Subgroups — D₅×#946 #11755269

Order 960.

644 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₃×D₅	C₂	1	1	yes	maximal, characteristic
3	#91835	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#946	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
6	D₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#91411	C₂	1	2	yes	maximal, complemented
8	#91410	C₂	1	2	yes	maximal, complemented
9	D₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
10	#91390	C₂	1	1	yes	maximal, characteristic, complemented
11	#91369	C₂	1	1	yes	maximal, characteristic, complemented
12	D₅×C₂²⋊Q₈	—	3	1	no	maximal, Hall
13	C₂²×D₃×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₃×Q₈×D₅	—	2	4	no	—
18	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₁₀	C₂²	1	2	yes	complemented
21	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
22	C₂×#1200	C₂²	1	1	yes	characteristic
23	C₂×D₃×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×D₅×Dic₃	C₂²	1	1	yes	characteristic
25	C₂×D₅×Dic₃	C₂²	1	2	yes	—
26	C₄×D₃×D₅	—	2	4	no	—
27	#6714	C₂²	1	2	yes	—
28	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic, complemented
29	#6712	C₂²	1	2	yes	complemented
30	C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
31	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
32	C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
33	Dic₃⋊Dic₅	C₂²	1	2	yes	complemented
34	#6669	C₂²	1	2	yes	complemented
35	#6668	C₂²	1	2	yes	complemented
36	#6667	C₂²	1	2	yes	—
37	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
38	C₂×#946	—	5	1	no	maximal, Hall
39	C₂×Q₈×D₅	D₃	1	1	yes	characteristic, complemented
40	C₂²×C₄×D₅	—	3	1	no	—
41	C₅×C₂²⋊Q₈	—	3	1	no	—
42	#4092	—	3	1	no	—
43	#4070	—	3	1	no	—
44	#4043	—	3	1	no	—
45	#4042	—	3	2	no	—
46	D₅×C₄⋊C₄	—	3	2	no	—
47	D₅×C₄⋊C₄	—	3	1	no	—
48	D₅×C₂²⋊C₄	—	3	2	no	—
49	#4024	—	3	2	no	—
50	C₂²×D₁₅	C₂³	1	1	yes	characteristic
51	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
52	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
53	C₂×D₃×D₅	—	2	2	no	—
54	C₂×D₃×D₅	—	2	2	no	—
55	C₂×D₃×D₅	Q₈	1	4	yes	complemented
56	C₃×C₅×Q₈	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	2	yes	—
60	C₂×Dic₁₅	C₂³	1	1	yes	characteristic
61	C₄×D₁₅	—	2	2	no	—
62	C₂×C₅×Dic₃	C₂³	1	2	yes	—
63	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₄×C₅×D₃	—	2	2	no	—
