Subgroups — #11987519

Order 1280.

462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#10600186	C₂	1	1	yes	maximal, characteristic
3	#10600184	C₂	1	1	yes	maximal, characteristic
4	#10598129	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅×Q₈	—	2	1	no	—
7	C₅×#391	—	2	1	no	—
8	C₅×#389	—	2	1	no	—
9	C₅×#389	C₂²	1	2	yes	—
10	C₅×#384	C₄	1	1	yes	characteristic
11	C₄×C₅×C₄⋊C₄	—	2	1	no	—
12	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
13	C₄³×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	—	2	1	no	—
15	C₂×#3935	—	2	1	no	—
16	#64744	—	2	1	no	—
17	#64744	—	2	1	no	—
18	#64744	C₂²	1	2	yes	—
19	#64744	C₂²	1	2	yes	—
20	#7625	—	5	1	no	maximal, Sylow
21	C₂²×C₅×Q₈	—	2	1	no	—
22	C₅×C₄⋊Q₈	—	2	4	no	—
23	C₄×C₅×Q₈	—	4	2	no	—
24	C₄×C₅×Q₈	—	4	2	no	—
25	C₂×C₅×C₄⋊C₄	—	2	1	no	—
26	C₂×C₅×C₄⋊C₄	—	2	1	no	—
27	C₂×C₅×C₄⋊C₄	—	2	1	no	—
28	C₂×C₅×C₄⋊C₄	—	2	2	no	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	—	2	2	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₂²×#168	—	2	2	no	—
40	C₂²×#168	—	2	2	no	—
41	C₅×#95	—	2	2	no	—
42	C₅×#95	—	2	2	no	—
43	C₄⋊#168	—	2	4	no	—
44	#3935	—	4	2	no	—
45	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
46	#1839	—	5	1	no	—
47	#1767	—	5	1	no	—
48	#1271	—	5	4	no	—
49	C₂×C₅×Q₈	—	2	4	no	—
50	C₂×C₅×Q₈	—	4	2	no	—
51	C₂×C₅×Q₈	—	4	2	no	—
52	C₂²×C₄×C₅	—	2	2	no	—
53	C₂²×C₄×C₅	—	2	2	no	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₅×C₄⋊C₄	—	2	4	no	—
62	C₅×C₄⋊C₄	—	2	2	no	—
63	C₅×C₄⋊C₄	—	4	2	no	—
64	C₅×C₄⋊C₄	—	4	2	no	—
65	C₅×C₄⋊C₄	—	4	2	no	—
66	C₅×C₄⋊C₄	—	4	2	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
69	C₄²×C₅	—	2	2	no	—
70	C₄²×C₅	—	2	2	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	4	1	no	—
74	C₄²×C₅	—	4	1	no	—
75	C₄²×C₅	—	4	4	no	—
76	C₄²×C₅	D₈	1	2	yes	—
77	C₄²×C₅	SD₁₆	1	2	yes	—
78	C₂×#168	—	2	4	no	—
79	C₂×#168	—	4	2	no	—
80	C₂×#168	—	4	2	no	—
81	C₂×#168	—	4	2	no	—
82	C₂×#168	—	4	2	no	—
83	C₂×#168	—	4	2	no	—
84	C₂×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
85	C₂×C₄×Q₈	—	2	1	no	—
86	C₂×C₄⋊C₈	—	10	1	no	—
87	C₂×#97	—	10	1	no	—
88	#391	—	2	1	no	—
89	#389	—	2	1	no	—
90	#389	C₂×D₅	1	2	yes	—
91	#384	Dic₅	1	1	yes	characteristic
92	C₄×C₄⋊C₄	—	2	1	no	—
93	C₄×C₄⋊C₄	Dic₅	1	2	yes	—
94	C₄³	C₂×D₅	1	1	yes	characteristic
95	#336	—	5	2	no	—
96	#336	—	5	2	no	—
97	#336	—	10	1	no	—
98	#336	—	10	1	no	—
99	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₅×Q₈	—	4	8	no	—
101	C₂×C₄×C₅	—	2	1	no	—
102	C₂×C₄×C₅	—	2	1	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	1	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	1	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	1	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	4	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	#101	1	2	yes	—
122	C₂×C₄×C₅	#102	1	2	yes	—
123	C₂×C₄×C₅	#102	1	2	yes	—
124	C₂×C₄×C₅	#104	1	2	yes	—
125	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₅	#137	1	2	yes	—
129	#168	—	4	4	no	—
