Subgroups — #11756384

Order 960.

384 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#91833	C₂	1	1	yes	maximal, characteristic, complemented
4	#91815	C₂	1	1	yes	maximal, characteristic
5	#91813	C₂	1	1	yes	maximal, characteristic
6	#91779	C₂	1	2	yes	maximal, complemented
7	#91779	C₂	1	1	yes	maximal, characteristic, complemented
8	#91778	C₂	1	2	yes	maximal, complemented
9	#91776	C₂	1	1	yes	maximal, characteristic, complemented
10	#91768	C₂	1	2	yes	maximal
11	#91768	C₂	1	2	yes	maximal
12	#65959	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	#6719	C₂²	1	2	yes	—
20	#6719	C₂²	1	1	yes	characteristic
21	#6719	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₁₅	C₂²	1	2	yes	—
23	C₄⋊Dic₁₅	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	#6712	C₂²	1	2	yes	—
28	#6712	C₂²	1	2	yes	—
29	#6712	C₂²	1	2	yes	—
30	#6712	C₂²	1	2	yes	—
31	#6712	C₂²	1	2	yes	—
32	C₄×Dic₁₅	C₂²	1	2	yes	—
33	C₄×Dic₁₅	C₂²	1	2	yes	—
34	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂²	1	1	yes	characteristic
36	#5625	—	5	1	no	maximal, Hall
37	C₅×C₂²⋊Q₈	D₃	1	1	yes	characteristic
38	Q₈×Dic₅	—	3	1	no	—
39	#4090	—	3	1	no	—
40	#4072	—	3	1	no	—
41	#4070	—	3	1	no	—
42	#4036	—	3	2	no	—
43	#4036	—	3	1	no	—
44	#4035	—	3	2	no	—
45	#4033	—	3	1	no	—
46	#4025	—	3	2	no	—
47	#4025	—	3	2	no	—
48	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₅×Q₈	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₅	C₂³	1	2	yes	—
59	C₂×Dic₁₅	C₂³	1	2	yes	—
60	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
61	Dic₃₀	—	2	2	no	—
62	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic
63	Q₈×Dic₃	—	5	1	no	—
64	#944	—	5	1	no	—
65	#926	—	5	1	no	—
66	#924	—	5	1	no	—
67	#890	—	5	2	no	—
68	#890	—	5	1	no	—
69	#889	—	5	2	no	—
70	#887	—	5	1	no	—
71	#879	—	5	2	no	—
72	#879	—	5	2	no	—
73	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
75	C₂×Dic₁₀	—	3	1	no	—
76	C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
78	C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
79	C₂²⋊Dic₅	—	3	2	no	—
80	C₂²⋊Dic₅	—	3	1	no	—
81	C₂²⋊Dic₅	—	3	1	no	—
82	C₄⋊Dic₅	—	3	2	no	—
83	C₄⋊Dic₅	—	3	2	no	—
84	C₄⋊Dic₅	—	3	1	no	—
85	C₄⋊Dic₅	—	3	1	no	—
86	C₄⋊Dic₅	—	3	1	no	—
87	#683	—	3	2	no	—
88	#683	—	3	2	no	—
89	#683	—	3	2	no	—
90	#683	—	3	2	no	—
91	#683	—	3	2	no	—
92	C₄×Dic₅	—	3	2	no	—
93	C₄×Dic₅	—	3	2	no	—
94	C₄×Dic₅	—	3	1	no	—
95	C₄×Dic₅	—	3	1	no	—
96	#541	—	15	1	no	Sylow
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	—	2	2	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	#41	1	2	yes	—
105	Dic₁₅	—	2	2	no	—
106	Dic₁₅	—	2	1	no	—
107	Dic₁₅	—	2	2	no	—
108	Dic₁₅	—	2	1	no	—
109	Dic₁₅	—	2	1	no	—
110	Dic₁₅	—	2	1	no	—
111	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
113	C₂×Dic₆	—	5	1	no	—
114	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
115	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
116	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
117	#217	—	5	2	no	—
118	#217	—	5	1	no	—
119	#217	—	5	1	no	—
120	C₄⋊Dic₃	—	5	2	no	—
121	C₄⋊Dic₃	—	5	2	no	—
122	C₄⋊Dic₃	—	5	1	no	—
123	C₄⋊Dic₃	—	5	1	no	—
124	C₄⋊Dic₃	—	5	1	no	—
125	Dic₃⋊C₄	—	5	2	no	—
126	Dic₃⋊C₄	—	5	2	no	—
127	Dic₃⋊C₄	—	5	2	no	—
128	Dic₃⋊C₄	—	5	2	no	—
129	Dic₃⋊C₄	—	5	2	no	—
130	C₄×Dic₃	—	5	2	no	—
131	C₄×Dic₃	—	5	2	no	—
132	C₄×Dic₃	—	5	1	no	—
133	C₄×Dic₃	—	5	1	no	—
134	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₅×Q₈	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
138	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
139	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂×Dic₅	—	3	1	no	—
142	C₂×Dic₅	—	3	1	no	—
143	C₂×Dic₅	—	3	1	no	—
