Subgroups — C₂×#10615059 #12963666

Order 1280.

1982 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	#10615059	C₂	1	16	yes	maximal, direct product factor
3	C₂×#65313	C₂	1	4	yes	maximal, complemented
4	C₂×Q₈×#168	C₂	1	4	yes	maximal, complemented
5	C₂×#65288	C₂	1	2	yes	maximal
6	C₂×#65286	C₂	1	2	yes	maximal
7	C₂×#65213	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic
9	C₅×#533	C₄	1	8	yes	—
10	C₅×#533	C₂²	1	8	yes	—
11	C₂×C₅×C₄⋊Q₈	C₄	1	2	yes	—
12	C₂×C₅×#125	C₄	1	2	yes	—
13	C₂×C₅×C₂²⋊Q₈	C₄	1	4	yes	—
14	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
15	C₂×C₅×#117	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₂²×#681	C₂²	1	2	yes	—
18	C₂³×#168	C₂²	1	1	yes	characteristic
19	C₂³×#168	C₂²	1	1	yes	characteristic
20	#65313	C₂²	1	32	yes	complemented
21	Q₈×#168	C₂²	1	32	yes	complemented
22	#65288	C₂²	1	16	yes	—
23	#65286	C₂²	1	16	yes	—
24	#65213	C₂²	1	8	yes	—
25	C₂×C₄⋊#168	C₂²	1	4	yes	—
26	C₂×C₄⋊#168	C₂²	1	8	yes	—
27	C₂×#3935	C₂²	1	4	yes	—
28	C₂×C₄×#168	C₂²	1	8	yes	—
29	C₂×C₄×#168	C₂²	1	2	yes	—
30	C₂×C₄×#168	C₂²	1	2	yes	—
31	C₂×C₄×#168	C₂²	1	4	yes	—
32	C₂×#2960	—	5	1	no	maximal, Sylow
33	C₂²×C₅×Q₈	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	C₂×C₄	1	16	yes	—
37	C₅×#125	C₂×C₄	1	16	yes	—
38	C₅×C₂²⋊Q₈	C₂×C₄	1	32	yes	—
39	C₄×C₅×Q₈	C₂×C₄	1	16	yes	—
40	C₄×C₅×Q₈	C₂³	1	16	yes	—
41	C₅×#117	C₂×C₄	1	8	yes	—
42	C₅×#117	C₂³	1	8	yes	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
46	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅	C₂³	1	2	yes	—
51	C₂×C₄²×C₅	C₂³	1	4	yes	—
52	C₂×C₄²×C₅	C₂³	1	4	yes	—
53	C₂×#681	—	2	8	no	—
54	C₂×#681	C₂³	1	8	yes	—
55	C₂²×#168	—	2	4	no	—
56	C₂²×#168	Q₈	1	8	yes	complemented
57	C₂²×#168	C₂³	1	4	yes	—
58	C₂²×#168	C₂³	1	2	yes	—
59	C₂²×#168	C₂³	1	4	yes	—
60	C₂²×#168	C₂³	1	1	yes	characteristic
61	C₂²×#168	C₂³	1	1	yes	characteristic
62	C₂²×#168	C₂³	1	4	yes	—
63	C₄⋊#168	C₂³	1	32	yes	—
64	C₄⋊#168	C₂³	1	16	yes	—
65	#3935	C₂³	1	16	yes	—
66	C₄×#168	C₂³	1	16	yes	—
67	C₄×#168	C₂³	1	8	yes	—
68	C₄×#168	C₂³	1	8	yes	—
69	C₂×#533	D₅	1	1	yes	characteristic
70	#2960	—	5	16	no	—
71	C₂×#446	—	5	4	no	—
72	C₂×C₈×Q₈	—	5	4	no	—
73	C₂×#424	—	5	2	no	—
74	C₂×#423	—	5	1	no	—
75	C₂×#405	—	5	2	no	—
76	C₂²×C₄×C₈	—	5	1	no	—
77	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₅×Q₈	—	2	16	no	—
79	C₂×C₅×Q₈	C₂²×C₄	1	16	yes	—
80	C₂²×C₄×C₅	—	2	8	no	—
81	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₅	#41	1	4	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
92	C₅×C₄⋊C₄	C₂²×C₄	1	32	yes	—
93	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
94	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
95	C₄²×C₅	C₂²×C₄	1	4	yes	—
96	C₄²×C₅	C₂²×C₄	1	4	yes	—
97	C₄²×C₅	C₂²×C₄	1	4	yes	—
98	C₄²×C₅	C₂²×C₄	1	8	yes	—
99	C₄²×C₅	C₂⁴	1	4	yes	—
100	C₄²×C₅	C₂⁴	1	8	yes	—
101	#681	—	2	16	no	—
102	C₂×#168	—	2	4	no	—
103	C₂×#168	—	2	8	no	—
104	C₂×#168	—	2	8	no	—
105	C₂×#168	C₂×Q₈	1	4	yes	—
106	C₂×#168	C₂×Q₈	1	4	yes	—
107	C₂×#168	C₂×Q₈	1	16	yes	complemented
108	C₂×#168	C₂⁴	1	4	yes	—
109	C₂×#168	C₂⁴	1	8	yes	—
110	C₂×#168	C₂⁴	1	2	yes	—
111	C₂×#168	C₂⁴	1	2	yes	—
112	C₂²×M₁₆	—	5	2	no	—
113	C₂³×C₈	—	5	1	no	—
114	C₂³×C₈	—	5	1	no	—
