Subgroups — #11998242

Order 1280.

434 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#10602417	C₂	1	2	yes	maximal
3	#10602414	C₂	1	2	yes	maximal
4	#10600522	C₂	1	1	yes	maximal, characteristic
5	#10600519	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
8	C₄×C₅×Q₁₆	—	2	1	no	—
9	C₄×C₅×Q₁₆	—	2	1	no	—
10	C₄×C₅×Q₁₆	—	2	1	no	—
11	C₄×C₅×Q₁₆	—	2	1	no	—
12	C₂×C₅×#107	C₄	1	1	yes	characteristic
13	C₂×C₅×#103	C₄	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#65214	C₂²	1	2	yes	—
16	#65214	C₂²	1	1	yes	characteristic
17	#65214	C₂²	1	1	yes	characteristic
18	Q₈⋊#168	—	2	4	no	—
19	#64748	—	2	4	no	—
20	#64678	—	2	1	no	—
21	#64677	—	2	1	no	—
22	#64674	—	2	1	no	—
23	#64674	—	2	1	no	—
24	#9671	—	5	1	no	maximal, Sylow
25	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
26	C₂×C₅×Q₁₆	—	2	4	no	—
27	C₂×C₅×Q₁₆	—	2	1	no	—
28	C₂×C₅×Q₁₆	—	2	1	no	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₄×C₅×Q₈	—	2	2	no	—
32	C₄×C₅×Q₈	—	2	2	no	—
33	C₄×C₅×Q₈	—	2	2	no	—
34	C₄×C₅×Q₈	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅	—	2	2	no	—
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×#107	—	2	1	no	—
40	C₅×#107	—	2	1	no	—
41	C₅×#103	—	2	2	no	—
42	C₅×#103	—	2	2	no	—
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²⋊#168	—	2	1	no	—
48	C₂²⋊#168	—	2	2	no	—
49	C₂²⋊#168	D₄	1	2	yes	—
50	C₄⋊#168	—	2	1	no	—
51	C₄⋊#168	—	2	2	no	—
52	C₄⋊#168	D₄	1	2	yes	—
53	C₄×#168	—	2	1	no	—
54	C₄×#168	—	2	1	no	—
55	C₄×#168	—	2	2	no	—
56	C₂×C₄×Q₁₆	D₅	1	1	yes	characteristic
57	#1564	—	5	1	no	—
58	#1479	—	5	2	no	—
59	#1461	—	5	2	no	—
60	#1452	—	5	1	no	—
61	C₂×C₅×Q₈	—	2	2	no	—
62	C₂×C₅×Q₈	—	2	2	no	—
63	C₂×C₅×Q₈	—	2	2	no	—
64	C₂×C₅×Q₈	—	2	2	no	—
65	C₂×C₅×Q₈	—	2	2	no	—
66	C₂×C₅×Q₈	M₁₆	1	4	yes	—
67	C₂²×C₄×C₅	—	2	2	no	—
68	C₂²×C₄×C₅	—	2	2	no	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	#41	1	1	yes	characteristic
72	C₅×Q₁₆	—	2	4	no	—
73	C₅×Q₁₆	—	2	4	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₄²×C₅	—	4	2	no	—
85	C₄²×C₅	—	4	2	no	—
86	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅	#41	1	1	yes	characteristic
90	C₂×#168	—	2	2	no	—
91	C₂×#168	—	2	2	no	—
92	C₂×#168	—	4	2	no	—
93	C₂²×Q₁₆	Dic₅	1	1	yes	characteristic
94	C₂×C₄×Q₈	C₂×D₅	1	2	yes	—
95	C₄×Q₁₆	—	2	1	no	—
96	C₄×Q₁₆	—	2	1	no	—
97	C₄×Q₁₆	—	2	1	no	—
98	C₄×Q₁₆	—	2	1	no	—
99	#431	—	5	1	no	—
100	#431	—	5	1	no	—
101	#431	—	5	2	no	—
102	C₂×#107	Dic₅	1	1	yes	characteristic
103	C₂×#103	Dic₅	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
105	#335	—	10	1	no	—
106	#334	—	10	1	no	—
107	Q₈⋊C₈	—	10	4	no	—
108	#324	—	10	4	no	—
109	#322	—	10	1	no	—
110	#322	—	10	1	no	—
111	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
112	C₅×Q₈	—	2	4	no	—
113	C₅×Q₈	—	2	4	no	—
114	C₅×Q₈	—	4	2	no	—
115	C₅×Q₈	—	4	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	4	1	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	—	8	2	no	—
123	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅	#121	1	1	yes	characteristic
126	C₂×C₄×C₅	#121	1	1	yes	characteristic
127	C₂×C₄×C₅	#124	1	1	yes	characteristic
