Subgroups — #11754018

Order 960.

516 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#92016	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#907	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4053	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#6651	C₂	1	1	yes	maximal, characteristic, complemented
6	#91286	C₂	1	1	yes	maximal, characteristic, complemented
7	#91283	C₂	1	1	yes	maximal, characteristic, complemented
8	#91268	C₂	1	2	yes	maximal, complemented
9	#91265	C₂	1	2	yes	maximal, complemented
10	D₃×#676	C₂	1	2	yes	maximal, complemented
11	D₅×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
12	#66084	—	3	1	no	maximal, Hall
13	C₂×C₄×D₁₅	C₂²	1	1	yes	characteristic
14	C₅×#235	C₂²	1	1	yes	characteristic
15	C₃×#709	C₂²	1	1	yes	characteristic
16	#6790	C₄	1	2	yes	—
17	#6780	C₄	1	2	yes	—
18	#6777	C₄	1	1	yes	characteristic
19	C₄×D₃×D₅	C₂²	1	2	yes	—
20	#6771	C₂²	1	2	yes	—
21	D₁₅⋊Q₈	C₄	1	1	yes	characteristic
22	#6768	C₄	1	1	yes	characteristic
23	#6766	C₄	1	1	yes	characteristic
24	C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
25	#6710	C₂²	1	1	yes	characteristic, complemented
26	#6705	C₂²	1	2	yes	—
27	C₈×D₁₅	C₂²	1	2	yes	complemented
28	C₂×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
29	C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
30	C₅×D₃×C₈	C₂²	1	2	yes	complemented
31	C₂×C₃×#168	C₂²	1	1	yes	characteristic, complemented
32	C₃×#676	C₂²	1	2	yes	complemented
33	C₃×C₈×D₅	C₂²	1	2	yes	complemented
34	#6651	C₂²	1	1	yes	characteristic
35	#6651	C₂²	1	1	yes	characteristic
36	#6651	C₂²	1	1	yes	characteristic, complemented
37	#6651	C₂²	1	1	yes	characteristic, complemented
38	#6650	C₂²	1	2	yes	complemented
39	#6649	C₂²	1	2	yes	complemented
40	D₃×#168	C₂²	1	2	yes	complemented
41	D₅×C₃⋊C₈	C₂²	1	2	yes	complemented
42	#5750	—	5	1	no	maximal, Hall
43	D₅×#41	—	3	1	no	—
44	C₅×#131	—	3	1	no	—
45	#4094	—	3	1	no	—
46	#4053	—	3	2	no	—
47	#4053	D₃	1	1	yes	characteristic, complemented
48	D₅×M₁₆	—	3	1	no	—
49	D₅×M₁₆	—	3	2	no	—
50	#4047	—	3	1	no	—
51	#4047	—	3	2	no	—
52	C₂×#676	—	3	1	no	—
53	C₂×#676	—	3	2	no	—
54	C₂²×D₁₅	C₂×C₄	1	1	yes	characteristic
55	C₂×D₃×D₅	C₂×C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×Dic₁₅	C₂×C₄	1	1	yes	characteristic
58	C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
59	C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
60	C₄×D₁₅	C₂³	1	1	yes	characteristic
