Subgroups — C₂×#10599844 #12251933

Order 1280.

854 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	C₂²×#3930	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#64978	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64722	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599844	C₂	1	8	yes	maximal, direct product factor
9	C₂²×#676	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₂×#4018	C₄	1	1	yes	characteristic
12	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3947	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂²×#672	C₂²	1	1	yes	characteristic
18	C₂²×#672	C₂²	1	1	yes	characteristic
19	C₂×#3935	C₄	1	1	yes	characteristic
20	C₂×#3930	C₄	1	8	yes	complemented
21	C₂×#3930	C₂²	1	4	yes	—
22	C₂×D₅×C₁₆	C₄	1	8	yes	complemented
23	C₂×D₅×C₁₆	C₂²	1	4	yes	—
24	#64978	C₄	1	4	yes	—
25	#64978	C₂²	1	4	yes	—
26	C₄×C₅×C₁₆	C₂²	1	4	yes	complemented
27	#64722	C₂²	1	4	yes	complemented
28	C₁₆×Dic₅	C₂²	1	4	yes	complemented
29	#64681	C₂²	1	4	yes	complemented
30	C₂×#2108	—	5	1	no	maximal, Sylow
31	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×#685	C₈	1	2	yes	—
36	C₂×#683	C₈	1	2	yes	—
37	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₂²×#168	C₂×C₄	1	1	yes	characteristic
39	C₂²×#168	C₂×C₄	1	1	yes	characteristic
40	C₂×#676	C₈	1	8	yes	—
41	C₂×#676	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
44	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
45	C₂×C₈×D₅	C₂³	1	2	yes	—
46	#4018	C₈	1	4	yes	—
47	#4018	C₂×C₄	1	4	yes	—
48	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	#3947	C₂×C₄	1	4	yes	—
55	C₈×Dic₅	C₂×C₄	1	2	yes	—
56	C₈×Dic₅	C₂³	1	2	yes	—
57	C₂×#672	C₂×C₄	1	4	yes	—
58	C₂×#672	C₂×C₄	1	4	yes	—
59	C₂×#672	C₂³	1	2	yes	—
60	C₂×#672	C₂³	1	2	yes	—
61	#3935	C₂×C₄	1	4	yes	—
62	#3930	C₂×C₄	1	16	yes	complemented
63	D₅×C₁₆	C₂×C₄	1	16	yes	complemented
64	C₂²×M₃₂	—	5	1	no	—
65	C₂³×C₁₆	—	5	1	no	—
66	C₂×#405	—	5	1	no	—
67	#2108	—	5	8	no	—
68	C₂×#346	—	5	1	no	—
69	C₂×#346	—	5	1	no	—
70	C₂×C₄×C₁₆	—	5	1	no	—
71	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic, complemented
72	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₅	C₂×C₈	1	2	yes	—
76	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅	C₄²	1	4	yes	—
78	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
80	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
87	C₄²×C₅	C₂×C₈	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	2	yes	—
89	#685	C₂×C₈	1	8	yes	—
90	#683	C₂×C₈	1	8	yes	—
91	C₄×Dic₅	C₂×C₈	1	2	yes	—
92	C₄×Dic₅	C₂²×C₄	1	2	yes	—
93	C₂×#168	C₄²	1	4	yes	—
94	C₂×#168	C₂×C₈	1	4	yes	—
95	C₂×#168	C₂²×C₄	1	2	yes	—
96	C₂×#168	C₂²×C₄	1	2	yes	—
97	#676	C₂×C₈	1	16	yes	—
98	C₈×D₅	C₄²	1	8	yes	—
99	C₈×D₅	C₂²×C₄	1	4	yes	—
100	C₈×D₅	C₂²×C₄	1	4	yes	—