128	C₂×C₄×C₅	#127	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₅	#131	1	2	yes	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	—	2	1	no	—
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	—	2	1	no	—
135	#168	—	4	2	no	—
136	#168	—	4	2	no	—
137	#168	—	8	2	no	—
138	C₂²×Q₈	C₂×Dic₅	1	2	yes	—
139	C₂×Q₁₆	—	2	4	no	—
140	C₂×Q₁₆	—	2	1	no	—
141	C₂×Q₁₆	—	2	1	no	—
142	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
152	#107	—	2	1	no	—
153	#107	—	2	1	no	—
154	C₄⋊C₈	—	5	2	no	—
155	C₄⋊C₈	—	10	1	no	—
156	C₄⋊C₈	—	10	2	no	—
157	#103	—	2	2	no	—
158	#103	—	2	2	no	—
159	C₂²⋊C₈	—	5	2	no	—
160	C₂²⋊C₈	—	10	1	no	—
161	C₂²⋊C₈	—	10	2	no	—
162	C₄×C₈	—	10	2	no	—
163	C₄×C₈	—	10	1	no	—
164	C₄×C₈	—	10	1	no	—
165	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
169	C₂²×C₅	—	2	1	no	—
170	C₂²×C₅	—	2	1	no	—
171	C₂²×C₅	#390	1	1	yes	characteristic
172	C₂²×C₅	#435	1	2	yes	—
173	C₄×C₅	—	2	1	no	—
174	C₄×C₅	—	2	1	no	—
175	C₄×C₅	—	2	1	no	—
176	C₄×C₅	—	4	4	no	—
177	C₄×C₅	—	8	2	no	—
178	C₄×C₅	#407	1	1	yes	characteristic
179	C₄×C₅	#408	1	1	yes	characteristic
180	C₄×C₅	#435	1	2	yes	—
181	C₄×C₅	#473	1	2	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	#681	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
191	C₂²×C₄	#711	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
193	Q₁₆	—	2	4	no	—
194	Q₁₆	—	2	4	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	10	2	no	—
198	C₂×C₈	—	10	2	no	—
199	C₂×C₈	—	20	2	no	—
200	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
203	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	2	no	—
210	C₄²	D₄×D₅	1	1	yes	characteristic
211	C₄²	#711	1	1	yes	characteristic
212	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₂×C₅	—	2	2	no	—
215	C₂×C₅	#1955	1	1	yes	characteristic
216	C₂×C₅	#1976	1	1	yes	characteristic
217	C₂×C₅	#1979	1	1	yes	characteristic
218	C₂³	D₄×Dic₅	1	1	yes	characteristic
219	Q₈	—	2	4	no	—
220	Q₈	—	2	4	no	—
221	Q₈	—	4	2	no	—
222	Q₈	—	4	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₂×#681	1	2	yes	—
231	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
232	C₂×C₄	#4082	1	1	yes	characteristic
233	C₂×C₄	#4084	1	1	yes	characteristic
234	C₂×C₄	#4084	1	1	yes	characteristic
235	C₂×C₄	#4086	1	1	yes	characteristic
236	C₂×C₄	#4094	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	20	2	no	—
243	C₈	—	20	2	no	—
244	C₈	—	40	2	no	—
245	C₅	#9671	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#65300	1	2	yes	—
249	C₂²	#65510	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	#65300	1	2	yes	—
256	C₄	#65481	1	2	yes	—
257	C₄	#65496	1	1	yes	characteristic
258	C₄	#65523	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#10602408	1	1	yes	characteristic
261	C₂	#10602862	1	1	yes	characteristic
262	C₂	#10602879	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4084
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₄×C₅×Q₁₆	C₂
socle	C₂²×C₅	#390
prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅
hypocenter	C₅	#9671
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
245. C₅	#9671	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₁₆	2
#10602417	2
#10602414	2
#10600522	2
#10600519	2
#9671	5