Subgroups — #11999324

Order 1280.

506 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#10600187	C₂	1	1	yes	maximal, characteristic
3	#10600181	C₂	1	1	yes	maximal, characteristic
4	#10598110	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	C₄	1	2	yes	—
10	C₅×#384	C₄	1	1	yes	characteristic
11	C₄×C₅×C₄⋊C₄	—	2	1	no	—
12	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄³×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#168	—	2	1	no	—
15	C₂×C₄⋊#168	—	2	1	no	—
16	C₂×C₄⋊#168	C₂²	1	2	yes	—
17	C₂×C₄×#168	C₂²	1	2	yes	—
18	#64744	—	2	1	no	—
19	#64744	—	2	1	no	—
20	#9874	—	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	C₄⋊#168	—	4	2	no	—
45	C₄⋊#168	—	4	2	no	—
46	C₄×#168	—	2	2	no	—
47	C₄×#168	Q₈	1	4	yes	complemented
48	C₄×C₄⋊Q₈	D₅	1	1	yes	characteristic
49	#1842	—	5	1	no	—
50	#1744	—	5	1	no	—
51	#1279	—	5	4	no	—
52	C₂×C₅×Q₈	—	2	4	no	—
53	C₂×C₅×Q₈	—	4	2	no	—
54	C₂×C₅×Q₈	—	4	2	no	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	—	2	2	no	—
57	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₅×C₄⋊C₄	—	2	4	no	—
65	C₅×C₄⋊C₄	—	2	2	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₄⋊C₄	—	4	4	no	—
69	C₅×C₄⋊C₄	—	4	2	no	—
70	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
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	2	no	—
76	C₄²×C₅	—	4	2	no	—
77	C₄²×C₅	D₈	1	2	yes	—
78	C₄²×C₅	SD₁₆	1	2	yes	—
79	C₄²×C₅	SD₁₆	1	2	yes	—
80	C₄²×C₅	Q₁₆	1	2	yes	—
81	C₄²×C₅	C₂×Q₈	1	2	yes	—
82	C₄²×C₅	#41	1	2	yes	—
83	C₂×#168	—	2	4	no	—
84	C₂×#168	—	4	2	no	—
85	C₂×#168	—	4	2	no	—
86	C₂×#168	—	4	2	no	—
87	C₂×#168	—	4	2	no	—
88	C₂×#168	—	4	2	no	—
89	C₂×C₄⋊Q₈	Dic₅	1	1	yes	characteristic
90	C₂×C₄×Q₈	—	2	1	no	—
91	C₂×C₄⋊C₈	—	5	2	no	—
92	C₂×C₄⋊C₈	—	10	1	no	—
93	C₂×C₄⋊C₈	—	10	1	no	—
94	C₂×C₄×C₈	—	5	2	no	—
95	#391	—	2	1	no	—
96	#389	—	2	1	no	—
97	#389	Dic₅	1	2	yes	—
98	#384	Dic₅	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	2	1	no	—
100	C₄×C₄⋊C₄	C₂×D₅	1	2	yes	—
101	C₄³	C₂×D₅	1	1	yes	characteristic
102	#336	—	10	1	no	—
103	#336	—	10	1	no	—
104	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₅×Q₈	—	4	8	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	1	no	—
116	C₂×C₄×C₅	—	2	4	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	—	4	2	no	—
122	C₂×C₄×C₅	#102	1	2	yes	—
123	C₂×C₄×C₅	#102	1	2	yes	—
124	C₂×C₄×C₅	#103	1	2	yes	—
125	C₂×C₄×C₅	#103	1	2	yes	—
126	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₅	#124	1	1	yes	characteristic
129	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₅	#131	1	2	yes	—
132	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
136	#168	—	4	4	no	—
137	#168	—	8	2	no	—
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×C₈	—	10	2	no	—
140	C₂²×C₈	—	10	2	no	—
141	C₄⋊Q₈	—	2	4	no	—
142	C₄×Q₈	—	4	2	no	—
143	C₄×Q₈	—	4	2	no	—
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
155	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₄⋊C₈	—	10	4	no	—
159	C₄⋊C₈	—	20	2	no	—
160	C₄⋊C₈	—	20	2	no	—
161	C₄×C₈	—	5	4	no	—
162	C₄×C₈	—	10	2	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	C₂²×C₅	#391	1	1	yes	characteristic