61	C₄×D₁₅	C₂³	1	1	yes	characteristic
62	C₅×#67	C₂×C₄	1	2	yes	—
63	C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
64	C₄×C₅×D₃	C₂³	1	2	yes	—
65	C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
66	C₃×C₂²⋊D₅	C₂×C₄	1	2	yes	—
67	C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
68	C₃×C₄×D₅	C₂³	1	2	yes	—
69	C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
70	#1204	C₂×C₄	1	2	yes	—
71	#1203	C₂×C₄	1	2	yes	—
72	#1202	C₂×C₄	1	2	yes	—
73	#1201	C₂×C₄	1	2	yes	—
74	#1200	C₂×C₄	1	2	yes	—
75	D₃×Dic₅	C₂×C₄	1	2	yes	—
76	D₅×Dic₃	C₂×C₄	1	2	yes	—
77	C₃×C₅×C₈	C₂³	1	2	yes	—
78	C₃⋊#168	C₂³	1	2	yes	—
79	C₃×#168	C₂³	1	1	yes	characteristic
80	C₃×#168	C₂³	1	1	yes	characteristic
81	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	D₃×#41	—	5	1	no	—
84	C₃×#131	—	5	1	no	—
85	#948	—	5	1	no	—
86	#907	—	5	2	no	—
87	#907	D₅	1	1	yes	characteristic, complemented
88	D₃×M₁₆	—	5	1	no	—
89	D₃×M₁₆	—	5	2	no	—
90	#901	—	5	1	no	—
91	#901	—	5	2	no	—
92	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
93	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
94	C₅×#41	—	3	1	no	—
95	#713	—	3	1	no	—
96	Q₈×D₅	—	3	1	no	—
97	#711	—	3	2	no	—
98	#711	—	3	1	no	—
99	D₄×D₅	—	3	2	no	—
100	D₄×D₅	—	3	1	no	—
101	#709	—	3	2	no	—
102	#709	C₂×D₃	1	1	yes	characteristic
103	C₂×C₄×D₅	—	3	1	no	—
104	C₂×C₄×D₅	—	3	2	no	—
105	C₅×M₁₆	—	3	2	no	—
106	C₅×M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
107	C₂×C₅×C₈	—	3	1	no	—
108	C₂×C₅×C₈	—	3	2	no	—
109	#681	—	3	1	no	—
110	#681	—	3	2	no	—
111	C₂×#168	—	3	2	no	—
112	C₂×#168	C₂×D₃	1	1	yes	characteristic, complemented
113	#676	—	3	2	no	—
114	#676	—	3	2	no	—
115	#676	—	3	1	no	—
116	#676	—	3	1	no	—
117	#676	—	3	1	no	—
118	#676	—	3	1	no	—
119	#676	C₂×D₃	1	2	yes	complemented
120	C₈×D₅	—	3	2	no	—
121	C₈×D₅	—	3	2	no	—
122	C₈×D₅	C₂×D₃	1	2	yes	complemented
123	#568	—	15	1	no	Sylow
124	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
125	C₂×D₁₅	—	4	1	no	—
126	C₂×D₁₅	C₂²×C₄	1	1	yes	characteristic
127	C₂×D₁₅	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₅×D₃	C₂²×C₄	1	2	yes	—
129	C₂×C₃×D₅	C₂²×C₄	1	2	yes	—
130	D₃×D₅	—	4	2	no	—
131	C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
132	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
133	Dic₁₅	C₂²×C₄	1	1	yes	characteristic
134	Dic₁₅	C₂²×C₄	1	1	yes	characteristic
135	C₃×Dic₅	C₂²×C₄	1	2	yes	—
136	C₅×Dic₃	C₂²×C₄	1	2	yes	—
137	C₃×#41	—	5	1	no	—
138	#239	—	5	1	no	—
