Subgroups — #12068707

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	#41×#168	C₂	1	2	yes	maximal, complemented
3	#10602431	C₂	1	2	yes	maximal, complemented
4	#10601695	C₂	1	2	yes	maximal, complemented
5	SD₁₆⋊#168	C₂	1	4	yes	maximal, complemented
6	#10601671	C₂	1	2	yes	maximal, complemented
7	#10600522	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#64675	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₄	1	1	yes	characteristic
10	C₄×C₅×#41	C₂²	1	2	yes	—
11	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
12	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₅×SD₁₆	C₄	1	2	yes	—
14	C₄×C₅×SD₁₆	C₂²	1	2	yes	—
15	C₄×C₅×D₈	C₄	1	1	yes	characteristic
16	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
17	C₅×#427	C₄	1	1	yes	characteristic
18	C₅×#416	C₄	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	Q₈×#168	C₂²	1	4	yes	—
21	D₄×#168	—	2	4	no	—
22	D₄×#168	C₂²	1	4	yes	—
23	#65214	—	2	2	no	—
24	#65214	C₂²	1	2	yes	—
25	C₂×C₄×#168	—	2	2	no	—
26	C₂×C₄×#168	C₂²	1	1	yes	characteristic
27	C₂×C₄×#168	C₂²	1	1	yes	characteristic
28	Q₈⋊#168	C₂²	1	4	yes	—
29	D₄⋊#168	C₂²	1	4	yes	—
30	#64678	C₂²	1	2	yes	—
31	#64677	C₂²	1	2	yes	—
32	#64675	C₂²	1	2	yes	—
33	#64675	C₂²	1	2	yes	—
34	#17849	—	5	1	no	maximal, Sylow
35	C₂×C₅×#41	C₂×C₄	1	2	yes	—
36	C₅×#135	C₈	1	8	yes	complemented
37	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
43	C₄×C₅×Q₈	C₂³	1	2	yes	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₄×C₅×D₄	—	2	2	no	—
46	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
47	C₄×C₅×D₄	C₂³	1	2	yes	—
48	C₅×#117	—	2	2	no	—
49	C₅×#117	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂²×#168	—	2	1	no	—
53	C₂²×#168	—	2	4	no	—
54	C₂²×#168	D₄	1	2	yes	—
55	C₅×#107	C₂×C₄	1	1	yes	characteristic
56	C₅×#107	C₂×C₄	1	1	yes	characteristic
57	C₅×#106	C₂×C₄	1	2	yes	—
58	C₅×#103	C₂×C₄	1	2	yes	—
59	C₅×#103	C₂×C₄	1	2	yes	—
60	C₅×#102	C₂×C₄	1	2	yes	—
61	C₅×#102	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂²⋊#168	—	2	2	no	—
67	C₂²⋊#168	—	2	4	no	—
68	C₄⋊#168	—	2	4	no	—
69	C₄⋊#168	C₂³	1	4	yes	—
70	C₄×#168	—	2	2	no	—
71	C₄×#168	—	2	1	no	—
72	C₄×#168	—	2	4	no	—
73	C₄×#168	D₄	1	2	yes	—
74	C₄×#168	C₂³	1	2	yes	—
75	C₄×#168	C₂³	1	2	yes	—
76	C₈×#41	—	5	2	no	—
77	C₄×#135	D₅	1	1	yes	characteristic
78	#1580	—	5	2	no	—
79	#1579	—	5	2	no	—
80	SD₁₆⋊C₈	—	5	4	no	—
81	#1564	—	5	1	no	—
82	#1476	—	5	2	no	—
83	C₂×#322	—	5	1	no	—
84	C₅×#41	—	2	4	no	—
85	C₅×#41	C₂×C₈	1	8	yes	complemented
86	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	#41	1	1	yes	characteristic
95	C₅×Q₁₆	C₂×C₈	1	4	yes	complemented
96	C₅×SD₁₆	C₂×C₈	1	8	yes	complemented
97	C₅×D₈	C₂×C₈	1	4	yes	complemented
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	C₂×C₈	1	4	yes	complemented
100	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
107	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
108	C₅×C₂²⋊C₄	—	2	2	no	—
109	C₅×C₂²⋊C₄	—	2	2	no	—
110	C₅×C₂²⋊C₄	—	2	2	no	—
111	C₄²×C₅	—	2	2	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	#41	1	1	yes	characteristic
117	C₄²×C₅	C₂⁴	1	1	yes	characteristic
118	C₂×#168	—	2	4	no	—
119	C₂×#168	—	2	2	no	—
120	C₂×#168	—	4	2	no	—
