Subgroups — #11756298

Order 960.

456 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	D₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	D₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic
4	#91813	C₂	1	2	yes	maximal, complemented
5	C₂×#6712	C₂	1	2	yes	maximal, complemented
6	#91779	C₂	1	1	yes	maximal, characteristic, complemented
7	#91777	C₂	1	1	yes	maximal, characteristic, complemented
8	#91767	C₂	1	2	yes	maximal, complemented
9	#91767	C₂	1	2	yes	maximal, complemented
10	#91759	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#65873	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	2	yes	—
15	C₂²×Dic₁₅	C₂²	1	2	yes	—
16	C₂²×Dic₁₅	C₂²	1	2	yes	—
17	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
18	C₂×Dic₃₀	C₂²	1	2	yes	—
19	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
22	#6719	C₂²	1	2	yes	—
23	#6719	C₂²	1	2	yes	—
24	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₁₅	C₂²	1	2	yes	—
28	#6712	C₂²	1	1	yes	characteristic
29	#6712	C₂²	1	1	yes	characteristic
30	#6712	C₂²	1	2	yes	—
31	#6712	C₂²	1	2	yes	—
32	#6712	C₂²	1	2	yes	—
33	#6712	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
36	#5539	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	D₃	1	1	yes	characteristic
38	D₄×Dic₅	—	3	1	no	—
39	D₄×Dic₅	—	3	1	no	—
40	#4070	—	3	2	no	—
41	C₂×#683	—	3	2	no	—
42	#4036	—	3	1	no	—
43	#4034	—	3	1	no	—
44	#4024	—	3	2	no	—
45	#4024	—	3	2	no	—
46	#4016	—	3	1	no	—
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₃×C₅×D₄	Q₈	1	4	yes	complemented
50	C₂×C₃×C₄×C₅	—	4	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	—	2	4	no	—
56	C₂×Dic₁₅	—	4	2	no	—
57	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂³	1	2	yes	—
61	C₂×Dic₁₅	C₂³	1	2	yes	—
62	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
63	Dic₃₀	—	2	2	no	—
64	Dic₃₀	—	4	2	no	—
65	C₃×C₄×D₄	D₅	1	1	yes	characteristic
66	D₄×Dic₃	—	5	1	no	—
67	D₄×Dic₃	—	5	1	no	—
68	#924	—	5	2	no	—
69	C₂×Dic₃⋊C₄	—	5	2	no	—
70	#890	—	5	1	no	—
71	#888	—	5	1	no	—
72	#878	—	5	2	no	—
73	#878	—	5	2	no	—
74	#870	—	5	1	no	—
75	C₄×Dic₆	—	5	1	no	—
76	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₃	1	2	yes	—
78	C₂²×Dic₅	—	3	2	no	—
79	C₂²×Dic₅	—	3	2	no	—
80	C₂×Dic₁₀	—	3	1	no	—
81	C₂×Dic₁₀	—	3	2	no	—
82	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
83	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
84	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
85	C₂²⋊Dic₅	—	3	2	no	—
86	C₂²⋊Dic₅	—	3	2	no	—
87	C₄⋊Dic₅	—	3	1	no	—
88	C₄⋊Dic₅	—	3	1	no	—
89	C₄⋊Dic₅	—	3	1	no	—
90	C₄⋊Dic₅	—	3	2	no	—
91	#683	—	3	1	no	—
92	#683	—	3	1	no	—
93	#683	—	3	2	no	—
94	#683	—	3	2	no	—
95	#683	—	3	2	no	—
96	#683	—	3	2	no	—