65	C₂×C₃×Dic₅	C₂³	1	2	yes	—
66	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
67	C₃×C₄×D₅	—	2	4	no	—
68	C₃×C₄×D₅	D₄	1	4	yes	—
69	C₃×Dic₁₀	—	2	2	no	—
70	C₃×Dic₁₀	—	2	4	no	—
71	#1200	—	2	2	no	—
72	D₃×Dic₅	—	2	2	no	—
73	D₅×Dic₃	—	2	2	no	—
74	D₅×Dic₃	—	2	4	no	—
75	C₂²×C₃×Q₈	—	5	1	no	—
76	C₂²×C₄×D₃	—	5	1	no	—
77	#946	—	5	1	no	—
78	#946	—	5	1	no	—
79	#946	—	5	2	no	—
80	#946	—	5	1	no	—
81	#946	—	5	2	no	—
82	#946	D₅	1	1	yes	characteristic, direct product factor
83	C₂×(C₂×D₃)⋊C₄	—	5	2	no	—
84	C₂×C₄⋊Dic₃	—	5	1	no	—
85	C₂×Dic₃⋊C₄	—	5	2	no	—
86	C₂³×D₅	—	3	1	no	—
87	C₂×C₅×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
88	C₂²×C₄×C₅	—	3	1	no	—
89	C₂²×Dic₅	—	3	1	no	—
90	Q₈×D₅	—	2	4	no	—
91	C₂×C₄×D₅	—	3	1	no	—
92	C₂×C₄×D₅	—	3	2	no	—
93	C₂×C₄×D₅	—	3	1	no	—
94	C₂×C₄×D₅	—	3	1	no	—
95	C₂×C₄×D₅	—	6	4	no	—
96	C₂×C₄×D₅	C₂×D₃	1	2	yes	—
97	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
98	C₂×Dic₁₀	C₂×D₃	1	2	yes	complemented
99	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic, complemented
100	C₅×C₄⋊C₄	—	3	2	no	—
101	C₅×C₄⋊C₄	—	3	1	no	—
102	C₅×C₂²⋊C₄	—	3	2	no	—
103	C₂²⋊Dic₅	—	3	2	no	—
104	#685	—	3	2	no	—
105	#685	—	3	2	no	—
106	C₄⋊Dic₅	—	3	1	no	—
107	C₄⋊Dic₅	—	3	2	no	—
108	#683	—	3	2	no	—
109	#683	—	3	2	no	—
110	#683	—	3	2	no	—
111	C₂×C₂²⋊Q₈	—	15	1	no	Sylow
112	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
113	C₂×D₁₅	—	2	1	no	—
114	C₂×D₁₅	—	2	1	no	—
115	C₂×D₁₅	C₂×Q₈	1	2	yes	complemented
116	C₂×C₅×D₃	—	2	1	no	—
117	C₂×C₅×D₃	—	2	1	no	—
118	C₂×C₅×D₃	C₂×Q₈	1	2	yes	complemented
119	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
120	C₂×C₃×D₅	C₂×Q₈	1	2	yes	—
121	C₂×C₃×D₅	#41	1	2	yes	—
122	D₃×D₅	—	2	8	no	—
123	C₃×C₄×C₅	—	2	2	no	—
124	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
125	Dic₁₅	—	2	1	no	—
126	Dic₁₅	—	2	2	no	—
127	C₃×Dic₅	—	2	2	no	—
128	C₃×Dic₅	C₂×D₄	1	2	yes	—
129	C₅×Dic₃	—	2	1	no	—
130	C₅×Dic₃	—	2	2	no	—
131	C₂³×D₃	—	5	1	no	—
132	C₂×C₃×Q₈	—	5	2	no	—
133	C₂×C₃×Q₈	—	5	1	no	—
134	C₂×C₃×Q₈	—	10	4	no	—
135	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
136	C₂²×C₃×C₄	—	5	1	no	—
137	C₂²×C₃×C₄	—	5	2	no	—
138	C₂²×Dic₃	—	5	1	no	—
139	C₂²×Dic₃	—	5	2	no	—
140	C₂×C₄×D₃	—	5	1	no	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₄×D₃	—	5	1	no	—
143	C₂×C₄×D₃	—	10	4	no	—
144	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