101	C₅×C₁₆	C₂²×C₄	1	4	yes	—
102	C₅×C₁₆	C₂²×C₄	1	4	yes	—
103	#672	C₂²×C₄	1	4	yes	—
104	#672	C₂²×C₄	1	4	yes	—
105	C₂²×M₁₆	—	5	1	no	—
106	C₂³×C₈	—	5	1	no	—
107	C₂×#117	—	5	1	no	—
108	C₂×M₃₂	—	5	8	no	—
109	C₂×M₃₂	—	5	4	no	—
110	C₂²×C₁₆	—	5	1	no	—
111	C₂²×C₁₆	—	5	1	no	—
112	C₂²×C₁₆	—	5	8	no	—
113	C₂²×C₁₆	—	5	4	no	—
114	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
115	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
116	#405	—	5	4	no	—
117	#405	—	5	4	no	—
118	C₂×#97	—	5	1	no	—
119	C₂×#97	—	5	1	no	—
120	C₂×C₄×C₈	—	5	1	no	—
121	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
122	#346	—	5	4	no	—
123	#346	—	5	4	no	—
124	C₄×C₁₆	—	5	4	no	—
125	C₄×C₁₆	C₂×D₅	1	4	yes	complemented
126	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂²×D₅	—	2	4	no	—
128	C₂²×D₅	C₄×C₈	1	4	yes	—
129	C₂²×D₅	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₅	—	2	4	no	—
137	C₂×Dic₅	C₄×C₈	1	4	yes	—
138	C₂×Dic₅	C₂²×C₈	1	4	yes	—
139	C₂×Dic₅	C₂²×C₈	1	2	yes	—
140	C₄×D₅	C₄×C₈	1	8	yes	—
141	C₄×D₅	C₂×C₄²	1	4	yes	—
142	C₄×D₅	C₂²×C₈	1	4	yes	—
143	C₅×C₈	C₂×C₄²	1	2	yes	—
144	C₅×C₈	C₂²×C₈	1	4	yes	—
145	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
146	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
147	#168	C₂×C₄²	1	4	yes	—
148	#168	C₂²×C₈	1	4	yes	—
149	C₂³×C₄	—	5	1	no	—
150	C₂×M₁₆	—	5	8	no	—
151	C₂×M₁₆	—	5	4	no	—
152	C₂²×C₈	—	5	1	no	—
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	5	2	no	—
155	C₂²×C₈	—	5	4	no	—
156	C₂²×C₈	—	5	4	no	—
157	C₂²×C₈	—	5	2	no	—
158	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
160	#117	—	5	4	no	—
161	#117	—	5	4	no	—
162	C₂×C₄⋊C₄	—	5	2	no	—
163	C₂×C₂²⋊C₄	—	5	2	no	—
164	C₂×C₄²	—	5	1	no	—
165	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
166	M₃₂	—	5	16	no	—
167	C₂×C₁₆	—	5	4	no	—
168	C₂×C₁₆	—	5	4	no	—
169	C₂×C₁₆	—	5	2	no	—
170	C₂×C₁₆	—	5	2	no	—
171	C₂×C₁₆	—	5	16	no	—
172	C₂×C₁₆	C₄×D₅	1	4	yes	—
173	C₂×C₁₆	C₄×D₅	1	4	yes	—
174	C₂×C₁₆	C₂²×D₅	1	2	yes	—
175	C₂×C₁₆	C₂²×D₅	1	2	yes	—
176	#97	—	5	4	no	—
177	#97	—	5	4	no	—
178	C₄×C₈	—	5	2	no	—
179	C₄×C₈	—	5	2	no	—
180	C₄×C₈	C₄×D₅	1	2	yes	—
181	C₄×C₈	C₂²×D₅	1	2	yes	—
182	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
183	C₂²×C₅	#504	1	4	yes	—
184	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
185	C₂×D₅	—	2	8	no	—
186	C₂×D₅	—	2	4	no	—
187	C₂×D₅	C₂×C₄×C₈	1	4	yes	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
190	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
191	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
192	Dic₅	—	2	4	no	—
193	Dic₅	C₂×C₄×C₈	1	4	yes	—
194	C₂⁴	—	5	1	no	—
195	C₂²×C₄	—	5	1	no	—
196	C₂²×C₄	—	5	2	no	—
197	C₂²×C₄	—	5	4	no	—
198	C₂²×C₄	—	5	2	no	—
199	C₂²×C₄	—	5	4	no	—
200	C₂²×C₄	—	5	2	no	—
201	C₂²×C₄	C₈×D₅	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
