Subgroups — #11753712

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×#887	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#91469	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#1202	C₂	1	1	yes	maximal, characteristic
6	C₄×#1202	C₂	1	2	yes	maximal
7	#91383	C₂	1	1	yes	maximal, characteristic, complemented
8	#91379	C₂	1	2	yes	maximal, complemented
9	#91377	C₂	1	2	yes	maximal, complemented
10	#91359	C₂	1	1	yes	maximal, characteristic, complemented
11	#91349	C₂	1	2	yes	maximal, complemented
12	#65913	—	3	1	no	maximal, Hall
13	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₅	C₂²	1	2	yes	—
15	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×#1204	C₂²	1	2	yes	—
19	C₂×#1202	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×Dic₃⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
25	C₄×C₅×Dic₃	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×#683	C₂²	1	2	yes	—
28	Dic₃⋊Dic₅	C₂²	1	2	yes	—
29	Dic₃⋊Dic₅	C₂²	1	1	yes	characteristic
30	#6669	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	#6667	C₂²	1	1	yes	characteristic
33	#6667	C₂²	1	2	yes	—
34	#6665	C₂²	1	1	yes	characteristic
35	#6665	C₂²	1	2	yes	—
36	#5511	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	—	3	1	no	—
38	#4092	—	3	1	no	—
39	#4092	—	3	1	no	—
40	#4042	—	3	2	no	—
41	#4042	—	3	2	no	—
42	D₅×C₄⋊C₄	—	3	1	no	—
43	D₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
44	C₄×D₂₀	—	3	2	no	—
45	C₄×D₂₀	—	3	1	no	—
46	#4016	—	3	2	no	—
47	#4015	—	3	1	no	—
48	C₂²×D₁₅	C₂³	1	1	yes	characteristic
49	C₂²×C₃×D₅	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	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₄×D₁₅	—	2	2	no	—
55	C₄×D₁₅	—	4	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₅×Dic₆	—	2	2	no	—
60	C₂×C₃×Dic₅	C₂³	1	2	yes	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₅	—	2	2	no	—
63	C₃×C₄×D₅	—	4	2	no	—
64	#1204	—	4	2	no	—
65	#1202	Q₈	1	4	yes	complemented
66	C₂×C₃×C₄⋊C₄	—	5	1	no	—
67	C₄×#67	—	5	1	no	—
68	C₄×#67	—	5	2	no	—
69	#924	—	5	1	no	—
70	#924	—	5	2	no	—
71	#897	—	5	1	no	—
72	#896	—	5	2	no	—
73	D₃×C₄⋊C₄	—	5	1	no	—
74	#889	—	5	2	no	—
75	#888	—	5	1	no	—
76	#887	D₅	1	1	yes	characteristic, complemented
77	C₂×C₅×Q₈	—	3	1	no	—
78	C₂×D₂₀	—	3	1	no	—
79	C₂×C₄×D₅	—	3	1	no	—
80	C₂×C₄×D₅	—	3	2	no	—
81	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
82	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
83	C₂×Dic₁₀	—	3	2	no	—
84	C₅×C₄⋊C₄	—	3	2	no	—
85	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
86	C₄²×C₅	—	3	1	no	—
87	C₄²×C₅	—	3	2	no	—
88	#685	—	3	1	no	—
89	#685	—	3	2	no	—
90	#685	—	3	1	no	—
91	#685	—	3	2	no	—
92	C₄⋊Dic₅	—	3	1	no	—
93	C₄⋊Dic₅	—	3	2	no	—
94	C₄⋊Dic₅	—	3	1	no	—
95	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
96	#683	—	3	2	no	—
97	#683	—	3	2	no	—
98	#683	—	3	2	no	—
99	#683	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×C₃×D₅	—	2	1	no	—
106	C₂×C₃×D₅	—	2	1	no	—
107	C₂×C₃×D₅	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	1	no	—
113	C₃×Dic₅	—	2	2	no	—
114	C₅×Dic₃	—	2	2	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	C₂×Q₈	1	2	yes	—
117	C₂²×C₃×C₄	—	5	1	no	—
118	C₂²×C₃×C₄	—	5	2	no	—
119	C₂×#67	—	5	1	no	—
120	C₂×C₄×D₃	—	5	1	no	—
121	C₂×C₄×D₃	—	5	2	no	—
122	C₂×Dic₆	—	5	2	no	—
123	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	—	5	1	no	—
126	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
127	#217	—	5	1	no	—
128	#217	—	5	2	no	—
129	(C₂×D₃)⋊C₄	—	5	1	no	—
130	(C₂×D₃)⋊C₄	—	5	2	no	—
131	C₄⋊Dic₃	—	5	1	no	—
132	C₄⋊Dic₃	—	5	2	no	—
133	Dic₃⋊C₄	—	5	2	no	—
134	Dic₃⋊C₄	—	5	2	no	—
135	Dic₃⋊C₄	—	5	2	no	—
136	Dic₃⋊C₄	—	5	1	no	—
137	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
138	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
139	C₄×Dic₃	C₂×D₅	1	2	yes	—
140	C₂²×D₅	—	3	1	no	—
141	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
142	C₅×Q₈	—	6	2	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	3	2	no	—
145	C₂×C₄×C₅	—	3	1	no	—
146	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
