Subgroups — #11753401

Order 960.

588 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6766	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4039	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#1202	C₂	1	2	yes	maximal, complemented
6	Dic₃×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
7	#91396	C₂	1	2	yes	maximal, complemented
8	#91393	C₂	1	2	yes	maximal, complemented
9	D₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
10	#91386	C₂	1	1	yes	maximal, characteristic, complemented
11	Dic₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#65908	—	3	1	no	maximal, Hall
13	C₂×C₄×D₁₅	C₂²	1	2	yes	—
14	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
18	C₂×#1202	C₂²	1	2	yes	—
19	C₂×D₅×Dic₃	C₂²	1	2	yes	—
20	C₂×D₅×Dic₃	C₂²	1	2	yes	—
21	#6766	C₄	1	8	yes	complemented
22	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
24	#6712	C₂²	1	2	yes	—
25	C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₄×C₅×Dic₃	C₂²	1	2	yes	—
27	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
28	C₃×#685	C₂²	1	2	yes	—
29	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	Dic₃⋊Dic₅	C₂²	1	2	yes	—
31	#6669	C₂²	1	1	yes	characteristic
32	#6669	C₂²	1	1	yes	characteristic
33	#6667	C₂²	1	1	yes	characteristic
34	#6667	C₂²	1	1	yes	characteristic
35	#6665	C₂²	1	2	yes	—
36	Dic₃×Dic₅	C₂²	1	2	yes	—
37	#5546	—	5	1	no	maximal, Hall
38	C₂×#713	—	3	1	no	—
39	C₄×C₅×Q₈	—	3	1	no	—
40	Q₈×Dic₅	—	3	1	no	—
41	#4039	—	3	2	no	—
42	#4039	D₃	1	1	yes	characteristic, complemented
43	D₅×C₄⋊C₄	—	3	2	no	—
44	D₅×C₄⋊C₄	—	3	1	no	—
45	C₄×D₂₀	—	3	1	no	—
46	C₄×D₂₀	—	3	2	no	—
47	#4018	—	3	2	no	—
48	#4018	—	3	1	no	—
49	C₂²×D₁₅	C₂³	1	1	yes	characteristic
50	C₂²×C₃×D₅	C₂³	1	2	yes	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
55	C₄×D₁₅	—	4	2	no	—
56	C₄×D₁₅	C₂×C₄	1	4	yes	complemented
57	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂×C₅×Dic₃	C₂³	1	2	yes	—
60	C₅×Dic₆	C₂×C₄	1	4	yes	complemented
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₃×D₂₀	C₂×C₄	1	4	yes	complemented
64	C₃×C₄×D₅	—	4	2	no	—
65	#1202	C₂×C₄	1	8	yes	complemented
66	D₅×Dic₃	—	4	2	no	—
67	D₅×Dic₃	C₂×C₄	1	8	yes	complemented
68	C₂×#237	—	5	1	no	—
69	C₃×C₄×D₄	—	5	1	no	—
70	D₄×Dic₃	—	5	1	no	—
71	C₄×#67	—	5	2	no	—
72	C₂×Dic₃⋊C₄	—	5	2	no	—
73	D₃×C₄⋊C₄	—	5	1	no	—
74	#887	D₅	1	1	yes	characteristic, complemented
75	#67⋊C₄	—	5	2	no	—
76	#877	—	5	2	no	—
77	#872	—	5	1	no	—
78	C₄×Dic₆	—	5	1	no	—
79	C₂×C₅×Q₈	—	3	1	no	—
80	#713	—	3	8	no	—
81	C₂×D₂₀	—	3	2	no	—
82	C₂×D₂₀	C₂×D₃	1	1	yes	characteristic
83	C₂×C₄×D₅	—	3	2	no	—
84	C₂×C₄×D₅	—	3	1	no	—
85	C₂×C₄×D₅	—	3	2	no	—
86	C₂×C₄×D₅	—	3	2	no	—
87	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
88	C₅×C₄⋊C₄	—	3	2	no	—
89	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
90	C₄²×C₅	—	3	2	no	—
91	C₄²×C₅	—	3	1	no	—
92	#685	—	3	2	no	—
93	#685	—	3	1	no	—
94	#685	—	3	1	no	—
95	#685	C₂×D₃	1	2	yes	—
96	C₄⋊Dic₅	—	3	1	no	—
97	C₄⋊Dic₅	—	3	2	no	—
98	#683	—	3	2	no	—
99	#683	—	3	1	no	—
100	#683	—	3	1	no	—
101	C₄×Dic₅	—	3	2	no	—
102	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
103	#520	—	15	1	no	Sylow
104	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
105	C₂×D₁₅	—	2	1	no	—
106	C₂×D₁₅	—	2	1	no	—