166	C₂²×C₅	#407	1	1	yes	characteristic
167	C₂²×C₅	#408	1	1	yes	characteristic
168	C₂²×C₅	C₂×#102	1	1	yes	characteristic
169	C₂²×C₅	C₂×#103	1	1	yes	characteristic
170	C₂²×C₅	#421	1	1	yes	characteristic
171	C₂²×C₅	#421	1	1	yes	characteristic
172	C₄×C₅	—	2	2	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	2	2	no	—
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	4	4	no	—
178	C₄×C₅	—	8	2	no	—
179	C₄×C₅	#446	1	4	yes	—
180	C₄×C₅	#486	1	1	yes	characteristic
181	C₄×C₅	#487	1	1	yes	characteristic
182	C₄×C₅	#498	1	1	yes	characteristic
183	C₄×C₅	#500	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
196	C₂×C₈	—	10	4	no	—
197	C₂×C₈	—	20	2	no	—
198	C₂×C₈	—	20	2	no	—
199	C₂×C₈	—	20	2	no	—
200	C₂×C₈	—	20	2	no	—
201	C₂×C₈	—	20	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	#681	1	4	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	#686	1	2	yes	—
216	C₄²	#687	1	2	yes	—
217	C₄²	#688	1	2	yes	—
218	C₄²	#689	1	2	yes	—
219	C₄²	Q₈×D₅	1	2	yes	—
220	C₄²	#713	1	2	yes	—
221	C₂×C₅	#1480	1	1	yes	characteristic
222	C₂×C₅	#1481	1	1	yes	characteristic
223	C₂×C₅	#1486	1	1	yes	characteristic
224	C₂×C₅	#1487	1	1	yes	characteristic
225	C₂×C₅	#1985	1	1	yes	characteristic
226	C₂×C₅	#1990	1	1	yes	characteristic
227	C₂×C₅	#1995	1	1	yes	characteristic
228	C₂³	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
229	Q₈	—	4	8	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	#3964	1	2	yes	—
247	C₂×C₄	#3964	1	2	yes	—
248	C₂×C₄	#3967	1	2	yes	—
249	C₂×C₄	#3967	1	2	yes	—
250	C₂×C₄	C₂×#681	1	2	yes	—
251	C₂×C₄	C₂×#686	1	1	yes	characteristic
252	C₂×C₄	C₂×#687	1	1	yes	characteristic
253	C₂×C₄	C₂×#688	1	1	yes	characteristic
254	C₂×C₄	C₂×#689	1	1	yes	characteristic
255	C₂×C₄	#4090	1	1	yes	characteristic
256	C₂×C₄	Q₈×Dic₅	1	2	yes	—
257	C₂×C₄	#4092	1	2	yes	—
258	C₂×C₄	#4093	1	1	yes	characteristic
259	C₂×C₄	#4094	1	2	yes	—
260	C₈	—	20	4	no	—
261	C₈	—	40	2	no	—
262	C₅	#9874	1	1	yes	characteristic, complemented, Sylow
263	C₂²	#65496	1	1	yes	characteristic
264	C₂²	C₂×#3964	1	1	yes	characteristic
265	C₂²	C₂×#3967	1	1	yes	characteristic
266	C₂²	#65519	1	1	yes	characteristic
267	C₂²	#65523	1	1	yes	characteristic
268	C₂²	#65525	1	1	yes	characteristic
269	C₂²	#65525	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	8	2	no	—
277	C₄	#65313	1	4	yes	—
278	C₄	#65367	1	1	yes	characteristic
279	C₄	#65368	1	1	yes	characteristic
280	C₄	#65377	1	1	yes	characteristic
281	C₄	#65382	1	1	yes	characteristic
282	C₂	#10602391	1	1	yes	characteristic
283	C₂	#10602415	1	1	yes	characteristic
284	C₂	#10602428	1	1	yes	characteristic
285	C₂	#10602580	1	1	yes	characteristic
286	C₂	#10602593	1	1	yes	characteristic
287	C₂	#10602617	1	1	yes	characteristic
288	C₂	#10602633	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
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	#9874
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
47. C₄×#168	Q₈	semidirect
262. C₅	#9874	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
#10600187	2
#10600181	2
#10598110	2
#9874	5