121	C₂×#168	—	4	8	no	—
122	C₂×#168	—	4	2	no	—
123	C₂×#168	C₂×D₄	1	4	yes	—
124	C₂×#135	Dic₅	1	1	yes	characteristic
125	C₄×#41	C₂×D₅	1	2	yes	—
126	C₈×Q₈	—	5	4	no	—
127	C₄×Q₁₆	Dic₅	1	1	yes	characteristic
128	C₄×Q₁₆	C₂×D₅	1	1	yes	characteristic
129	C₄×SD₁₆	Dic₅	1	2	yes	—
130	C₄×SD₁₆	C₂×D₅	1	2	yes	—
131	C₄×D₈	Dic₅	1	1	yes	characteristic
132	C₄×D₈	C₂×D₅	1	1	yes	characteristic
133	C₈×D₄	—	5	4	no	—
134	C₈×D₄	—	10	4	no	—
135	#431	—	5	2	no	—
136	#431	—	10	2	no	—
137	#427	Dic₅	1	1	yes	characteristic
138	#416	Dic₅	1	2	yes	—
139	C₂×C₄×C₈	—	5	1	no	—
140	C₂×C₄×C₈	—	5	1	no	—
141	C₂×C₄×C₈	—	10	2	no	—
142	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
143	#335	—	5	2	no	—
144	#334	—	5	2	no	—
145	Q₈⋊C₈	—	5	4	no	—
146	D₄⋊C₈	—	5	4	no	—
147	#322	—	5	2	no	—
148	#322	—	5	2	no	—
149	C₂³×C₅	—	2	2	no	—
150	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	C₂²×C₈	1	4	yes	—
153	C₅×D₄	—	2	8	no	—
154	C₅×D₄	—	2	2	no	—
155	C₅×D₄	C₂²×C₈	1	4	yes	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	8	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	4	4	no	—
163	C₂×C₄×C₅	—	4	4	no	—
164	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
168	C₂×C₄×C₅	#131	1	2	yes	—
169	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
172	C₅×C₈	—	2	1	no	—
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	C₂²×C₈	1	2	yes	—
175	C₅×C₈	C₂²×C₈	1	2	yes	—
176	#168	—	2	4	no	—
177	#168	—	4	2	no	—
178	#168	—	4	4	no	—
179	C₂×#41	C₂×Dic₅	1	2	yes	—
180	#135	#168	1	8	yes	complemented
181	C₂×Q₁₆	C₂×Dic₅	1	1	yes	characteristic
182	C₂×SD₁₆	C₂×Dic₅	1	2	yes	—
183	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
184	C₂²×C₈	—	5	2	no	—
185	C₂²×C₈	—	10	4	no	—
186	C₂²×C₈	—	10	1	no	—
187	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
189	C₄×Q₈	C₂×Dic₅	1	2	yes	—
190	C₄×Q₈	C₂²×D₅	1	2	yes	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	C₂×Dic₅	1	2	yes	—
194	C₄×D₄	C₂²×D₅	1	2	yes	—
195	#117	—	2	2	no	—
196	#117	C₂×Dic₅	1	2	yes	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
199	#107	C₂×Dic₅	1	1	yes	characteristic
200	#107	C₂×Dic₅	1	1	yes	characteristic
201	#106	C₂×Dic₅	1	2	yes	—
202	C₄⋊C₈	—	5	4	no	—
203	C₄⋊C₈	—	10	4	no	—
204	#103	C₂×Dic₅	1	2	yes	—
205	#103	C₂×Dic₅	1	2	yes	—
206	#102	C₂×Dic₅	1	2	yes	—
207	#102	C₂×Dic₅	1	2	yes	—
208	C₂²⋊C₈	—	10	4	no	—
209	C₂²⋊C₈	—	10	2	no	—
210	C₄×C₈	—	5	2	no	—
211	C₄×C₈	—	5	2	no	—
212	C₄×C₈	—	5	2	no	—
213	C₄×C₈	—	10	4	no	—
214	C₄×C₈	—	10	2	no	—
215	C₄×C₈	—	10	1	no	—
216	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
217	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	4	no	—
223	C₂²×C₅	—	4	2	no	—
224	C₂²×C₅	—	4	2	no	—
225	C₂²×C₅	C₈×D₄	1	2	yes	—
226	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	2	1	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	C₈×D₄	1	2	yes	—
233	C₄×C₅	#444	1	2	yes	—
234	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
235	C₄×C₅	C₂×#131	1	1	yes	characteristic
236	C₄×C₅	#575	1	2	yes	—
237	#41	—	2	4	no	—
238	#41	C₂×#168	1	8	yes	complemented
239	C₂×Q₈	C₂²×Dic₅	1	2	yes	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	C₂²×Dic₅	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