107	C₂×D₁₅	C₂²×C₄	1	2	yes	—
108	C₂×C₃×D₅	—	2	2	no	—
109	C₂×C₃×D₅	—	2	2	no	—
110	C₂×C₃×D₅	C₂²×C₄	1	4	yes	—
111	C₃×C₄×C₅	—	2	2	no	—
112	C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
113	Dic₁₅	—	2	2	no	—
114	Dic₁₅	C₂²×C₄	1	2	yes	—
115	C₃×Dic₅	—	2	1	no	—
116	C₃×Dic₅	—	2	1	no	—
117	C₅×Dic₃	—	2	1	no	—
118	C₅×Dic₃	—	2	1	no	—
119	C₅×Dic₃	C₂²×C₄	1	4	yes	—
120	C₂×C₃×D₄	—	5	1	no	—
121	C₂²×C₃×C₄	—	5	2	no	—
122	C₂×#67	—	5	2	no	—
123	C₂²×Dic₃	—	5	2	no	—
124	C₂²×Dic₃	—	5	2	no	—
125	#237	—	5	8	no	—
126	C₂×C₄×D₃	—	5	1	no	—
127	C₂×C₄×D₃	—	5	2	no	—
128	C₂×Dic₆	C₂×D₅	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
130	C₃×C₂²⋊C₄	—	5	2	no	—
131	C₃×C₄²	—	5	1	no	—
132	#217	—	5	2	no	—
133	(C₂×D₃)⋊C₄	—	5	1	no	—
134	(C₂×D₃)⋊C₄	—	5	1	no	—
135	C₄⋊Dic₃	—	5	1	no	—
136	Dic₃⋊C₄	—	5	2	no	—
137	Dic₃⋊C₄	—	5	1	no	—
138	Dic₃⋊C₄	—	5	1	no	—
139	Dic₃⋊C₄	—	5	2	no	—
140	Dic₃⋊C₄	C₂×D₅	1	2	yes	—
141	C₄×Dic₃	—	5	2	no	—
142	C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
143	C₄×Dic₃	C₂×D₅	1	2	yes	—
144	C₂²×D₅	—	3	1	no	—
145	C₂²×D₅	C₂²×D₃	1	2	yes	—
146	C₅×Q₈	—	3	4	no	—
147	C₂×C₄×C₅	—	3	2	no	—
148	C₂×C₄×C₅	—	3	1	no	—
149	C₂×C₄×C₅	—	3	1	no	—
150	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
151	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
152	C₂×Dic₅	—	3	1	no	—
153	C₂×Dic₅	—	3	2	no	—
154	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
155	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
156	D₂₀	—	3	8	no	—
157	D₂₀	C₄×D₃	1	4	yes	complemented
158	C₄×D₅	—	3	8	no	—
159	C₄×D₅	—	3	4	no	—
160	C₄×D₅	—	4	2	no	—
161	C₄×D₅	—	12	2	no	—
162	C₄×D₅	—	12	2	no	—
163	C₂×#41	—	15	1	no	—
164	C₄×Q₈	—	3	1	no	—
165	C₄×Q₈	—	15	1	no	—
166	C₄×D₄	—	5	1	no	—
167	C₄×D₄	—	15	2	no	—
168	C₄×D₄	—	15	1	no	—
169	C₄×D₄	—	15	2	no	—
170	#117	—	15	1	no	—
171	#117	—	15	2	no	—
172	C₂×C₄⋊C₄	—	15	2	no	—
173	C₂×C₄⋊C₄	—	15	1	no	—
174	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
177	D₁₅	—	2	2	no	—
178	C₃×D₅	—	2	4	no	—
179	C₂³×C₃	—	5	2	no	—
180	C₂²×D₃	—	5	1	no	—
181	C₃×D₄	—	5	4	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	20	2	no	—
185	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
187	#67	—	5	8	no	—
188	C₂×Dic₃	—	5	1	no	—
189	C₂×Dic₃	—	5	8	no	—
190	C₂×Dic₃	—	5	2	no	—
191	C₂×Dic₃	—	20	2	no	—
192	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
193	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
194	C₂×Dic₃	C₂²×D₅	1	2	yes	—
195	C₄×D₃	—	5	4	no	—
196	C₄×D₃	—	20	2	no	—
197	Dic₆	C₄×D₅	1	4	yes	complemented
198	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×D₅	—	2	2	no	—
200	C₂×D₅	—	2	2	no	—
201	C₂×D₅	—	3	2	no	—
202	C₂×D₅	—	6	1	no	—
203	C₂×D₅	—	6	1	no	—
204	C₂×D₅	C₂×C₄×D₃	1	4	yes	—
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	3	4	no	—
207	C₄×C₅	—	6	1	no	—
208	C₄×C₅	—	6	1	no	—
209	C₄×C₅	C₂×C₄×D₃	1	2	yes	—
210	Dic₅	—	2	1	no	—
211	Dic₅	—	2	1	no	—
212	Dic₅	—	3	2	no	—
213	Dic₅	—	6	2	no	—
214	#41	—	15	8	no	—
215	C₂×Q₈	—	3	1	no	—
216	C₂×D₄	—	5	1	no	—
217	C₂×D₄	—	15	2	no	—
218	C₂²×C₄	—	5	2	no	—
219	C₂²×C₄	—	15	2	no	—
220	C₂²×C₄	—	15	2	no	—
221	C₂²×C₄	—	15	1	no	—
222	C₂²×C₄	—	15	2	no	—
223	C₄⋊C₄	—	3	2	no	—
224	C₄⋊C₄	—	15	1	no	—
225	C₄⋊C₄	—	15	2	no	—
226	C₄⋊C₄	—	15	1	no	—
227	C₄⋊C₄	—	15	2	no	—
