Subgroups — C₂²×C₅×#349 #11956055

Order 1280.

1262 conjugacy classes of subgroups.

The subgroup lattice has 250 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	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#349	C₂	1	12	yes	maximal, direct product factor
4	C₂⁴×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂²×C₅×M₁₆	—	2	6	no	—
6	C₂²×C₅×M₁₆	—	2	2	no	—
7	C₂²×C₅×M₁₆	C₄	1	6	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	6	yes	—
9	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×M₃₂	—	2	6	no	—
12	C₂×C₅×M₃₂	C₂²	1	12	yes	complemented
13	C₂²×C₅×C₁₆	—	2	2	no	—
14	C₅×#349	C₂²	1	16	yes	direct product factor
15	C₂²×#349	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵×C₅	C₈	1	1	yes	characteristic
17	C₂³×C₄×C₅	—	2	3	no	—
18	C₂³×C₄×C₅	—	2	1	no	—
19	C₂³×C₄×C₅	—	2	3	no	—
20	C₂³×C₄×C₅	—	2	1	no	—
21	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
22	C₂³×C₄×C₅	C₈	1	6	yes	—
23	C₂³×C₄×C₅	C₂×C₄	1	6	yes	—
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×M₁₆	—	2	12	no	—
26	C₂×C₅×M₁₆	—	2	12	no	—
27	C₂×C₅×M₁₆	—	2	12	no	—
28	C₂×C₅×M₁₆	—	2	12	no	—
29	C₂×C₅×M₁₆	C₂×C₄	1	12	yes	—
30	C₂×C₅×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₅×C₈	—	2	3	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
35	C₂²×C₅×C₈	D₄	1	6	yes	—
36	C₂²×C₅×C₈	D₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	3	yes	—
38	C₅×M₃₂	—	2	12	no	—
39	C₅×M₃₂	C₂³	1	8	yes	complemented
40	C₂×C₅×C₁₆	—	2	12	no	—
41	C₂³×M₁₆	C₂×C₅	1	1	yes	characteristic
42	C₂²×M₃₂	C₂×C₅	1	2	yes	complemented
43	C₂×#349	C₂×C₅	1	12	yes	direct product factor
44	C₂⁴×C₅	—	2	3	no	—
45	C₂⁴×C₅	—	2	1	no	—
46	C₂⁴×C₅	—	4	4	no	—
47	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
48	C₂⁴×C₅	C₂×C₈	1	6	yes	—
49	C₂²×C₄×C₅	—	2	3	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	12	no	—
52	C₂²×C₄×C₅	—	2	6	no	—
53	C₂²×C₄×C₅	—	2	6	no	—
54	C₂²×C₄×C₅	—	2	6	no	—
55	C₂²×C₄×C₅	—	2	6	no	—
56	C₂²×C₄×C₅	—	2	6	no	—
57	C₂²×C₄×C₅	—	2	6	no	—
58	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
61	C₂²×C₄×C₅	C₂×C₈	1	12	yes	—
62	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
63	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
64	C₂²×C₄×C₅	M₁₆	1	3	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
67	C₅×M₁₆	—	2	24	no	—
68	C₅×M₁₆	—	2	8	no	—
69	C₂×C₅×C₈	—	2	6	no	—
70	C₂×C₅×C₈	—	2	6	no	—
71	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
72	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
73	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	6	yes	—
77	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
78	C₅×C₁₆	—	2	8	no	—
79	C₂⁴×C₄	C₄×C₅	1	1	yes	characteristic
80	C₂²×M₁₆	—	2	2	no	—
81	C₂²×M₁₆	—	2	6	no	—
82	C₂²×M₁₆	C₄×C₅	1	6	yes	—
83	C₂²×M₁₆	C₂²×C₅	1	6	yes	—
84	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
85	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×M₃₂	—	2	6	no	—
87	C₂×M₃₂	C₂²×C₅	1	12	yes	complemented
88	C₂²×C₁₆	—	2	2	no	—
89	#349	C₂²×C₅	1	16	yes	direct product factor
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	—	2	6	no	—
92	C₂³×C₅	—	2	6	no	—
93	C₂³×C₅	—	4	12	no	—
94	C₂³×C₅	—	4	12	no	—
95	C₂³×C₅	—	4	4	no	—
96	C₂³×C₅	C₂²⋊C₈	1	3	yes	—
97	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
98	C₂³×C₅	C₂²×C₈	1	3	yes	—
99	C₂³×C₅	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₅	—	2	6	no	—
101	C₂×C₄×C₅	—	2	6	no	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	12	no	—
105	C₂×C₄×C₅	—	2	12	no	—
106	C₂×C₄×C₅	C₂²⋊C₈	1	6	yes	—
107	C₂×C₄×C₅	C₂²⋊C₈	1	3	yes	—
108	C₂×C₄×C₅	C₂²⋊C₈	1	3	yes	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