139	D₃×Q₈	—	5	1	no	—
140	#237	—	5	2	no	—
141	#237	—	5	1	no	—
142	D₃×D₄	—	5	2	no	—
143	D₃×D₄	—	5	1	no	—
144	#235	—	5	2	no	—
145	#235	C₂×D₅	1	1	yes	characteristic
146	C₂×C₄×D₃	—	5	1	no	—
147	C₂×C₄×D₃	—	5	2	no	—
148	C₃×M₁₆	—	5	2	no	—
149	C₃×M₁₆	C₂×D₅	1	1	yes	characteristic, complemented
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	5	2	no	—
152	#208	—	5	1	no	—
153	#208	—	5	2	no	—
154	C₂×C₃⋊C₈	—	5	2	no	—
155	C₂×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
156	ΓL(1,𝔽₂₅)	—	5	2	no	—
157	ΓL(1,𝔽₂₅)	—	5	2	no	—
158	ΓL(1,𝔽₂₅)	—	5	1	no	—
159	ΓL(1,𝔽₂₅)	—	5	1	no	—
160	ΓL(1,𝔽₂₅)	—	5	1	no	—
161	ΓL(1,𝔽₂₅)	—	5	1	no	—
162	ΓL(1,𝔽₂₅)	C₂×D₅	1	2	yes	complemented
163	D₃×C₈	—	5	2	no	—
164	D₃×C₈	—	5	2	no	—
165	D₃×C₈	C₂×D₅	1	2	yes	complemented
166	C₂²×D₅	—	3	1	no	—
167	C₂²×D₅	—	3	2	no	—
168	C₅×Q₈	—	3	1	no	—
169	C₅×D₄	—	3	2	no	—
170	C₅×D₄	—	3	1	no	—
171	C₂×C₄×C₅	—	3	2	no	—
172	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
173	C₂²⋊D₅	—	3	2	no	—
174	C₂²⋊D₅	—	3	2	no	—
175	C₂²⋊D₅	C₄×D₃	1	2	yes	—
176	C₂×Dic₅	—	3	1	no	—
177	C₂×Dic₅	—	3	2	no	—
178	D₂₀	—	3	2	no	—
179	D₂₀	C₄×D₃	1	1	yes	characteristic
180	C₄×D₅	—	3	2	no	—
181	C₄×D₅	—	3	1	no	—
182	C₄×D₅	—	3	1	no	—
183	C₄×D₅	—	3	1	no	—
184	C₄×D₅	—	3	1	no	—
185	C₄×D₅	—	3	2	no	—
186	C₄×D₅	C₂²×D₃	1	2	yes	—
187	Dic₁₀	—	3	2	no	—
188	Dic₁₀	C₄×D₃	1	1	yes	characteristic
189	C₅×C₈	—	3	1	no	—
190	C₅×C₈	—	3	1	no	—
191	C₅×C₈	C₂²×D₃	1	2	yes	—
192	#168	—	3	2	no	—
193	#168	C₂²×D₃	1	1	yes	characteristic
194	#168	C₂²×D₃	1	1	yes	characteristic
195	C₂×#41	—	15	1	no	—
196	#131	—	3	1	no	—
197	#131	—	5	1	no	—
198	#131	—	15	2	no	—
199	#131	—	15	1	no	—
200	#131	—	15	1	no	—
201	#131	—	15	2	no	—
202	C₂×M₁₆	—	15	1	no	—
203	C₂×M₁₆	—	15	2	no	—
204	C₂×M₁₆	—	15	2	no	—
205	C₂×M₁₆	—	15	1	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
208	D₁₅	—	2	1	no	—
209	D₁₅	—	2	1	no	—
210	C₃×D₅	—	2	2	no	—
211	C₅×D₃	—	2	2	no	—
212	C₂²×D₃	—	5	1	no	—
213	C₂²×D₃	—	5	2	no	—
214	C₃×Q₈	—	5	1	no	—
215	C₃×D₄	—	5	2	no	—
216	C₃×D₄	—	5	1	no	—
217	C₂×C₃×C₄	—	5	2	no	—
218	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
219	#67	—	5	2	no	—
220	#67	—	5	2	no	—
221	#67	C₄×D₅	1	2	yes	—
222	C₂×Dic₃	—	5	1	no	—
223	C₂×Dic₃	—	5	2	no	—
224	D₁₂	—	5	2	no	—
225	D₁₂	C₄×D₅	1	1	yes	characteristic
226	C₄×D₃	—	5	2	no	—
227	C₄×D₃	—	5	1	no	—
228	C₄×D₃	—	5	1	no	—
229	C₄×D₃	—	5	1	no	—
230	C₄×D₃	—	5	1	no	—
231	C₄×D₃	—	5	2	no	—
232	C₄×D₃	C₂²×D₅	1	2	yes	—
