Subgroups — C₂²×C₅×#341 #11943069

Order 1280.

2238 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×M₁₆	C₂	1	2	yes	maximal
2	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#341	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
5	C₂²×C₅×M₁₆	—	2	8	no	—
6	C₂²×C₅×M₁₆	—	2	6	no	—
7	C₂²×C₅×M₁₆	C₄	1	16	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	12	yes	—
9	C₂³×C₅×C₈	—	2	8	no	—
10	C₂³×C₅×C₈	—	2	2	no	—
11	C₂³×C₅×C₈	C₄	1	2	yes	—
12	C₂³×C₅×C₈	C₄	1	6	yes	—
13	C₂³×C₅×C₈	C₂²	1	6	yes	—
14	C₅×#341	C₂²	1	16	yes	direct product factor
15	C₂²×#341	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵×C₅	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₅	C₂×C₄	1	3	yes	—
19	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₅	D₄	1	8	yes	—
21	C₂³×C₄×C₅	D₄	1	3	yes	—
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	Q₈	1	3	yes	—
24	C₂³×C₄×C₅	C₂³	1	3	yes	—
25	C₂×C₅×M₁₆	—	2	48	no	—
26	C₂×C₅×M₁₆	—	2	24	no	—
27	C₂×C₅×M₁₆	—	2	12	no	—
28	C₂×C₅×M₁₆	C₂×C₄	1	48	yes	—
29	C₂×C₅×M₁₆	C₂³	1	8	yes	—
30	C₂²×C₅×C₈	—	2	24	no	—
31	C₂²×C₅×C₈	—	2	8	no	—
32	C₂²×C₅×C₈	—	2	16	no	—
33	C₂²×C₅×C₈	—	2	24	no	—
34	C₂²×C₅×C₈	—	2	12	no	—
35	C₂²×C₅×C₈	C₂×C₄	1	12	yes	—
36	C₂²×C₅×C₈	C₂×C₄	1	12	yes	—
37	C₂²×C₅×C₈	C₂³	1	4	yes	—
38	C₂³×M₁₆	C₂×C₅	1	2	yes	—
39	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic
40	C₂×#341	C₂×C₅	1	12	yes	direct product factor
41	C₂⁴×C₅	—	2	8	no	—
42	C₂⁴×C₅	C₄⋊C₄	1	8	yes	—
43	C₂⁴×C₅	C₄⋊C₄	1	3	yes	—
44	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
45	C₂⁴×C₅	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₅	C₄²	1	12	yes	—
47	C₂²×C₄×C₅	C₄²	1	4	yes	—
48	C₂²×C₄×C₅	C₂²⋊C₄	1	12	yes	—
49	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅	C₂²⋊C₄	1	8	yes	—
51	C₂²×C₄×C₅	C₂²⋊C₄	1	24	yes	—
52	C₂²×C₄×C₅	C₄⋊C₄	1	6	yes	—
53	C₂²×C₄×C₅	C₄⋊C₄	1	24	yes	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	3	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₅×M₁₆	—	2	48	no	—
65	C₅×M₁₆	—	2	8	no	—
66	C₅×M₁₆	C₂²×C₄	1	16	yes	—
67	C₂×C₅×C₈	—	2	24	no	—
68	C₂×C₅×C₈	—	2	24	no	—
69	C₂×C₅×C₈	—	2	48	no	—
70	C₂×C₅×C₈	—	2	8	no	—
71	C₂×C₅×C₈	—	2	8	no	—
72	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
73	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
74	C₂²×M₁₆	—	2	8	no	—
75	C₂²×M₁₆	—	2	6	no	—
76	C₂²×M₁₆	C₄×C₅	1	16	yes	—
77	C₂²×M₁₆	C₂²×C₅	1	12	yes	—
78	C₂³×C₈	—	2	8	no	—
79	C₂³×C₈	—	2	2	no	—
80	C₂³×C₈	C₄×C₅	1	6	yes	—
81	C₂³×C₈	C₄×C₅	1	2	yes	—
82	C₂³×C₈	C₂²×C₅	1	6	yes	—
83	#341	C₂²×C₅	1	16	yes	direct product factor
84	C₂³×C₅	—	2	32	no	—
85	C₂³×C₅	—	2	12	no	—
86	C₂³×C₅	—	2	12	no	—
87	C₂³×C₅	#95	1	12	yes	—
88	C₂³×C₅	#95	1	4	yes	—
89	C₂³×C₅	#108	1	8	yes	—
90	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
91	C₂³×C₅	C₂×C₄⋊C₄	1	12	yes	—
92	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
