Finite Group Databasebeta

Subgroups — #13085679

Order 1440.

326 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 58 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	#10624205	C₂	1	2	yes	maximal, complemented
2	#10623822	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3713	C₂	1	1	yes	maximal, characteristic, complemented
4	#10623753	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#6654	C₂	1	2	yes	maximal, complemented
6	#91481	—	3	2	no	maximal
7	C₂×D₃×D₁₅	—	2	2	no	—
8	C₄×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
9	C₃×D₆₀	C₂²	1	2	yes	complemented
10	C₃×C₅×D₁₂	C₂²	1	2	yes	complemented
11	#67150	—	2	2	no	—
12	C₅×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic, complemented
13	C₃²×#168	C₂²	1	1	yes	characteristic, complemented
14	#63797	—	5	1	no	maximal, Hall
15	D₄×D₁₅	—	3	2	no	—
16	#6777	—	3	2	no	—
17	C₅×#213	—	3	2	no	—
18	C₃×#686	—	3	2	no	—
19	#6654	D₃	1	2	yes	complemented
20	#6653	—	3	2	no	—
21	#6648	—	3	2	no	—
22	#6648	—	6	1	no	—
23	C₂×C₃⋊D₁₅	D₄	1	1	yes	characteristic
24	C₂×C₃×D₁₅	—	2	2	no	—
25	C₂×C₃×C₅×D₃	—	2	2	no	—
26	D₃×D₁₅	—	4	4	no	—
27	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₃⋊Dic₁₅	D₄	1	1	yes	characteristic
29	D₅×D₈	—	9	1	no	Hall
30	#3802	—	5	2	no	—
31	C₈×C₃⋊D₃	—	5	1	no	—
32	C₃×D₂₄	—	5	2	no	—
33	#3713	—	5	1	no	—
34	#3713	D₅	1	1	yes	characteristic, complemented
35	C₂²×D₁₅	—	6	2	no	—
36	C₂×D₃×D₅	—	6	2	no	—
37	C₃×C₅×D₄	—	3	2	no	—
38	#1220	—	6	2	no	—
39	D₆₀	C₂×D₃	1	2	yes	complemented
40	C₄×D₁₅	—	3	2	no	—
41	C₄×D₁₅	—	6	1	no	—
42	C₅×D₁₂	C₂×D₃	1	2	yes	complemented
43	C₃×D₂₀	—	3	2	no	—
44	#1201	—	6	2	no	—
45	C₃×#168	—	2	1	no	—
46	C₃×#168	C₂×D₃	1	2	yes	complemented
47	C₅×C₃⋊C₈	—	3	2	no	—
48	C₅×C₃⋊C₈	—	6	1	no	—
49	D₃×D₈	—	15	2	no	—
50	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
51	C₃⋊D₁₅	D₈	1	2	yes	complemented
52	C₃×D₁₅	—	4	2	no	—
53	C₃×C₅×D₃	—	4	2	no	—
54	D₄×D₅	—	9	2	no	—
55	C₅×D₈	—	9	1	no	—
56	#686	—	3	2	no	—
57	D₄₀	—	9	1	no	—
58	C₈×D₅	—	9	1	no	—
59	C₂×D₃²	—	10	2	no	—
60	C₄×C₃⋊D₃	—	5	1	no	—
61	C₃×D₁₂	—	5	2	no	—
62	C₃×D₁₂	C₂×D₅	1	2	yes	complemented
63	#625	—	10	2	no	—
64	C₃²×C₈	—	5	1	no	—
65	C₃⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic, complemented
66	C₂²×C₃×C₅	—	6	2	no	—
67	C₂×D₁₅	—	2	2	no	—
68	C₂×D₁₅	—	3	2	no	—
69	C₂×D₁₅	—	6	1	no	—
70	C₂×D₁₅	—	12	4	no	—
71	C₂×C₅×D₃	—	2	2	no	—
72	C₂×C₃×D₅	—	6	2	no	—
73	D₃×D₅	—	12	4	no	—
74	C₃×C₄×C₅	—	2	1	no	—
75	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
76	Dic₁₅	—	3	2	no	—
77	Dic₁₅	—	6	1	no	—
78	D₃×D₄	—	15	2	no	—
79	D₃×D₄	—	15	2	no	—
80	C₃×D₈	—	15	2	no	—
81	#213	—	3	2	no	—
82	#213	—	15	2	no	—
83	D₂₄	—	5	2	no	—
84	D₃×C₈	—	15	2	no	—
85	D₃×C₈	—	30	1	no	—
86	C₃²×C₅	C₂×D₈	1	1	yes	characteristic, complemented, Hall
87	C₂²×D₅	—	18	2	no	—
88	C₅×D₄	—	3	2	no	—
89	C₂²⋊D₅	—	18	2	no	—
90	D₂₀	—	3	2	no	—
91	C₄×D₅	—	9	1	no	—
92	C₅×C₈	—	9	1	no	—
93	#168	D₃²	1	1	yes	characteristic, complemented
94	C₂×C₃⋊D₃	—	5	1	no	—
95	C₂×C₃×D₃	—	2	2	no	—
96	C₂×C₃×D₃	—	10	2	no	—
97	D₃²	—	20	4	no	—
98	C₃²×C₄	C₂²×D₅	1	1	yes	characteristic
99	C₃⋊Dic₃	—	5	1	no	—
100	C₂×D₈	—	45	1	no	Sylow