203	M₁₆	—	5	16	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	5	4	no	—
207	C₂×C₈	—	5	4	no	—
208	C₂×C₈	—	5	4	no	—
209	C₂×C₈	—	5	4	no	—
210	C₂×C₈	—	5	8	no	—
211	C₂×C₈	C₈×D₅	1	4	yes	—
212	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
216	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
217	C₄⋊C₄	—	5	8	no	—
218	C₂²⋊C₄	—	5	8	no	—
219	C₄²	—	5	2	no	—
220	C₄²	—	5	2	no	—
221	C₄²	C₈×D₅	1	2	yes	—
222	C₄²	C₂×C₄×D₅	1	2	yes	—
223	C₁₆	—	5	4	no	—
224	C₁₆	—	5	4	no	—
225	C₁₆	C₂×C₄×D₅	1	4	yes	—
226	C₁₆	C₂×C₄×D₅	1	4	yes	—
227	C₂×C₅	#2108	1	4	yes	complemented
228	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
229	C₂×C₅	C₂×#504	1	2	yes	—
230	D₅	—	2	4	no	—
231	C₂³	—	5	2	no	—
232	C₂³	—	5	4	no	—
233	C₂³	—	10	4	no	—
234	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	5	2	no	—
237	C₂×C₄	—	5	4	no	—
238	C₂×C₄	—	5	4	no	—
239	C₂×C₄	—	5	4	no	—
240	C₂×C₄	—	5	8	no	—
241	C₂×C₄	—	5	4	no	—
242	C₂×C₄	—	10	4	no	—
243	C₂×C₄	C₄²×D₅	1	2	yes	—
244	C₂×C₄	C₂×C₈×D₅	1	4	yes	—
245	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
246	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
248	C₈	—	5	4	no	—
249	C₈	—	5	4	no	—
250	C₈	C₄²×D₅	1	2	yes	—
251	C₈	C₂×C₈×D₅	1	4	yes	—
252	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
253	C₈	C₂²×C₄×D₅	1	1	yes	characteristic
254	C₅	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
255	C₂²	—	5	4	no	—
256	C₂²	—	10	8	no	—
257	C₂²	—	10	4	no	—
258	C₂²	C₄×C₈×D₅	1	2	yes	—
259	C₂²	#65190	1	4	yes	—
260	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	5	4	no	—
263	C₄	—	10	4	no	—
264	C₄	C₄×C₈×D₅	1	2	yes	—
265	C₄	C₂×C₄²×D₅	1	1	yes	characteristic
266	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
267	C₂	—	10	4	no	—
268	C₂	#10599844	1	4	yes	direct product factor
269	C₂	C₂×C₄×C₈×D₅	1	1	yes	characteristic
270	C₂	C₂×#65190	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	C₂×#2108
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×C₄×C₅×C₁₆	C₂	semidirect
5. C₂×#64722	C₂	semidirect
6. C₂×C₁₆×Dic₅	C₂	semidirect
7. C₂×#64681	C₂	semidirect
8. #10599844	C₂	direct (N × H)
20. C₂×#3930	C₄	semidirect
22. C₂×D₅×C₁₆	C₄	semidirect
26. C₄×C₅×C₁₆	C₂²	semidirect
27. #64722	C₂²	semidirect
28. C₁₆×Dic₅	C₂²	semidirect
29. #64681	C₂²	semidirect
62. #3930	C₂×C₄	semidirect
63. D₅×C₁₆	C₂×C₄	semidirect
71. C₂×C₄×C₁₆	D₅	semidirect
125. C₄×C₁₆	C₂×D₅	semidirect
227. C₂×C₅	#2108	semidirect
254. C₅	C₂×#2108	semidirect
268. C₂	#10599844	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#3930	2
C₂²×D₅×C₁₆	2
C₂×#64978	2
C₂×C₄×C₅×C₁₆	2
C₂×#64722	2
C₂×C₁₆×Dic₅	2
C₂×#64681	2
#10599844	2
C₂×#2108	5

Subgroups — C2×#10599844 #12251933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10599844 #12251933