Subgroups — #36374138

Order 22336.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150233	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14584095	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013562	C₂	1	2	yes	maximal, complemented
4	#26013559	C₂	1	2	yes	maximal, complemented
5	#117×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150233	—	2	2	no	—
7	#19150233	C₄	1	4	yes	complemented
8	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
9	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₃₄₉	—	2	1	no	—
11	C₂×C₄×D₃₄₉	—	2	1	no	—
12	C₂×#14584095	—	2	2	no	—
13	C₂×#14584095	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₃₄₉	—	2	1	no	—
17	C₄²×C₃₄₉	—	2	1	no	—
18	C₂²×D₃₄₉	—	2	1	no	—
19	Dic₆₉₈	—	2	1	no	—
20	Dic₆₉₈	C₂×C₄	1	2	yes	—
21	D₁₃₉₆	—	2	1	no	—
22	D₁₃₉₆	C₂×C₄	1	2	yes	—
23	C₂²⋊D₃₄₉	—	2	4	no	—
24	C₂×Dic₃₄₉	—	2	1	no	—
25	C₄×D₃₄₉	—	2	4	no	—
26	#14584095	—	2	2	no	—
27	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₄₉	—	2	2	no	—
29	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
32	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
33	C₂²×C₃₄₉	—	2	2	no	—
34	C₂²×C₃₄₉	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
36	C₂×D₃₄₉	—	2	2	no	—
37	C₂×D₃₄₉	—	4	2	no	—
38	C₄×C₃₄₉	—	4	2	no	—
39	C₄×C₃₄₉	C₂²⋊C₄	1	2	yes	—
40	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
42	Dic₃₄₉	—	2	2	no	—
43	C₂×C₃₄₉	—	2	2	no	—
44	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃₄₉	#135	1	2	yes	—
46	D₃₄₉	—	4	2	no	—
47	C₃₄₉	#416	1	1	yes	characteristic, complemented, Sylow
48	#416	—	349	1	no	maximal, Sylow
49	C₂×#41	—	349	1	no	—
50	C₂²×C₈	—	349	1	no	—
51	#117	D₃₄₉	1	1	yes	characteristic, complemented
52	#103	—	349	2	no	—
53	#102	—	349	2	no	—
54	#41	—	349	4	no	—
55	#41	—	698	2	no	—
56	C₂×Q₈	—	349	1	no	—
57	C₂×D₄	—	349	1	no	—
58	C₂×D₄	—	698	1	no	—
59	C₂²×C₄	—	698	1	no	—
60	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
61	C₂×C₈	—	349	2	no	—
62	C₂×C₈	—	698	2	no	—
63	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	—	2	1	no	—
66	C₂³	—	698	1	no	—
67	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
68	Q₈	—	349	2	no	—
69	Q₈	—	698	1	no	—
70	D₄	—	349	2	no	—
71	D₄	—	698	4	no	—
72	D₄	—	698	1	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	698	1	no	—
75	C₂×C₄	—	698	4	no	—
76	C₂×C₄	C₄×D₃₄₉	1	2	yes	—
77	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
78	C₂×C₄	D₁₃₉₆	1	2	yes	—
79	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
80	C₈	—	698	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	1396	2	no	—
83	C₂²	—	698	2	no	—
84	C₂²	#19150210	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	—	698	2	no	—
88	C₄	#19150210	1	2	yes	—
89	C₄	C₂×C₄×D₃₄₉	1	1	yes	characteristic
90	C₄	C₂×D₁₃₉₆	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	1396	2	no	—
93	C₂	C₂×#19150210	1	1	yes	characteristic
94	C₂	#26013659	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₃₄₉
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#117×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂×#19150233	C₂
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	#117	D₃₄₉
hypocenter	C₃₄₉	#416
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂²×#14584095	C₂	semidirect
3. #26013562	C₂	semidirect
4. #26013559	C₂	semidirect
5. #117×C₃₄₉	C₂	semidirect
7. #19150233	C₄	semidirect
47. C₃₄₉	#416	semidirect
51. #117	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150233	2
C₂²×#14584095	2
#26013562	2
#26013559	2
#117×C₃₄₉	2
#416	349