Subgroups — C₃×#10602579 #14305930

Order 1920.

616 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3967	C₂	1	4	yes	maximal
3	C₃×#65225	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
5	#10602579	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂²×C₃×#168	C₂²	1	2	yes	—
13	C₃×#3967	—	2	8	no	—
14	C₃×#3963	—	2	2	no	—
15	C₃×#3935	—	2	2	no	—
16	C₃×#1960	—	5	1	no	maximal, Hall
17	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂×#3967	C₂×C₃	1	4	yes	—
19	#65225	C₂×C₃	1	1	yes	characteristic
20	C₂×#3935	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×C₅×Q₈	—	2	2	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₅	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	—	2	8	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄⋊Dic₅	—	2	4	no	—
33	C₂×C₃×#168	—	2	2	no	—
34	C₂×C₃×#168	—	2	2	no	—
35	C₂×C₃×#168	D₄	1	4	yes	—
36	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
37	C₂×C₃×#103	—	5	4	no	—
38	C₂×C₃×#97	—	5	1	no	—
39	C₃×#382	—	5	1	no	—
40	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
41	C₅×C₄⋊Q₈	—	2	2	no	—
42	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
46	C₂²×#168	C₂²×C₃	1	2	yes	—
47	#3967	—	2	8	no	—
48	#3963	—	2	2	no	—
49	#3935	—	2	2	no	—
50	#1960	—	5	1	no	Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	2	8	no	—
53	C₃×C₅×Q₈	—	4	4	no	—
54	C₂×C₃×C₄×C₅	—	2	4	no	—
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×C₃×Dic₅	—	4	2	no	—
64	C₂×C₃×Dic₅	—	4	4	no	—
65	C₃×#168	—	2	4	no	—
66	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
67	C₂²×C₃×C₈	—	5	2	no	—
68	C₃×C₄⋊Q₈	—	2	2	no	—
69	C₃×C₄⋊Q₈	Dic₅	1	4	yes	complemented
70	C₂×C₃×C₄⋊C₄	—	2	2	no	—
71	C₂×C₃×C₄⋊C₄	—	5	2	no	—
72	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
73	C₃×#103	—	10	8	no	—
74	C₃×#97	—	10	2	no	—
75	C₃×#95	—	10	2	no	—
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×Q₈	—	2	4	no	—
78	C₂×C₅×Q₈	—	2	4	no	—
79	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
83	C₂²×Dic₅	—	2	2	no	—
84	C₅×C₄⋊C₄	—	2	8	no	—
85	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
86	C₄²×C₅	C₃×D₄	1	2	yes	—
87	C₄⋊Dic₅	—	2	4	no	—
88	C₂×#168	—	2	2	no	—
89	C₂×#168	—	2	2	no	—
90	C₂×#168	C₃×D₄	1	4	yes	—
91	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic
92	C₂×#103	—	5	4	no	—
93	C₂×#97	—	5	1	no	—
94	#382	—	5	1	no	—
95	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₃×C₅	#124	1	1	yes	characteristic
97	C₂²×C₃×C₅	#127	1	1	yes	characteristic
98	C₂²×C₃×C₅	#137	1	4	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	—	4	4	no	—
101	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
102	C₃×C₄×C₅	#121	1	2	yes	—
103	C₃×Dic₅	—	8	2	no	—
104	C₂×C₃×Q₈	—	2	2	no	—
105	C₂×C₃×Q₈	—	2	4	no	—
106	C₂×C₃×Q₈	—	2	4	no	—
107	C₂×C₃×Q₈	C₂×Dic₅	1	4	yes	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	10	2	no	—
110	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
111	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
112	C₂×C₃×C₈	—	5	4	no	—
113	C₂×C₃×C₈	—	10	2	no	—
114	C₂×C₃×C₈	—	10	2	no	—
115	C₃×C₄⋊C₄	—	2	8	no	—
116	C₃×C₄⋊C₄	—	10	4	no	—
117	C₃×C₄²	C₂×Dic₅	1	2	yes	—
118	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
119	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
120	C₅×Q₈	—	2	8	no	—
121	C₅×Q₈	—	4	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	4	2	no	—
125	C₂×C₄×C₅	—	4	2	no	—
126	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅	C₃×#41	1	2	yes	—
131	C₂×Dic₅	—	4	2	no	—
132	C₂×Dic₅	—	4	4	no	—
133	#168	—	2	4	no	—
134	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
135	C₂²×C₈	—	5	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	C₃×Dic₅	1	4	yes	complemented
