Subgroups — C₂×#26044217 #37554639

Order 23488.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#26044217	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₃₆₇⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊Dic₃₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₃₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₄×C₃₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₆₇	C₄	1	8	yes	complemented
8	C₂²×Dic₃₆₇	C₂²	1	2	yes	—
9	C₂²×Dic₃₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₆₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃₆₇	C₂²	1	4	yes	—
12	Dic₃₆₇⋊C₄	C₂²	1	8	yes	complemented
13	C₂²⋊Dic₃₆₇	C₂²	1	4	yes	—
14	C₄×Dic₃₆₇	C₂²	1	8	yes	complemented
15	C₂⁴×C₃₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₆₇	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₃₆₇	C₂²	1	4	yes	—
18	C₂×Dic₃₆₇	—	2	4	no	—
19	C₂×Dic₃₆₇	C₂×C₄	1	4	yes	—
20	C₂×Dic₃₆₇	C₂×C₄	1	4	yes	—
21	C₂×Dic₃₆₇	C₂×C₄	1	16	yes	complemented
22	C₂×Dic₃₆₇	C₂³	1	4	yes	—
23	C₂×Dic₃₆₇	C₂³	1	2	yes	—
24	C₂×Dic₃₆₇	C₂³	1	2	yes	—
25	C₂³×C₃₆₇	—	2	4	no	—
26	C₂³×C₃₆₇	C₂×C₄	1	4	yes	—
27	C₂³×C₃₆₇	C₂³	1	2	yes	—
28	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₃₆₇	—	2	4	no	—
30	C₂×C₄×C₃₆₇	C₂³	1	4	yes	—
31	C₂²×C₃₆₇	—	2	8	no	—
32	C₂²×C₃₆₇	—	2	4	no	—
33	C₂²×C₃₆₇	C₂²×C₄	1	4	yes	—
34	C₂²×C₃₆₇	C₂²×C₄	1	2	yes	—
35	C₂²×C₃₆₇	#41	1	4	yes	—
36	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
37	C₄×C₃₆₇	—	2	4	no	—
38	Dic₃₆₇	—	2	4	no	—
39	Dic₃₆₇	C₂²×C₄	1	4	yes	—
40	Dic₃₆₇	C₂²×C₄	1	4	yes	—
41	C₂×C₃₆₇	—	2	4	no	—
42	C₂×C₃₆₇	#117	1	4	yes	complemented
43	C₂×C₃₆₇	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₃₆₇	C₂×#41	1	2	yes	—
45	C₃₆₇	C₂×#117	1	1	yes	characteristic, complemented, Sylow
46	C₂×#117	—	367	1	no	maximal, Sylow
47	C₂³×C₄	—	367	1	no	—
48	#117	—	367	8	no	—
49	C₂×C₄⋊C₄	—	367	2	no	—
50	C₂×C₂²⋊C₄	—	367	1	no	—
51	C₂×C₂²⋊C₄	D₃₆₇	1	1	yes	characteristic
52	C₂×C₄²	—	367	2	no	—
53	C₂⁴	C₂×D₃₆₇	1	1	yes	characteristic
54	C₂²×C₄	—	367	8	no	—
55	C₂²×C₄	—	367	4	no	—
56	C₂²×C₄	—	367	1	no	—
57	C₂²×C₄	—	367	1	no	—
58	C₂²×C₄	—	367	2	no	—
59	C₂²×C₄	C₂×D₃₆₇	1	2	yes	—
60	C₄⋊C₄	—	367	8	no	—
61	C₂²⋊C₄	—	367	4	no	—
62	C₂²⋊C₄	C₂×D₃₆₇	1	4	yes	—
63	C₄²	—	367	8	no	—
64	C₂³	—	2	4	no	—
65	C₂³	C₄×D₃₆₇	1	4	yes	—
66	C₂³	C₂²×D₃₆₇	1	2	yes	—
67	C₂³	C₂²×D₃₆₇	1	1	yes	characteristic
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	367	4	no	—
70	C₂×C₄	—	367	4	no	—
71	C₂×C₄	—	367	16	no	—
72	C₂×C₄	—	367	2	no	—
73	C₂×C₄	—	367	2	no	—
74	C₂×C₄	—	367	4	no	—
75	C₂×C₄	—	734	4	no	—
76	C₂×C₄	C₂²×D₃₆₇	1	4	yes	—
77	C₂²	—	2	8	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂×C₄×D₃₆₇	1	4	yes	—
80	C₂²	C₂×C₄×D₃₆₇	1	2	yes	—
81	C₂²	#19154576	1	4	yes	—
82	C₂²	C₂³×D₃₆₇	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	367	4	no	—
85	C₄	—	367	4	no	—
86	C₄	—	734	4	no	—
87	C₂	—	2	4	no	—
88	C₂	#26044217	1	4	yes	direct product factor
89	C₂	C₂²×C₄×D₃₆₇	1	1	yes	characteristic
90	C₂	C₂×#19154576	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₆₇
commutator	C₂×C₃₆₇	C₂³×C₄
Frattini	C₂²	C₂³×D₃₆₇
Fitting	C₂×C₂²⋊C₄×C₃₆₇	C₂
socle	C₂³×C₃₆₇	C₂³
prime radical	C₂⁴×C₃₆₇	C₂²
cosocle	C₂²×C₃₆₇	C₂⁴
hypercenter	C₂×C₂²⋊C₄	D₃₆₇
hypocenter	C₃₆₇	C₂×#117
LCS	G ▷ C₂×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #26044217	C₂	direct (N × H)
3. C₂×Dic₃₆₇⋊C₄	C₂	semidirect
5. C₂×C₄×Dic₃₆₇	C₂	semidirect
7. C₂²×Dic₃₆₇	C₄	semidirect
12. Dic₃₆₇⋊C₄	C₂²	semidirect
14. C₄×Dic₃₆₇	C₂²	semidirect
21. C₂×Dic₃₆₇	C₂×C₄	semidirect
42. C₂×C₃₆₇	#117	semidirect
45. C₃₆₇	C₂×#117	semidirect
88. C₂	#26044217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₆₇	2
#26044217	2
C₂×Dic₃₆₇⋊C₄	2
C₂×C₂²⋊Dic₃₆₇	2
C₂×C₄×Dic₃₆₇	2
C₂×C₂²⋊C₄×C₃₆₇	2
C₂×#117	367

Subgroups — C2×#26044217 #37554639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26044217 #37554639