂²×C₄×C₃₄₉	C₂²
hypercenter	C₂×C₄⋊C₄	D₃₄₉
hypocenter	C₃₄₉	#340
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂×#14584095	C₄	semidirect
9. C₄⋊Dic₃₄₉	C₄	semidirect
13. C₄⋊C₄×C₃₄₉	C₄	semidirect
48. C₃₄₉	#340	semidirect
62. C₄⋊C₄	Dic₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14584095	2
C₂×C₄⋊Dic₃₄₉	2
C₂×C₄⋊C₄×C₃₄₉	2
#340	349