228	C₄⋊C₄	—	15	1	no	—
229	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
230	C₂²⋊C₄	—	5	2	no	—
231	C₂²⋊C₄	—	15	1	no	—
232	C₂²⋊C₄	—	15	1	no	—
233	C₂²⋊C₄	—	15	2	no	—
234	C₄²	—	3	2	no	—
235	C₄²	—	3	1	no	—
236	C₄²	—	5	1	no	—
237	C₄²	—	15	2	no	—
238	C₃×C₅	#520	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	5	4	no	—
240	C₂²×C₃	—	10	2	no	—
241	C₂²×C₃	—	10	2	no	—
242	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
243	C₂×D₃	—	5	2	no	—
244	C₂×D₃	—	10	1	no	—
245	C₂×D₃	—	10	1	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	10	1	no	—
248	C₃×C₄	—	10	1	no	—
249	C₃×C₄	C₂×C₄×D₅	1	2	yes	—
250	Dic₃	—	2	1	no	—
251	Dic₃	—	2	1	no	—
252	Dic₃	—	5	2	no	—
253	Dic₃	—	10	2	no	—
254	Dic₃	C₂×C₄×D₅	1	4	yes	—
255	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
256	C₂×C₅	#1004	1	1	yes	characteristic
257	C₂×C₅	#1010	1	1	yes	characteristic
258	D₅	—	2	4	no	—
259	D₅	—	6	2	no	—
260	C₂³	—	5	2	no	—
261	C₂³	—	15	1	no	—
262	Q₈	—	3	4	no	—
263	D₄	—	5	4	no	—
264	D₄	—	15	8	no	—
265	C₂×C₄	—	3	2	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	1	no	—
270	C₂×C₄	—	15	8	no	—
271	C₂×C₄	—	15	1	no	—
272	C₂×C₄	—	15	4	no	—
273	C₂×C₄	—	15	2	no	—
274	C₂×C₄	—	20	2	no	—
275	C₂×C₄	—	60	2	no	—
276	C₂×C₄	—	60	2	no	—
277	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
278	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
279	C₂×C₃	—	10	4	no	—
280	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
281	C₂×C₃	#4147	1	1	yes	characteristic
282	C₂×C₃	#4149	1	1	yes	characteristic
283	D₃	—	10	2	no	—
284	C₅	#5546	1	1	yes	characteristic, complemented, Sylow
285	C₂²	—	5	4	no	—
286	C₂²	—	10	2	no	—
287	C₂²	—	10	2	no	—
288	C₂²	—	15	2	no	—
289	C₂²	—	30	1	no	—
290	C₂²	—	30	1	no	—
291	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	3	4	no	—
294	C₄	—	6	1	no	—
295	C₄	—	6	1	no	—
296	C₄	—	10	1	no	—
297	C₄	—	10	1	no	—
298	C₄	—	15	2	no	—
299	C₄	—	30	2	no	—
300	C₄	C₄×D₃×D₅	1	2	yes	—
301	C₃	#65908	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	10	4	no	—
303	C₂	—	30	2	no	—
304	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic
305	C₂	#92026	1	1	yes	characteristic
306	C₂	#92028	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂³×C₄
Fitting	C₃×C₅×C₄⋊C₄	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂×#6766	C₂
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#520
LCS	G ▷ C₂×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
2. C₄⋊C₄×D₁₅	C₂	semidirect
3. C₅×#887	C₂	semidirect
4. C₃×#4039	C₂	semidirect
5. C₄×#1202	C₂	semidirect
6. Dic₃×D₂₀	C₂	semidirect
7. #91396	C₂	semidirect
8. #91393	C₂	semidirect
9. D₅×Dic₃⋊C₄	C₂	semidirect
10. #91386	C₂	semidirect
11. Dic₅×Dic₆	C₂	semidirect
21. #6766	C₄	semidirect
42. #4039	D₃	semidirect
56. C₄×D₁₅	C₂×C₄	semidirect
60. C₅×Dic₆	C₂×C₄	semidirect
63. C₃×D₂₀	C₂×C₄	semidirect
65. #1202	C₂×C₄	semidirect
67. D₅×Dic₃	C₂×C₄	semidirect
74. #887	D₅	semidirect
157. D₂₀	C₄×D₃	semidirect
197. Dic₆	C₄×D₅	semidirect
238. C₃×C₅	#520	semidirect
284. C₅	#5546	semidirect
301. C₃	#65908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6766	2
C₄⋊C₄×D₁₅	2
C₅×#887	2
C₃×#4039	2
C₄×#1202	2
Dic₃×D₂₀	2
#91396	2
#91393	2
D₅×Dic₃⋊C₄	2
#91386	2
Dic₅×Dic₆	2
#65908	3
#5546	5