Subgroups — C₂×C₅×#1694 #11941868

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 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×#462	C₂	1	2	yes	maximal
3	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
4	C₅×#1694	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#332	C₂	1	2	yes	maximal
7	C₂²×C₅×Q₁₆	—	2	2	no	—
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₅×#500	—	2	2	no	—
11	C₅×#500	C₂²	1	4	yes	—
12	C₅×#462	—	2	4	no	—
13	C₅×#462	C₂²	1	8	yes	—
14	C₅×#446	—	2	2	no	—
15	C₅×#446	C₂²	1	4	yes	—
16	C₂×C₅×#107	—	2	1	no	—
17	C₂×C₅×#107	—	2	1	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×#332	C₂²	1	8	yes	—
26	C₂×#1694	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	C₂×C₅×Q₁₆	—	2	8	no	—
30	C₂×C₅×Q₁₆	—	4	8	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₅×C₄⋊Q₈	—	2	2	no	—
36	C₅×C₄⋊Q₈	—	2	2	no	—
37	C₅×C₄⋊Q₈	—	2	2	no	—
38	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₄×C₅×Q₈	—	2	1	no	—
40	C₄×C₅×Q₈	—	2	1	no	—
41	C₄×C₅×Q₈	D₄	1	2	yes	—
42	C₄×C₅×Q₈	C₂³	1	2	yes	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	—	2	1	no	—
48	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×#107	—	2	4	no	—
50	C₅×#107	—	2	4	no	—
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×C₄⋊C₈	—	2	1	no	—
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×C₄⋊C₈	C₂³	1	2	yes	—
55	C₅×C₄⋊C₈	C₂³	1	4	yes	—
56	C₅×#103	—	2	8	no	—
57	C₅×#97	—	2	4	no	—
58	C₄×C₅×C₈	D₄	1	2	yes	—
59	C₄×C₅×C₈	C₂³	1	2	yes	—
60	C₂×#500	C₂×C₅	1	1	yes	characteristic
61	C₂×#462	C₂×C₅	1	2	yes	—
62	C₂×#446	C₂×C₅	1	1	yes	characteristic
63	#1694	C₂×C₅	1	8	yes	direct product factor
64	C₂×#334	C₂×C₅	1	1	yes	characteristic
65	C₂×#332	C₂×C₅	1	2	yes	—
66	C₂×C₅×Q₈	—	2	1	no	—
67	C₂×C₅×Q₈	—	2	4	no	—
68	C₂×C₅×Q₈	—	2	4	no	—
69	C₂×C₅×Q₈	—	4	2	no	—
70	C₂×C₅×Q₈	—	4	4	no	—
71	C₂×C₅×Q₈	—	4	4	no	—
72	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	#41	1	1	yes	characteristic
79	C₅×Q₁₆	—	4	16	no	—
80	C₂×C₅×C₈	—	2	2	no	—
81	C₂×C₅×C₈	—	2	4	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	4	4	no	—
84	C₂×C₅×C₈	—	4	2	no	—
85	C₂×C₅×C₈	Q₁₆	1	4	yes	complemented
86	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	1	no	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	4	2	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₅	#41	1	1	yes	characteristic
101	C₄²×C₅	C₂⁴	1	1	yes	characteristic
102	C₂²×Q₁₆	—	2	2	no	—
103	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
104	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
105	#500	—	2	2	no	—
106	#500	C₂²×C₅	1	4	yes	—
107	#462	—	2	4	no	—
108	#462	C₂²×C₅	1	8	yes	—
109	#446	—	2	2	no	—
110	#446	C₂²×C₅	1	4	yes	—
111	C₂×#107	—	2	1	no	—
112	C₂×#107	—	2	1	no	—
113	C₂×C₄⋊C₈	—	2	1	no	—
114	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
116	C₂×#103	—	2	2	no	—
117	C₂×#97	—	2	1	no	—
118	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
119	#334	C₂²×C₅	1	4	yes	—
120	#332	C₂²×C₅	1	8	yes	—
121	C₂³×C₅	#121	1	1	yes	characteristic
122	C₅×Q₈	—	2	4	no	—
123	C₅×Q₈	—	4	2	no	—
124	C₅×Q₈	—	4	4	no	—
125	C₅×Q₈	—	4	4	no	—
126	C₅×Q₈	—	8	4	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	4	2	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	8	4	no	—
134	C₂×C₄×C₅	#121	1	1	yes	characteristic
135	C₂×C₄×C₅	#121	1	1	yes	characteristic
136	C₂×C₄×C₅	#121	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
139	C₂×C₄×C₅	#137	1	2	yes	—
140	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	4	4	no	—
145	C₅×C₈	—	4	2	no	—
146	C₅×C₈	C₂×Q₁₆	1	4	yes	complemented
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
