Subgroups — #12009105

Order 1280.

362 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₅	C₂	1	1	yes	maximal, characteristic
2	#10600077	C₂	1	1	yes	maximal, characteristic
3	C₄×#3930	C₂	1	1	yes	maximal, characteristic
4	C₅×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10598049	C₂	1	1	yes	maximal, characteristic, complemented
7	#10597997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₅	C₄	1	1	yes	characteristic
9	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
10	Q₈×#168	C₄	1	1	yes	characteristic
11	C₂×#3930	—	2	1	no	—
12	C₂×#3930	—	2	1	no	—
13	D₅×C₄⋊C₈	—	2	1	no	—
14	D₅×C₄⋊C₈	C₄	1	1	yes	characteristic
15	#65119	—	2	1	no	—
16	#65119	C₄	1	1	yes	characteristic
17	C₄×C₈×D₅	—	2	1	no	—
18	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₀	—	2	1	no	—
20	C₈×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
23	#64722	—	2	1	no	—
24	#64721	—	2	1	no	—
25	C₄⋊#672	C₂²	1	1	yes	characteristic
26	C₄×#672	C₂²	1	1	yes	characteristic
27	#11607	—	5	1	no	maximal, Sylow
28	C₂×Q₈×D₅	C₈	1	1	yes	characteristic
29	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	Q₈×Dic₅	C₈	1	1	yes	characteristic
31	C₂×C₈×D₅	—	2	1	no	—
32	C₂×C₈×D₅	D₄	1	1	yes	characteristic
33	C₂×C₈×D₅	D₄	1	1	yes	characteristic
34	D₅×C₄⋊C₄	—	2	1	no	—
35	D₅×C₄⋊C₄	C₈	1	1	yes	characteristic
36	#4033	—	2	1	no	—
37	#4033	C₈	1	1	yes	characteristic
38	C₄²×D₅	—	2	1	no	—
39	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₀	—	2	1	no	—
41	C₄×Dic₁₀	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	1	no	—
43	C₂×C₅×C₁₆	—	2	1	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₅×C₈	—	2	1	no	—
47	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
48	#3946	—	2	1	no	—
49	#3946	—	2	1	no	—
50	#3946	—	2	1	no	—
51	C₈×Dic₅	—	2	1	no	—
52	C₈×Dic₅	—	2	1	no	—
53	C₈×Dic₅	D₄	1	1	yes	characteristic
54	C₈×Dic₅	D₄	1	1	yes	characteristic
55	C₂×#672	—	2	1	no	—
56	C₂×#672	—	2	1	no	—
57	C₄⋊#168	—	2	1	no	—
58	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
59	C₄×#168	—	2	1	no	—
60	C₄×#168	C₂×C₄	1	1	yes	characteristic
61	#3930	—	2	4	no	—
62	#3930	—	4	2	no	—
63	C₂×C₈×Q₈	—	5	1	no	—
64	#2150	—	5	1	no	—
65	C₄×M₃₂	—	5	1	no	—
66	Q₈⋊C₁₆	—	5	1	no	—
67	Q₈⋊C₁₆	—	5	1	no	—
68	Q₈⋊C₁₆	—	5	1	no	—
69	Q₈⋊C₁₆	D₅	1	1	yes	characteristic, complemented
70	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
71	Q₈×D₅	—	2	2	no	—
72	Q₈×D₅	—	2	2	no	—
73	C₂×C₄×D₅	—	2	1	no	—
74	C₂×C₄×D₅	—	2	1	no	—
75	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅	M₁₆	1	1	yes	characteristic
78	C₂×Dic₁₀	—	2	1	no	—
79	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	2	1	no	—
84	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₅	—	2	1	no	—
86	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₄⋊Dic₅	—	2	1	no	—
88	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
89	#683	—	2	1	no	—
90	#683	—	2	1	no	—
91	#683	—	2	1	no	—
92	C₄×Dic₅	—	2	1	no	—
93	C₄×Dic₅	—	2	1	no	—
94	C₄×Dic₅	—	2	1	no	—
95	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₅	M₁₆	1	1	yes	characteristic
