Subgroups — #11754014

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	D₅×#235	C₂	1	1	yes	maximal, characteristic
2	#91796	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#907	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#6649	C₂	1	1	yes	maximal, characteristic, complemented
6	#91291	C₂	1	1	yes	maximal, characteristic, complemented
7	#91288	C₂	1	1	yes	maximal, characteristic, complemented
8	#91266	C₂	1	2	yes	maximal, complemented
9	#91265	C₂	1	2	yes	maximal, complemented
10	#91246	C₂	1	2	yes	maximal, complemented
11	D₃×#676	C₂	1	2	yes	maximal, complemented
12	#66084	—	3	1	no	maximal, Hall
13	#6817	C₂²	1	1	yes	characteristic
14	C₅×#235	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
16	D₅×#67	C₄	1	2	yes	—
17	#6782	C₄	1	2	yes	—
18	D₅×D₁₂	C₄	1	1	yes	characteristic
19	C₄×D₃×D₅	C₂²	1	2	yes	—
20	#6773	C₄	1	1	yes	characteristic
21	#6772	C₄	1	1	yes	characteristic
22	#6771	C₂²	1	2	yes	—
23	D₅×Dic₆	C₄	1	1	yes	characteristic
24	C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃⋊#168	C₂²	1	1	yes	characteristic, complemented
26	#6705	C₂²	1	2	yes	complemented
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	C₃×#681	C₂²	1	1	yes	characteristic, complemented
32	C₃×#676	C₂²	1	2	yes	—
33	C₃×C₈×D₅	C₂²	1	2	yes	complemented
34	#6651	C₂²	1	2	yes	complemented
35	#6650	C₂²	1	2	yes	complemented
36	#6649	C₂²	1	1	yes	characteristic
37	#6649	C₂²	1	1	yes	characteristic
38	#6649	C₂²	1	1	yes	characteristic, complemented
39	#6649	C₂²	1	1	yes	characteristic, complemented
40	#6648	C₂²	1	2	yes	complemented
41	D₃×#168	C₂²	1	2	yes	complemented
42	#5745	—	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	1	no	—
47	#4053	—	3	2	no	—
48	D₅×M₁₆	—	3	2	no	—
49	D₅×M₁₆	D₃	1	1	yes	characteristic, complemented
50	#4047	—	3	1	no	—
51	#4047	—	3	2	no	—
52	C₂×#676	—	3	1	no	—
53	C₂×#676	—	3	2	no	—
54	C₂²×C₃×D₅	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	#1220	C₂×C₄	1	2	yes	—
58	D₆₀	C₂×C₄	1	1	yes	characteristic
59	C₄×D₁₅	C₂³	1	2	yes	—
60	Dic₃₀	C₂×C₄	1	1	yes	characteristic
61	C₅×#67	C₂×C₄	1	2	yes	—
62	C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×D₃	C₂³	1	2	yes	—
64	C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
65	C₂×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
66	C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
67	C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
68	C₃×C₄×D₅	C₂³	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂³	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	1	yes	characteristic
79	C₃⋊#168	C₂³	1	1	yes	characteristic
80	C₃×#168	C₂³	1	2	yes	—
81	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×#235	—	5	1	no	—
84	C₂×C₃×M₁₆	—	5	1	no	—
85	C₂×#208	—	5	1	no	—
86	#907	—	5	1	no	—
87	#907	—	5	1	no	—
88	#907	—	5	1	no	—
89	#907	D₅	1	1	yes	characteristic, complemented
90	D₃×M₁₆	—	5	2	no	—
91	D₃×M₁₆	—	5	2	no	—
92	#901	—	5	2	no	—
93	#901	—	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	1	no	—
102	#709	—	3	2	no	—
103	C₂×C₄×D₅	—	3	2	no	—
104	C₂×C₄×D₅	C₂×D₃	1	1	yes	characteristic
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	2	no	—
110	#681	C₂×D₃	1	1	yes	characteristic, complemented
111	C₂×#168	—	3	1	no	—
112	C₂×#168	—	3	2	no	—
113	#676	—	3	1	no	—
114	#676	—	3	1	no	—
115	#676	—	3	1	no	—
116	#676	—	3	1	no	—
117	#676	—	3	2	no	—
118	#676	—	3	2	no	—
