Subgroups — #11752711

Order 960.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
4	#91468	C₂	1	1	yes	maximal, characteristic, complemented
5	#91464	C₂	1	2	yes	maximal, complemented
6	#91452	C₂	1	2	yes	maximal, complemented
7	#91449	C₂	1	1	yes	maximal, characteristic
8	#91373	C₂	1	1	yes	maximal, characteristic, complemented
9	#91367	C₂	1	2	yes	maximal, complemented
10	#91360	C₂	1	1	yes	maximal, characteristic, complemented
11	#91352	C₂	1	2	yes	maximal, complemented
12	#65895	—	3	1	no	maximal, Hall
13	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
14	C₂×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×#1204	C₂²	1	2	yes	—
18	C₂×#1201	C₂²	1	2	yes	—
19	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
21	#6712	C₂²	1	2	yes	—
22	C₄×Dic₁₅	C₂²	1	2	yes	—
23	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₃×#685	C₂²	1	1	yes	characteristic
26	C₃×#685	C₂²	1	1	yes	characteristic
27	C₃×#683	C₂²	1	1	yes	characteristic
28	C₃×#683	C₂²	1	1	yes	characteristic
29	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
30	Dic₃⋊Dic₅	C₂²	1	2	yes	—
31	#6668	C₂²	1	2	yes	—
32	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
33	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
34	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
35	#6665	C₂²	1	2	yes	—
36	#5534	—	5	1	no	maximal, Hall
37	C₄×C₅×D₄	—	3	1	no	—
38	#4084	—	3	1	no	—
39	#4082	—	3	1	no	—
40	#4075	—	3	2	no	—
41	#4070	—	3	2	no	—
42	#4043	—	3	1	no	—
43	#4034	—	3	1	no	—
44	#4031	—	3	2	no	—
45	#4025	—	3	2	no	—
46	#4018	D₃	1	1	yes	characteristic, complemented
47	C₄×Dic₁₀	—	3	1	no	—
48	C₂²×C₅×D₃	C₂³	1	2	yes	—
49	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₅	C₂³	1	2	yes	—
54	C₂×Dic₁₅	C₂³	1	2	yes	—
55	Dic₃₀	—	2	2	no	—
56	C₂×C₅×Dic₃	C₂³	1	2	yes	—
57	C₅×D₁₂	—	2	2	no	—
58	C₄×C₅×D₃	—	4	2	no	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₃×C₄×D₅	—	2	2	no	—
63	#1204	—	4	2	no	—
64	#1201	—	4	2	no	—
65	C₃×#117	—	5	1	no	—
66	#930	—	5	1	no	—
67	#924	—	5	1	no	—
68	#898	—	5	2	no	—
69	#896	—	5	2	no	—
70	#885	—	5	2	no	—
71	#883	—	5	2	no	—
72	#875	—	5	1	no	—
73	C₄×D₁₂	D₅	1	1	yes	characteristic, complemented
74	#869	—	5	1	no	—
75	C₄×Dic₆	—	5	1	no	—
76	C₂×C₅×D₄	—	3	1	no	—
77	C₂²×C₄×C₅	—	3	2	no	—
78	C₂×C₂²⋊D₅	—	3	2	no	—
79	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
80	C₂×Dic₁₀	—	3	1	no	—
81	C₂×Dic₁₀	—	3	2	no	—
82	C₅×C₄⋊C₄	—	3	1	no	—
83	C₅×C₂²⋊C₄	—	3	2	no	—
84	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
85	C₂²⋊Dic₅	—	3	2	no	—
86	C₂²⋊Dic₅	—	3	2	no	—
87	C₂²⋊Dic₅	—	3	2	no	—
88	#685	—	3	2	no	—
89	#685	C₂×D₃	1	1	yes	characteristic
90	#685	C₂×D₃	1	1	yes	characteristic
91	C₄⋊Dic₅	—	3	1	no	—
92	C₄⋊Dic₅	—	3	2	no	—
93	#683	—	3	2	no	—
94	#683	—	3	2	no	—
95	#683	C₂×D₃	1	1	yes	characteristic
96	#683	C₂×D₃	1	1	yes	characteristic
97	C₄×Dic₅	—	3	2	no	—
98	C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