98	C₂×#168	—	2	1	no	—
99	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
101	C₈×D₅	—	2	2	no	—
102	C₈×D₅	—	4	2	no	—
103	C₈×D₅	SD₁₆	1	2	yes	—
104	C₈×D₅	Q₁₆	1	2	yes	—
105	C₅×C₁₆	—	2	2	no	—
106	C₅×C₁₆	—	4	1	no	—
107	#672	—	2	2	no	—
108	#672	—	4	1	no	—
109	C₂×C₄×Q₈	—	5	1	no	—
110	C₂×M₃₂	—	10	1	no	—
111	C₂×M₃₂	—	10	1	no	—
112	C₈×Q₈	—	5	1	no	—
113	C₈×Q₈	—	5	1	no	—
114	C₈×Q₈	—	5	1	no	—
115	C₈×Q₈	—	10	1	no	—
116	C₈×Q₈	—	10	1	no	—
117	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄⋊C₈	—	5	1	no	—
119	C₂×C₄⋊C₈	—	10	1	no	—
120	C₂×C₄×C₈	—	5	1	no	—
121	C₂×C₄×C₈	—	10	1	no	—
122	C₄⋊C₁₆	—	5	1	no	—
123	C₄⋊C₁₆	—	10	1	no	—
124	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
125	#346	—	10	1	no	—
126	C₄×C₁₆	—	5	1	no	—
127	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
128	C₂²×D₅	C₂²⋊C₈	1	1	yes	characteristic
129	C₅×Q₈	—	2	1	no	—
130	C₅×Q₈	M₃₂	1	2	yes	complemented
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₅	—	2	1	no	—
137	C₂×Dic₅	—	2	1	no	—
138	C₂×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
139	C₂×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
140	C₂×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
141	C₄×D₅	—	2	4	no	—
142	C₄×D₅	—	2	1	no	—
143	C₄×D₅	—	2	1	no	—
144	C₄×D₅	—	4	1	no	—
145	C₄×D₅	—	4	2	no	—
146	C₄×D₅	#103	1	2	yes	—
147	C₄×D₅	#104	1	2	yes	—
148	Dic₁₀	—	2	4	no	—
149	Dic₁₀	—	2	1	no	—
150	Dic₁₀	M₃₂	1	2	yes	complemented
151	C₅×C₈	—	2	1	no	—
152	C₅×C₈	—	4	1	no	—
153	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
154	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
155	#168	—	2	1	no	—
156	#168	—	4	1	no	—
157	#168	#103	1	2	yes	—
158	C₂²×Q₈	—	5	1	no	—
159	C₂²×C₈	—	5	1	no	—
160	C₂²×C₈	—	5	1	no	—
161	C₂²×C₈	—	10	1	no	—
162	C₄×Q₈	—	5	1	no	—
163	C₄×Q₈	—	5	1	no	—
164	C₄×Q₈	—	5	1	no	—
165	C₄×Q₈	—	10	1	no	—
166	C₄×Q₈	—	10	1	no	—
167	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	5	1	no	—
169	C₂×C₄⋊C₄	—	10	1	no	—
170	C₂×C₄²	—	5	1	no	—
171	C₂×C₄²	—	10	1	no	—
172	M₃₂	—	10	4	no	—
173	M₃₂	—	20	2	no	—
174	C₂×C₁₆	—	2	1	no	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	10	1	no	—
177	C₂×C₁₆	—	10	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	5	1	no	—
180	C₄⋊C₈	—	10	1	no	—
181	C₄⋊C₈	—	10	1	no	—
182	C₄⋊C₈	—	10	1	no	—
183	C₄⋊C₈	—	10	1	no	—
184	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₄×C₈	—	5	1	no	—
189	C₄×C₈	—	10	1	no	—
190	C₄×C₈	—	10	1	no	—
191	C₄×C₈	—	10	1	no	—
192	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
193	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
194	C₂×D₅	—	2	1	no	—
195	C₂×D₅	—	2	1	no	—
196	C₂×D₅	Q₈⋊C₈	1	2	yes	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	C₂×#103	1	1	yes	characteristic
201	C₄×C₅	C₂×#104	1	1	yes	characteristic
202	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
203	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
204	Dic₅	—	2	1	no	—
205	Dic₅	—	2	2	no	—
206	Dic₅	—	2	1	no	—
207	Dic₅	—	4	1	no	—
208	Dic₅	Q₈⋊C₈	1	2	yes	—
