Subgroups — C₂×C₅×#1696 #11941869

Order 1280.

618 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×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#477	C₂	1	2	yes	maximal
3	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
4	C₅×#1696	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#333	C₂	1	2	yes	maximal
7	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₅×#500	—	2	2	no	—
10	C₅×#500	C₂²	1	4	yes	—
11	C₅×#477	—	2	4	no	—
12	C₅×#477	C₂²	1	8	yes	—
13	C₅×#446	—	2	2	no	—
14	C₅×#446	C₂²	1	4	yes	—
15	C₂×C₅×#107	—	2	1	no	—
16	C₂×C₅×#107	—	2	1	no	—
17	C₂×C₅×#107	—	2	2	no	—
18	C₂×C₅×C₄⋊C₈	—	2	1	no	—
19	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₅×#103	—	2	2	no	—
22	C₂×C₅×#97	—	2	1	no	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#334	C₂²	1	4	yes	—
25	C₅×#333	C₂²	1	8	yes	—
26	C₂×#1696	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅×Q₈	—	2	2	no	—
28	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	—	2	1	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₅×C₄⋊Q₈	—	2	2	no	—
34	C₅×C₄⋊Q₈	—	2	2	no	—
35	C₅×C₄⋊Q₈	—	2	2	no	—
36	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
37	C₄×C₅×Q₈	—	2	1	no	—
38	C₄×C₅×Q₈	—	2	1	no	—
39	C₄×C₅×Q₈	D₄	1	2	yes	—
40	C₄×C₅×Q₈	C₂³	1	2	yes	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	—	2	1	no	—
46	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₅×#107	—	2	4	no	—
48	C₅×#107	—	2	4	no	—
49	C₅×#107	—	2	8	no	—
50	C₅×C₄⋊C₈	—	2	4	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	—	2	2	no	—
53	C₅×C₄⋊C₈	C₂³	1	2	yes	—
54	C₅×C₄⋊C₈	C₂³	1	4	yes	—
55	C₅×#103	—	2	8	no	—
56	C₅×#97	—	2	4	no	—
57	C₄×C₅×C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	C₂×#500	C₂×C₅	1	1	yes	characteristic
60	C₂×#477	C₂×C₅	1	2	yes	—
61	C₂×#446	C₂×C₅	1	1	yes	characteristic
62	#1696	C₂×C₅	1	8	yes	direct product factor
63	C₂×#334	C₂×C₅	1	1	yes	characteristic
64	C₂×#333	C₂×C₅	1	2	yes	—
65	C₂×C₅×Q₈	—	2	1	no	—
66	C₂×C₅×Q₈	—	2	4	no	—
67	C₂×C₅×Q₈	—	2	4	no	—
68	C₂×C₅×Q₈	—	4	2	no	—
69	C₂×C₅×Q₈	—	8	4	no	—
70	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	2	1	no	—
73	C₂²×C₄×C₅	—	4	2	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	—	4	4	no	—
82	C₂×C₅×C₈	Q₁₆	1	4	yes	—
83	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
85	C₅×C₄⋊C₄	—	2	2	no	—
86	C₅×C₄⋊C₄	—	2	1	no	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	2	4	no	—
90	C₅×C₄⋊C₄	—	4	2	no	—
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	#41	1	1	yes	characteristic
98	C₄²×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
100	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
101	#500	—	2	2	no	—
102	#500	C₂²×C₅	1	4	yes	—
103	#477	—	2	4	no	—
104	#477	C₂²×C₅	1	8	yes	—
105	#446	—	2	2	no	—
106	#446	C₂²×C₅	1	4	yes	—
107	C₂×#107	—	2	1	no	—
108	C₂×#107	—	2	1	no	—
109	C₂×#107	—	2	2	no	—
110	C₂×C₄⋊C₈	—	2	1	no	—
111	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
113	C₂×#103	—	2	2	no	—
114	C₂×#97	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	#334	C₂²×C₅	1	4	yes	—
117	#333	C₂²×C₅	1	8	yes	—
118	C₂³×C₅	#121	1	1	yes	characteristic
119	C₅×Q₈	—	2	4	no	—
120	C₅×Q₈	—	4	2	no	—
121	C₅×Q₈	—	8	4	no	—
122	C₅×Q₈	—	8	4	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	4	2	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	4	no	—
129	C₂×C₄×C₅	—	8	4	no	—
130	C₂×C₄×C₅	#121	1	1	yes	characteristic
