Subgroups — #11753618

Order 960.

444 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#91777	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#892	C₂	1	1	yes	maximal, characteristic
3	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
4	#91470	C₂	1	1	yes	maximal, characteristic, complemented
5	#91436	C₂	1	2	yes	maximal
6	#91432	C₂	1	2	yes	maximal
7	C₄×D₃×Dic₅	C₂	1	1	yes	maximal, characteristic
8	#91388	C₂	1	1	yes	maximal, characteristic, complemented
9	#91340	C₂	1	2	yes	maximal, complemented
10	#91338	C₂	1	1	yes	maximal, characteristic
11	#91324	C₂	1	2	yes	maximal, complemented
12	#65851	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
15	C₂×#1204	C₂²	1	2	yes	—
16	C₂×D₃×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×D₃×Dic₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
20	#6712	C₂²	1	2	yes	—
21	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
22	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₄×C₅×Dic₃	C₂²	1	2	yes	—
25	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₃×#683	C₂²	1	2	yes	—
28	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
29	Dic₃⋊Dic₅	C₂²	1	2	yes	—
30	#6669	C₂²	1	2	yes	—
31	#6669	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
33	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
34	Dic₃×Dic₅	C₂²	1	1	yes	characteristic
35	Dic₃×Dic₅	C₂²	1	2	yes	—
36	#5688	—	5	1	no	maximal, Hall
37	C₅×#117	—	3	1	no	—
38	#4072	—	3	1	no	—
39	C₂×C₄×Dic₅	—	3	1	no	—
40	#4036	—	3	1	no	—
41	#4036	D₃	1	1	yes	characteristic, complemented
42	#4034	—	3	1	no	—
43	#4034	—	3	1	no	—
44	#4024	—	3	2	no	—
45	#4024	—	3	2	no	—
46	C₄×Dic₁₀	—	3	2	no	—
47	C₄×Dic₁₀	—	3	2	no	—
48	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×Dic₁₅	C₂³	1	2	yes	—
52	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
54	Dic₃₀	—	2	4	no	—
55	C₂×C₅×Dic₃	C₂³	1	2	yes	—
56	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
57	C₄×C₅×D₃	Q₈	1	4	yes	complemented
58	C₂×C₃×Dic₅	C₂³	1	2	yes	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	#1204	—	2	4	no	—
62	D₃×Dic₅	—	2	2	no	—
63	D₃×Dic₅	—	2	2	no	—
64	C₃×#125	—	5	1	no	—
65	#896	—	5	2	no	—
66	#896	—	5	2	no	—
67	#892	—	5	1	no	—
68	#892	D₅	1	1	yes	characteristic
69	#888	—	5	1	no	—
70	#888	—	5	1	no	—
71	#887	—	5	2	no	—
72	#887	—	5	2	no	—
73	C₄²×D₃	—	5	1	no	—
74	#870	—	5	1	no	—
75	C₂²×C₄×C₅	—	3	1	no	—
76	C₂²×Dic₅	—	3	1	no	—
77	C₂²×Dic₅	—	3	1	no	—
78	C₂×Dic₁₀	—	3	2	no	—
79	C₂×Dic₁₀	—	3	2	no	—
80	C₅×C₄⋊C₄	—	3	1	no	—
81	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
82	C₅×C₂²⋊C₄	—	3	2	no	—
83	C₄²×C₅	—	3	2	no	—
84	C₂²⋊Dic₅	—	3	2	no	—
85	C₄⋊Dic₅	—	3	1	no	—
86	C₄⋊Dic₅	—	3	2	no	—
87	C₄⋊Dic₅	C₂×D₃	1	2	yes	—
88	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
89	#683	—	3	2	no	—