209	C₂×Q₈	—	5	1	no	—
210	C₂×Q₈	—	10	1	no	—
211	C₂×Q₈	—	10	2	no	—
212	C₂×Q₈	—	10	2	no	—
213	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
214	C₂²×C₄	—	5	1	no	—
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	5	1	no	—
217	C₂²×C₄	—	10	1	no	—
218	C₂²×C₄	—	10	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	5	1	no	—
221	C₂×C₈	—	5	2	no	—
222	C₂×C₈	—	5	2	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	10	2	no	—
225	C₂×C₈	—	10	1	no	—
226	C₂×C₈	—	20	2	no	—
227	C₂×C₈	D₄×D₅	1	1	yes	characteristic
228	C₂×C₈	D₄×D₅	1	1	yes	characteristic
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	5	1	no	—
231	C₄⋊C₄	—	10	1	no	—
232	C₄⋊C₄	—	10	1	no	—
233	C₄⋊C₄	—	10	1	no	—
234	C₄⋊C₄	—	10	1	no	—
235	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
236	C₄²	—	2	1	no	—
237	C₄²	—	5	1	no	—
238	C₄²	—	5	1	no	—
239	C₄²	—	5	1	no	—
240	C₄²	—	10	1	no	—
241	C₄²	—	10	1	no	—
242	C₄²	—	10	1	no	—
243	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
244	C₁₆	—	2	2	no	—
245	C₁₆	—	4	1	no	—
246	C₁₆	—	10	2	no	—
247	C₁₆	—	20	1	no	—
248	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
249	C₂×C₅	#2112	1	1	yes	characteristic
250	C₂×C₅	#2117	1	1	yes	characteristic
251	D₅	—	2	2	no	—
252	C₂³	—	5	1	no	—
253	Q₈	—	2	1	no	—
254	Q₈	—	5	2	no	—
255	Q₈	—	10	4	no	—
256	Q₈	—	10	1	no	—
257	Q₈	#3930	1	2	yes	complemented
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	10	4	no	—
266	C₂×C₄	—	10	1	no	—
267	C₂×C₄	—	10	1	no	—
268	C₂×C₄	—	10	1	no	—
269	C₂×C₄	—	10	1	no	—
270	C₂×C₄	—	20	1	no	—
271	C₂×C₄	—	20	2	no	—
272	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
274	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	4	1	no	—
277	C₈	—	5	2	no	—
278	C₈	—	10	1	no	—
279	C₈	—	20	1	no	—
280	C₈	D₅×SD₁₆	1	1	yes	characteristic
281	C₈	D₅×Q₁₆	1	1	yes	characteristic
282	C₅	#11607	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	5	2	no	—
284	C₂²	—	10	1	no	—
285	C₂²	—	10	1	no	—
286	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	1	no	—
290	C₄	—	5	2	no	—
291	C₄	—	10	1	no	—
292	C₄	—	10	2	no	—
293	C₄	—	10	1	no	—
294	C₄	—	20	1	no	—
295	C₄	D₅×#103	1	1	yes	characteristic
296	C₄	D₅×#104	1	1	yes	characteristic
297	C₄	C₂×#3930	1	1	yes	characteristic
298	C₄	D₅×M₃₂	1	1	yes	characteristic
299	C₂	—	10	2	no	—
300	C₂	D₅×Q₈⋊C₈	1	1	yes	characteristic
301	C₂	#10599899	1	1	yes	characteristic
302	C₂	#10599918	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×Q₈⋊C₁₆	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂²×D₅	C₂²⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	Q₈⋊C₁₆	D₅
hypocenter	C₅	#11607
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. C₅×Q₈⋊C₁₆	C₂	semidirect
5. Q₈⋊#672	C₂	semidirect
6. #10598049	C₂	semidirect
7. #10597997	C₂	semidirect
69. Q₈⋊C₁₆	D₅	semidirect
130. C₅×Q₈	M₃₂	semidirect
150. Dic₁₀	M₃₂	semidirect
257. Q₈	#3930	semidirect
282. C₅	#11607	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₅	2
#10600077	2
C₄×#3930	2
C₅×Q₈⋊C₁₆	2
Q₈⋊#672	2
#10598049	2
#10597997	2
#11607	5