Subgroups — #11756591

Order 960.

620 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#6817	C₂	1	1	yes	maximal, characteristic, complemented
3	#91836	C₂	1	1	yes	maximal, characteristic, complemented
4	#91833	C₂	1	1	yes	maximal, characteristic, complemented
5	#91829	C₂	1	1	yes	maximal, characteristic, complemented
6	#91828	C₂	1	4	yes	maximal, complemented
7	#91825	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#1220	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
10	#66166	—	3	1	no	maximal, Hall
11	C₃×C₅×#41	—	2	4	no	—
12	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	2	yes	—
15	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
16	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂×#1220	C₂²	1	4	yes	—
18	C₂×#1220	C₂²	1	2	yes	—
19	C₂²×Dic₁₅	C₂²	1	2	yes	—
20	#6817	—	2	4	no	—
21	C₂×D₆₀	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₁₅	C₂²	1	2	yes	—
23	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
24	#6719	C₂²	1	4	yes	—
25	#6714	C₂²	1	4	yes	—
26	#6712	C₂²	1	4	yes	—
27	C₄×Dic₁₅	C₂²	1	2	yes	—
28	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
30	#5832	—	5	1	no	maximal, Hall
31	C₂×C₅×#41	D₃	1	1	yes	characteristic, complemented
32	C₂×#709	—	3	1	no	—
33	#4093	—	3	1	no	—
34	#4090	—	3	1	no	—
35	#4086	—	3	1	no	—
36	#4085	—	3	4	no	—
37	#4082	—	3	1	no	—
38	C₄×C₂²⋊D₅	—	3	4	no	—
39	C₂×C₄×Dic₅	—	3	1	no	—
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
42	C₂²×D₁₅	C₂³	1	2	yes	—
43	C₃×C₅×Q₈	—	2	2	no	—
44	C₃×C₅×D₄	—	2	4	no	—
45	C₃×C₅×D₄	—	2	2	no	—
46	C₂×C₃×C₄×C₅	—	2	4	no	—
47	C₂×C₃×C₄×C₅	D₄	1	4	yes	complemented
48	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
49	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	#1220	—	2	4	no	—
52	#1220	—	2	8	no	—
53	C₂×Dic₁₅	—	2	4	no	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	C₂×Dic₁₅	C₂³	1	2	yes	—
56	C₂×Dic₁₅	C₂³	1	2	yes	—
57	D₆₀	—	2	2	no	—
58	C₄×D₁₅	—	2	4	no	—
59	Dic₃₀	—	2	2	no	—
60	C₂×C₃×#41	D₅	1	1	yes	characteristic, complemented
61	C₂×#235	—	5	1	no	—
62	#947	—	5	1	no	—
63	#944	—	5	1	no	—
64	#940	—	5	1	no	—
65	#939	—	5	4	no	—
66	#936	—	5	1	no	—
67	C₄×#67	—	5	4	no	—
68	C₂×C₄×Dic₃	—	5	1	no	—
69	C₅×#41	—	2	4	no	—
70	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
72	C₂×C₅×D₄	C₂×D₃	1	2	yes	—
73	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
74	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
75	C₂×C₂²⋊D₅	—	3	2	no	—
76	C₂×C₂²⋊D₅	—	3	4	no	—
77	C₂²×Dic₅	—	3	2	no	—
78	#709	—	6	4	no	—
79	C₂×D₂₀	—	3	1	no	—
80	C₂×C₄×D₅	—	3	2	no	—
81	C₂×Dic₁₀	—	3	1	no	—
82	C₂²⋊Dic₅	—	3	4	no	—
83	#685	—	3	4	no	—
84	#683	—	3	4	no	—
85	C₄×Dic₅	—	3	2	no	—
86	C₄×Dic₅	—	3	1	no	—
87	C₄×Dic₅	—	3	1	no	—
88	#532	—	15	1	no	Sylow
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	4	no	—
92	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
93	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×D₁₅	—	2	2	no	—