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	#683	C₂×D₃	1	2	yes	—
93	C₄×Dic₅	—	3	1	no	—
94	C₄×Dic₅	—	3	1	no	—
95	C₄×Dic₅	—	3	1	no	—
96	C₄×Dic₅	—	3	2	no	—
97	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
98	#533	—	15	1	no	Sylow
99	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
100	C₂×C₅×D₃	—	2	1	no	—
101	C₂×C₅×D₃	—	2	1	no	—
102	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
105	Dic₁₅	—	2	2	no	—
106	Dic₁₅	#41	1	2	yes	—
107	Dic₁₅	#41	1	2	yes	—
108	C₃×Dic₅	—	2	1	no	—
109	C₃×Dic₅	—	2	1	no	—
110	C₃×Dic₅	—	2	2	no	—
111	C₅×Dic₃	—	2	2	no	—
112	C₅×Dic₃	C₂×Q₈	1	2	yes	—
113	C₂×C₄×D₃	—	5	1	no	—
114	C₂×C₄×D₃	—	5	1	no	—
115	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
116	C₂×Dic₆	—	5	2	no	—
117	C₂×Dic₆	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	2	no	—
119	C₃×C₄⋊C₄	—	5	2	no	—
120	C₃×C₄⋊C₄	—	5	1	no	—
121	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
122	C₃×C₄²	—	5	1	no	—
123	(C₂×D₃)⋊C₄	—	5	2	no	—
124	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
125	C₄⋊Dic₃	—	5	1	no	—
126	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
127	Dic₃⋊C₄	—	5	2	no	—
128	Dic₃⋊C₄	—	5	2	no	—
129	Dic₃⋊C₄	—	5	2	no	—
130	Dic₃⋊C₄	—	5	2	no	—
131	C₄×Dic₃	—	5	1	no	—
132	C₄×Dic₃	—	5	1	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₄×Dic₃	—	5	2	no	—
135	C₄×Dic₃	C₂×D₅	1	2	yes	—
136	C₂³×C₅	—	3	1	no	—
137	C₂×C₄×C₅	—	3	1	no	—
138	C₂×C₄×C₅	—	3	4	no	—
139	C₂×C₄×C₅	—	3	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	2	no	—
145	C₂×Dic₅	—	6	2	no	—
146	C₂×Dic₅	—	6	2	no	—
147	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
148	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
149	C₂×Dic₅	C₂²×D₃	1	2	yes	—
150	Dic₁₀	—	6	4	no	—
151	Dic₁₀	—	6	4	no	—
152	C₄⋊Q₈	—	15	1	no	—
153	C₄⋊Q₈	—	15	1	no	—
154	#125	—	5	1	no	—
155	#125	—	15	1	no	—
156	C₂²⋊Q₈	—	15	2	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	C₄×Q₈	—	15	2	no	—
159	C₄×Q₈	—	15	2	no	—
160	#117	—	3	1	no	—
161	#117	—	15	1	no	—
162	C₂×C₄²	—	15	1	no	—
163	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₅×D₃	—	2	2	no	—
167	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
173	C₂×Dic₃	—	5	1	no	—
174	C₂×Dic₃	—	5	1	no	—
175	C₂×Dic₃	—	5	2	no	—
176	C₂×Dic₃	C₂²×D₅	1	2	yes	—
177	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
178	C₄×D₃	—	10	2	no	—
179	C₄×D₃	—	10	2	no	—
180	C₄×D₃	Dic₁₀	1	4	yes	complemented
181	Dic₆	—	10	4	no	—
182	Dic₆	—	10	4	no	—
183	C₂²×C₅	—	3	2	no	—
184	C₂²×C₅	—	6	1	no	—
185	C₂²×C₅	—	6	1	no	—
186	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	3	2	no	—
189	C₄×C₅	—	6	2	no	—
190	C₄×C₅	D₃×Q₈	1	2	yes	—
191	Dic₅	—	2	2	no	—
192	Dic₅	—	2	1	no	—
193	Dic₅	—	2	1	no	—
194	Dic₅	—	3	2	no	—
195	Dic₅	—	3	2	no	—
196	Dic₅	—	6	2	no	—
197	C₂×Q₈	—	15	2	no	—
198	C₂×Q₈	—	15	2	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	15	1	no	—
