Subgroups — #14313967

Order 1920.

440 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	C₂²×Dic₆₀	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic
4	#11746313	C₂	1	4	yes	maximal, complemented
5	#10601447	—	3	1	no	maximal, Hall
6	C₂²×Dic₃₀	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	—	2	1	no	—
9	C₂×Dic₆₀	—	2	1	no	—
10	C₂×Dic₆₀	—	2	2	no	—
11	C₂×Dic₆₀	C₄	1	4	yes	complemented
12	C₂×Dic₆₀	C₂²	1	2	yes	—
13	C₃×C₅×#107	C₂²	1	2	yes	—
14	C₃×C₅×#106	—	2	1	no	—
15	#91096	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	—
17	#70654	—	5	1	no	maximal, Hall
18	C₂²×Dic₂₀	—	3	1	no	—
19	C₅×#427	D₃	1	1	yes	characteristic
20	C₂×#3944	—	3	1	no	—
21	#64712	—	3	4	no	—
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×Dic₁₅	—	4	1	no	—
24	C₂×Dic₃₀	—	2	4	no	—
25	C₂×Dic₃₀	—	2	2	no	—
26	C₂×Dic₃₀	—	4	1	no	—
27	C₂×Dic₃₀	—	4	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
34	C₃×C₄²×C₅	—	4	1	no	—
35	Dic₆₀	—	2	4	no	—
36	Dic₆₀	—	2	2	no	—
37	Dic₆₀	C₂×C₄	1	4	yes	—
38	C₃⋊#672	—	2	2	no	—
39	C₂²×Dic₁₂	—	5	1	no	—
40	C₃×#427	D₅	1	1	yes	characteristic
41	C₂×#812	—	5	1	no	—
42	#4492	—	5	4	no	—
43	C₂²×Dic₁₀	—	6	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₅×#117	—	2	1	no	—
46	C₂×Dic₂₀	—	3	4	no	—
47	C₂×Dic₂₀	—	3	2	no	—
48	C₂×Dic₂₀	—	6	2	no	—
49	C₂×Dic₂₀	—	6	1	no	—
50	C₅×#107	C₂×D₃	1	2	yes	—
51	C₅×#106	—	2	1	no	—
52	#3944	—	6	2	no	—
53	C₂×#672	—	3	2	no	—
54	#2141	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₁₅	—	4	4	no	—
62	C₂×Dic₁₅	—	4	2	no	—
63	Dic₃₀	—	2	4	no	—
64	Dic₃₀	—	4	4	no	—
65	Dic₃₀	—	4	2	no	—
66	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
67	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₆	—	10	1	no	—
70	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
71	C₃×#117	—	2	1	no	—
72	C₂×Dic₁₂	—	5	4	no	—
73	C₂×Dic₁₂	—	5	2	no	—
74	C₂×Dic₁₂	—	10	2	no	—
75	C₂×Dic₁₂	—	10	1	no	—
76	C₃×#107	C₂×D₅	1	2	yes	—
77	C₃×#106	—	2	1	no	—
78	#812	—	10	2	no	—
79	C₂×C₃⋊C₁₆	—	5	2	no	—
80	C₂²×C₄×C₅	#67	1	1	yes	characteristic
81	C₂²×Dic₅	—	12	1	no	—
82	C₂×Dic₁₀	—	6	4	no	—
83	C₂×Dic₁₀	—	6	2	no	—
84	C₂×Dic₁₀	—	12	1	no	—
85	C₂×Dic₁₀	—	12	2	no	—
86	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
87	C₂×C₅×C₈	D₁₂	1	2	yes	—
88	C₂×C₅×C₈	#67	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₂²⋊C₄	—	4	1	no	—
92	C₄²×C₅	—	4	1	no	—
93	Dic₂₀	—	3	4	no	—
94	Dic₂₀	—	6	4	no	—
95	Dic₂₀	—	6	2	no	—
96	#672	—	6	2	no	—
97	C₂²×Q₁₆	—	15	1	no	—
98	C₂×M₃₂	—	15	1	no	—
99	#427	D₁₅	1	1	yes	characteristic
100	#358	—	15	4	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	#102	1	2	yes	—
103	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
104	C₃×C₄×C₅	—	8	2	no	—
105	C₃×C₄×C₅	#102	1	2	yes	—
106	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
108	Dic₁₅	—	4	4	no	—
109	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
110	C₂²×Dic₃	—	20	1	no	—
111	C₂×Dic₆	—	10	4	no	—
112	C₂×Dic₆	—	10	2	no	—
113	C₂×Dic₆	—	20	1	no	—
114	C₂×Dic₆	—	20	2	no	—
115	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
116	C₂×C₃×C₈	D₂₀	1	2	yes	—
117	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₂²⋊C₄	—	4	1	no	—
121	C₃×C₄²	—	4	1	no	—
122	Dic₁₂	—	5	4	no	—
123	Dic₁₂	—	10	4	no	—
124	Dic₁₂	—	10	2	no	—
125	C₃⋊C₁₆	—	10	2	no	—
126	C₂³×C₅	#213	1	1	yes	characteristic
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	#213	1	1	yes	characteristic
130	C₂×C₄×C₅	#215	1	2	yes	—
131	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
132	C₂×Dic₅	—	12	4	no	—
133	C₂×Dic₅	—	12	2	no	—
134	Dic₁₀	—	6	4	no	—
135	Dic₁₀	—	12	4	no	—
136	Dic₁₀	—	12	2	no	—
137	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
