Subgroups — C₃×#66151 #11755828

Order 960.

580 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4093	C₂	1	3	yes	maximal, complemented
4	C₃×#4092	C₂	1	3	yes	maximal, complemented
5	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4076	C₂	1	3	yes	maximal, complemented
7	C₃×C₄×C₂²⋊D₅	C₂	1	3	yes	maximal, complemented
8	#66151	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₈	—	2	4	no	—
10	C₂×C₃×C₅×Q₈	C₂²	1	3	yes	—
11	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₅	C₂²	1	3	yes	—
13	C₂×C₃×C₂²⋊D₅	C₂²	1	3	yes	—
14	C₃×#713	—	2	4	no	—
15	C₂×C₃×D₂₀	C₂²	1	3	yes	—
16	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
17	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
18	C₃×#685	C₂²	1	6	yes	—
19	C₃×#685	C₂²	1	3	yes	—
20	C₃×C₄⋊Dic₅	C₂²	1	3	yes	—
21	C₃×#683	C₂²	1	3	yes	—
22	C₃×C₄×Dic₅	C₂²	1	3	yes	—
23	C₃×#548	—	5	1	no	maximal, Hall
24	C₂²×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#713	C₂×C₃	1	1	yes	characteristic, complemented
26	#4093	C₂×C₃	1	3	yes	complemented
27	#4092	C₂×C₃	1	3	yes	complemented
28	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
29	#4076	C₂×C₃	1	3	yes	complemented
30	C₄×C₂²⋊D₅	C₂×C₃	1	3	yes	complemented
31	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₃×D₅	C₂³	1	3	yes	—
33	C₃×C₅×Q₈	—	2	6	no	—
34	C₃×C₅×Q₈	D₄	1	4	yes	complemented
35	C₂×C₃×C₄×C₅	—	2	6	no	—
36	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂×C₃×C₄×C₅	C₂³	1	3	yes	—
38	C₃×C₂²⋊D₅	—	2	6	no	—
39	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
40	C₂×C₃×Dic₅	C₂³	1	3	yes	—
41	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
42	C₃×D₂₀	—	2	6	no	—
43	C₃×C₄×D₅	—	2	6	no	—
44	C₂×C₃×#41	—	5	1	no	—
45	C₂²×C₃×Q₈	D₅	1	1	yes	characteristic, complemented
46	C₃×#124	—	5	3	no	—
47	C₃×C₂²⋊Q₈	—	5	3	no	—
48	C₃×#121	—	5	3	no	—
49	C₃×C₄×Q₈	—	5	1	no	—
50	C₃×C₄×D₄	—	5	3	no	—
51	C₂×C₅×Q₈	—	2	4	no	—
52	C₂×C₅×Q₈	C₂²×C₃	1	3	yes	—
53	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂²×C₃	1	3	yes	—
55	C₂×C₂²⋊D₅	C₂²×C₃	1	3	yes	—
56	#713	—	2	4	no	—
57	C₂×D₂₀	C₂²×C₃	1	3	yes	—
58	C₂×C₄×D₅	C₂²×C₃	1	3	yes	—
59	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
60	#685	C₂²×C₃	1	3	yes	—
61	#685	C₂²×C₃	1	6	yes	—
62	C₄⋊Dic₅	C₂²×C₃	1	3	yes	—
63	#683	C₂²×C₃	1	3	yes	—
64	C₄×Dic₅	C₂²×C₃	1	3	yes	—
65	#548	—	5	1	no	Sylow
66	C₂²×C₃×C₅	—	2	1	no	—
67	C₂²×C₃×C₅	—	2	1	no	—
68	C₂²×C₃×C₅	#41	1	2	yes	complemented
69	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
70	C₂×C₃×D₅	—	2	3	no	—
71	C₂×C₃×D₅	—	2	3	no	—
72	C₂×C₃×D₅	—	2	3	no	—
73	C₃×C₄×C₅	—	2	3	no	—
74	C₃×C₄×C₅	C₂×D₄	1	6	yes	—
75	C₃×Dic₅	—	2	1	no	—
76	C₃×Dic₅	—	2	3	no	—
