Subgroups — #12015223

Order 1280.

478 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	C₂×#65390	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic
4	C₂×#64721	C₂	1	2	yes	maximal, complemented
5	#10598295	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×#4072	C₄	1	1	yes	characteristic
8	C₂²×#681	C₄	1	1	yes	characteristic
9	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
10	#65390	—	2	4	no	—
11	C₂×#3947	C₄	1	1	yes	characteristic
12	C₂×#3947	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×#672	C₂²	1	2	yes	—
16	C₅×C₂²⋊C₁₆	—	2	2	no	—
17	C₂²⋊#672	—	2	2	no	—
18	#64721	—	2	4	no	—
19	#12784	—	5	1	no	maximal, Sylow
20	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	2	yes	—
25	C₂²×C₅×C₈	Q₈	1	2	yes	—
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊Dic₅	C₈	1	2	yes	—
28	#4072	—	2	2	no	—
29	#4072	—	2	2	no	—
30	C₂×C₄⋊Dic₅	C₈	1	2	yes	—
31	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₂×#681	—	2	2	no	—
34	C₂×#681	—	2	2	no	—
35	C₂×#681	—	2	2	no	—
36	C₂²×#168	C₂×C₄	1	1	yes	characteristic
37	C₂²×#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₁₆	—	2	2	no	—
39	C₂×C₅×C₁₆	—	2	2	no	—
40	C₂×C₅×C₁₆	—	2	2	no	—
41	#3947	—	2	2	no	—
42	#3947	—	2	2	no	—
43	C₈×Dic₅	—	2	2	no	—
44	C₈×Dic₅	D₄	1	4	yes	—
45	C₂×#672	—	2	2	no	—
46	C₂×#672	—	2	2	no	—
47	C₂×#672	—	2	2	no	—
48	C₂×#405	—	5	1	no	—
49	C₂×C₄⋊C₁₆	—	5	2	no	—
50	C₂×C₂²⋊C₁₆	—	5	1	no	—
51	C₂×C₂²⋊C₁₆	D₅	1	1	yes	characteristic
52	#1374	—	5	2	no	—
53	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	M₁₆	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅	C₂×C₈	1	2	yes	—
63	C₂²×Dic₅	C₂×C₈	1	2	yes	—
64	C₂×C₅×C₈	—	2	1	no	—
65	C₂×C₅×C₈	—	2	1	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₅×C₈	#41	1	2	yes	—
73	C₂²⋊Dic₅	—	2	4	no	—
74	C₄⋊Dic₅	—	2	4	no	—
75	C₄×Dic₅	—	2	1	no	—
76	C₄×Dic₅	—	2	1	no	—
77	C₄×Dic₅	C₂²⋊C₄	1	2	yes	—
78	C₄×Dic₅	M₁₆	1	2	yes	—
79	#681	—	2	8	no	—
80	C₂×#168	—	2	1	no	—
81	C₂×#168	—	2	1	no	—
82	C₂×#168	—	2	1	no	—
83	C₂×#168	—	2	1	no	—
84	C₂×#168	C₂²⋊C₄	1	2	yes	—
85	C₂×#168	M₁₆	1	2	yes	—
86	C₅×C₁₆	—	4	2	no	—
87	#672	—	4	2	no	—
88	C₂²×M₁₆	—	5	1	no	—
89	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
90	C₂×#117	—	5	1	no	—
91	C₂²×C₁₆	—	5	2	no	—
92	C₂²×C₁₆	C₂×D₅	1	2	yes	—
93	#405	—	10	4	no	—
94	C₂×#97	—	5	1	no	—
95	C₂×#97	—	5	1	no	—
96	C₂×C₄×C₈	—	5	1	no	—
97	C₂×C₄×C₈	—	5	1	no	—
98	C₄⋊C₁₆	—	10	4	no	—
99	C₂²⋊C₁₆	—	2	2	no	—
100	C₂²⋊C₁₆	—	10	2	no	—
101	C₂³×C₅	—	2	1	no	—
102	C₂³×C₅	—	2	1	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	—	2	2	no	—
105	C₂³×C₅	C₄⋊C₈	1	2	yes	—
106	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	#117	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
120	C₂×Dic₅	—	2	2	no	—
121	C₂×Dic₅	—	2	2	no	—
122	C₂×Dic₅	—	2	2	no	—
123	C₂×Dic₅	—	2	2	no	—
124	C₂×Dic₅	—	2	2	no	—
125	C₂×Dic₅	C₂²⋊C₈	1	4	yes	—
126	C₅×C₈	—	2	2	no	—
127	C₅×C₈	#121	1	2	yes	—
128	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
129	#168	—	2	2	no	—
130	#168	—	2	2	no	—
131	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
132	C₂×M₁₆	—	10	2	no	—
133	C₂×M₁₆	—	10	2	no	—
134	C₂×M₁₆	—	10	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	5	1	no	—
138	C₂²×C₈	—	5	1	no	—
139	C₂²×C₈	Dic₁₀	1	2	yes	—
140	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
141	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
142	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
143	#117	—	10	2	no	—
144	#117	—	10	2	no	—
145	C₂×C₄⋊C₄	—	5	2	no	—
146	C₂×C₂²⋊C₄	—	5	2	no	—
147	C₂×C₄²	—	5	1	no	—