131	C₂×C₄×C₅	#121	1	1	yes	characteristic
132	C₂×C₄×C₅	#121	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₅	#137	1	2	yes	—
135	C₂×C₄×C₅	#137	1	4	yes	—
136	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	—	4	4	no	—
142	C₅×C₈	C₂×Q₁₆	1	4	yes	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
148	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	C₅×D₄	1	2	yes	—
156	C₄×Q₈	C₂³×C₅	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
163	#107	—	2	4	no	—
164	#107	—	2	4	no	—
165	#107	—	2	8	no	—
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	2	1	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	C₂³×C₅	1	2	yes	—
170	C₄⋊C₈	C₂³×C₅	1	4	yes	—
171	#103	—	2	8	no	—
172	#97	—	2	4	no	—
173	C₄×C₈	C₅×D₄	1	2	yes	—
174	C₄×C₈	C₂³×C₅	1	2	yes	—
175	C₂²×C₅	#462	1	2	yes	—
176	C₂²×C₅	#470	1	2	yes	—
177	C₂²×C₅	#472	1	2	yes	—
178	C₂²×C₅	C₂×#121	1	1	yes	characteristic
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	4	2	no	—
182	C₄×C₅	—	4	2	no	—
183	C₄×C₅	—	8	4	no	—
184	C₄×C₅	#462	1	2	yes	—
185	C₄×C₅	#470	1	2	yes	—
186	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
187	C₄×C₅	C₂×#137	1	1	yes	characteristic
188	C₄×C₅	C₂×#137	1	2	yes	—
189	C₂×Q₈	—	2	1	no	—
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×Q₈	—	8	4	no	—
194	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂²×C₄	C₅×#41	1	1	yes	characteristic
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₅×Q₁₆	1	4	yes	—
207	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
220	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
221	C₄²	C₅×#41	1	1	yes	characteristic
222	C₄²	C₂⁴×C₅	1	1	yes	characteristic
223	C₂×C₅	#1696	1	4	yes	direct product factor
224	C₂×C₅	C₂×#462	1	1	yes	characteristic
225	C₂×C₅	C₂×#470	1	1	yes	characteristic
226	C₂×C₅	C₂×#472	1	1	yes	characteristic
227	C₂³	C₅×#121	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	8	4	no	—
231	Q₈	—	8	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	C₅×#121	1	1	yes	characteristic
240	C₂×C₄	C₅×#121	1	1	yes	characteristic
241	C₂×C₄	C₅×#121	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
243	C₂×C₄	C₅×#137	1	2	yes	—
244	C₂×C₄	C₅×#137	1	4	yes	—
245	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
247	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	4	no	—
251	C₈	C₂×C₅×Q₁₆	1	4	yes	—
252	C₅	C₂×#1696	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	C₅×#462	1	2	yes	—
254	C₂²	C₅×#470	1	2	yes	—
255	C₂²	C₅×#472	1	2	yes	—
256	C₂²	C₂×C₅×#121	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	8	4	no	—
262	C₄	C₅×#462	1	2	yes	—
263	C₄	C₅×#470	1	2	yes	—
264	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
265	C₄	C₂×C₅×#137	1	1	yes	characteristic
266	C₄	C₂×C₅×#137	1	2	yes	—
267	C₂	C₅×#1696	1	4	yes	direct product factor
268	C₂	C₂×C₅×#462	1	1	yes	characteristic
269	C₂	C₂×C₅×#470	1	1	yes	characteristic
270	C₂	C₂×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1696	C₂	direct (N × H)
26. C₂×#1696	C₅	direct (N × H)
62. #1696	C₂×C₅	direct (N × H)
223. C₂×C₅	#1696	direct (N × H)
252. C₅	C₂×#1696	direct (N × H)
267. C₂	C₅×#1696	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×#500	2
C₂×C₅×#477	2
C₂×C₅×#446	2
C₅×#1696	2
C₂×C₅×#334	2
C₂×C₅×#333	2
C₂×#1696	5

Subgroups — C2×C5×#1696 #11941869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1696 #11941869