77	C₃×Dic₅	—	2	1	no	—
78	C₃×#41	—	10	4	no	—
79	C₂×C₃×Q₈	—	2	4	no	—
80	C₂×C₃×Q₈	C₂×D₅	1	3	yes	—
81	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
82	C₂×C₃×D₄	—	5	3	no	—
83	C₂×C₃×D₄	—	5	3	no	—
84	C₂²×C₃×C₄	—	5	3	no	—
85	C₂²×C₃×C₄	C₂×D₅	1	3	yes	—
86	C₃×C₄⋊C₄	—	5	3	no	—
87	C₃×C₄⋊C₄	—	5	3	no	—
88	C₃×C₂²⋊C₄	—	5	1	no	—
89	C₃×C₂²⋊C₄	—	5	3	no	—
90	C₃×C₂²⋊C₄	—	5	6	no	—
91	C₃×C₄²	—	5	3	no	—
92	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₂²×D₅	C₂³×C₃	1	3	yes	—
94	C₅×Q₈	—	2	6	no	—
95	C₅×Q₈	C₃×D₄	1	4	yes	complemented
96	C₂×C₄×C₅	—	2	6	no	—
97	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
98	C₂×C₄×C₅	C₂³×C₃	1	3	yes	—
99	C₂²⋊D₅	—	2	6	no	—
100	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
101	C₂×Dic₅	C₂³×C₃	1	3	yes	—
102	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
103	D₂₀	—	2	6	no	—
104	C₄×D₅	—	2	6	no	—
105	C₂×#41	—	5	1	no	—
106	C₂²×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
107	#124	—	5	3	no	—
108	C₂²⋊Q₈	—	5	3	no	—
109	#121	—	5	3	no	—
110	C₄×Q₈	—	5	1	no	—
111	C₄×D₄	—	5	3	no	—
112	C₂×C₃×C₅	—	2	2	no	—
113	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
115	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
116	C₃×D₅	—	4	3	no	—
117	C₂³×C₃	—	5	3	no	—
118	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
119	C₃×Q₈	—	2	6	no	—
120	C₃×Q₈	C₂²⋊D₅	1	4	yes	complemented
121	C₃×D₄	—	10	6	no	—
122	C₃×D₄	—	10	6	no	—
123	C₂×C₃×C₄	—	2	6	no	—
124	C₂×C₃×C₄	—	5	1	no	—
125	C₂×C₃×C₄	—	5	3	no	—
126	C₂×C₃×C₄	—	5	1	no	—
127	C₂×C₃×C₄	—	10	6	no	—
128	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
129	C₂×C₃×C₄	C₂²×D₅	1	3	yes	—
130	C₂²×C₅	—	2	1	no	—
131	C₂²×C₅	—	2	1	no	—
132	C₂²×C₅	C₃×#41	1	2	yes	complemented
133	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
134	C₂×D₅	—	2	3	no	—
135	C₂×D₅	—	2	3	no	—
136	C₂×D₅	—	2	3	no	—
137	C₄×C₅	—	2	3	no	—
138	C₄×C₅	C₂×C₃×D₄	1	6	yes	—
139	Dic₅	—	2	1	no	—
140	Dic₅	—	2	3	no	—
141	Dic₅	—	2	1	no	—
142	#41	—	10	4	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	C₂×C₃×D₅	1	3	yes	—
145	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
146	C₂×D₄	—	5	3	no	—
147	C₂×D₄	—	5	3	no	—
148	C₂²×C₄	—	5	3	no	—
149	C₂²×C₄	C₂×C₃×D₅	1	3	yes	—
150	C₄⋊C₄	—	5	3	no	—
151	C₄⋊C₄	—	5	3	no	—
152	C₂²⋊C₄	—	5	1	no	—
153	C₂²⋊C₄	—	5	3	no	—
154	C₂²⋊C₄	—	5	6	no	—
155	C₄²	—	5	3	no	—
156	C₃×C₅	#548	1	1	yes	characteristic, complemented, Hall
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	2	1	no	—
159	C₂²×C₃	—	10	3	no	—
160	C₂²×C₃	—	10	3	no	—
161	C₂²×C₃	—	10	3	no	—
162	C₂²×C₃	#713	1	2	yes	complemented
163	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
164	C₃×C₄	—	2	3	no	—