95	C₂×D₁₅	—	2	4	no	—
96	C₃×C₄×C₅	—	2	2	no	—
97	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
99	Dic₁₅	—	2	2	no	—
100	Dic₁₅	—	2	2	no	—
101	Dic₁₅	#41	1	4	yes	complemented
102	C₃×#41	—	2	4	no	—
103	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
104	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×D₅	1	2	yes	—
106	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
107	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
108	C₂×#67	—	5	4	no	—
109	C₂×#67	—	5	2	no	—
110	C₂²×Dic₃	—	5	2	no	—
111	#235	—	10	4	no	—
112	C₂×D₁₂	—	5	1	no	—
113	C₂×C₄×D₃	—	5	2	no	—
114	C₂×Dic₆	—	5	1	no	—
115	#217	—	5	4	no	—
116	(C₂×D₃)⋊C₄	—	5	4	no	—
117	Dic₃⋊C₄	—	5	4	no	—
118	C₄×Dic₃	—	5	2	no	—
119	C₄×Dic₃	—	5	1	no	—
120	C₄×Dic₃	—	5	1	no	—
121	C₂³×C₅	C₂²×D₃	1	2	yes	—
122	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
123	C₂²×D₅	—	3	2	no	—
124	C₅×Q₈	—	2	2	no	—
125	C₅×D₄	—	2	4	no	—
126	C₅×D₄	—	2	2	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	#67	1	4	yes	complemented
129	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
130	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂²⋊D₅	—	6	8	no	—
133	C₂²⋊D₅	—	6	4	no	—
134	C₂×Dic₅	—	3	2	no	—
135	C₂×Dic₅	—	3	2	no	—
136	C₂×Dic₅	—	3	2	no	—
137	C₂×Dic₅	—	6	4	no	—
138	D₂₀	—	6	2	no	—
139	C₄×D₅	—	6	4	no	—
140	Dic₁₀	—	6	2	no	—
141	C₂×#41	—	15	1	no	—
142	C₂×#41	D₁₅	1	1	yes	characteristic, complemented
143	C₄⋊Q₈	—	15	1	no	—
144	#127	—	15	1	no	—
145	#124	—	15	1	no	—
146	#124	—	15	1	no	—
147	#121	—	15	4	no	—
148	C₄×D₄	—	15	4	no	—
149	C₂×C₄²	—	15	1	no	—
150	C₂×C₃×C₅	—	2	2	no	—
151	C₂×C₃×C₅	—	4	2	no	—
152	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₃×C₅	C₂×#41	1	2	yes	—
154	D₁₅	—	4	2	no	—
155	C₂³×C₃	C₂²×D₅	1	2	yes	—
156	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
157	C₂²×D₃	—	5	2	no	—
158	C₃×Q₈	—	2	2	no	—
159	C₃×D₄	—	2	4	no	—
160	C₃×D₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	4	no	—
162	C₂×C₃×C₄	C₂²⋊D₅	1	4	yes	complemented
163	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
164	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
166	#67	—	10	8	no	—
167	#67	—	10	4	no	—
168	C₂×Dic₃	—	5	2	no	—
169	C₂×Dic₃	—	5	2	no	—
170	C₂×Dic₃	—	5	2	no	—
171	C₂×Dic₃	—	10	4	no	—
172	D₁₂	—	10	2	no	—
173	C₄×D₃	—	10	4	no	—
174	Dic₆	—	10	2	no	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	—	2	4	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	C₂×#67	1	2	yes	—
179	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
180	C₂×D₅	—	6	4	no	—
181	C₂×D₅	—	6	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	C₂×#67	1	2	yes	—
184	C₄×C₅	C₂×#67	1	2	yes	—
185	Dic₅	—	3	4	no	—
186	Dic₅	—	6	2	no	—
187	Dic₅	—	6	2	no	—
188	#41	—	2	4	no	—
189	#41	—	30	4	no	—
190	C₂×Q₈	—	15	1	no	—
191	C₂×Q₈	C₂×D₁₅	1	1	yes	characteristic
192	C₂×D₄	—	15	4	no	—
193	C₂×D₄	—	15	2	no	—