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	10	2	no	—
153	C₂×C₁₆	—	10	2	no	—
154	C₂×C₁₆	—	10	2	no	—
155	#97	—	10	2	no	—
156	#97	—	10	2	no	—
157	C₄×C₈	—	5	4	no	—
158	C₄×C₈	—	10	2	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	4	2	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₂²×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
166	C₂²×C₅	#431	1	1	yes	characteristic
167	C₂²×C₅	#504	1	2	yes	—
168	C₂²×C₅	#504	1	2	yes	—
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	#380	1	1	yes	characteristic
171	C₄×C₅	#407	1	1	yes	characteristic
172	C₄×C₅	#423	1	1	yes	characteristic
173	C₄×C₅	#432	1	1	yes	characteristic
174	Dic₅	—	2	4	no	—
175	Dic₅	—	4	2	no	—
176	C₂⁴	C₈×D₅	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	5	2	no	—
182	C₂²×C₄	—	5	2	no	—
183	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
184	C₂²×C₄	#676	1	2	yes	—
185	C₂²×C₄	#683	1	2	yes	—
186	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
187	M₁₆	—	10	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	5	2	no	—
193	C₂×C₈	—	5	2	no	—
194	C₂×C₈	—	10	1	no	—
195	C₂×C₈	—	10	1	no	—
196	C₂×C₈	—	10	1	no	—
197	C₂×C₈	—	10	1	no	—
198	C₂×C₈	#683	1	2	yes	—
199	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
200	C₂×C₈	D₄×D₅	1	2	yes	—
201	C₂×C₈	#711	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
203	C₄⋊C₄	—	10	4	no	—
204	C₂²⋊C₄	—	10	4	no	—
205	C₄²	—	5	2	no	—
206	C₄²	—	5	2	no	—
207	C₄²	—	10	1	no	—
208	C₄²	—	10	1	no	—
209	C₁₆	—	4	2	no	—
210	C₁₆	—	20	2	no	—
211	C₂×C₅	—	4	2	no	—
212	C₂×C₅	#1797	1	1	yes	characteristic
213	C₂×C₅	#2113	1	2	yes	—
214	C₂×C₅	#2151	1	1	yes	characteristic
215	C₂×C₅	#2151	1	1	yes	characteristic
216	C₂×C₅	#2164	1	2	yes	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	#3946	1	2	yes	—
222	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	5	4	no	—
230	C₂×C₄	—	10	2	no	—
231	C₂×C₄	—	10	2	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	—	10	2	no	—
234	C₂×C₄	—	10	2	no	—
235	C₂×C₄	#3946	1	2	yes	—
236	C₂×C₄	#4023	1	1	yes	characteristic
237	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄	C₂×#676	1	1	yes	characteristic
239	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
240	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
241	C₂×C₄	C₂×#683	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	10	2	no	—
244	C₈	—	10	2	no	—
245	C₈	#4024	1	2	yes	—
246	C₈	#4085	1	2	yes	—
247	C₅	#12784	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	#65007	1	1	yes	characteristic
254	C₂²	D₅×C₂²⋊C₈	1	1	yes	characteristic
255	C₂²	#65190	1	2	yes	—
256	C₂²	#65196	1	2	yes	—
257	C₂²	C₂×#3946	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	10	4	no	—
260	C₄	—	20	2	no	—
261	C₄	#65008	1	1	yes	characteristic
262	C₄	#65015	1	1	yes	characteristic
263	C₄	#65231	1	1	yes	characteristic
264	C₄	#65407	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	#10599887	1	2	yes	—
267	C₂	#10599900	1	2	yes	—
268	C₂	#10600701	1	1	yes	characteristic
269	C₂	#10602194	1	1	yes	characteristic
270	C₂	#10602214	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle	C₂³×C₅	C₂²×C₈
Fitting	C₂×C₅×C₂²⋊C₁₆	C₂
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅
hypocenter	C₅	#12784
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×#64721	C₂	semidirect
5. #10598295	C₂	semidirect
247. C₅	#12784	semidirect

Maximal subgroups

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

subgroup	index
C₂×#65390	2
C₂×C₅×C₂²⋊C₁₆	2
C₂×C₂²⋊#672	2
C₂×#64721	2
#10598295	2
#12784	5