115	#533	Dic₅	1	8	yes	—
116	#533	C₂×D₅	1	8	yes	—
117	C₂×C₄⋊Q₈	Dic₅	1	2	yes	—
118	C₂×#125	Dic₅	1	2	yes	—
119	C₂×C₂²⋊Q₈	Dic₅	1	4	yes	—
120	C₂×C₄×Q₈	C₂×D₅	1	4	yes	—
121	C₂×#117	C₂×D₅	1	2	yes	—
122	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
123	#446	—	5	32	no	—
124	C₈×Q₈	—	5	32	no	—
125	#424	—	5	16	no	—
126	#423	—	5	8	no	—
127	C₂×C₄⋊C₈	—	5	4	no	—
128	C₂×C₄⋊C₈	—	5	8	no	—
129	#405	—	5	16	no	—
130	C₂×#97	—	5	4	no	—
131	C₂×C₄×C₈	—	5	2	no	—
132	C₂×C₄×C₈	—	5	2	no	—
133	C₂×C₄×C₈	—	5	8	no	—
134	C₂×C₄×C₈	—	5	4	no	—
135	C₂³×C₅	—	2	4	no	—
136	C₂³×C₅	C₄×Q₈	1	4	yes	—
137	C₂³×C₅	C₂³×C₄	1	2	yes	—
138	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₅×Q₈	—	2	32	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	16	no	—
142	C₂×C₄×C₅	—	2	16	no	—
143	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
144	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
145	C₂×C₄×C₅	#131	1	8	yes	—
146	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
147	C₂×C₄×C₅	C₂³×C₄	1	16	yes	—
148	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
150	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
151	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
152	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
153	C₂×C₄×C₅	C₂×#41	1	2	yes	—
154	C₂×C₄×C₅	C₂×#41	1	4	yes	—
155	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
156	#168	—	2	4	no	—
157	#168	—	2	8	no	—
158	#168	C₂²×Q₈	1	4	yes	—
159	#168	C₂²×Q₈	1	4	yes	—
160	C₂²×Q₈	C₂×Dic₅	1	4	yes	—
161	C₂³×C₄	C₂×Dic₅	1	2	yes	—
162	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
163	C₂×M₁₆	—	5	8	no	—
164	C₂×M₁₆	—	10	8	no	—
165	C₂²×C₈	—	5	8	no	—
166	C₂²×C₈	—	5	4	no	—
167	C₂²×C₈	—	5	1	no	—
168	C₂²×C₈	—	5	1	no	—
169	C₂²×C₈	—	5	4	no	—
170	C₂²×C₈	—	5	2	no	—
171	C₂²×C₈	—	5	4	no	—
172	C₂²×C₈	—	10	4	no	—
173	C₄⋊Q₈	C₂×Dic₅	1	16	yes	—
174	#125	C₂×Dic₅	1	16	yes	—
175	C₂²⋊Q₈	C₂×Dic₅	1	32	yes	—
176	C₄×Q₈	C₂×Dic₅	1	16	yes	—
177	C₄×Q₈	C₂²×D₅	1	16	yes	—
178	#117	C₂×Dic₅	1	8	yes	—
179	#117	C₂²×D₅	1	8	yes	—
180	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×Dic₅	1	8	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
183	C₂×C₂²⋊C₄	C₂×Dic₅	1	4	yes	—
184	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
185	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₅	1	4	yes	—
187	C₂×C₄²	C₂²×D₅	1	4	yes	—
188	C₂×C₄²	C₂²×D₅	1	4	yes	—
189	C₂×C₄²	C₂²×D₅	1	2	yes	—
190	C₄⋊C₈	—	5	16	no	—
191	C₄⋊C₈	—	5	32	no	—
192	#97	—	5	16	no	—
193	C₄×C₈	—	5	8	no	—
194	C₄×C₈	—	5	8	no	—
195	C₄×C₈	—	5	16	no	—
196	C₂²×C₅	—	2	8	no	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	C₂×C₄×Q₈	1	4	yes	—
199	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
200	C₂²×C₅	C₂×#131	1	4	yes	—
201	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
202	C₄×C₅	—	2	4	no	—
203	C₄×C₅	—	2	16	no	—
204	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
205	C₄×C₅	C₂×C₄×Q₈	1	4	yes	—
206	C₄×C₅	C₂×#131	1	8	yes	—
207	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
208	C₄×C₅	C₂²×#41	1	1	yes	characteristic
209	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
210	C₂×Q₈	—	2	16	no	—
211	C₂×Q₈	C₂²×Dic₅	1	16	yes	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	Q₈×D₅	1	4	yes	—