97	C₄×Dic₅	—	3	1	no	—
98	C₄×Dic₅	—	3	1	no	—
99	#554	—	15	1	no	Sylow
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	C₂×Q₈	1	4	yes	—
103	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
108	Dic₁₅	—	2	1	no	—
109	Dic₁₅	—	2	2	no	—
110	Dic₁₅	—	2	1	no	—
111	Dic₁₅	—	2	2	no	—
112	Dic₁₅	—	2	1	no	—
113	Dic₁₅	#41	1	2	yes	—
114	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×D₅	1	2	yes	—
116	C₂²×Dic₃	—	5	2	no	—
117	C₂²×Dic₃	—	5	2	no	—
118	C₂×Dic₆	—	5	1	no	—
119	C₂×Dic₆	—	5	2	no	—
120	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
121	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
122	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
123	#217	—	5	2	no	—
124	#217	—	5	2	no	—
125	C₄⋊Dic₃	—	5	1	no	—
126	C₄⋊Dic₃	—	5	1	no	—
127	C₄⋊Dic₃	—	5	1	no	—
128	C₄⋊Dic₃	—	5	2	no	—
129	Dic₃⋊C₄	—	5	1	no	—
130	Dic₃⋊C₄	—	5	1	no	—
131	Dic₃⋊C₄	—	5	2	no	—
132	Dic₃⋊C₄	—	5	2	no	—
133	Dic₃⋊C₄	—	5	2	no	—
134	Dic₃⋊C₄	—	5	2	no	—
135	C₄×Dic₃	—	5	1	no	—
136	C₄×Dic₃	—	5	1	no	—
137	C₂³×C₅	C₂²×D₃	1	2	yes	—
138	C₅×D₄	Dic₆	1	4	yes	complemented
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
143	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₂×Dic₅	—	3	1	no	—
147	C₂×Dic₅	—	3	1	no	—
148	C₂×Dic₅	—	3	2	no	—
149	C₂×Dic₅	—	3	2	no	—
150	C₂×Dic₅	—	6	4	no	—
151	C₂×Dic₅	—	12	2	no	—
152	Dic₁₀	—	6	2	no	—
153	Dic₁₀	—	12	2	no	—
154	C₄⋊Q₈	—	15	1	no	—
155	#125	—	15	1	no	—
156	#125	—	15	1	no	—
157	C₂²⋊Q₈	—	15	2	no	—
158	C₂²⋊Q₈	—	15	2	no	—
159	C₂²⋊Q₈	—	15	2	no	—
160	C₄×Q₈	—	15	1	no	—
161	C₄×D₄	—	15	1	no	—
162	C₄×D₄	—	15	1	no	—
163	C₄×D₄	D₁₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	15	2	no	—
165	C₂×C₃×C₅	—	2	4	no	—
166	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂³×C₃	C₂²×D₅	1	2	yes	—
170	C₃×D₄	Dic₁₀	1	4	yes	complemented
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
176	C₂×Dic₃	—	5	1	no	—
177	C₂×Dic₃	—	5	1	no	—
178	C₂×Dic₃	—	5	1	no	—
179	C₂×Dic₃	—	5	1	no	—
180	C₂×Dic₃	—	5	2	no	—
181	C₂×Dic₃	—	5	2	no	—
182	C₂×Dic₃	—	10	4	no	—
183	C₂×Dic₃	—	20	2	no	—
184	Dic₆	—	10	2	no	—
185	Dic₆	—	20	2	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	C₂×Dic₆	1	4	yes	—
189	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	1	no	—
192	C₄×C₅	—	2	1	no	—
193	C₄×C₅	C₂×Dic₆	1	2	yes	—
194	Dic₅	—	3	2	no	—
195	Dic₅	—	6	1	no	—
196	Dic₅	—	6	2	no	—
197	Dic₅	—	6	1	no	—
198	Dic₅	—	6	1	no	—
199	Dic₅	—	6	2	no	—
200	C₂×Q₈	—	15	1	no	—
201	C₂×Q₈	—	15	2	no	—
202	C₂×D₄	C₂×D₁₅	1	1	yes	characteristic
203	C₂²×C₄	—	15	2	no	—
204	C₂²×C₄	—	15	2	no	—
205	C₂²×C₄	C₂×D₁₅	1	2	yes	—
206	C₄⋊C₄	—	15	1	no	—
207	C₄⋊C₄	—	15	2	no	—
208	C₄⋊C₄	—	15	2	no	—
209	C₄⋊C₄	—	15	2	no	—
210	C₄⋊C₄	—	15	1	no	—
211	C₄⋊C₄	—	15	1	no	—
