Subgroups — #12009109

Order 1280.

334 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10611648	C₂	1	1	yes	maximal, characteristic
2	#10600077	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅×C₁₆	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	#65912	C₄	1	1	yes	characteristic
9	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
10	#65313	C₄	1	1	yes	characteristic
11	C₂×#3930	—	2	1	no	—
12	C₂×D₅×C₁₆	—	2	1	no	—
13	#65128	—	2	1	no	—
14	#65126	C₄	1	1	yes	characteristic
15	#65120	C₄	1	1	yes	characteristic
16	#65117	—	2	1	no	—
17	#64978	—	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	#64721	—	2	1	no	—
24	C₁₆×Dic₅	—	2	1	no	—
25	C₄⋊#672	C₂²	1	1	yes	characteristic
26	C₄×#672	C₂²	1	1	yes	characteristic
27	#11608	—	5	1	no	maximal, Sylow
28	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
29	#4092	C₈	1	1	yes	characteristic
30	#4090	C₈	1	1	yes	characteristic
31	C₂×#676	—	2	1	no	—
32	C₂×C₈×D₅	D₄	1	1	yes	characteristic
33	C₂×C₈×D₅	D₄	1	1	yes	characteristic
34	#4043	C₈	1	1	yes	characteristic
35	#4042	—	2	1	no	—
36	#4035	—	2	1	no	—
37	#4034	C₈	1	1	yes	characteristic
38	#4018	—	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	#3947	—	2	1	no	—
49	#3946	—	2	1	no	—
50	#3946	—	2	1	no	—
51	#3946	—	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	#3935	—	2	1	no	—
60	C₄×#168	C₂×C₄	1	1	yes	characteristic
61	#3930	—	4	2	no	—
62	D₅×C₁₆	—	2	4	no	—
63	#2960	—	5	1	no	—
64	#2150	—	5	1	no	—
65	C₂×C₄×C₁₆	—	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	C₂×C₄×D₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅	M₁₆	1	1	yes	characteristic
74	C₂×Dic₁₀	—	2	1	no	—
75	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	1	no	—
80	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
83	#685	—	2	1	no	—
84	#685	—	2	1	no	—
85	C₄⋊Dic₅	—	2	1	no	—
86	C₄⋊Dic₅	C₂×C₈	1	1	yes	characteristic
87	#683	—	2	1	no	—
88	#683	—	2	1	no	—
89	#683	—	2	1	no	—
90	#683	—	2	1	no	—
91	#683	—	2	1	no	—
92	C₄×Dic₅	—	2	1	no	—
93	C₄×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₅	M₁₆	1	1	yes	characteristic
96	C₂×#168	—	2	1	no	—
97	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
99	#676	—	4	2	no	—
100	C₈×D₅	—	2	2	no	—
101	C₈×D₅	SD₁₆	1	2	yes	—
102	C₈×D₅	Q₁₆	1	2	yes	—
103	C₅×C₁₆	—	2	2	no	—
104	C₅×C₁₆	—	4	1	no	—
105	#672	—	2	2	no	—
106	#672	—	4	1	no	—
107	#533	—	5	1	no	—
108	C₂×M₃₂	—	10	1	no	—
109	C₂²×C₁₆	—	10	1	no	—
110	#446	—	5	1	no	—
111	#446	—	5	1	no	—
112	#446	—	10	1	no	—
113	C₈×Q₈	—	5	1	no	—
114	C₈×Q₈	—	10	1	no	—
115	C₈×Q₈	C₂×D₅	1	1	yes	characteristic
116	#424	—	10	1	no	—
117	#423	—	5	1	no	—
118	#405	—	10	1	no	—
119	C₂×C₄×C₈	—	5	1	no	—
120	C₄⋊C₁₆	—	5	1	no	—
121	C₄⋊C₁₆	—	10	1	no	—
122	C₄⋊C₁₆	C₂×D₅	1	1	yes	characteristic
123	C₄×C₁₆	—	5	1	no	—
124	C₄×C₁₆	—	10	1	no	—
125	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
126	C₂²×D₅	C₂²⋊C₈	1	1	yes	characteristic
127	C₅×Q₈	—	2	1	no	—
128	C₅×Q₈	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
134	C₂×Dic₅	—	2	1	no	—
135	C₂×Dic₅	—	2	1	no	—
136	C₂×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
137	C₂×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
138	C₂×Dic₅	C₂²⋊C₈	1	1	yes	characteristic
139	C₄×D₅	—	2	1	no	—
140	C₄×D₅	—	2	1	no	—
141	C₄×D₅	—	2	2	no	—
142	C₄×D₅	#103	1	2	yes	—
143	C₄×D₅	#104	1	2	yes	—
144	Dic₁₀	—	2	1	no	—
145	Dic₁₀	—	2	1	no	—
146	Dic₁₀	—	4	2	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	4	1	no	—
149	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
150	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
151	#168	—	2	1	no	—
152	#168	—	4	1	no	—
153	#168	#103	1	2	yes	—
154	C₂×M₁₆	—	10	1	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	C₄⋊Q₈	—	5	1	no	—
158	C₄⋊Q₈	—	5	1	no	—
159	#125	—	10	1	no	—
160	C₂²⋊Q₈	—	5	1	no	—
161	C₂²⋊Q₈	—	5	1	no	—
162	C₂²⋊Q₈	—	10	1	no	—
163	C₄×Q₈	—	5	1	no	—