233	Dic₆	—	5	2	no	—
234	Dic₆	C₄×D₅	1	1	yes	characteristic
235	C₃×C₈	—	5	1	no	—
236	C₃×C₈	—	5	1	no	—
237	C₃×C₈	C₂²×D₅	1	2	yes	—
238	C₃⋊C₈	—	5	2	no	—
239	C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
240	C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
241	C₂²×C₅	—	3	2	no	—
242	C₂²×C₅	C₂×C₄×D₃	1	1	yes	characteristic
243	C₂×D₅	—	3	1	no	—
244	C₂×D₅	—	3	1	no	—
245	C₂×D₅	—	12	2	no	—
246	C₂×D₅	—	12	1	no	—
247	C₂×D₅	C₂×C₄×D₃	1	2	yes	—
248	C₄×C₅	—	3	2	no	—
249	C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
250	C₄×C₅	C₂³×D₃	1	1	yes	characteristic
251	Dic₅	—	3	1	no	—
252	Dic₅	—	3	1	no	—
253	Dic₅	C₂×C₄×D₃	1	2	yes	—
254	#41	—	3	1	no	—
255	#41	—	5	1	no	—
256	#41	—	15	2	no	—
257	#41	—	15	2	no	—
258	#41	—	15	1	no	—
259	#41	—	15	1	no	—
260	C₂×Q₈	—	15	1	no	—
261	C₂×D₄	—	15	2	no	—
262	C₂×D₄	—	15	1	no	—
263	C₂²×C₄	—	15	2	no	—
264	C₂²×C₄	—	15	1	no	—
265	M₁₆	—	3	2	no	—
266	M₁₆	—	5	2	no	—
267	M₁₆	—	15	1	no	—
268	M₁₆	—	15	1	no	—
269	M₁₆	—	15	1	no	—
270	M₁₆	—	15	1	no	—
271	M₁₆	—	15	2	no	—
272	M₁₆	—	15	2	no	—
273	M₁₆	—	15	1	no	—
274	M₁₆	—	15	2	no	—
275	M₁₆	D₃×D₅	1	1	yes	characteristic, complemented
276	C₂×C₈	—	3	1	no	—
277	C₂×C₈	—	3	2	no	—
278	C₂×C₈	—	5	1	no	—
279	C₂×C₈	—	5	2	no	—
280	C₂×C₈	—	15	2	no	—
281	C₂×C₈	—	15	2	no	—
282	C₂×C₈	—	15	2	no	—
283	C₃×C₅	#568	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	—	5	2	no	—
285	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
286	C₂×D₃	—	5	1	no	—
287	C₂×D₃	—	5	1	no	—
288	C₂×D₃	—	20	2	no	—
289	C₂×D₃	—	20	1	no	—
290	C₂×D₃	C₂×C₄×D₅	1	2	yes	—
291	C₃×C₄	—	5	2	no	—
292	C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
293	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
294	Dic₃	—	5	1	no	—
295	Dic₃	—	5	1	no	—
296	Dic₃	C₂×C₄×D₅	1	2	yes	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	6	2	no	—
299	C₂×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
300	D₅	—	2	2	no	—
301	D₅	—	6	1	no	—
302	D₅	—	6	1	no	—
303	C₂³	—	15	2	no	—
304	C₂³	—	15	1	no	—
305	Q₈	—	3	1	no	—
306	Q₈	—	5	1	no	—
307	Q₈	—	15	2	no	—
308	D₄	—	3	2	no	—
309	D₄	—	3	1	no	—
310	D₄	—	5	2	no	—
311	D₄	—	5	1	no	—
312	D₄	—	15	2	no	—
313	D₄	—	15	2	no	—
314	D₄	—	15	2	no	—
315	C₂×C₄	—	3	2	no	—
316	C₂×C₄	—	5	2	no	—
317	C₂×C₄	—	15	2	no	—
318	C₂×C₄	—	15	2	no	—
319	C₂×C₄	—	15	2	no	—
320	C₂×C₄	—	15	1	no	—
321	C₂×C₄	—	15	1	no	—
322	C₂×C₄	—	15	1	no	—
323	C₂×C₄	—	15	1	no	—
324	C₂×C₄	—	15	1	no	—
325	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
