Subgroups — #11753715

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
4	#91468	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#1203	C₂	1	1	yes	maximal, characteristic
6	C₄×#1203	C₂	1	2	yes	maximal
7	#91390	C₂	1	1	yes	maximal, characteristic, complemented
8	#91378	C₂	1	2	yes	maximal, complemented
9	#91377	C₂	1	2	yes	maximal, complemented
10	#91369	C₂	1	1	yes	maximal, characteristic, complemented
11	#91352	C₂	1	2	yes	maximal, complemented
12	#65845	—	3	1	no	maximal, Hall
13	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₅	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×#1204	C₂²	1	2	yes	—
19	C₂×#1203	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
22	#6712	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₃×#683	C₂²	1	2	yes	—
26	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
27	C₃×C₄×Dic₅	C₂²	1	2	yes	—
28	Dic₃⋊Dic₅	C₂²	1	2	yes	—
29	#6669	C₂²	1	2	yes	—
30	#6668	C₂²	1	2	yes	—
31	#6668	C₂²	1	1	yes	characteristic
32	#6667	C₂²	1	1	yes	characteristic
33	#6667	C₂²	1	2	yes	—
34	(C₂×D₃)⋊Dic₅	C₂²	1	1	yes	characteristic
35	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
36	#5579	—	5	1	no	maximal, Hall
37	C₂×C₅×C₄⋊C₄	—	3	1	no	—
38	C₄×C₂²⋊D₅	—	3	1	no	—
39	C₄×C₂²⋊D₅	—	3	2	no	—
40	#4070	—	3	1	no	—
41	#4070	—	3	2	no	—
42	#4043	—	3	1	no	—
43	#4042	—	3	2	no	—
44	D₅×C₄⋊C₄	—	3	1	no	—
45	#4035	—	3	2	no	—
46	#4034	—	3	1	no	—
47	#4033	D₃	1	1	yes	characteristic, complemented
48	C₂²×D₁₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂³	1	2	yes	—
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	C₄×D₁₅	—	2	2	no	—
55	C₄×D₁₅	—	4	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₂×C₅×Dic₃	C₂³	1	2	yes	—
58	C₄×C₅×D₃	—	2	2	no	—
59	C₄×C₅×D₃	—	4	2	no	—
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₃×Dic₁₀	—	2	2	no	—
64	#1204	—	4	2	no	—
65	#1203	Q₈	1	4	yes	complemented
66	C₃×C₄×Q₈	—	5	1	no	—
67	#946	—	5	1	no	—
68	#946	—	5	1	no	—
69	#896	—	5	2	no	—
70	#896	—	5	2	no	—
71	D₃×C₄⋊C₄	—	5	1	no	—
72	D₃×C₄⋊C₄	D₅	1	1	yes	characteristic, complemented
73	C₄×D₁₂	—	5	2	no	—
74	C₄×D₁₂	—	5	1	no	—
75	#870	—	5	2	no	—
76	#869	—	5	1	no	—
77	C₂²×C₄×C₅	—	3	1	no	—
78	C₂²×C₄×C₅	—	3	2	no	—
79	C₂×C₂²⋊D₅	—	3	1	no	—
80	C₂×C₄×D₅	—	3	1	no	—
81	C₂×C₄×D₅	—	3	2	no	—
82	C₂×Dic₁₀	—	3	2	no	—
83	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	—	3	1	no	—
86	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
87	C₂²⋊Dic₅	—	3	1	no	—
88	C₂²⋊Dic₅	—	3	2	no	—
89	#685	—	3	1	no	—
90	#685	—	3	2	no	—
91	C₄⋊Dic₅	—	3	1	no	—
92	C₄⋊Dic₅	—	3	2	no	—
93	#683	—	3	2	no	—
94	#683	—	3	2	no	—
95	#683	—	3	1	no	—
96	#683	—	3	2	no	—
97	#683	C₂×D₃	1	2	yes	—
98	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
99	C₄×Dic₅	C₂×D₃	1	2	yes	—
100	#554	—	15	1	no	Sylow
101	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×D₁₅	—	2	1	no	—
103	C₂×D₁₅	—	2	1	no	—
104	C₂×D₁₅	C₂×Q₈	1	2	yes	—
105	C₂×C₅×D₃	—	2	1	no	—
106	C₂×C₅×D₃	—	2	1	no	—