138	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
139	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
140	C₂²×Q₈	—	30	1	no	—
141	C₂×Q₁₆	—	15	4	no	—
142	C₂×Q₁₆	—	15	2	no	—
143	C₂×Q₁₆	—	30	2	no	—
144	C₂×Q₁₆	—	30	1	no	—
145	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
146	#117	—	2	1	no	—
147	M₃₂	—	30	2	no	—
148	C₂×C₁₆	—	15	2	no	—
149	#107	C₂×D₁₅	1	2	yes	—
150	#106	—	2	1	no	—
151	C₂×C₃×C₅	—	2	2	no	—
152	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
153	C₂×C₃×C₅	#510	1	2	yes	—
154	C₂³×C₃	#686	1	1	yes	characteristic
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	#685	1	2	yes	—
157	C₂×C₃×C₄	#686	1	1	yes	characteristic
158	C₂×C₃×C₄	#688	1	2	yes	—
159	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
160	C₂×Dic₃	—	20	4	no	—
161	C₂×Dic₃	—	20	2	no	—
162	Dic₆	—	10	4	no	—
163	Dic₆	—	20	4	no	—
164	Dic₆	—	20	2	no	—
165	C₃×C₈	#685	1	2	yes	—
166	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
167	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	#809	1	2	yes	—
170	C₂²×C₅	C₂×#213	1	1	yes	characteristic
171	C₄×C₅	—	8	2	no	—
172	C₄×C₅	#809	1	2	yes	—
173	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₄×C₅	C₂×#215	1	1	yes	characteristic
175	Dic₅	—	12	4	no	—
176	C₂×Q₈	—	30	4	no	—
177	C₂×Q₈	—	30	2	no	—
178	C₂×Q₈	—	60	1	no	—
179	C₂×Q₈	—	60	2	no	—
180	C₂²×C₄	—	60	1	no	—
181	C₂²×C₄	#1220	1	1	yes	characteristic
182	Q₁₆	—	15	4	no	—
183	Q₁₆	—	30	4	no	—
184	Q₁₆	—	30	2	no	—
185	C₂×C₈	C₄×D₁₅	1	2	yes	—
186	C₂×C₈	D₆₀	1	2	yes	—
187	C₂×C₈	#1220	1	1	yes	characteristic
188	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	4	1	no	—
191	C₄²	—	4	1	no	—
192	C₁₆	—	30	2	no	—
193	C₃×C₅	#2141	1	1	yes	characteristic, complemented, Hall
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	#3941	1	2	yes	—
196	C₂²×C₃	C₂×#686	1	1	yes	characteristic
197	C₃×C₄	—	8	2	no	—
198	C₃×C₄	#3941	1	2	yes	—
199	C₃×C₄	C₂×#685	1	1	yes	characteristic
200	C₃×C₄	C₂×#688	1	1	yes	characteristic
201	Dic₃	—	20	4	no	—
202	C₂×C₅	—	2	2	no	—
203	C₂×C₅	C₂×#809	1	1	yes	characteristic
204	C₂×C₅	#5195	1	2	yes	—
205	C₂³	#6715	1	1	yes	characteristic
206	Q₈	—	30	4	no	—
207	Q₈	—	60	2	no	—
208	Q₈	—	60	4	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	60	2	no	—
211	C₂×C₄	—	60	4	no	—
212	C₂×C₄	#6714	1	2	yes	—
213	C₂×C₄	#6715	1	1	yes	characteristic
214	C₂×C₄	#6717	1	2	yes	—
215	C₂×C₄	C₂×#1220	1	1	yes	characteristic
216	C₈	#6714	1	2	yes	—
217	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
218	C₈	C₂×D₆₀	1	1	yes	characteristic
219	C₂×C₃	—	2	2	no	—
220	C₂×C₃	C₂×#3941	1	1	yes	characteristic
221	C₂×C₃	#65484	1	2	yes	—
222	C₅	#70654	1	1	yes	characteristic, complemented, Sylow
223	C₂²	—	2	2	no	—
224	C₂²	#91093	1	2	yes	—
225	C₂²	C₂×#6715	1	1	yes	characteristic
226	C₄	—	8	2	no	—
227	C₄	—	60	4	no	—
228	C₄	#91093	1	2	yes	—
229	C₄	C₂×#6714	1	1	yes	characteristic
230	C₄	C₂×#6717	1	1	yes	characteristic
231	C₃	#10601447	1	1	yes	characteristic, complemented, Sylow
232	C₂	—	2	2	no	—
233	C₂	C₂×#91093	1	1	yes	characteristic
234	C₂	#11751296	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#6715
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂³×C₃×C₅	D₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	#427	D₁₅
hypocenter	C₃×C₅	#2141
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

Complements

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

N	H ≅ G/N	type
4. #11746313	C₂	semidirect
11. C₂×Dic₆₀	C₄	semidirect
193. C₃×C₅	#2141	semidirect
222. C₅	#70654	semidirect
231. C₃	#10601447	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₀	2
C₃×C₅×#427	2
C₂×#91096	2
#11746313	2
#10601447	3
#70654	5