165	C₃×C₄	—	10	3	no	—
166	C₃×C₄	—	10	1	no	—
167	C₃×C₄	—	10	1	no	—
168	C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
169	C₂×C₅	—	2	2	no	—
170	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
171	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
172	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
173	D₅	—	4	3	no	—
174	C₂³	—	5	3	no	—
175	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
176	Q₈	—	2	6	no	—
177	Q₈	C₃×C₂²⋊D₅	1	4	yes	complemented
178	D₄	—	10	6	no	—
179	D₄	—	10	6	no	—
180	C₂×C₄	—	2	6	no	—
181	C₂×C₄	—	5	1	no	—
182	C₂×C₄	—	5	3	no	—
183	C₂×C₄	—	5	1	no	—
184	C₂×C₄	—	10	6	no	—
185	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
186	C₂×C₄	C₂²×C₃×D₅	1	3	yes	—
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	20	3	no	—
189	C₂×C₃	C₂×#713	1	1	yes	characteristic
190	C₂×C₃	#4147	1	1	yes	characteristic
191	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
192	C₅	C₃×#548	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	10	3	no	—
196	C₂²	—	10	3	no	—
197	C₂²	—	10	3	no	—
198	C₂²	C₃×#713	1	2	yes	complemented
199	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
200	C₄	—	2	3	no	—
201	C₄	—	10	3	no	—
202	C₄	—	10	1	no	—
203	C₄	—	10	1	no	—
204	C₄	C₂×C₃×C₂²⋊D₅	1	6	yes	—
205	C₃	#66151	1	1	yes	characteristic, direct product factor, Sylow
206	C₂	—	2	2	no	—
207	C₂	—	20	3	no	—
208	C₂	C₂×C₃×#713	1	1	yes	characteristic
209	C₂	C₃×#4147	1	1	yes	characteristic
210	C₂	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₂²×C₃×Q₈	D₅
hypocenter	C₅	C₃×#548
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×Q₈	C₂	semidirect
2. C₂×C₃×#713	C₂	semidirect
3. C₃×#4093	C₂	semidirect
4. C₃×#4092	C₂	semidirect
5. C₃×Q₈×Dic₅	C₂	semidirect
6. C₃×#4076	C₂	semidirect
7. C₃×C₄×C₂²⋊D₅	C₂	semidirect
8. #66151	C₃	direct (N × H)
24. C₂²×C₅×Q₈	C₂×C₃	semidirect
25. C₂×#713	C₂×C₃	semidirect
26. #4093	C₂×C₃	semidirect
27. #4092	C₂×C₃	semidirect
28. Q₈×Dic₅	C₂×C₃	semidirect
29. #4076	C₂×C₃	semidirect
30. C₄×C₂²⋊D₅	C₂×C₃	semidirect
34. C₃×C₅×Q₈	D₄	semidirect
45. C₂²×C₃×Q₈	D₅	semidirect
68. C₂²×C₃×C₅	#41	semidirect
95. C₅×Q₈	C₃×D₄	semidirect
106. C₂²×Q₈	C₃×D₅	semidirect
120. C₃×Q₈	C₂²⋊D₅	semidirect
132. C₂²×C₅	C₃×#41	semidirect
156. C₃×C₅	#548	semidirect
162. C₂²×C₃	#713	semidirect
177. Q₈	C₃×C₂²⋊D₅	semidirect
192. C₅	C₃×#548	semidirect
198. C₂²	C₃×#713	semidirect
205. C₃	#66151	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×C₃×C₅×Q₈	2
C₂×C₃×#713	2
C₃×#4093	2
C₃×#4092	2
C₃×Q₈×Dic₅	2
C₃×#4076	2
C₃×C₄×C₂²⋊D₅	2
#66151	3
C₃×#548	5

Subgroups — C3×#66151 #11755828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66151 #11755828