212	C₄⋊C₄	—	15	1	no	—
213	C₄⋊C₄	—	15	1	no	—
214	C₄⋊C₄	—	15	2	no	—
215	C₄⋊C₄	—	15	2	no	—
216	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
217	C₂²⋊C₄	—	15	2	no	—
218	C₂²⋊C₄	—	15	2	no	—
219	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
220	C₄²	—	15	1	no	—
221	C₄²	—	15	1	no	—
222	C₄²	C₂×D₁₅	1	1	yes	characteristic
223	C₃×C₅	#554	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	2	2	no	—
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	C₂×Dic₁₀	1	4	yes	—
227	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	C₂×Dic₁₀	1	2	yes	—
232	Dic₃	—	5	2	no	—
233	Dic₃	—	10	1	no	—
234	Dic₃	—	10	2	no	—
235	Dic₃	—	10	1	no	—
236	Dic₃	—	10	1	no	—
237	Dic₃	—	10	2	no	—
238	C₂×C₅	—	2	4	no	—
239	C₂×C₅	C₂²×Dic₆	1	1	yes	characteristic
240	C₂×C₅	#1004	1	1	yes	characteristic
241	C₂×C₅	D₃×#41	1	1	yes	characteristic
242	C₂³	C₂²×D₁₅	1	2	yes	—
243	Q₈	—	30	2	no	—
244	Q₈	—	60	2	no	—
245	D₄	Dic₃₀	1	4	yes	complemented
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	15	1	no	—
248	C₂×C₄	—	15	1	no	—
249	C₂×C₄	—	15	1	no	—
250	C₂×C₄	—	15	1	no	—
251	C₂×C₄	—	15	2	no	—
252	C₂×C₄	—	15	2	no	—
253	C₂×C₄	—	30	4	no	—
254	C₂×C₄	—	60	2	no	—
255	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₁₅	1	2	yes	—
258	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
259	C₂×C₃	—	2	4	no	—
260	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
261	C₂×C₃	#4144	1	1	yes	characteristic
262	C₂×C₃	D₅×#41	1	1	yes	characteristic
263	C₅	#5539	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	C₂×Dic₃₀	1	4	yes	—
267	C₂²	C₂³×D₁₅	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	1	no	—
271	C₄	—	15	2	no	—
272	C₄	—	30	1	no	—
273	C₄	—	30	2	no	—
274	C₄	—	30	1	no	—
275	C₄	—	30	2	no	—
276	C₄	—	30	1	no	—
277	C₄	C₂×Dic₃₀	1	2	yes	—
278	C₃	#65873	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	4	no	—
280	C₂	C₂²×Dic₃₀	1	1	yes	characteristic
281	C₂	#92095	1	1	yes	characteristic
282	C₂	#41×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄×C₅×D₄	C₂
prime radical	C₂×C₃×C₅×D₄	C₂²
hypercenter	C₄×D₄	D₁₅
hypocenter	C₃×C₅	#554
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
4. #91813	C₂	semidirect
5. C₂×#6712	C₂	semidirect
6. #91779	C₂	semidirect
7. #91777	C₂	semidirect
8. #91767	C₂	semidirect
9. #91767	C₂	semidirect
10. #91759	C₂	semidirect
11. C₄×Dic₃₀	C₂	semidirect
49. C₃×C₅×D₄	Q₈	semidirect
138. C₅×D₄	Dic₆	semidirect
170. C₃×D₄	Dic₁₀	semidirect
223. C₃×C₅	#554	semidirect
245. D₄	Dic₃₀	semidirect
263. C₅	#5539	semidirect
278. C₃	#65873	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×D₄	2
D₄×Dic₁₅	2
#91813	2
C₂×#6712	2
#91779	2
#91777	2
#91767	2
#91759	2
C₄×Dic₃₀	2
#65873	3
#5539	5