101	C₂×C₃×C₅	—	2	1	no	—
102	C₂×C₃×C₅	—	12	2	no	—
103	C₂×C₃×C₅	D₃×D₄	1	2	yes	—
104	D₁₅	—	3	4	no	—
105	D₁₅	—	4	2	no	—
106	D₁₅	—	6	2	no	—
107	C₃×D₅	—	12	2	no	—
108	C₅×D₃	—	4	2	no	—
109	C₂²×D₃	—	30	2	no	—
110	C₂²×D₃	—	30	2	no	—
111	C₃×D₄	—	3	2	no	—
112	C₃×D₄	—	15	2	no	—
113	#67	—	30	2	no	—
114	#67	—	30	2	no	—
115	D₁₂	—	5	2	no	—
116	D₁₂	D₃×D₅	1	2	yes	complemented
117	C₄×D₃	—	15	2	no	—
118	C₄×D₃	—	30	1	no	—
119	C₃×C₈	—	5	2	no	—
120	C₃×C₈	—	10	1	no	—
121	C₃⋊C₈	—	3	2	no	—
122	C₃⋊C₈	—	6	1	no	—
123	C₂²×C₅	—	6	2	no	—
124	C₂×D₅	—	6	2	no	—
125	C₂×D₅	—	9	1	no	—
126	C₂×D₅	—	36	4	no	—
127	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
128	Dic₅	—	9	1	no	—
129	C₂×C₃²	D₄×D₅	1	1	yes	characteristic
130	C₃⋊D₃	—	5	2	no	—
131	C₃×D₃	—	4	2	no	—
132	C₃×D₃	—	20	2	no	—
133	C₂×D₄	—	45	2	no	—
134	D₈	—	9	1	no	—
135	D₈	—	15	2	no	—
136	D₈	—	45	1	no	—
137	C₂×C₈	—	45	1	no	—
138	C₃×C₅	—	2	1	no	—
139	C₃×C₅	D₃×D₈	1	2	yes	complemented
140	C₂²×C₃	—	6	2	no	—
141	C₂²×C₃	—	30	2	no	—
142	C₂×D₃	—	2	2	no	—
143	C₂×D₃	—	10	2	no	—
144	C₂×D₃	—	15	2	no	—
145	C₂×D₃	—	30	1	no	—
146	C₂×D₃	—	60	4	no	—
147	C₂×D₃	—	60	4	no	—
148	C₃×C₄	—	2	1	no	—
149	C₃×C₄	C₂×D₃×D₅	1	2	yes	—
150	Dic₃	—	15	2	no	—
151	Dic₃	—	30	1	no	—
152	C₂×C₅	—	12	2	no	—
153	C₂×C₅	#3802	1	1	yes	characteristic
154	D₅	—	9	2	no	—
155	D₅	—	12	2	no	—
156	C₃²	D₅×D₈	1	1	yes	characteristic, complemented, Sylow
157	C₂³	—	90	2	no	—
158	D₄	—	3	2	no	—
159	D₄	—	15	2	no	—
160	D₄	—	90	2	no	—
161	C₂×C₄	—	45	1	no	—
162	C₈	—	5	1	no	—
163	C₈	—	9	1	no	—
164	C₂×C₃	—	2	1	no	—
165	C₂×C₃	—	12	2	no	—
166	C₂×C₃	—	60	2	no	—
167	C₂×C₃	#6790	1	2	yes	—
168	D₃	—	4	2	no	—
169	D₃	—	15	4	no	—
170	D₃	—	20	2	no	—
171	D₃	—	30	2	no	—
172	C₅	#63797	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	6	2	no	—
174	C₂²	—	30	2	no	—
175	C₂²	—	45	1	no	—
176	C₂²	—	180	4	no	—
177	C₄	—	45	1	no	—
178	C₄	D₃²×D₅	1	1	yes	characteristic
179	C₃	—	2	1	no	—
180	C₃	#91481	1	2	yes	complemented
181	C₂	—	12	2	no	—
182	C₂	—	45	2	no	—
183	C₂	—	60	2	no	—
184	C₂	#10624035	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10624035
commutator, Fitting, cosocle	C₃²×C₄×C₅	C₂³
Frattini, hypercenter	C₄	D₃²×D₅
socle	C₂×C₃²×C₅	C₂×D₄
hypocenter	C₃²×C₅	C₂×D₈
LCS	G ▷ C₃²×C₄×C₅ ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. #10624205	C₂	semidirect
2. #10623822	C₂	semidirect
3. C₅×#3713	C₂	semidirect
4. #10623753	C₂	semidirect
5. C₃×#6654	C₂	semidirect
9. C₃×D₆₀	C₂²	semidirect
10. C₃×C₅×D₁₂	C₂²	semidirect
12. C₅×C₃⋊(C₃⋊C₈)	C₂²	semidirect
13. C₃²×#168	C₂²	semidirect
19. #6654	D₃	semidirect
34. #3713	D₅	semidirect
39. D₆₀	C₂×D₃	semidirect
42. C₅×D₁₂	C₂×D₃	semidirect
46. C₃×#168	C₂×D₃	semidirect
51. C₃⋊D₁₅	D₈	semidirect
62. C₃×D₁₂	C₂×D₅	semidirect
65. C₃⋊(C₃⋊C₈)	C₂×D₅	semidirect
86. C₃²×C₅	C₂×D₈	semidirect
93. #168	D₃²	semidirect
116. D₁₂	D₃×D₅	semidirect
139. C₃×C₅	D₃×D₈	semidirect
156. C₃²	D₅×D₈	semidirect
172. C₅	#63797	semidirect
180. C₃	#91481	semidirect

Maximal subgroups

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

subgroup	index
#10624205	2
#10623822	2
C₅×#3713	2
#10623753	2
C₃×#6654	2
#91481	3
#63797	5