93	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅	#95	1	24	yes	—
95	C₂×C₄×C₅	#95	1	12	yes	—
96	C₂×C₄×C₅	#95	1	12	yes	—
97	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
98	C₂×C₄×C₅	C₂×C₄²	1	6	yes	—
99	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
100	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
101	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
102	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	12	yes	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
105	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
106	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
110	C₅×C₈	—	2	8	no	—
111	C₅×C₈	—	2	16	no	—
112	C₂⁵	C₅×Q₈	1	1	yes	characteristic
113	C₂³×C₄	C₂×C₄×C₅	1	3	yes	—
114	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂³×C₄	C₂×C₄×C₅	1	8	yes	—
116	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
117	C₂³×C₄	C₅×D₄	1	8	yes	—
118	C₂³×C₄	C₅×D₄	1	3	yes	—
119	C₂³×C₄	C₅×Q₈	1	3	yes	—
120	C₂³×C₄	C₂³×C₅	1	3	yes	—
121	C₂×M₁₆	—	2	48	no	—
122	C₂×M₁₆	—	2	24	no	—
123	C₂×M₁₆	—	2	12	no	—
124	C₂×M₁₆	C₂×C₄×C₅	1	48	yes	—
125	C₂×M₁₆	C₂³×C₅	1	8	yes	—
126	C₂²×C₈	—	2	24	no	—
127	C₂²×C₈	—	2	8	no	—
128	C₂²×C₈	—	2	16	no	—
129	C₂²×C₈	—	2	24	no	—
130	C₂²×C₈	—	2	12	no	—
131	C₂²×C₈	C₂×C₄×C₅	1	12	yes	—
132	C₂²×C₈	C₂×C₄×C₅	1	12	yes	—
133	C₂²×C₈	C₂³×C₅	1	4	yes	—
134	C₂²×C₅	—	2	6	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	2	24	no	—
137	C₂²×C₅	—	2	24	no	—
138	C₂²×C₅	#341	1	16	yes	direct product factor
139	C₂²×C₅	C₂×#95	1	6	yes	—
140	C₂²×C₅	C₂×#95	1	6	yes	—
141	C₂²×C₅	C₂×#108	1	12	yes	—
142	C₂²×C₅	C₂²×C₄⋊C₄	1	2	yes	—
143	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₅	C₂×#95	1	6	yes	—
145	C₄×C₅	C₂×#95	1	6	yes	—
146	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₄×C₅	C₂²×C₂²⋊C₄	1	2	yes	—
149	C₂⁴	—	2	8	no	—
150	C₂⁴	C₅×C₄⋊C₄	1	8	yes	—
151	C₂⁴	C₅×C₄⋊C₄	1	3	yes	—
152	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂⁴	C₂×C₅×Q₈	1	3	yes	—
154	C₂²×C₄	C₄²×C₅	1	12	yes	—
155	C₂²×C₄	C₄²×C₅	1	4	yes	—
156	C₂²×C₄	C₅×C₂²⋊C₄	1	12	yes	—
157	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
158	C₂²×C₄	C₅×C₂²⋊C₄	1	8	yes	—
159	C₂²×C₄	C₅×C₂²⋊C₄	1	24	yes	—
160	C₂²×C₄	C₅×C₄⋊C₄	1	6	yes	—
161	C₂²×C₄	C₅×C₄⋊C₄	1	24	yes	—
162	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
163	C₂²×C₄	C₅×C₄⋊C₄	1	3	yes	—
164	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
165	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
166	C₂²×C₄	C₂²×C₄×C₅	1	12	yes	—
167	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
168	C₂²×C₄	C₂×C₅×D₄	1	12	yes	—
169	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
170	C₂²×C₄	C₂×C₅×Q₈	1	3	yes	—
171	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
172	M₁₆	—	2	48	no	—
173	M₁₆	—	2	8	no	—
174	M₁₆	C₂²×C₄×C₅	1	16	yes	—
175	C₂×C₈	—	2	24	no	—