119	#676	C₂×D₃	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
126	C₂×C₅×D₃	C₂²×C₄	1	2	yes	—
127	C₂×C₃×D₅	—	4	1	no	—
128	C₂×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
134	C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
135	C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
136	C₅×Dic₃	C₂²×C₄	1	2	yes	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂×#67	—	5	2	no	—
139	#235	—	5	1	no	—
140	#235	—	5	2	no	—
141	#235	—	5	1	no	—
142	#235	—	5	2	no	—
143	#235	—	5	1	no	—
144	#235	C₂×D₅	1	1	yes	characteristic
145	C₂×D₁₂	—	5	1	no	—
146	C₂×C₄×D₃	—	5	2	no	—
147	C₂×Dic₆	—	5	1	no	—
148	C₃×M₁₆	—	5	1	no	—
149	C₃×M₁₆	—	5	2	no	—
150	C₃×M₁₆	C₂×D₅	1	1	yes	characteristic, complemented
151	C₂×C₃×C₈	—	5	2	no	—
152	#208	—	5	1	no	—
153	#208	—	5	1	no	—
154	#208	—	5	1	no	—
155	#208	—	5	1	no	—
156	C₂×C₃⋊C₈	—	5	1	no	—
157	C₂×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic, complemented
158	ΓL(1,𝔽₂₅)	—	5	2	no	—
159	ΓL(1,𝔽₂₅)	—	5	2	no	—
160	ΓL(1,𝔽₂₅)	—	5	2	no	—
161	ΓL(1,𝔽₂₅)	C₂×D₅	1	2	yes	complemented
162	D₃×C₈	—	5	2	no	—
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	2	no	—
167	C₂²×D₅	C₄×D₃	1	1	yes	characteristic
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	—	3	2	no	—
176	C₂×Dic₅	—	3	2	no	—
177	C₂×Dic₅	C₄×D₃	1	1	yes	characteristic
178	D₂₀	—	3	2	no	—
179	D₂₀	—	3	1	no	—
180	C₄×D₅	—	3	2	no	—
181	C₄×D₅	—	3	2	no	—
182	C₄×D₅	—	3	2	no	—
183	C₄×D₅	C₄×D₃	1	1	yes	characteristic
184	C₄×D₅	C₄×D₃	1	1	yes	characteristic
185	C₄×D₅	C₂²×D₃	1	1	yes	characteristic
186	C₄×D₅	C₂²×D₃	1	1	yes	characteristic
187	Dic₁₀	—	3	1	no	—
188	Dic₁₀	—	3	2	no	—
189	C₅×C₈	—	3	1	no	—
190	C₅×C₈	—	3	1	no	—
191	C₅×C₈	C₂²×D₃	1	2	yes	—
192	#168	—	3	1	no	—
193	#168	—	3	1	no	—
194	#168	C₂²×D₃	1	2	yes	—
195	C₂×#41	—	15	1	no	—
196	#131	—	3	1	no	—
197	#131	—	15	2	no	—
198	#131	—	15	1	no	—
199	#131	—	15	2	no	—
200	#131	—	15	1	no	—
201	#131	—	15	1	no	—
202	C₂×M₁₆	—	5	1	no	—
203	C₂×M₁₆	—	15	2	no	—
204	C₂×M₁₆	—	15	1	no	—
205	C₂×M₁₆	—	15	2	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
208	D₁₅	—	2	2	no	—
209	C₃×D₅	—	2	1	no	—
210	C₃×D₅	—	2	1	no	—
211	C₅×D₃	—	2	2	no	—
212	C₂³×C₃	—	5	1	no	—
213	C₂²×D₃	—	5	2	no	—
214	C₂×C₃×C₄	—	5	1	no	—
215	C₂×C₃×C₄	—	5	1	no	—
216	C₂×C₃×C₄	—	5	1	no	—
217	C₂×C₃×C₄	—	5	1	no	—
218	C₂×C₃×C₄	—	5	1	no	—
219	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
220	#67	—	5	2	no	—
221	#67	—	5	2	no	—
222	#67	—	5	2	no	—
223	#67	C₄×D₅	1	2	yes	—
224	C₂×Dic₃	—	5	2	no	—
225	D₁₂	—	5	2	no	—
226	D₁₂	—	5	1	no	—
227	D₁₂	C₄×D₅	1	1	yes	characteristic
228	C₄×D₃	—	5	2	no	—
229	C₄×D₃	—	5	2	no	—
230	C₄×D₃	—	5	2	no	—
231	C₄×D₃	C₂²×D₅	1	2	yes	—
232	Dic₆	—	5	1	no	—
233	Dic₆	—	5	2	no	—
234	Dic₆	C₄×D₅	1	1	yes	characteristic
235	C₃×C₈	—	5	2	no	—
236	C₃×C₈	C₂²×D₅	1	2	yes	—
237	C₃⋊C₈	—	5	1	no	—
238	C₃⋊C₈	—	5	1	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	2	no	—
244	C₂×D₅	—	4	1	no	—
245	C₂×D₅	—	12	2	no	—
246	C₂×D₅	C₂×C₄×D₃	1	1	yes	characteristic
247	C₂×D₅	C₂×C₄×D₃	1	1	yes	characteristic
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	2	no	—
252	Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
253	Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
254	#41	—	3	1	no	—
255	#41	—	15	2	no	—
256	#41	—	15	2	no	—
257	#41	—	15	1	no	—
258	#41	—	15	1	no	—
259	#41	—	15	1	no	—
260	C₂×Q₈	—	15	1	no	—
261	C₂×D₄	—	15	1	no	—