145	(C₂×D₃)⋊C₄	—	5	2	no	—
146	(C₂×D₃)⋊C₄	—	5	2	no	—
147	(C₂×D₃)⋊C₄	—	5	2	no	—
148	(C₂×D₃)⋊C₄	C₂×D₅	1	2	yes	—
149	C₄⋊Dic₃	—	5	1	no	—
150	C₄⋊Dic₃	—	5	2	no	—
151	C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic, complemented
152	Dic₃⋊C₄	—	5	2	no	—
153	Dic₃⋊C₄	—	5	2	no	—
154	Dic₃⋊C₄	—	5	2	no	—
155	Dic₃⋊C₄	C₂×D₅	1	2	yes	complemented
156	C₂³×C₅	—	3	1	no	—
157	C₂²×D₅	—	3	1	no	—
158	C₂²×D₅	—	3	4	no	—
159	C₂²×D₅	—	6	2	no	—
160	C₂²×D₅	—	6	2	no	—
161	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
162	C₅×Q₈	—	2	2	no	—
163	C₂×C₄×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	2	no	—
165	C₂×C₄×C₅	—	6	2	no	—
166	C₂×C₄×C₅	C₂²×D₃	1	2	yes	—
167	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
168	C₂×Dic₅	—	3	2	no	—
169	C₂×Dic₅	—	3	1	no	—
170	C₂×Dic₅	—	6	2	no	—
171	C₂×Dic₅	C₂²×D₃	1	2	yes	—
172	C₂×Dic₅	C₂²×D₃	1	1	yes	characteristic
173	C₄×D₅	—	2	4	no	—
174	C₄×D₅	—	6	4	no	—
175	C₄×D₅	—	6	2	no	—
176	C₄×D₅	—	6	2	no	—
177	C₄×D₅	—	6	2	no	—
178	C₄×D₅	#67	1	4	yes	—
179	Dic₁₀	—	2	2	no	—
180	Dic₁₀	—	2	4	no	—
181	C₂²×Q₈	—	5	1	no	—
182	C₂³×C₄	—	15	1	no	—
183	C₂²⋊Q₈	—	3	1	no	—
184	C₂²⋊Q₈	—	15	2	no	—
185	C₂²⋊Q₈	—	15	2	no	—
186	C₂²⋊Q₈	—	15	1	no	—
187	C₂²⋊Q₈	—	15	1	no	—
188	C₂²⋊Q₈	—	15	1	no	—
189	C₂×C₄⋊C₄	—	15	2	no	—
190	C₂×C₄⋊C₄	—	15	1	no	—
191	C₂×C₂²⋊C₄	—	15	2	no	—
192	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
195	D₁₅	—	2	2	no	—
196	C₃×D₅	C₂²⋊Q₈	1	4	yes	complemented
197	C₅×D₃	—	2	2	no	—
198	C₂³×C₃	—	5	1	no	—
199	C₂²×D₃	—	5	1	no	—
200	C₂²×D₃	—	5	4	no	—
201	C₂²×D₃	—	10	2	no	—
202	C₂²×D₃	—	10	2	no	—
203	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
204	C₃×Q₈	—	2	2	no	—
205	C₃×Q₈	—	10	4	no	—
206	C₃×Q₈	—	10	2	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	4	no	—
209	C₂×C₃×C₄	—	5	2	no	—
210	C₂×C₃×C₄	—	10	4	no	—
211	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
212	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
213	C₂×Dic₃	—	5	1	no	—
214	C₂×Dic₃	—	5	2	no	—
215	C₂×Dic₃	—	10	2	no	—
216	C₂×Dic₃	—	10	4	no	—
217	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
218	C₂×Dic₃	C₂²×D₅	1	2	yes	—
219	C₄×D₃	—	2	2	no	—
220	C₄×D₃	—	10	2	no	—
221	C₄×D₃	—	10	2	no	—
222	C₄×D₃	—	10	2	no	—
223	C₂²×C₅	—	3	2	no	—
224	C₂²×C₅	—	6	1	no	—
225	C₂²×C₅	—	6	1	no	—
226	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
227	C₂×D₅	—	3	2	no	—
228	C₂×D₅	—	6	1	no	—