130	#168	—	8	2	no	—
131	C₂²×Q₈	—	2	1	no	—
132	C₂²×C₈	—	10	2	no	—
133	C₂²×C₈	—	10	2	no	—
134	C₄⋊Q₈	—	2	4	no	—
135	C₄×Q₈	—	4	2	no	—
136	C₄×Q₈	—	4	2	no	—
137	C₂×C₄⋊C₄	—	2	1	no	—
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
149	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
150	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
151	C₄⋊C₈	—	10	4	no	—
152	#97	—	20	2	no	—
153	#95	—	2	2	no	—
154	#95	—	2	2	no	—
155	C₂²×C₅	#407	1	1	yes	characteristic
156	C₂²×C₅	#410	1	1	yes	characteristic
157	C₂²×C₅	C₂×#101	1	1	yes	characteristic
158	C₂²×C₅	C₂×#102	1	1	yes	characteristic
159	C₂²×C₅	#419	1	1	yes	characteristic
160	C₂²×C₅	C₂×#104	1	1	yes	characteristic
161	C₂²×C₅	#421	1	1	yes	characteristic
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	2	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	—	4	4	no	—
171	C₄×C₅	—	8	2	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
178	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
179	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
184	C₂×C₈	—	10	4	no	—
185	C₂×C₈	—	20	2	no	—
186	C₂×C₈	—	20	2	no	—
187	C₂×C₈	—	20	2	no	—
188	C₂×C₈	—	20	2	no	—
189	C₂×C₈	—	20	2	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	#681	1	4	yes	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	4	1	no	—
203	C₄²	—	4	1	no	—
204	C₄²	—	4	4	no	—
205	C₄²	#686	1	2	yes	—
206	C₄²	#687	1	2	yes	—
207	C₂×C₅	#1467	1	1	yes	characteristic
208	C₂×C₅	#1486	1	1	yes	characteristic
209	C₂×C₅	#1491	1	1	yes	characteristic
210	C₂×C₅	#1502	1	1	yes	characteristic
211	C₂×C₅	#1515	1	1	yes	characteristic
212	C₂×C₅	#1531	1	1	yes	characteristic
213	C₂×C₅	#1546	1	1	yes	characteristic
214	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
215	Q₈	—	4	8	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	#3964	1	2	yes	—
237	C₂×C₄	#3964	1	2	yes	—
238	C₂×C₄	#3968	1	2	yes	—
239	C₂×C₄	#3969	1	2	yes	—
240	C₂×C₄	C₂×#681	1	2	yes	—
241	C₂×C₄	C₂×#686	1	1	yes	characteristic
242	C₂×C₄	C₂×#687	1	1	yes	characteristic
243	C₂×C₄	#4088	1	2	yes	—
244	C₈	—	20	4	no	—
245	C₈	—	40	2	no	—
246	C₅	#7625	1	1	yes	characteristic, complemented, Sylow
247	C₂²	#65496	1	1	yes	characteristic
248	C₂²	C₂×#3964	1	1	yes	characteristic
249	C₂²	#65514	1	1	yes	characteristic
250	C₂²	C₂×#3968	1	1	yes	characteristic
251	C₂²	C₂×#3969	1	1	yes	characteristic
252	C₂²	#65525	1	1	yes	characteristic
253	C₂²	#65529	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₂	#10602391	1	1	yes	characteristic
265	C₂	#10602416	1	1	yes	characteristic
266	C₂	#10602418	1	1	yes	characteristic
267	C₂	#10602566	1	1	yes	characteristic
268	C₂	#10602604	1	1	yes	characteristic
269	C₂	#10602615	1	1	yes	characteristic
270	C₂	#10602630	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅
hypocenter	C₅	#7625
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
246. C₅	#7625	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₄⋊Q₈	2
#10600186	2
#10600184	2
#10598129	2
#7625	5