176	C₂×C₈	—	2	24	no	—
177	C₂×C₈	—	2	48	no	—
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
181	C₂×C₅	—	2	2	no	—
182	C₂×C₅	—	2	6	no	—
183	C₂×C₅	C₂×#341	1	12	yes	direct product factor
184	C₂×C₅	C₂²×#95	1	1	yes	characteristic
185	C₂×C₅	C₂²×#108	1	2	yes	—
186	C₂³	—	2	32	no	—
187	C₂³	—	2	12	no	—
188	C₂³	—	2	12	no	—
189	C₂³	C₅×#95	1	12	yes	—
190	C₂³	C₅×#95	1	4	yes	—
191	C₂³	C₅×#108	1	8	yes	—
192	C₂³	C₂×C₅×C₄⋊C₄	1	3	yes	—
193	C₂³	C₂×C₅×C₄⋊C₄	1	12	yes	—
194	C₂³	C₂×C₅×C₄⋊C₄	1	3	yes	—
195	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
196	C₂×C₄	C₅×#95	1	24	yes	—
197	C₂×C₄	C₅×#95	1	12	yes	—
198	C₂×C₄	C₅×#95	1	12	yes	—
199	C₂×C₄	C₂×C₄²×C₅	1	6	yes	—
200	C₂×C₄	C₂×C₄²×C₅	1	6	yes	—
201	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
202	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
203	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
204	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	12	yes	—
205	C₂×C₄	C₂×C₅×C₄⋊C₄	1	12	yes	—
206	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
207	C₂×C₄	C₂×C₅×C₄⋊C₄	1	3	yes	—
208	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
209	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
210	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
212	C₈	—	2	8	no	—
213	C₈	—	2	16	no	—
214	C₅	C₂²×#341	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	2	6	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	24	no	—
218	C₂²	—	2	24	no	—
219	C₂²	C₅×#341	1	16	yes	direct product factor
220	C₂²	C₂×C₅×#95	1	6	yes	—
221	C₂²	C₂×C₅×#95	1	6	yes	—
222	C₂²	C₂×C₅×#108	1	12	yes	—
223	C₂²	C₂²×C₅×C₄⋊C₄	1	2	yes	—
224	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
225	C₄	C₂×C₅×#95	1	6	yes	—
226	C₄	C₂×C₅×#95	1	6	yes	—
227	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
228	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
229	C₄	C₂²×C₅×C₂²⋊C₄	1	2	yes	—
230	C₂	—	2	6	no	—
231	C₂	—	2	2	no	—
232	C₂	C₂×C₅×#341	1	12	yes	direct product factor
233	C₂	C₂²×C₅×#95	1	1	yes	characteristic
234	C₂	C₂²×C₅×#108	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	D₄
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂⁴×C₅	C₄⋊C₄
prime radical	C₂⁵×C₅	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂⁴×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#341	C₂	direct (N × H)
14. C₅×#341	C₂²	direct (N × H)
15. C₂²×#341	C₅	direct (N × H)
40. C₂×#341	C₂×C₅	direct (N × H)
83. #341	C₂²×C₅	direct (N × H)
138. C₂²×C₅	#341	direct (N × H)
183. C₂×C₅	C₂×#341	direct (N × H)
214. C₅	C₂²×#341	direct (N × H)
219. C₂²	C₅×#341	direct (N × H)
232. C₂	C₂×C₅×#341	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×M₁₆	2
C₂⁴×C₅×C₈	2
C₂×C₅×#341	2
C₂²×#341	5

Subgroups — C22×C5×#341 #11943069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#341 #11943069