229	C₂×D₅	—	6	1	no	—
230	C₂×D₅	—	6	8	no	—
231	C₂×D₅	D₃×Q₈	1	2	yes	—
232	C₂×D₅	#239	1	2	yes	—
233	C₂×D₅	C₂×#67	1	2	yes	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	6	2	no	—
236	C₄×C₅	—	6	1	no	—
237	C₄×C₅	C₂×#67	1	2	yes	—
238	Dic₅	—	2	2	no	—
239	Dic₅	—	6	2	no	—
240	Dic₅	—	6	1	no	—
241	Dic₅	C₂×#67	1	2	yes	—
242	C₂⁴	—	15	1	no	—
243	C₂×Q₈	—	5	2	no	—
244	C₂×Q₈	—	5	1	no	—
245	C₂×Q₈	—	10	4	no	—
246	C₂×Q₈	D₃×D₅	1	1	yes	characteristic, complemented
247	C₂²×C₄	—	3	1	no	—
248	C₂²×C₄	—	5	1	no	—
249	C₂²×C₄	—	5	2	no	—
250	C₂²×C₄	—	15	1	no	—
251	C₂²×C₄	—	15	2	no	—
252	C₂²×C₄	—	15	1	no	—
253	C₂²×C₄	—	15	1	no	—
254	C₂²×C₄	—	15	1	no	—
255	C₂²×C₄	—	30	4	no	—
256	C₄⋊C₄	—	3	2	no	—
257	C₄⋊C₄	—	3	1	no	—
258	C₄⋊C₄	—	15	2	no	—
259	C₄⋊C₄	—	15	2	no	—
260	C₄⋊C₄	—	15	2	no	—
261	C₄⋊C₄	—	15	2	no	—
262	C₄⋊C₄	—	15	1	no	—
263	C₂²⋊C₄	—	3	2	no	—
264	C₂²⋊C₄	—	15	2	no	—
265	C₂²⋊C₄	—	15	2	no	—
266	C₂²⋊C₄	—	15	2	no	—
267	C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	5	2	no	—
269	C₂²×C₃	—	5	2	no	—
270	C₂²×C₃	—	5	2	no	—
271	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
272	C₂×D₃	—	2	1	no	—
273	C₂×D₃	—	2	1	no	—
274	C₂×D₃	—	5	2	no	—
275	C₂×D₃	—	10	8	no	—
276	C₂×D₃	—	10	1	no	—
277	C₂×D₃	—	10	1	no	—
278	C₂×D₃	Q₈×D₅	1	2	yes	complemented
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	5	2	no	—
281	C₃×C₄	—	10	2	no	—
282	C₃×C₄	D₄×D₅	1	2	yes	—
283	Dic₃	—	2	1	no	—
284	Dic₃	—	2	2	no	—
285	Dic₃	—	10	1	no	—
286	Dic₃	—	10	2	no	—
287	C₂×C₅	—	6	2	no	—
288	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
289	C₂×C₅	C₂×#239	1	1	yes	characteristic
290	C₂×C₅	C₂²×#67	1	1	yes	characteristic
291	D₅	—	6	2	no	—
292	D₅	#946	1	4	yes	direct product factor
293	C₂³	—	3	1	no	—
294	C₂³	—	5	1	no	—
295	C₂³	—	15	4	no	—
296	C₂³	—	15	1	no	—
297	C₂³	—	30	2	no	—
298	C₂³	—	30	2	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	10	4	no	—
301	Q₈	—	10	2	no	—
302	C₂×C₄	—	3	2	no	—
303	C₂×C₄	—	3	1	no	—
304	C₂×C₄	—	5	1	no	—
305	C₂×C₄	—	5	4	no	—
306	C₂×C₄	—	5	2	no	—
307	C₂×C₄	—	6	2	no	—
308	C₂×C₄	—	10	4	no	—
309	C₂×C₄	—	15	2	no	—
310	C₂×C₄	—	15	1	no	—
311	C₂×C₄	—	30	2	no	—
312	C₂×C₄	—	30	2	no	—
313	C₂×C₄	—	30	2	no	—
314	C₂×C₄	—	30	4	no	—
315	C₂×C₄	—	30	2	no	—
316	C₂×C₄	C₂×D₃×D₅	1	2	yes	—
317	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
318	C₂×C₃	—	5	4	no	—
319	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