149	C₂×Q₁₆	—	2	8	no	—
150	C₂×Q₁₆	—	4	8	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
154	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
155	C₄⋊Q₈	—	2	2	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
159	C₄×Q₈	—	2	1	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₅×D₄	1	2	yes	—
162	C₄×Q₈	C₂³×C₅	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	2	no	—
166	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
169	#107	—	2	4	no	—
170	#107	—	2	4	no	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	C₂³×C₅	1	2	yes	—
175	C₄⋊C₈	C₂³×C₅	1	4	yes	—
176	#103	—	2	8	no	—
177	#97	—	2	4	no	—
178	C₄×C₈	C₅×D₄	1	2	yes	—
179	C₄×C₈	C₂³×C₅	1	2	yes	—
180	C₂²×C₅	#462	1	2	yes	—
181	C₂²×C₅	#469	1	2	yes	—
182	C₂²×C₅	#473	1	2	yes	—
183	C₂²×C₅	C₂×#121	1	1	yes	characteristic
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	4	2	no	—
187	C₄×C₅	—	4	2	no	—
188	C₄×C₅	—	8	4	no	—
189	C₄×C₅	#462	1	2	yes	—
190	C₄×C₅	#469	1	2	yes	—
191	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
192	C₄×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
193	C₄×C₅	C₂×#137	1	1	yes	characteristic
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	—	4	4	no	—
200	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₅×#41	1	1	yes	characteristic
207	Q₁₆	—	4	16	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	C₅×Q₁₆	1	4	yes	complemented
214	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
215	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	1	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
227	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
228	C₄²	C₅×#41	1	1	yes	characteristic
229	C₄²	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×C₅	#1694	1	4	yes	direct product factor
231	C₂×C₅	C₂×#462	1	1	yes	characteristic
232	C₂×C₅	C₂×#469	1	1	yes	characteristic
233	C₂×C₅	C₂×#473	1	1	yes	characteristic
234	C₂³	C₅×#121	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	4	2	no	—
237	Q₈	—	4	4	no	—
238	Q₈	—	4	4	no	—
239	Q₈	—	8	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₅×#121	1	1	yes	characteristic
248	C₂×C₄	C₅×#121	1	1	yes	characteristic
249	C₂×C₄	C₅×#121	1	1	yes	characteristic
250	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
251	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
252	C₂×C₄	C₅×#137	1	2	yes	—
253	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	2	no	—
259	C₈	C₂×C₅×Q₁₆	1	4	yes	complemented
260	C₅	C₂×#1694	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²	C₅×#462	1	2	yes	—
262	C₂²	C₅×#469	1	2	yes	—
263	C₂²	C₅×#473	1	2	yes	—
264	C₂²	C₂×C₅×#121	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₅×#462	1	2	yes	—
271	C₄	C₅×#469	1	2	yes	—
272	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
273	C₄	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
274	C₄	C₂×C₅×#137	1	1	yes	characteristic
275	C₂	C₅×#1694	1	4	yes	direct product factor
276	C₂	C₂×C₅×#462	1	1	yes	characteristic
277	C₂	C₂×C₅×#469	1	1	yes	characteristic
278	C₂	C₂×C₅×#473	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×#1694	C₂	direct (N × H)
26. C₂×#1694	C₅	direct (N × H)
63. #1694	C₂×C₅	direct (N × H)
85. C₂×C₅×C₈	Q₁₆	semidirect
146. C₅×C₈	C₂×Q₁₆	semidirect
213. C₂×C₈	C₅×Q₁₆	semidirect
230. C₂×C₅	#1694	direct (N × H)
259. C₈	C₂×C₅×Q₁₆	semidirect
260. C₅	C₂×#1694	direct (N × H)
275. C₂	C₅×#1694	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×#462	2
C₂×C₅×#446	2
C₅×#1694	2
C₂×C₅×#334	2
C₂×C₅×#332	2
C₂×#1694	5

Subgroups — C2×C5×#1694 #11941868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1694 #11941868