147	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
148	C₂×Dic₅	—	3	1	no	—
149	C₂×Dic₅	—	3	2	no	—
150	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×Dic₅	C₂²×D₃	1	2	yes	—
152	D₂₀	—	3	4	no	—
153	C₄×D₅	—	2	2	no	—
154	C₄×D₅	—	4	2	no	—
155	C₄×D₅	—	6	2	no	—
156	C₄×D₅	—	12	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	2	no	—
162	C₂²⋊Q₈	—	15	1	no	—
163	C₂²⋊Q₈	—	15	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×D₄	—	15	2	no	—
166	C₄×D₄	—	15	1	no	—
167	C₂×C₄⋊C₄	—	5	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	C₃×D₅	—	2	2	no	—
174	C₂³×C₃	—	5	1	no	—
175	C₂²×D₃	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	2	no	—
178	C₂×C₃×C₄	—	10	2	no	—
179	C₂×C₃×C₄	—	20	2	no	—
180	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	#67	—	5	4	no	—
183	C₂×Dic₃	—	5	1	no	—
184	C₂×Dic₃	—	5	2	no	—
185	C₂×Dic₃	C₂²×D₅	1	2	yes	—
186	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
187	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
188	C₄×D₃	—	10	2	no	—
189	C₄×D₃	—	20	2	no	—
190	Dic₆	—	2	2	no	—
191	Dic₆	—	20	2	no	—
192	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
193	C₂×D₅	—	2	1	no	—
194	C₂×D₅	—	2	1	no	—
195	C₂×D₅	—	3	2	no	—
196	C₂×D₅	—	6	1	no	—
197	C₂×D₅	—	6	1	no	—
198	C₂×D₅	D₃×Q₈	1	2	yes	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	3	2	no	—
201	C₄×C₅	—	6	2	no	—
202	C₄×C₅	—	6	1	no	—
203	C₄×C₅	#235	1	2	yes	—
204	Dic₅	—	2	1	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	6	1	no	—
207	Dic₅	—	6	2	no	—
208	C₂×Q₈	—	3	1	no	—
209	C₂×Q₈	—	15	2	no	—
210	C₂×D₄	—	15	1	no	—
211	C₂²×C₄	—	5	1	no	—
212	C₂²×C₄	—	5	2	no	—
213	C₂²×C₄	—	15	1	no	—
214	C₂²×C₄	—	15	2	no	—
215	C₄⋊C₄	—	3	2	no	—
216	C₄⋊C₄	—	5	2	no	—
217	C₄⋊C₄	—	5	1	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	—	3	2	no	—
230	C₄²	—	3	1	no	—
231	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	5	2	no	—
233	C₂²×C₃	—	10	1	no	—
234	C₂²×C₃	—	10	1	no	—
235	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
236	C₂×D₃	—	5	2	no	—
237	C₂×D₃	—	10	1	no	—
238	C₂×D₃	—	10	1	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	10	1	no	—
241	C₃×C₄	—	10	2	no	—
242	C₃×C₄	#709	1	2	yes	—
243	Dic₃	—	2	2	no	—
244	Dic₃	—	2	1	no	—
245	Dic₃	—	10	1	no	—
246	Dic₃	—	10	2	no	—
247	Dic₃	Q₈×D₅	1	2	yes	—
248	C₂×C₅	C₂×#235	1	1	yes	characteristic
249	C₂×C₅	#1004	1	1	yes	characteristic
250	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
251	D₅	—	2	2	no	—
252	D₅	—	6	2	no	—
253	C₂³	—	5	1	no	—
254	C₂³	—	15	1	no	—
255	Q₈	—	6	2	no	—
256	Q₈	—	60	2	no	—
257	D₄	—	15	4	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	2	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	10	2	no	—
264	C₂×C₄	—	15	1	no	—
265	C₂×C₄	—	15	2	no	—
266	C₂×C₄	—	20	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	—	10	2	no	—
272	C₂×C₃	C₂×#709	1	1	yes	characteristic
273	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
274	C₂×C₃	#4149	1	1	yes	characteristic
275	D₃	—	10	2	no	—
276	C₅	#5511	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	5	2	no	—
278	C₂²	—	10	1	no	—
279	C₂²	—	10	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	—	3	2	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	1	no	—
288	C₄	—	10	1	no	—
289	C₄	—	10	2	no	—
290	C₄	—	30	1	no	—
291	C₄	—	30	2	no	—
292	C₄	#6771	1	2	yes	—
293	C₃	#65913	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	10	2	no	—
295	C₂	—	30	2	no	—
296	C₂	C₂×#6771	1	1	yes	characteristic
297	C₂	#92026	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×#1202	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×#887	C₂	semidirect
3. C₃×D₅×C₄⋊C₄	C₂	semidirect
4. #91469	C₂	semidirect
7. #91383	C₂	semidirect
8. #91379	C₂	semidirect
9. #91377	C₂	semidirect
10. #91359	C₂	semidirect
11. #91349	C₂	semidirect
43. D₅×C₄⋊C₄	D₃	semidirect
65. #1202	Q₈	semidirect
76. #887	D₅	semidirect
231. C₃×C₅	#554	semidirect
276. C₅	#5511	semidirect
293. C₃	#65913	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×#887	2
C₃×D₅×C₄⋊C₄	2
#91469	2
C₄×#1202	2
#91383	2
#91379	2
#91377	2
#91359	2
#91349	2
#65913	3
#5511	5