99	#542	—	15	1	no	Sylow
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₂×C₅×D₃	—	2	2	no	—
102	C₂×C₅×D₃	—	2	2	no	—
103	C₂×C₅×D₃	—	2	2	no	—
104	C₂×C₃×D₅	—	2	1	no	—
105	C₂×C₃×D₅	—	2	1	no	—
106	C₂×C₃×D₅	—	2	1	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	#41	1	2	yes	—
110	Dic₁₅	—	2	2	no	—
111	Dic₁₅	—	2	2	no	—
112	C₃×Dic₅	—	2	1	no	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	2	1	no	—
115	C₅×Dic₃	—	2	2	no	—
116	C₂²×C₃×C₄	—	5	1	no	—
117	C₂×#67	—	5	2	no	—
118	C₂×D₁₂	C₂×D₅	1	1	yes	characteristic
119	C₂×C₄×D₃	C₂×D₅	1	2	yes	—
120	C₂×Dic₆	—	5	1	no	—
121	C₂×Dic₆	—	5	2	no	—
122	C₃×C₄⋊C₄	—	5	1	no	—
123	C₃×C₄⋊C₄	—	5	1	no	—
124	C₃×C₂²⋊C₄	—	5	1	no	—
125	C₃×C₂²⋊C₄	—	5	1	no	—
126	C₃×C₄²	—	5	1	no	—
127	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
128	#217	—	5	2	no	—
129	(C₂×D₃)⋊C₄	—	5	2	no	—
130	(C₂×D₃)⋊C₄	—	5	2	no	—
131	(C₂×D₃)⋊C₄	—	5	2	no	—
132	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
133	C₄⋊Dic₃	—	5	1	no	—
134	C₄⋊Dic₃	—	5	2	no	—
135	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
136	Dic₃⋊C₄	—	5	2	no	—
137	Dic₃⋊C₄	—	5	2	no	—
138	C₄×Dic₃	—	5	2	no	—
139	C₂³×C₅	—	3	2	no	—
140	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
141	C₅×D₄	—	6	2	no	—
142	C₂×C₄×C₅	—	3	2	no	—
143	C₂×C₄×C₅	—	12	2	no	—
144	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
147	C₂²⋊D₅	—	12	2	no	—
148	C₂×Dic₅	—	3	2	no	—
149	C₂×Dic₅	—	3	2	no	—
150	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
151	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
152	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
153	C₄×D₅	—	2	2	no	—
154	Dic₁₀	—	6	2	no	—
155	Dic₁₀	—	12	2	no	—
156	C₄⋊Q₈	—	15	1	no	—
157	#126	—	15	2	no	—
158	#124	—	15	1	no	—
159	#124	—	15	2	no	—
160	#123	—	15	2	no	—
161	C₂²⋊Q₈	—	15	1	no	—
162	C₂²⋊Q₈	—	15	2	no	—
163	#121	—	15	1	no	—
164	C₄×Q₈	—	15	1	no	—
165	C₄×D₄	—	3	1	no	—
166	#117	—	5	1	no	—
167	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
170	C₃×D₅	—	4	1	no	—
171	C₅×D₃	—	4	2	no	—
172	C₂³×C₃	—	5	1	no	—
173	C₂²×D₃	C₂²×D₅	1	2	yes	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	10	2	no	—
178	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	#67	—	20	2	no	—
182	C₂×Dic₃	—	5	2	no	—
183	C₂×Dic₃	—	5	2	no	—
184	C₂×Dic₃	C₂²×D₅	1	2	yes	—
185	D₁₂	—	2	2	no	—
186	C₄×D₃	—	4	2	no	—
187	Dic₆	—	10	2	no	—
188	Dic₆	—	20	2	no	—
189	C₂²×C₅	—	6	2	no	—
190	C₂²×C₅	—	6	2	no	—
191	C₂²×C₅	—	6	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	—	2	1	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	6	2	no	—
199	C₄×C₅	#235	1	2	yes	—
200	Dic₅	—	2	1	no	—
201	Dic₅	—	2	1	no	—
202	Dic₅	—	2	1	no	—
203	Dic₅	—	6	2	no	—
204	Dic₅	—	6	2	no	—
205	C₂×Q₈	—	15	2	no	—
206	C₂×Q₈	—	15	1	no	—
207	C₂×D₄	—	3	1	no	—
208	C₂×D₄	—	15	2	no	—
209	C₂²×C₄	—	3	2	no	—
210	C₂²×C₄	—	5	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	5	1	no	—