262	C₂×D₄	—	15	2	no	—
263	C₂²×C₄	—	5	1	no	—
264	C₂²×C₄	—	15	2	no	—
265	M₁₆	—	3	2	no	—
266	M₁₆	—	5	1	no	—
267	M₁₆	—	5	2	no	—
268	M₁₆	—	15	2	no	—
269	M₁₆	—	15	1	no	—
270	M₁₆	—	15	1	no	—
271	M₁₆	—	15	1	no	—
272	M₁₆	—	15	1	no	—
273	M₁₆	—	15	2	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	2	no	—
279	C₂×C₈	—	15	2	no	—
280	C₂×C₈	—	15	2	no	—
281	C₂×C₈	—	15	1	no	—
282	C₂×C₈	—	15	2	no	—
283	C₃×C₅	#568	1	1	yes	characteristic, complemented, Hall
284	C₂²×C₃	—	5	1	no	—
285	C₂²×C₃	—	5	1	no	—
286	C₂²×C₃	—	20	1	no	—
287	C₂²×C₃	C₂×C₄×D₅	1	1	yes	characteristic
288	C₂×D₃	—	5	2	no	—
289	C₂×D₃	—	20	2	no	—
290	C₂×D₃	C₂×C₄×D₅	1	2	yes	—
291	C₃×C₄	—	5	1	no	—
292	C₃×C₄	—	5	1	no	—
293	C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
294	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
295	Dic₃	—	5	2	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	1	no	—
301	D₅	—	2	1	no	—
302	D₅	—	6	2	no	—
303	C₂³	—	5	1	no	—
304	C₂³	—	15	2	no	—
305	Q₈	—	3	1	no	—
306	Q₈	—	15	2	no	—
307	Q₈	—	15	1	no	—
308	D₄	—	3	2	no	—
309	D₄	—	3	1	no	—
310	D₄	—	15	2	no	—
311	D₄	—	15	2	no	—
312	D₄	—	15	2	no	—
313	D₄	—	15	2	no	—
314	D₄	—	15	1	no	—
315	C₂×C₄	—	3	2	no	—
316	C₂×C₄	—	5	1	no	—
317	C₂×C₄	—	5	1	no	—
318	C₂×C₄	—	5	1	no	—
319	C₂×C₄	—	5	1	no	—
320	C₂×C₄	—	5	1	no	—
321	C₂×C₄	—	15	2	no	—
322	C₂×C₄	—	15	2	no	—
323	C₂×C₄	—	15	2	no	—
324	C₂×C₄	—	15	2	no	—
325	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
326	C₈	—	3	1	no	—
327	C₈	—	3	1	no	—
328	C₈	—	5	2	no	—
329	C₈	—	15	1	no	—
330	C₈	—	15	1	no	—
331	C₈	C₂×D₃×D₅	1	2	yes	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	10	1	no	—
334	C₂×C₃	—	10	1	no	—
335	C₂×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
336	D₃	—	2	2	no	—
337	D₃	—	10	2	no	—
338	C₅	#5745	1	1	yes	characteristic, complemented, Sylow
339	C₂²	—	3	2	no	—
340	C₂²	—	5	1	no	—
341	C₂²	—	5	1	no	—
342	C₂²	—	15	2	no	—
343	C₂²	—	20	1	no	—
344	C₂²	—	60	2	no	—
345	C₂²	C₄×D₃×D₅	1	1	yes	characteristic
346	C₄	—	3	2	no	—
347	C₄	—	5	1	no	—
348	C₄	—	5	1	no	—
349	C₄	—	15	2	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	1	no	—
356	C₂	—	10	1	no	—
357	C₂	—	30	2	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	D₅×#235	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. #91796	C₂	semidirect
3. C₅×#907	C₂	semidirect
4. C₃×D₅×M₁₆	C₂	semidirect
5. C₂×#6649	C₂	semidirect
6. #91291	C₂	semidirect
7. #91288	C₂	semidirect
8. #91266	C₂	semidirect
9. #91265	C₂	semidirect
10. #91246	C₂	semidirect
11. D₃×#676	C₂	semidirect
24. C₃×C₅×M₁₆	C₂²	semidirect
25. C₂×C₃⋊#168	C₂²	semidirect
26. #6705	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. C₃×#681	C₂²	semidirect
33. C₃×C₈×D₅	C₂²	semidirect
34. #6651	C₂²	semidirect
35. #6650	C₂²	semidirect
38. #6649	C₂²	semidirect
39. #6649	C₂²	semidirect
40. #6648	C₂²	semidirect
41. D₃×#168	C₂²	semidirect
49. D₅×M₁₆	D₃	semidirect
89. #907	D₅	semidirect
106. C₅×M₁₆	C₂×D₃	semidirect
110. #681	C₂×D₃	semidirect
122. C₈×D₅	C₂×D₃	semidirect
150. C₃×M₁₆	C₂×D₅	semidirect
157. C₂×C₃⋊C₈	C₂×D₅	semidirect
161. ΓL(1,𝔽₂₅)	C₂×D₅	semidirect
165. D₃×C₈	C₂×D₅	semidirect
275. M₁₆	D₃×D₅	semidirect
283. C₃×C₅	#568	semidirect
338. C₅	#5745	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
D₅×#235	2
#91796	2
C₅×#907	2
C₃×D₅×M₁₆	2
C₂×#6649	2
#91291	2
#91288	2
#91266	2
#91265	2
#91246	2
D₃×#676	2
#66084	3
#5745	5