194	C₂×D₄	—	15	1	no	—
195	C₂×D₄	C₂×D₁₅	1	2	yes	—
196	C₂×D₄	C₂×D₁₅	1	1	yes	characteristic
197	C₂²×C₄	—	15	2	no	—
198	C₂²×C₄	—	15	2	no	—
199	C₂²×C₄	C₂×D₁₅	1	2	yes	—
200	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
201	C₄⋊C₄	—	15	4	no	—
202	C₂²⋊C₄	—	15	4	no	—
203	C₂²⋊C₄	—	15	4	no	—
204	C₄²	—	15	1	no	—
205	C₄²	—	15	1	no	—
206	C₄²	—	15	2	no	—
207	C₃×C₅	#532	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	C₂×C₂²⋊D₅	1	2	yes	—
212	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
213	C₂×D₃	—	10	4	no	—
214	C₂×D₃	—	10	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
217	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
218	Dic₃	—	5	4	no	—
219	Dic₃	—	10	2	no	—
220	Dic₃	—	10	2	no	—
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	D₃×#41	1	2	yes	—
224	C₂×C₅	C₂²×#67	1	1	yes	characteristic
225	D₅	—	12	2	no	—
226	C₂³	—	15	2	no	—
227	C₂³	C₂²×D₁₅	1	2	yes	—
228	C₂³	C₂²×D₁₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	Q₈	—	30	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	D₄	—	30	8	no	—
234	D₄	—	30	4	no	—
235	D₄	—	30	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	15	2	no	—
238	C₂×C₄	—	15	2	no	—
239	C₂×C₄	—	15	2	no	—
240	C₂×C₄	—	30	4	no	—
241	C₂×C₄	—	30	4	no	—
242	C₂×C₄	#1220	1	4	yes	complemented
243	C₂×C₄	C₂²×D₁₅	1	2	yes	—
244	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
245	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	4	2	no	—
248	C₂×C₃	D₅×#41	1	2	yes	—
249	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
250	D₃	—	20	2	no	—
251	C₅	#5832	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	30	4	no	—
256	C₂²	—	30	2	no	—
257	C₂²	C₂×#1220	1	2	yes	—
258	C₂²	C₂³×D₁₅	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	15	4	no	—
261	C₄	—	30	2	no	—
262	C₄	—	30	2	no	—
263	C₄	C₂×#1220	1	2	yes	—
264	C₄	C₂×#1220	1	2	yes	—
265	C₃	#66166	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	2	2	no	—
267	C₂	—	4	2	no	—
268	C₂	—	60	2	no	—
269	C₂	#41×D₁₅	1	2	yes	—
270	C₂	C₂²×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Frattini	C₂²	C₂³×D₁₅
Fitting	C₂×C₃×C₅×#41	C₂
hypercenter	C₂×#41	D₁₅
hypocenter	C₃×C₅	#532
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#41	C₂	semidirect
2. C₂×#6817	C₂	semidirect
3. #91836	C₂	semidirect
4. #91833	C₂	semidirect
5. #91829	C₂	semidirect
6. #91828	C₂	semidirect
7. #91825	C₂	semidirect
8. C₄×#1220	C₂	semidirect
9. C₂×C₄×Dic₁₅	C₂	semidirect
31. C₂×C₅×#41	D₃	semidirect
47. C₂×C₃×C₄×C₅	D₄	semidirect
60. C₂×C₃×#41	D₅	semidirect
101. Dic₁₅	#41	semidirect
128. C₂×C₄×C₅	#67	semidirect
142. C₂×#41	D₁₅	semidirect
162. C₂×C₃×C₄	C₂²⋊D₅	semidirect
207. C₃×C₅	#532	semidirect
242. C₂×C₄	#1220	semidirect
251. C₅	#5832	semidirect
265. C₃	#66166	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#41	2
C₂×#6817	2
#91836	2
#91833	2
#91829	2
#91828	2
#91825	2
C₄×#1220	2
C₂×C₄×Dic₁₅	2
#66166	3
#5832	5