213	C₄⋊C₄	—	5	1	no	—
214	C₄⋊C₄	—	15	2	no	—
215	C₄⋊C₄	—	15	2	no	—
216	C₄⋊C₄	—	15	1	no	—
217	C₄⋊C₄	—	15	2	no	—
218	C₂²⋊C₄	—	3	2	no	—
219	C₂²⋊C₄	—	5	1	no	—
220	C₂²⋊C₄	—	5	1	no	—
221	C₂²⋊C₄	—	15	2	no	—
222	C₂²⋊C₄	—	15	2	no	—
223	C₂²⋊C₄	—	15	2	no	—
224	C₂²⋊C₄	—	15	2	no	—
225	C₄²	—	5	1	no	—
226	C₄²	—	15	2	no	—
227	C₄²	D₃×D₅	1	1	yes	characteristic
228	C₃×C₅	#542	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	10	1	no	—
230	C₂²×C₃	—	10	1	no	—
231	C₂²×C₃	—	10	1	no	—
232	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
233	C₂×D₃	—	2	2	no	—
234	C₂×D₃	—	2	2	no	—
235	C₂×D₃	—	2	2	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	1	no	—
238	C₃×C₄	—	10	1	no	—
239	C₃×C₄	—	10	1	no	—
240	C₃×C₄	—	10	1	no	—
241	C₃×C₄	#709	1	2	yes	—
242	Dic₃	—	2	2	no	—
243	Dic₃	—	10	2	no	—
244	Dic₃	—	10	2	no	—
245	C₂×C₅	—	12	2	no	—
246	C₂×C₅	C₂×#235	1	1	yes	characteristic
247	C₂×C₅	#1009	1	1	yes	characteristic
248	C₂×C₅	#1010	1	1	yes	characteristic
249	D₅	—	4	1	no	—
250	C₂³	—	3	2	no	—
251	C₂³	—	5	1	no	—
252	Q₈	—	30	2	no	—
253	Q₈	—	60	2	no	—
254	D₄	—	6	2	no	—
255	D₄	—	60	2	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	10	2	no	—
261	C₂×C₄	—	12	2	no	—
262	C₂×C₄	—	15	2	no	—
263	C₂×C₄	—	15	2	no	—
264	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
265	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
266	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
267	C₂×C₃	—	20	1	no	—
268	C₂×C₃	C₂×#709	1	1	yes	characteristic
269	C₂×C₃	#4144	1	1	yes	characteristic
270	C₂×C₃	#4150	1	1	yes	characteristic
271	D₃	—	4	2	no	—
272	C₅	#5534	1	1	yes	characteristic, complemented, Sylow
273	C₂²	—	6	2	no	—
274	C₂²	—	6	2	no	—
275	C₂²	—	6	2	no	—
276	C₂²	—	10	1	no	—
277	C₂²	—	10	1	no	—
278	C₂²	—	10	1	no	—
279	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	6	2	no	—
283	C₄	—	10	1	no	—
284	C₄	—	10	1	no	—
285	C₄	—	10	1	no	—
286	C₄	—	30	2	no	—
287	C₄	—	30	2	no	—
288	C₄	#6771	1	2	yes	—
289	C₃	#65895	1	1	yes	characteristic, complemented, Sylow
290	C₂	—	12	2	no	—
291	C₂	—	20	1	no	—
292	C₂	#92001	1	1	yes	characteristic
293	C₂	C₂×#6771	1	1	yes	characteristic
294	C₂	#92018	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₄²×C₅	C₂²
prime radical	#91449	C₂
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#542
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₄×Dic₃₀	C₂	semidirect
2. C₄×C₅×D₁₂	C₂	semidirect
3. C₃×#4018	C₂	semidirect
4. #91468	C₂	semidirect
5. #91464	C₂	semidirect
6. #91452	C₂	semidirect
8. #91373	C₂	semidirect
9. #91367	C₂	semidirect
10. #91360	C₂	semidirect
11. #91352	C₂	semidirect
46. #4018	D₃	semidirect
73. C₄×D₁₂	D₅	semidirect
228. C₃×C₅	#542	semidirect
272. C₅	#5534	semidirect
289. C₃	#65895	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₀	2
C₄×C₅×D₁₂	2
C₃×#4018	2
#91468	2
#91464	2
#91452	2
#91449	2
#91373	2
#91367	2
#91360	2
#91352	2
#65895	3
#5534	5