326	C₈	—	3	1	no	—
327	C₈	—	3	1	no	—
328	C₈	—	5	1	no	—
329	C₈	—	5	1	no	—
330	C₈	—	15	2	no	—
331	C₈	C₂×D₃×D₅	1	2	yes	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	10	2	no	—
334	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
335	D₃	—	2	2	no	—
336	D₃	—	10	1	no	—
337	D₃	—	10	1	no	—
338	C₅	#5750	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	3	2	no	—
340	C₂²	—	5	2	no	—
341	C₂²	—	15	1	no	—
342	C₂²	—	15	1	no	—
343	C₂²	—	60	1	no	—
344	C₂²	—	60	2	no	—
345	C₂²	C₄×D₃×D₅	1	1	yes	characteristic
346	C₄	—	3	2	no	—
347	C₄	—	5	2	no	—
348	C₄	—	15	1	no	—
349	C₄	—	15	1	no	—
350	C₄	C₄×D₃×D₅	1	1	yes	characteristic
351	C₄	C₂²×D₃×D₅	1	1	yes	characteristic
352	C₃	#66084	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	1	no	—
354	C₂	—	6	2	no	—
355	C₂	—	10	2	no	—
356	C₂	—	30	1	no	—
357	C₂	—	30	1	no	—
358	C₂	C₂×C₄×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×D₃×D₅
commutator, socle	C₂×C₃×C₅	C₂³×C₄
Fitting	C₃×C₅×M₁₆	C₂²
prime radical	#92016	C₂
cosocle	C₃×C₄×C₅	C₂⁴
hypercenter	M₁₆	D₃×D₅
hypocenter	C₃×C₅	#568
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₁₅	C₂	semidirect
3. C₅×#907	C₂	semidirect
4. C₃×#4053	C₂	semidirect
5. C₂×#6651	C₂	semidirect
6. #91286	C₂	semidirect
7. #91283	C₂	semidirect
8. #91268	C₂	semidirect
9. #91265	C₂	semidirect
10. D₃×#676	C₂	semidirect
11. D₅×ΓL(1,𝔽₂₅)	C₂	semidirect
24. C₃×C₅×M₁₆	C₂²	semidirect
25. #6710	C₂²	semidirect
27. C₈×D₁₅	C₂²	semidirect
28. C₂×C₅×C₃⋊C₈	C₂²	semidirect
29. C₅×ΓL(1,𝔽₂₅)	C₂²	semidirect
30. C₅×D₃×C₈	C₂²	semidirect
31. C₂×C₃×#168	C₂²	semidirect
32. C₃×#676	C₂²	semidirect
33. C₃×C₈×D₅	C₂²	semidirect
36. #6651	C₂²	semidirect
37. #6651	C₂²	semidirect
38. #6650	C₂²	semidirect
39. #6649	C₂²	semidirect
40. D₃×#168	C₂²	semidirect
41. D₅×C₃⋊C₈	C₂²	semidirect
47. #4053	D₃	semidirect
87. #907	D₅	semidirect
106. C₅×M₁₆	C₂×D₃	semidirect
112. C₂×#168	C₂×D₃	semidirect
119. #676	C₂×D₃	semidirect
122. C₈×D₅	C₂×D₃	semidirect
149. C₃×M₁₆	C₂×D₅	semidirect
155. C₂×C₃⋊C₈	C₂×D₅	semidirect
162. ΓL(1,𝔽₂₅)	C₂×D₅	semidirect
165. D₃×C₈	C₂×D₅	semidirect
275. M₁₆	D₃×D₅	semidirect
283. C₃×C₅	#568	semidirect
338. C₅	#5750	semidirect
352. C₃	#66084	semidirect

Maximal subgroups

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

subgroup	index
#92016	2
M₁₆×D₁₅	2
C₅×#907	2
C₃×#4053	2
C₂×#6651	2
#91286	2
#91283	2
#91268	2
#91265	2
D₃×#676	2
D₅×ΓL(1,𝔽₂₅)	2
#66084	3
#5750	5