214	C₂²×C₄	#713	1	4	yes	—
215	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
216	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
217	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
218	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
221	C₂²×C₄	C₂³×D₅	1	2	yes	—
222	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
223	M₁₆	—	10	16	no	—
224	C₂×C₈	—	5	4	no	—
225	C₂×C₈	—	5	4	no	—
226	C₂×C₈	—	5	16	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	5	2	no	—
229	C₂×C₈	—	5	4	no	—
230	C₂×C₈	—	5	8	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	—	10	8	no	—
233	C₂×C₈	—	10	8	no	—
234	C₄⋊C₄	C₂²×Dic₅	1	16	yes	—
235	C₄⋊C₄	C₂²×Dic₅	1	32	yes	—
236	C₄⋊C₄	C₂²×Dic₅	1	16	yes	—
237	C₂²⋊C₄	C₂²×Dic₅	1	16	yes	—
238	C₄²	C₂²×Dic₅	1	4	yes	—
239	C₄²	C₂²×Dic₅	1	4	yes	—
240	C₄²	C₂²×Dic₅	1	4	yes	—
241	C₄²	C₂²×Dic₅	1	8	yes	—
242	C₄²	C₂³×D₅	1	4	yes	—
243	C₄²	C₂³×D₅	1	8	yes	—
244	C₂×C₅	—	2	4	no	—
245	C₂×C₅	#2960	1	4	yes	complemented
246	C₂×C₅	C₂²×C₄×Q₈	1	1	yes	characteristic
247	C₂×C₅	C₂²×#131	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	Q₈×Dic₅	1	4	yes	—
250	C₂³	C₂³×Dic₅	1	2	yes	—
251	C₂³	C₂³×Dic₅	1	1	yes	characteristic
252	Q₈	—	2	32	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	16	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	Q₈×Dic₅	1	8	yes	—
257	C₂×C₄	Q₈×Dic₅	1	4	yes	—
258	C₂×C₄	#4094	1	8	yes	—
259	C₂×C₄	C₂×Q₈×D₅	1	4	yes	—
260	C₂×C₄	C₂×Q₈×D₅	1	2	yes	—
261	C₂×C₄	C₂×#713	1	2	yes	—
262	C₂×C₄	C₂×#713	1	4	yes	—
263	C₂×C₄	C₂³×Dic₅	1	4	yes	—
264	C₂×C₄	C₂³×Dic₅	1	16	yes	—
265	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
266	C₂×C₄	C₂³×Dic₅	1	2	yes	—
267	C₂×C₄	C₂³×Dic₅	1	4	yes	—
268	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
269	C₈	—	5	4	no	—
270	C₈	—	5	4	no	—
271	C₈	—	10	4	no	—
272	C₈	—	10	8	no	—
273	C₅	C₂×#2960	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	4	no	—
275	C₂²	—	2	8	no	—
276	C₂²	C₂×Q₈×Dic₅	1	4	yes	—
277	C₂²	C₂×Q₈×Dic₅	1	2	yes	—
278	C₂²	C₂×#4094	1	4	yes	—
279	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	16	no	—
282	C₄	C₂×Q₈×Dic₅	1	4	yes	—
283	C₄	C₂×Q₈×Dic₅	1	2	yes	—
284	C₄	C₂×#4094	1	8	yes	—
285	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
286	C₄	C₂²×#713	1	1	yes	characteristic
287	C₂	—	2	4	no	—
288	C₂	#10615059	1	4	yes	direct product factor
289	C₂	C₂²×Q₈×Dic₅	1	1	yes	characteristic
290	C₂	C₂²×#4094	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂×C₅×#533	C₂
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂×#533	D₅
hypocenter	C₅	C₂×#2960
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
2. #10615059	C₂	direct (N × H)
3. C₂×#65313	C₂	semidirect
4. C₂×Q₈×#168	C₂	semidirect
20. #65313	C₂²	semidirect
21. Q₈×#168	C₂²	semidirect
56. C₂²×#168	Q₈	semidirect
107. C₂×#168	C₂×Q₈	semidirect
245. C₂×C₅	#2960	semidirect
273. C₅	C₂×#2960	semidirect
288. C₂	#10615059	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#533	2
#10615059	2
C₂×#65313	2
C₂×Q₈×#168	2
C₂×#65288	2
C₂×#65286	2
C₂×#65213	2
C₂²×C₄×#168	2
C₂×#2960	5

Subgroups — C2×#10615059 #12963666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10615059 #12963666