144	C₂×Dic₅	—	3	2	no	—
145	C₂×Dic₅	—	3	2	no	—
146	C₂×Dic₅	—	3	1	no	—
147	Dic₁₀	—	6	2	no	—
148	C₄⋊Q₈	—	15	1	no	—
149	#126	—	15	2	no	—
150	#126	—	15	2	no	—
151	#125	—	15	2	no	—
152	#125	—	15	1	no	—
153	#125	—	15	2	no	—
154	C₂²⋊Q₈	—	15	1	no	—
155	C₂²⋊Q₈	D₁₅	1	1	yes	characteristic
156	C₄×Q₈	—	15	1	no	—
157	C₄×Q₈	—	15	1	no	—
158	#117	—	15	1	no	—
159	C₂×C₃×C₅	—	4	1	no	—
160	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
162	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
163	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
164	C₃×Q₈	—	2	2	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
167	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
168	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×Dic₃	—	5	1	no	—
171	C₂×Dic₃	—	5	1	no	—
172	C₂×Dic₃	—	5	1	no	—
173	C₂×Dic₃	—	5	2	no	—
174	C₂×Dic₃	—	5	2	no	—
175	C₂×Dic₃	—	5	1	no	—
176	Dic₆	—	10	2	no	—
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	#237	1	2	yes	—
185	Dic₅	—	6	2	no	—
186	Dic₅	—	6	1	no	—
187	Dic₅	—	6	2	no	—
188	Dic₅	—	6	1	no	—
189	Dic₅	—	6	1	no	—
190	Dic₅	—	6	1	no	—
191	C₂×Q₈	—	15	1	no	—
192	C₂×Q₈	C₂×D₁₅	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₁₅	1	1	yes	characteristic
194	C₄⋊C₄	—	15	1	no	—
195	C₄⋊C₄	—	15	1	no	—
196	C₄⋊C₄	—	15	1	no	—
197	C₄⋊C₄	—	15	2	no	—
198	C₄⋊C₄	—	15	2	no	—
199	C₄⋊C₄	—	15	2	no	—
200	C₄⋊C₄	—	15	2	no	—
201	C₄⋊C₄	—	15	2	no	—
202	C₄⋊C₄	—	15	2	no	—
203	C₄⋊C₄	—	15	2	no	—
204	C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
205	C₄⋊C₄	C₂×D₁₅	1	2	yes	—
206	C₂²⋊C₄	—	15	1	no	—
207	C₂²⋊C₄	—	15	1	no	—
208	C₂²⋊C₄	—	15	2	no	—
209	C₂²⋊C₄	C₂×D₁₅	1	2	yes	—
210	C₄²	—	15	1	no	—
211	C₄²	—	15	1	no	—
212	C₄²	—	15	2	no	—
213	C₄²	—	15	2	no	—
214	C₃×C₅	#541	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	#711	1	2	yes	—
223	Dic₃	—	10	2	no	—
224	Dic₃	—	10	1	no	—
225	Dic₃	—	10	2	no	—
226	Dic₃	—	10	1	no	—
227	Dic₃	—	10	1	no	—
228	Dic₃	—	10	1	no	—
229	C₂×C₅	—	4	1	no	—
230	C₂×C₅	C₂×#237	1	1	yes	characteristic
231	C₂×C₅	#1007	1	1	yes	characteristic
232	C₂×C₅	#1010	1	1	yes	characteristic
233	C₂³	C₂²×D₁₅	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	30	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	15	1	no	—
238	C₂×C₄	—	15	1	no	—
239	C₂×C₄	—	15	1	no	—
240	C₂×C₄	—	15	2	no	—
241	C₂×C₄	—	15	1	no	—
242	C₂×C₄	—	15	2	no	—
243	C₂×C₄	C₂²×D₁₅	1	2	yes	—
244	C₂×C₄	C₂²×D₁₅	1	2	yes	—
245	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₁₅	1	1	yes	characteristic
247	C₂×C₃	—	4	1	no	—
248	C₂×C₃	C₂×#711	1	1	yes	characteristic
249	C₂×C₃	#4147	1	1	yes	characteristic
250	C₂×C₃	#4150	1	1	yes	characteristic
251	C₅	#5625	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	C₂³×D₁₅	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	30	2	no	—
260	C₄	—	30	1	no	—
261	C₄	—	30	1	no	—
262	C₄	—	30	2	no	—
263	C₄	—	30	1	no	—
264	C₄	—	30	1	no	—
265	C₄	#6819	1	2	yes	—
266	C₃	#65959	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	4	1	no	—
268	C₂	C₂×#6819	1	1	yes	characteristic
269	C₂	#92098	1	1	yes	characteristic
270	C₂	#92101	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₃×C₅×C₂²⋊Q₈	C₂
prime radical	C₂³×C₃×C₅	C₂³
hypercenter	C₂²⋊Q₈	D₁₅
hypocenter	C₃×C₅	#541
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₅	C₂	semidirect
3. #91833	C₂	semidirect
6. #91779	C₂	semidirect
7. #91779	C₂	semidirect
8. #91778	C₂	semidirect
9. #91776	C₂	semidirect
214. C₃×C₅	#541	semidirect
251. C₅	#5625	semidirect
266. C₃	#65959	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₂²⋊Q₈	2
Q₈×Dic₁₅	2
#91833	2
#91815	2
#91813	2
#91779	2
#91778	2
#91776	2
#91768	2
#65959	3
#5625	5