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	5	2	no	—
140	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
141	#103	—	10	8	no	—
142	#97	—	10	2	no	—
143	#95	—	10	2	no	—
144	C₂×C₃×C₅	#390	1	1	yes	characteristic
145	C₂×C₃×C₅	#419	1	2	yes	—
146	C₂×C₃×C₅	#491	1	2	yes	—
147	C₂×C₃×C₅	#497	1	2	yes	—
148	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	C₃×Q₈	—	2	8	no	—
150	C₃×Q₈	—	4	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	4	2	no	—
154	C₂×C₃×C₄	—	4	2	no	—
155	C₂×C₃×C₄	—	20	2	no	—
156	C₂×C₃×C₄	—	20	4	no	—
157	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
158	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
159	C₂×C₃×C₄	#711	1	2	yes	—
160	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₈	—	10	4	no	—
163	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₅	C₃×#124	1	1	yes	characteristic
165	C₂²×C₅	C₃×#127	1	1	yes	characteristic
166	C₂²×C₅	C₃×#137	1	4	yes	—
167	C₄×C₅	—	2	4	no	—
168	C₄×C₅	—	4	4	no	—
169	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
170	C₄×C₅	C₃×#121	1	2	yes	—
171	Dic₅	—	8	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	4	no	—
175	C₂×Q₈	C₂×C₃×Dic₅	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	10	2	no	—
178	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
179	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
180	C₂×C₈	—	5	4	no	—
181	C₂×C₈	—	10	2	no	—
182	C₂×C₈	—	10	2	no	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	10	4	no	—
185	C₄²	C₂×C₃×Dic₅	1	2	yes	—
186	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
187	C₃×C₅	#1960	1	1	yes	characteristic, complemented, Hall
188	C₂²×C₃	#4082	1	1	yes	characteristic
189	C₂²×C₃	#4086	1	1	yes	characteristic
190	C₂²×C₃	#4088	1	4	yes	—
191	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	4	4	no	—
194	C₃×C₄	—	40	2	no	—
195	C₃×C₄	D₄×Dic₅	1	2	yes	—
196	C₃×C₄	#4084	1	2	yes	—
197	C₂×C₅	C₃×#390	1	1	yes	characteristic
198	C₂×C₅	C₃×#419	1	2	yes	—
199	C₂×C₅	C₃×#491	1	2	yes	—
200	C₂×C₅	C₃×#497	1	2	yes	—
201	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
202	Q₈	—	2	8	no	—
203	Q₈	—	4	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	20	2	no	—
209	C₂×C₄	—	20	4	no	—
210	C₂×C₄	C₃×C₂²⋊Dic₅	1	4	yes	—
211	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
212	C₂×C₄	C₃×#711	1	2	yes	—
213	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
215	C₈	—	10	4	no	—
216	C₂×C₃	#65371	1	2	yes	—
217	C₂×C₃	#65375	1	2	yes	—
218	C₂×C₃	#65510	1	1	yes	characteristic
219	C₂×C₃	#65514	1	2	yes	—
220	C₅	C₃×#1960	1	1	yes	characteristic, complemented, Sylow
221	C₂²	C₃×#4082	1	1	yes	characteristic
222	C₂²	C₃×#4086	1	1	yes	characteristic
223	C₂²	C₃×#4088	1	4	yes	—
224	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	4	4	no	—
227	C₄	—	40	2	no	—
228	C₄	C₃×D₄×Dic₅	1	2	yes	—
229	C₄	C₃×#4084	1	2	yes	—
230	C₃	#10602579	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	C₃×#65371	1	2	yes	—
232	C₂	C₃×#65375	1	2	yes	—
233	C₂	C₃×#65510	1	1	yes	characteristic
234	C₂	C₃×#65514	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#1960
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
5. #10602579	C₃	direct (N × H)
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
42. C₅×C₄⋊Q₈	C₃×C₄	semidirect
69. C₃×C₄⋊Q₈	Dic₅	semidirect
137. C₄⋊Q₈	C₃×Dic₅	semidirect
187. C₃×C₅	#1960	semidirect
220. C₅	C₃×#1960	semidirect
230. C₃	#10602579	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×C₅×C₄⋊Q₈	2
C₂×C₃×#3967	2
C₃×#65225	2
C₂×C₃×#3935	2
#10602579	3
C₃×#1960	5

Subgroups — C3×#10602579 #14305930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602579 #14305930