246	C₂²×C₄	#711	1	1	yes	characteristic
247	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
248	Q₁₆	C₂×#168	1	4	yes	complemented
249	SD₁₆	C₂×#168	1	8	yes	complemented
250	D₈	C₂×#168	1	4	yes	complemented
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	5	4	no	—
253	C₂×C₈	—	10	4	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	—	20	2	no	—
256	C₂×C₈	—	20	8	no	—
257	C₂×C₈	—	20	2	no	—
258	C₂×C₈	C₂×#168	1	4	yes	complemented
259	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
261	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
262	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
266	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	D₄×D₅	1	1	yes	characteristic
274	C₄²	#711	1	1	yes	characteristic
275	C₄²	C₂²×Dic₅	1	1	yes	characteristic
276	C₄²	C₂³×D₅	1	1	yes	characteristic
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	4	4	no	—
279	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
280	C₂×C₅	#2943	1	1	yes	characteristic
281	C₂×C₅	C₂×#444	1	1	yes	characteristic
282	C₂³	—	2	2	no	—
283	C₂³	D₄×Dic₅	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	C₂²×#168	1	4	yes	—
286	D₄	—	2	8	no	—
287	D₄	—	2	2	no	—
288	D₄	C₂²×#168	1	4	yes	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	C₂²×#168	1	2	yes	—
298	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
299	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
300	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
301	C₂×C₄	#4094	1	2	yes	—
302	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
303	C₂×C₄	C₂×#711	1	1	yes	characteristic
304	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	10	4	no	—
308	C₈	—	20	2	no	—
309	C₈	—	20	4	no	—
310	C₈	C₂²×#168	1	2	yes	—
311	C₈	C₂²×#168	1	2	yes	—
312	C₅	#17849	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	D₄×#168	1	2	yes	—
319	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	D₄×#168	1	2	yes	—
326	C₄	#65486	1	2	yes	—
327	C₄	#66102	1	2	yes	—
328	C₄	C₂³×#168	1	1	yes	characteristic
329	C₄	C₂×#4094	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	4	4	no	—
332	C₂	C₂×D₄×#168	1	1	yes	characteristic
333	C₂	#10615628	1	1	yes	characteristic
334	C₂	C₂×#65486	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₄²	C₂³×D₅
Fitting	C₄×C₅×#135	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂×C₅×#135	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₄×#135	D₅
hypocenter	C₅	#17849
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
2. #41×#168	C₂	semidirect
3. #10602431	C₂	semidirect
4. #10601695	C₂	semidirect
5. SD₁₆⋊#168	C₂	semidirect
6. #10601671	C₂	semidirect
7. #10600522	C₂	semidirect
8. C₂×#64675	C₂	semidirect
36. C₅×#135	C₈	semidirect
85. C₅×#41	C₂×C₈	semidirect
95. C₅×Q₁₆	C₂×C₈	semidirect
96. C₅×SD₁₆	C₂×C₈	semidirect
97. C₅×D₈	C₂×C₈	semidirect
99. C₂×C₅×C₈	C₂×C₈	semidirect
180. #135	#168	semidirect
238. #41	C₂×#168	semidirect
248. Q₁₆	C₂×#168	semidirect
249. SD₁₆	C₂×#168	semidirect
250. D₈	C₂×#168	semidirect
258. C₂×C₈	C₂×#168	semidirect
312. C₅	#17849	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#135	2
#41×#168	2
#10602431	2
#10601695	2
SD₁₆⋊#168	2
#10601671	2
#10600522	2
C₂×#64675	2
#17849	5