201	C₂²×C₄	—	15	1	no	—
202	C₄⋊C₄	—	3	1	no	—
203	C₄⋊C₄	—	5	1	no	—
204	C₄⋊C₄	—	5	2	no	—
205	C₄⋊C₄	—	5	2	no	—
206	C₄⋊C₄	—	15	2	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	1	no	—
210	C₄⋊C₄	—	15	2	no	—
211	C₄⋊C₄	D₃×D₅	1	1	yes	characteristic
212	C₂²⋊C₄	—	3	2	no	—
213	C₂²⋊C₄	—	15	2	no	—
214	C₄²	—	3	2	no	—
215	C₄²	—	5	1	no	—
216	C₄²	—	15	1	no	—
217	C₄²	—	15	1	no	—
218	C₄²	—	15	2	no	—
219	C₄²	—	15	1	no	—
220	C₃×C₅	#533	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
222	C₂×D₃	—	2	1	no	—
223	C₂×D₃	—	2	1	no	—
224	C₂×D₃	C₂×Dic₁₀	1	2	yes	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	10	2	no	—
227	C₃×C₄	—	10	1	no	—
228	C₃×C₄	—	10	1	no	—
229	C₃×C₄	C₂×Dic₁₀	1	2	yes	—
230	Dic₃	—	2	2	no	—
231	Dic₃	—	5	2	no	—
232	Dic₃	—	5	2	no	—
233	Dic₃	—	10	2	no	—
234	Dic₃	C₂×Dic₁₀	1	2	yes	—
235	C₂×C₅	—	6	2	no	—
236	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
237	C₂×C₅	D₃×#41	1	1	yes	characteristic
238	C₂×C₅	D₃×#41	1	1	yes	characteristic
239	C₂³	—	3	1	no	—
240	Q₈	—	30	4	no	—
241	Q₈	—	30	4	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	4	no	—
244	C₂×C₄	—	3	2	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	5	1	no	—
247	C₂×C₄	—	5	2	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	—	15	1	no	—
250	C₂×C₄	—	15	2	no	—
251	C₂×C₄	—	30	2	no	—
252	C₂×C₄	—	30	2	no	—
253	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
254	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
256	C₂×C₃	D₅×#41	1	1	yes	characteristic
257	C₂×C₃	D₅×#41	1	1	yes	characteristic
258	D₃	—	2	2	no	—
259	C₅	#5688	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	3	2	no	—
261	C₂²	—	6	1	no	—
262	C₂²	—	6	1	no	—
263	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	3	2	no	—
266	C₄	—	6	2	no	—
267	C₄	—	10	1	no	—
268	C₄	—	10	1	no	—
269	C₄	—	10	2	no	—
270	C₄	—	15	2	no	—
271	C₄	—	15	2	no	—
272	C₄	—	30	2	no	—
273	C₄	D₃×Dic₁₀	1	2	yes	—
274	C₃	#65851	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	6	2	no	—
276	C₂	C₂×D₃×Dic₁₀	1	1	yes	characteristic
277	C₂	#92024	1	1	yes	characteristic
278	C₂	#92034	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	C₂²×C₅×D₃	C₂³
hypercenter	C₄⋊C₄	D₃×D₅
hypocenter	C₃×C₅	#533
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. #91777	C₂	semidirect
3. C₃×#4036	C₂	semidirect
4. #91470	C₂	semidirect
8. #91388	C₂	semidirect
9. #91340	C₂	semidirect
11. #91324	C₂	semidirect
41. #4036	D₃	semidirect
57. C₄×C₅×D₃	Q₈	semidirect
180. C₄×D₃	Dic₁₀	semidirect
220. C₃×C₅	#533	semidirect
259. C₅	#5688	semidirect
274. C₃	#65851	semidirect

Maximal subgroups

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

subgroup	index
#91777	2
C₅×#892	2
C₃×#4036	2
#91470	2
#91436	2
#91432	2
C₄×D₃×Dic₅	2
#91388	2
#91340	2
#91338	2
#91324	2
#65851	3
#5688	5