164	C₄×Q₈	—	10	1	no	—
165	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
166	#117	—	10	1	no	—
167	C₂×C₄²	—	5	1	no	—
168	M₃₂	—	20	2	no	—
169	C₂×C₁₆	—	2	1	no	—
170	C₂×C₁₆	—	2	1	no	—
171	C₂×C₁₆	—	10	4	no	—
172	C₂×C₁₆	—	10	1	no	—
173	C₂×C₁₆	—	10	1	no	—
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	5	1	no	—
176	C₄⋊C₈	—	10	1	no	—
177	C₄⋊C₈	—	10	1	no	—
178	C₄⋊C₈	—	10	1	no	—
179	C₄⋊C₈	—	10	1	no	—
180	C₄⋊C₈	C₄×D₅	1	1	yes	characteristic
181	#97	—	10	1	no	—
182	#97	—	10	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	5	1	no	—
185	C₄×C₈	—	5	1	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	10	1	no	—
188	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
189	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
190	C₂×D₅	—	2	1	no	—
191	C₂×D₅	—	2	1	no	—
192	C₂×D₅	Q₈⋊C₈	1	2	yes	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	1	no	—
195	C₄×C₅	—	4	1	no	—
196	C₄×C₅	C₂×#103	1	1	yes	characteristic
197	C₄×C₅	C₂×#104	1	1	yes	characteristic
198	C₄×C₅	#504	1	1	yes	characteristic
199	C₄×C₅	#504	1	1	yes	characteristic
200	Dic₅	—	2	1	no	—
201	Dic₅	—	4	1	no	—
202	Dic₅	—	4	1	no	—
203	Dic₅	Q₈⋊C₈	1	2	yes	—
204	Dic₅	#350	1	2	yes	—
205	C₂×Q₈	—	5	1	no	—
206	C₂×Q₈	—	10	1	no	—
207	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
208	C₂²×C₄	—	5	1	no	—
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	5	1	no	—
211	M₁₆	—	20	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	5	1	no	—
214	C₂×C₈	—	5	2	no	—
215	C₂×C₈	—	5	2	no	—
216	C₂×C₈	—	5	1	no	—
217	C₂×C₈	—	10	2	no	—
218	C₂×C₈	—	10	1	no	—
219	C₂×C₈	D₄×D₅	1	1	yes	characteristic
220	C₂×C₈	D₄×D₅	1	1	yes	characteristic
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	5	1	no	—
223	C₄⋊C₄	—	10	1	no	—
224	C₄⋊C₄	—	10	1	no	—
225	C₄⋊C₄	—	10	1	no	—
226	C₄⋊C₄	—	10	1	no	—
227	C₄⋊C₄	—	10	1	no	—
228	C₄⋊C₄	—	10	1	no	—
229	C₄⋊C₄	C₈×D₅	1	1	yes	characteristic
230	C₂²⋊C₄	—	10	1	no	—
231	C₂²⋊C₄	—	10	1	no	—
232	C₄²	—	2	1	no	—
233	C₄²	—	5	1	no	—
234	C₄²	—	5	1	no	—
235	C₄²	—	5	1	no	—
236	C₄²	—	10	1	no	—
237	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
238	C₁₆	—	2	2	no	—
239	C₁₆	—	4	1	no	—
240	C₁₆	—	10	2	no	—
241	C₁₆	—	20	1	no	—
242	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
243	C₂×C₅	#2113	1	1	yes	characteristic
244	C₂×C₅	C₂×#350	1	1	yes	characteristic
245	D₅	—	2	2	no	—
246	C₂³	—	5	1	no	—
247	Q₈	—	2	1	no	—
248	Q₈	—	2	1	no	—
249	Q₈	—	10	1	no	—
250	Q₈	—	10	1	no	—
251	Q₈	—	20	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	2	no	—
257	C₂×C₄	—	5	2	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	10	2	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	—	10	1	no	—
262	C₂×C₄	—	10	1	no	—
263	C₂×C₄	—	10	1	no	—
264	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
266	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	4	1	no	—
269	C₈	—	5	2	no	—
270	C₈	—	10	1	no	—
271	C₈	—	20	1	no	—
272	C₈	D₅×SD₁₆	1	1	yes	characteristic
273	C₈	D₅×Q₁₆	1	1	yes	characteristic
274	C₅	#11608	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	5	2	no	—
276	C₂²	—	10	1	no	—
277	C₂²	—	10	1	no	—
278	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	—	5	2	no	—
283	C₄	—	5	2	no	—
284	C₄	—	10	1	no	—
285	C₄	—	20	1	no	—
286	C₄	—	20	1	no	—
287	C₄	D₅×#103	1	1	yes	characteristic
288	C₄	D₅×#104	1	1	yes	characteristic
289	C₄	#65190	1	1	yes	characteristic
290	C₄	#65196	1	1	yes	characteristic
291	C₂	—	10	2	no	—
292	C₂	D₅×Q₈⋊C₈	1	1	yes	characteristic
293	C₂	#10599900	1	1	yes	characteristic
294	C₂	D₅×#350	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	#11608
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
274. C₅	#11608	semidirect

Maximal subgroups

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

subgroup	index
#10611648	2
#10600077	2
C₄×D₅×C₁₆	2
C₅×Q₈⋊C₁₆	2
Q₈⋊#672	2
#10598049	2
#10597997	2
#11608	5