Finite Group Databasebeta

← C₂×#5968 #87569

Subgroups — C₂×#5968 #87569

Order 384.

5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#5968	C₂	1	32	yes	maximal, direct product factor
3	C₂×#1009	C₂	1	10	yes	maximal, complemented
4	C₂×D₃×#41	C₂	1	10	yes	maximal, complemented
5	C₂×#1007	C₂	1	5	yes	maximal, complemented
6	C₂²×#239	C₂	1	5	yes	maximal, complemented
7	C₂×#585	—	3	1	no	maximal, Sylow
8	C₃×2₋¹⁺⁴	C₂²	1	16	yes	complemented
9	C₂×C₃×#41	C₂²	1	10	yes	complemented
10	C₂²×C₃×Q₈	C₂²	1	5	yes	complemented
11	#1009	C₂²	1	160	yes	complemented
12	D₃×#41	C₂²	1	160	yes	complemented
13	#1007	C₂²	1	80	yes	complemented
14	C₂×#239	C₂²	1	10	yes	complemented
15	C₂×#239	C₂²	1	20	yes	complemented
16	C₂×#239	C₂²	1	80	yes	complemented
17	C₂×D₃×Q₈	C₂²	1	10	yes	complemented
18	C₂×#237	C₂²	1	10	yes	complemented
19	C₂×D₃×D₄	C₂²	1	30	yes	complemented
20	C₂×#235	C₂²	1	30	yes	complemented
21	C₂²×D₁₂	C₂²	1	15	yes	complemented
22	C₂²×C₄×D₃	C₂²	1	15	yes	complemented
23	#585	—	3	32	no	—
24	C₂×2₋¹⁺⁴	—	3	5	no	—
25	C₂×2₋¹⁺⁴	D₃	1	1	yes	characteristic, complemented
26	C₂×H(5,𝔽₂)	—	3	10	no	—
27	C₂²×#41	—	3	10	no	—
28	C₂²×#41	—	3	5	no	—
29	C₂³×D₃	C₂³	1	15	yes	—
30	C₃×#41	C₂³	1	80	yes	complemented
31	C₂×C₃×Q₈	C₂³	1	10	yes	—
32	C₂×C₃×Q₈	C₂³	1	40	yes	complemented
33	C₂×C₃×D₄	C₂³	1	10	yes	—
34	C₂²×C₃×C₄	C₂³	1	15	yes	—
35	C₂×#67	C₂³	1	20	yes	—
36	C₂²×Dic₃	C₂³	1	5	yes	—
37	#239	C₂³	1	80	yes	complemented
38	#239	C₂³	1	160	yes	complemented
39	D₃×Q₈	C₂³	1	80	yes	complemented
40	#237	C₂³	1	80	yes	complemented
41	D₃×D₄	C₂³	1	240	yes	complemented
42	#235	C₂³	1	240	yes	complemented
43	C₂×D₁₂	C₂³	1	30	yes	—
44	C₂×D₁₂	C₂³	1	120	yes	complemented
45	C₂×C₄×D₃	C₂³	1	10	yes	—
46	C₂×C₄×D₃	C₂³	1	30	yes	—
47	C₂×C₄×D₃	C₂³	1	120	yes	complemented
48	C₂×Dic₆	C₂³	1	10	yes	—
49	2₋¹⁺⁴	—	3	80	no	—
50	2₋¹⁺⁴	C₂×D₃	1	16	yes	complemented
51	H(5,𝔽₂)	—	3	160	no	—
52	C₂×#41	—	3	160	no	—
53	C₂×#41	—	3	10	no	—
54	C₂×#41	—	3	20	no	—
55	C₂×#41	—	3	80	no	—
56	C₂×#41	—	3	10	no	—
57	C₂×#41	—	3	30	no	—
58	C₂×#41	C₂×D₃	1	10	yes	complemented
59	C₂²×Q₈	—	3	10	no	—
60	C₂²×Q₈	C₂×D₃	1	5	yes	complemented
61	C₂²×D₄	—	3	30	no	—
62	C₂²×D₄	—	3	15	no	—
63	C₂³×C₄	—	3	15	no	—
64	C₂³×C₃	C₂⁴	1	5	yes	—
65	C₂²×D₃	—	4	30	no	—
66	C₂²×D₃	C₂⁴	1	10	yes	—
67	C₂²×D₃	C₂⁴	1	60	yes	—
68	C₃×Q₈	C₂⁴	1	40	yes	—
69	C₃×D₄	C₂⁴	1	40	yes	—
70	C₂×C₃×C₄	C₂⁴	1	10	yes	—
71	C₂×C₃×C₄	C₂⁴	1	60	yes	—
72	#67	C₂⁴	1	80	yes	—
73	C₂×Dic₃	C₂⁴	1	5	yes	—
74	C₂×Dic₃	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₃	C₂⁴	1	20	yes	—
76	D₁₂	C₂⁴	1	120	yes	—
77	C₄×D₃	C₂⁴	1	40	yes	—
78	C₄×D₃	C₂⁴	1	120	yes	—
79	Dic₆	C₂⁴	1	40	yes	—
80	C₂⁴	—	3	15	no	—
81	#41	—	3	80	no	—
82	#41	—	3	160	no	—
83	#41	—	3	80	no	—
84	#41	—	3	240	no	—
85	#41	C₂²×D₃	1	80	yes	complemented
86	C₂×Q₈	—	3	80	no	—
87	C₂×Q₈	—	3	10	no	—
88	C₂×Q₈	C₂²×D₃	1	10	yes	—
