Subgroups — C₂×#26013559 #36374130

Order 22336.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₃₉₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14584095	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013559	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₄₉	—	2	1	no	—
6	C₂×D₁₃₉₆	—	2	1	no	—
7	C₂×D₁₃₉₆	—	2	2	no	—
8	C₂×D₁₃₉₆	C₄	1	4	yes	complemented
9	C₂×D₁₃₉₆	C₂²	1	2	yes	—
10	C₂×#14584095	—	2	2	no	—
11	C₂×#14584095	C₂²	1	2	yes	complemented
12	C₂²×C₄×C₃₄₉	—	2	1	no	—
13	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₄₉	—	2	1	no	—
15	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	complemented
16	C₂²×D₃₄₉	—	2	4	no	—
17	C₂²×D₃₄₉	—	2	2	no	—
18	C₂²×D₃₄₉	—	4	2	no	—
19	C₂²×D₃₄₉	—	4	2	no	—
20	D₁₃₉₆	—	2	4	no	—
21	D₁₃₉₆	—	2	2	no	—
22	D₁₃₉₆	C₂×C₄	1	4	yes	complemented
23	#14584095	—	2	2	no	—
24	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	—	2	2	no	—
26	C₂×C₄×C₃₄₉	—	4	2	no	—
27	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
30	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
31	C₂²×C₃₄₉	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₃₄₉	D₈	1	2	yes	—
33	C₂²×C₃₄₉	SD₁₆	1	2	yes	—
34	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
35	C₂×D₃₄₉	—	2	4	no	—
36	C₂×D₃₄₉	—	4	8	no	—
37	C₂×D₃₄₉	—	4	2	no	—
38	C₂×D₃₄₉	—	4	2	no	—
39	C₄×C₃₄₉	—	4	2	no	—
40	C₄×C₃₄₉	C₂²⋊C₄	1	2	yes	—
41	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃₄₉	#102	1	4	yes	complemented
44	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₃₄₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₃₄₉	C₂×SD₁₆	1	1	yes	characteristic
47	D₃₄₉	—	4	4	no	—
48	C₃₄₉	C₂×#102	1	1	yes	characteristic, complemented, Sylow
49	C₂×#102	—	349	1	no	maximal, Sylow
50	C₂²×D₄	—	349	1	no	—
51	C₂²×C₈	—	349	1	no	—
52	C₂×C₄⋊C₄	D₃₄₉	1	1	yes	characteristic, complemented
53	#102	—	349	4	no	—
54	C₂⁴	—	698	1	no	—
55	C₂×D₄	—	349	4	no	—
56	C₂×D₄	—	349	2	no	—
57	C₂×D₄	—	698	1	no	—
58	C₂×D₄	—	698	2	no	—
59	C₂²×C₄	—	2	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	—	2	1	no	—
64	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	complemented
65	C₂³	—	1396	2	no	—
66	C₂³	—	1396	2	no	—
67	C₂³	—	698	4	no	—
68	C₂³	—	698	2	no	—
69	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
70	D₄	—	349	4	no	—
71	D₄	—	698	4	no	—
72	D₄	—	698	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	C₄×D₃₄₉	1	2	yes	—
76	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
77	C₂×C₄	D₁₃₉₆	1	2	yes	—
78	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
79	C₈	—	698	2	no	—
80	C₂²	—	1396	8	no	—
81	C₂²	—	1396	2	no	—
82	C₂²	—	1396	2	no	—
83	C₂²	—	698	4	no	—
84	C₂²	#19150210	1	2	yes	—
85	C₂²	#19150218	1	2	yes	—
86	C₂²	#19150220	1	2	yes	—
87	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
88	C₄	—	4	2	no	—
89	C₄	#19150210	1	2	yes	—
90	C₄	C₂×C₄×D₃₄₉	1	1	yes	characteristic
91	C₄	C₂×D₁₃₉₆	1	1	yes	characteristic
92	C₂	—	1396	4	no	—
93	C₂	#26013559	1	4	yes	direct product factor
94	C₂	C₂×#19150210	1	1	yes	characteristic
95	C₂	C₂×#19150218	1	1	yes	characteristic
96	C₂	C₂×#19150220	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₄₉
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	C₂×C₄⋊C₄×C₃₄₉	C₂
socle	C₂³×C₃₄₉	D₄
prime radical	C₂²×D₁₃₉₆	C₂
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₃₄₉
hypocenter	C₃₄₉	C₂×#102
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
2. C₂²×#14584095	C₂	semidirect
3. #26013559	C₂	direct (N × H)
4. C₂×C₄⋊C₄×C₃₄₉	C₂	semidirect
8. C₂×D₁₃₉₆	C₄	semidirect
11. C₂×#14584095	C₂²	semidirect
15. C₄⋊C₄×C₃₄₉	C₂²	semidirect
22. D₁₃₉₆	C₂×C₄	semidirect
43. C₂×C₃₄₉	#102	semidirect
48. C₃₄₉	C₂×#102	semidirect
52. C₂×C₄⋊C₄	D₃₄₉	semidirect
64. C₄⋊C₄	C₂×D₃₄₉	semidirect
93. C₂	#26013559	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₁₃₉₆	2
C₂²×#14584095	2
#26013559	2
C₂×C₄⋊C₄×C₃₄₉	2
C₂×#102	349

Subgroups — C2×#26013559 #36374130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013559 #36374130