107	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	Dic₁₅	—	2	1	no	—
111	Dic₁₅	—	2	2	no	—
112	C₃×Dic₅	—	2	2	no	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	C₂×Q₈	1	2	yes	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	—	2	2	no	—
117	C₂×C₃×Q₈	—	5	1	no	—
118	C₂×D₁₂	—	5	1	no	—
119	C₂×C₄×D₃	—	5	1	no	—
120	C₂×C₄×D₃	—	5	2	no	—
121	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
122	C₂×C₄×D₃	C₂×D₅	1	2	yes	—
123	C₂×Dic₆	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
126	C₃×C₄²	—	5	1	no	—
127	C₃×C₄²	—	5	2	no	—
128	(C₂×D₃)⋊C₄	—	5	1	no	—
129	(C₂×D₃)⋊C₄	—	5	2	no	—
130	(C₂×D₃)⋊C₄	—	5	1	no	—
131	(C₂×D₃)⋊C₄	—	5	2	no	—
132	C₄⋊Dic₃	—	5	1	no	—
133	C₄⋊Dic₃	—	5	2	no	—
134	C₄⋊Dic₃	—	5	1	no	—
135	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
136	Dic₃⋊C₄	—	5	2	no	—
137	Dic₃⋊C₄	—	5	2	no	—
138	Dic₃⋊C₄	—	5	2	no	—
139	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
140	C₂³×C₅	—	3	1	no	—
141	C₂²×D₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	2	no	—
144	C₂×C₄×C₅	—	6	2	no	—
145	C₂×C₄×C₅	—	12	2	no	—
146	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
147	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
148	C₂²⋊D₅	—	3	4	no	—
149	C₂×Dic₅	—	3	1	no	—
150	C₂×Dic₅	—	3	2	no	—
151	C₂×Dic₅	C₂²×D₃	1	2	yes	—
152	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
153	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
154	C₄×D₅	—	6	2	no	—
155	C₄×D₅	—	12	2	no	—
156	Dic₁₀	—	2	2	no	—
157	Dic₁₀	—	12	2	no	—
158	C₄⋊Q₈	—	15	1	no	—
159	#125	—	15	2	no	—
160	C₂²⋊Q₈	—	15	2	no	—
161	C₂²⋊Q₈	—	15	1	no	—
162	C₂²⋊Q₈	—	15	2	no	—
163	C₂²⋊Q₈	—	15	1	no	—
164	C₄×Q₈	—	5	1	no	—
165	C₄×D₄	—	15	2	no	—
166	C₄×D₄	—	15	1	no	—
167	C₂×C₄⋊C₄	—	3	1	no	—
168	C₂×C₄⋊C₄	—	15	1	no	—
169	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
170	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
172	D₁₅	—	2	2	no	—
173	C₅×D₃	—	2	2	no	—
174	C₂²×D₃	—	5	1	no	—
175	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
176	C₃×Q₈	—	10	2	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×Dic₃	—	5	1	no	—
183	C₂×Dic₃	—	5	2	no	—
184	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
185	C₂×Dic₃	C₂²×D₅	1	2	yes	—
186	D₁₂	—	5	4	no	—
187	C₄×D₃	—	2	2	no	—
188	C₄×D₃	—	4	2	no	—
189	C₄×D₃	—	10	2	no	—
190	C₄×D₃	—	20	2	no	—
191	Dic₆	—	20	2	no	—
192	C₂²×C₅	—	3	2	no	—
193	C₂²×C₅	—	6	1	no	—
194	C₂²×C₅	—	6	1	no	—
195	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₂×D₅	—	3	2	no	—
197	C₂×D₅	—	6	1	no	—
198	C₂×D₅	—	6	1	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	6	1	no	—
201	C₄×C₅	—	6	2	no	—
202	C₄×C₅	#235	1	2	yes	—
203	Dic₅	—	2	2	no	—
204	Dic₅	—	2	1	no	—
205	Dic₅	—	6	1	no	—
206	Dic₅	—	6	2	no	—
207	Dic₅	D₃×Q₈	1	2	yes	—
208	C₂×Q₈	—	5	1	no	—
209	C₂×Q₈	—	15	2	no	—
210	C₂×D₄	—	15	1	no	—
211	C₂²×C₄	—	3	1	no	—
212	C₂²×C₄	—	3	2	no	—
213	C₂²×C₄	—	15	1	no	—
214	C₂²×C₄	—	15	2	no	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	5	2	no	—