89	C₂×Q₈	C₂²×D₃	1	40	yes	complemented
90	C₂×D₄	—	3	20	no	—
91	C₂×D₄	—	3	240	no	—
92	C₂×D₄	—	3	30	no	—
93	C₂×D₄	—	3	120	no	—
94	C₂×D₄	C₂²×D₃	1	10	yes	—
95	C₂²×C₄	—	3	5	no	—
96	C₂²×C₄	—	3	10	no	—
97	C₂²×C₄	—	3	30	no	—
98	C₂²×C₄	—	3	120	no	—
99	C₂²×C₄	C₂²×D₃	1	15	yes	—
100	C₂²×C₃	—	2	10	no	—
101	C₂²×C₃	C₂⁵	1	10	yes	—
102	C₂²×C₃	C₂⁵	1	1	yes	characteristic
103	C₂×D₃	—	2	20	no	—
104	C₂×D₃	—	4	60	no	—
105	C₂×D₃	C₂⁵	1	20	yes	—
106	C₃×C₄	C₂⁵	1	20	yes	—
107	Dic₃	C₂⁵	1	10	yes	—
108	Dic₃	C₂⁵	1	2	yes	—
109	C₂³	—	3	10	no	—
110	C₂³	—	3	60	no	—
111	C₂³	—	12	30	no	—
112	C₂³	C₂³×D₃	1	5	yes	—
113	Q₈	—	3	40	no	—
114	Q₈	C₂³×D₃	1	40	yes	—
115	D₄	—	3	80	no	—
116	D₄	—	3	120	no	—
117	D₄	C₂³×D₃	1	40	yes	—
118	C₂×C₄	—	3	20	no	—
119	C₂×C₄	—	3	1	no	—
120	C₂×C₄	—	3	5	no	—
121	C₂×C₄	—	3	40	no	—
122	C₂×C₄	—	3	120	no	—
123	C₂×C₄	C₂³×D₃	1	60	yes	—
124	C₂×C₄	C₂³×D₃	1	10	yes	—
125	C₂×C₃	—	2	10	no	—
126	C₂×C₃	#585	1	2	yes	complemented
127	C₂×C₃	C₂⁶	1	1	yes	characteristic
128	D₃	—	2	20	no	—
129	C₂²	—	2	10	no	—
130	C₂²	—	3	20	no	—
131	C₂²	—	6	20	no	—
132	C₂²	—	12	60	no	—
133	C₂²	C₂⁴×D₃	1	10	yes	—
134	C₂²	C₂⁴×D₃	1	1	yes	characteristic
135	C₄	—	3	2	no	—
136	C₄	—	3	10	no	—
137	C₄	C₂⁴×D₃	1	20	yes	—
138	C₃	C₂×#585	1	1	yes	characteristic, complemented, Sylow
139	C₂	—	2	10	no	—
140	C₂	—	6	20	no	—
141	C₂	#5968	1	2	yes	direct product factor
142	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂⁴×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂×C₃×2₋¹⁺⁴	C₂
socle	C₂²×C₃	C₂⁵
hypercenter	C₂×2₋¹⁺⁴	D₃
hypocenter	C₃	C₂×#585
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×2₋¹⁺⁴	C₂	semidirect
2. #5968	C₂	direct (N × H)
3. C₂×#1009	C₂	semidirect
4. C₂×D₃×#41	C₂	semidirect
5. C₂×#1007	C₂	semidirect
6. C₂²×#239	C₂	semidirect
8. C₃×2₋¹⁺⁴	C₂²	semidirect
9. C₂×C₃×#41	C₂²	semidirect
10. C₂²×C₃×Q₈	C₂²	semidirect
11. #1009	C₂²	semidirect
12. D₃×#41	C₂²	semidirect
13. #1007	C₂²	semidirect
14. C₂×#239	C₂²	semidirect
15. C₂×#239	C₂²	semidirect
16. C₂×#239	C₂²	semidirect
17. C₂×D₃×Q₈	C₂²	semidirect
18. C₂×#237	C₂²	semidirect
19. C₂×D₃×D₄	C₂²	semidirect
20. C₂×#235	C₂²	semidirect
21. C₂²×D₁₂	C₂²	semidirect
22. C₂²×C₄×D₃	C₂²	semidirect
25. C₂×2₋¹⁺⁴	D₃	semidirect
30. C₃×#41	C₂³	semidirect
32. C₂×C₃×Q₈	C₂³	semidirect
37. #239	C₂³	semidirect
38. #239	C₂³	semidirect
39. D₃×Q₈	C₂³	semidirect
40. #237	C₂³	semidirect
41. D₃×D₄	C₂³	semidirect
42. #235	C₂³	semidirect
44. C₂×D₁₂	C₂³	semidirect
47. C₂×C₄×D₃	C₂³	semidirect
50. 2₋¹⁺⁴	C₂×D₃	semidirect
58. C₂×#41	C₂×D₃	semidirect
60. C₂²×Q₈	C₂×D₃	semidirect
85. #41	C₂²×D₃	semidirect
89. C₂×Q₈	C₂²×D₃	semidirect
126. C₂×C₃	#585	semidirect
138. C₃	C₂×#585	semidirect
141. C₂	#5968	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×2₋¹⁺⁴	2
#5968	2
C₂×#1009	2
C₂×D₃×#41	2
C₂×#1007	2
C₂²×#239	2
C₂×#585	3