Subgroups — #36374473

Order 22336.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013615	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₃₄₉	C₂	1	1	yes	maximal, characteristic
3	#26013571	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013570	C₂	1	1	yes	maximal, characteristic, complemented
5	#26013563	C₂	1	1	yes	maximal, characteristic, complemented
6	#26013551	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₄₉	—	2	1	no	—
9	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
10	C₂×#14584095	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₄₉	C₂²	1	1	yes	characteristic
13	#19150212	—	2	1	no	—
14	#19150212	—	2	1	no	—
15	#19150210	—	2	1	no	—
16	C₂×Q₈×C₃₄₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₄₉	C₂²	1	1	yes	characteristic
19	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
20	C₂×Dic₃₄₉	—	2	1	no	—
21	C₂×Dic₃₄₉	—	2	1	no	—
22	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
23	C₄×D₃₄₉	—	2	1	no	—
24	C₄×D₃₄₉	—	2	1	no	—
25	C₄×D₃₄₉	—	4	2	no	—
26	#14584095	—	2	1	no	—
27	Q₈×C₃₄₉	—	2	1	no	—
28	Q₈×C₃₄₉	—	2	1	no	—
29	C₂×C₄×C₃₄₉	—	2	1	no	—
30	C₂×C₄×C₃₄₉	—	2	1	no	—
31	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
32	C₈×C₃₄₉	—	2	1	no	—
33	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
34	C₂×D₃₄₉	—	2	1	no	—
35	C₂×D₃₄₉	—	2	1	no	—
36	C₂×D₃₄₉	SD₁₆	1	2	yes	—
37	C₄×C₃₄₉	—	4	1	no	—
38	C₄×C₃₄₉	—	4	1	no	—
39	C₄×C₃₄₉	#41	1	1	yes	characteristic
40	C₄×C₃₄₉	#41	1	1	yes	characteristic
41	Dic₃₄₉	—	2	1	no	—
42	Dic₃₄₉	—	4	1	no	—
43	Dic₃₄₉	—	4	1	no	—
44	C₂×C₃₄₉	#123	1	1	yes	characteristic
45	C₂×C₃₄₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₃₄₉	#137	1	1	yes	characteristic
47	D₃₄₉	—	2	2	no	—
48	C₃₄₉	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	349	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	349	1	no	—
51	C₂×C₄⋊C₄	—	349	1	no	—
52	#106	—	349	1	no	—
53	#106	—	349	1	no	—
54	#103	—	349	1	no	—
55	#103	D₃₄₉	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	—	349	1	no	—
57	C₂×Q₈	C₂×D₃₄₉	1	1	yes	characteristic
58	C₂²×C₄	—	349	1	no	—
59	C₂²×C₄	—	698	1	no	—
60	C₂×C₈	—	349	1	no	—
61	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
62	C₄⋊C₄	—	349	1	no	—
63	C₄⋊C₄	—	349	1	no	—
64	C₄⋊C₄	—	698	1	no	—
65	C₄⋊C₄	—	698	1	no	—
66	C₄⋊C₄	C₂×D₃₄₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	698	1	no	—
68	C₂³	—	349	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	349	1	no	—
74	C₂×C₄	—	1396	2	no	—
75	C₂×C₄	—	698	1	no	—
76	C₂×C₄	—	698	1	no	—
77	C₂×C₄	—	698	1	no	—
78	C₂×C₄	—	698	1	no	—
79	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	698	1	no	—
82	C₂²	—	349	2	no	—
83	C₂²	—	698	1	no	—
84	C₂²	—	698	1	no	—
85	C₂²	D₄×D₃₄₉	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	1396	1	no	—
89	C₄	—	1396	1	no	—
90	C₄	—	698	1	no	—
91	C₄	#19150232	1	1	yes	characteristic
92	C₄	#19150233	1	1	yes	characteristic
93	C₂	—	698	2	no	—
94	C₂	#26013621	1	1	yes	characteristic
95	C₂	SD₁₆×D₃₄₉	1	1	yes	characteristic
96	C₂	#26013671	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₄₉
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#103×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂²×D₃₄₉	D₄
hypercenter	#103	D₃₄₉
hypocenter	C₃₄₉	#483
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. #26013571	C₂	semidirect
4. #26013570	C₂	semidirect
5. #26013563	C₂	semidirect
7. #103×C₃₄₉	C₂	semidirect
48. C₃₄₉	#483	semidirect
55. #103	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013615	2
C₄⋊C₄×D₃₄₉	2
#26013571	2
#26013570	2
#26013563	2
#26013551	2
#103×C₃₄₉	2
#483	349