218	C₄⋊C₄	—	15	1	no	—
219	C₄⋊C₄	—	15	2	no	—
220	C₄⋊C₄	—	15	2	no	—
221	C₄⋊C₄	—	15	2	no	—
222	C₄⋊C₄	—	15	2	no	—
223	C₄⋊C₄	—	15	1	no	—
224	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
225	C₂²⋊C₄	—	15	1	no	—
226	C₂²⋊C₄	—	15	2	no	—
227	C₂²⋊C₄	—	15	1	no	—
228	C₂²⋊C₄	—	15	2	no	—
229	C₄²	—	5	2	no	—
230	C₄²	—	5	1	no	—
231	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
233	C₂×D₃	—	2	1	no	—
234	C₂×D₃	—	2	1	no	—
235	C₂×D₃	—	5	2	no	—
236	C₂×D₃	—	10	1	no	—
237	C₂×D₃	—	10	1	no	—
238	C₂×D₃	Q₈×D₅	1	2	yes	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	5	2	no	—
241	C₃×C₄	—	10	2	no	—
242	C₃×C₄	—	10	1	no	—
243	C₃×C₄	#709	1	2	yes	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	2	2	no	—
246	Dic₃	—	10	1	no	—
247	Dic₃	—	10	2	no	—
248	C₂×C₅	—	6	2	no	—
249	C₂×C₅	C₂×#235	1	1	yes	characteristic
250	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
251	C₂×C₅	#1009	1	1	yes	characteristic
252	D₅	—	6	2	no	—
253	C₂³	—	3	1	no	—
254	C₂³	—	15	1	no	—
255	Q₈	—	10	2	no	—
256	Q₈	—	60	2	no	—
257	D₄	—	15	4	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	2	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	6	2	no	—
264	C₂×C₄	—	12	2	no	—
265	C₂×C₄	—	15	1	no	—
266	C₂×C₄	—	15	2	no	—
267	C₂×C₄	—	30	2	no	—
268	C₂×C₄	—	60	2	no	—
269	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
270	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₃	C₂×#709	1	1	yes	characteristic
272	C₂×C₃	#4144	1	1	yes	characteristic
273	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
274	D₃	—	2	2	no	—
275	D₃	—	10	2	no	—
276	C₅	#5579	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	3	2	no	—
278	C₂²	—	6	1	no	—
279	C₂²	—	6	1	no	—
280	C₂²	—	15	2	no	—
281	C₂²	—	30	1	no	—
282	C₂²	—	30	1	no	—
283	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	5	2	no	—
286	C₄	—	6	1	no	—
287	C₄	—	6	2	no	—
288	C₄	—	10	2	no	—
289	C₄	—	10	1	no	—
290	C₄	—	30	1	no	—
291	C₄	—	30	2	no	—
292	C₄	#6771	1	2	yes	—
293	C₃	#65845	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	6	2	no	—
295	C₂	—	30	2	no	—
296	C₂	C₂×#6771	1	1	yes	characteristic
297	C₂	#92027	1	1	yes	characteristic
298	C₂	D₃×Q₈×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₄⋊C₄	C₂²
prime radical	C₂×#1203	C₂²
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#554
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊C₄×D₁₅	C₂	semidirect
2. C₅×D₃×C₄⋊C₄	C₂	semidirect
3. C₃×#4033	C₂	semidirect
4. #91468	C₂	semidirect
7. #91390	C₂	semidirect
8. #91378	C₂	semidirect
9. #91377	C₂	semidirect
10. #91369	C₂	semidirect
11. #91352	C₂	semidirect
47. #4033	D₃	semidirect
65. #1203	Q₈	semidirect
72. D₃×C₄⋊C₄	D₅	semidirect
231. C₃×C₅	#554	semidirect
276. C₅	#5579	semidirect
293. C₃	#65845	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×D₁₅	2
C₅×D₃×C₄⋊C₄	2
C₃×#4033	2
#91468	2
C₄×#1203	2
#91390	2
#91378	2
#91377	2
#91369	2
#91352	2
#65845	3
#5579	5