320	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
321	C₂×C₃	D₅×#41	1	1	yes	characteristic
322	D₃	—	2	2	no	—
323	D₃	—	10	2	no	—
324	C₅	C₂×#946	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	3	2	no	—
326	C₂²	—	5	2	no	—
327	C₂²	—	5	2	no	—
328	C₂²	—	5	2	no	—
329	C₂²	—	6	1	no	—
330	C₂²	—	6	1	no	—
331	C₂²	—	15	2	no	—
332	C₂²	—	30	8	no	—
333	C₂²	—	30	1	no	—
334	C₂²	—	30	1	no	—
335	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	5	2	no	—
338	C₄	—	6	2	no	—
339	C₄	—	6	1	no	—
340	C₄	—	10	2	no	—
341	C₄	—	30	2	no	—
342	C₄	—	30	1	no	—
343	C₄	D₅×#67	1	2	yes	—
344	C₃	D₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
345	C₂	—	5	4	no	—
346	C₂	—	6	2	no	—
347	C₂	—	30	2	no	—
348	C₂	D₃×Q₈×D₅	1	1	yes	characteristic
349	C₂	D₅×#239	1	1	yes	characteristic
350	C₂	C₂×D₅×#67	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂×C₃×C₅×Q₈	C₂²
prime radical	C₂²×D₃×D₅	C₂²
hypercenter	C₂×Q₈	D₃×D₅
hypocenter	C₃×C₅	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×D₅	C₂	semidirect
3. #91835	C₂	semidirect
4. C₅×#946	C₂	semidirect
6. D₅×C₄⋊Dic₃	C₂	semidirect
7. #91411	C₂	semidirect
8. #91410	C₂	semidirect
9. D₅×Dic₃⋊C₄	C₂	semidirect
10. #91390	C₂	semidirect
11. #91369	C₂	semidirect
14. C₂×C₃×C₅×Q₈	C₂²	semidirect
20. C₂×C₃×Dic₁₀	C₂²	semidirect
21. C₂×C₃×Dic₁₀	C₂²	semidirect
28. C₄⋊Dic₁₅	C₂²	semidirect
29. #6712	C₂²	semidirect
31. C₅×C₄⋊Dic₃	C₂²	semidirect
32. C₅×Dic₃⋊C₄	C₂²	semidirect
33. Dic₃⋊Dic₅	C₂²	semidirect
34. #6669	C₂²	semidirect
35. #6668	C₂²	semidirect
39. C₂×Q₈×D₅	D₃	semidirect
55. C₂×D₃×D₅	Q₈	semidirect
82. #946	D₅	direct (N × H)
87. C₂×C₅×Q₈	C₂×D₃	semidirect
98. C₂×Dic₁₀	C₂×D₃	semidirect
99. C₂×Dic₁₀	C₂×D₃	semidirect
115. C₂×D₁₅	C₂×Q₈	semidirect
118. C₂×C₅×D₃	C₂×Q₈	semidirect
135. C₂×C₃×Q₈	C₂×D₅	semidirect
151. C₄⋊Dic₃	C₂×D₅	semidirect
155. Dic₃⋊C₄	C₂×D₅	semidirect
196. C₃×D₅	C₂²⋊Q₈	semidirect
246. C₂×Q₈	D₃×D₅	semidirect
267. C₃×C₅	C₂×C₂²⋊Q₈	semidirect
278. C₂×D₃	Q₈×D₅	semidirect
292. D₅	#946	direct (N × H)
324. C₅	C₂×#946	semidirect
344. C₃	D₅×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₈×D₅	2
C₂×C₄×D₃×D₅	2
#91835	2
C₅×#946	2
D₅×(C₂×D₃)⋊C₄	2
D₅×C₄⋊Dic₃	2
#91411	2
#91410	2
D₅×Dic₃⋊C₄	2
#91390	2
#91369	2
D₅×C₂²⋊Q₈	3
C₂×#946	5

Subgroups — D5×#946 #11755269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×#946 #11755269