110	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
113	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
114	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
115	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
116	C₅×C₈	—	2	4	no	—
117	C₅×C₈	C₂×C₂²⋊C₄	1	6	yes	—
118	C₅×C₈	C₂²×D₄	1	2	yes	—
119	C₂⁵	C₅×C₈	1	1	yes	characteristic
120	C₂³×C₄	—	2	3	no	—
121	C₂³×C₄	—	2	1	no	—
122	C₂³×C₄	—	2	3	no	—
123	C₂³×C₄	—	2	1	no	—
124	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
125	C₂³×C₄	C₅×C₈	1	6	yes	—
126	C₂³×C₄	C₂×C₄×C₅	1	6	yes	—
127	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×M₁₆	—	2	12	no	—
129	C₂×M₁₆	—	2	12	no	—
130	C₂×M₁₆	—	2	12	no	—
131	C₂×M₁₆	—	2	12	no	—
132	C₂×M₁₆	C₂×C₄×C₅	1	12	yes	—
133	C₂×M₁₆	C₂³×C₅	1	4	yes	—
134	C₂²×C₈	—	2	3	no	—
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	C₂×C₄×C₅	1	6	yes	—
137	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
138	C₂²×C₈	C₅×D₄	1	2	yes	—
139	C₂²×C₈	C₅×D₄	1	6	yes	—
140	C₂²×C₈	C₂³×C₅	1	3	yes	—
141	M₃₂	—	2	12	no	—
142	M₃₂	C₂³×C₅	1	8	yes	complemented
143	C₂×C₁₆	—	2	12	no	—
144	C₂²×C₅	—	2	6	no	—
145	C₂²×C₅	—	2	6	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	4	4	no	—
148	C₂²×C₅	—	4	12	no	—
149	C₂²×C₅	—	4	12	no	—
150	C₂²×C₅	#349	1	4	yes	direct product factor
151	C₂²×C₅	C₂×C₂²⋊C₈	1	3	yes	—
152	C₂²×C₅	C₂×C₂²⋊C₈	1	3	yes	—
153	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	C₂×C₂²⋊C₈	1	3	yes	—
156	C₄×C₅	C₂×C₂²⋊C₈	1	3	yes	—
157	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
159	C₂⁴	—	2	3	no	—
160	C₂⁴	—	2	1	no	—
161	C₂⁴	—	4	4	no	—
162	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
163	C₂⁴	C₂×C₅×C₈	1	6	yes	—
164	C₂²×C₄	—	2	3	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	2	6	no	—
169	C₂²×C₄	—	2	6	no	—
170	C₂²×C₄	—	2	12	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
174	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₅×C₈	1	6	yes	—
176	C₂²×C₄	C₂×C₅×C₈	1	12	yes	—
177	C₂²×C₄	C₂×C₅×C₈	1	3	yes	—
178	C₂²×C₄	C₅×M₁₆	1	3	yes	—
179	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
181	C₂²×C₄	C₂²×C₄×C₅	1	3	yes	—
182	M₁₆	—	2	24	no	—
183	M₁₆	—	2	8	no	—
184	C₂×C₈	—	2	6	no	—
185	C₂×C₈	—	2	6	no	—
186	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
187	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
188	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
189	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
190	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
191	C₂×C₈	C₂×C₅×D₄	1	6	yes	—
192	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
193	C₁₆	—	2	8	no	—
194	C₂×C₅	—	2	1	no	—
195	C₂×C₅	—	2	3	no	—
196	C₂×C₅	—	4	4	no	—
197	C₂×C₅	C₂×#349	1	6	yes	direct product factor
198	C₂×C₅	C₂²×C₂²⋊C₈	1	1	yes	characteristic
199	C₂³	—	2	4	no	—
200	C₂³	—	2	6	no	—
201	C₂³	—	2	6	no	—
202	C₂³	—	4	12	no	—
203	C₂³	—	4	12	no	—
204	C₂³	—	4	4	no	—
205	C₂³	C₅×C₂²⋊C₈	1	3	yes	—
206	C₂³	C₅×C₂²⋊C₈	1	1	yes	characteristic
207	C₂³	C₂²×C₅×C₈	1	3	yes	—
208	C₂³	C₂²×C₅×C₈	1	4	yes	—
209	C₂×C₄	—	2	6	no	—
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	2	12	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	C₅×C₂²⋊C₈	1	6	yes	—
216	C₂×C₄	C₅×C₂²⋊C₈	1	3	yes	—
217	C₂×C₄	C₅×C₂²⋊C₈	1	3	yes	—
218	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
219	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
220	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
221	C₂×C₄	C₂²×C₅×C₈	1	3	yes	—
222	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
223	C₂×C₄	C₂×C₅×M₁₆	1	3	yes	—
224	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
225	C₈	—	2	4	no	—
226	C₈	C₂×C₅×C₂²⋊C₄	1	6	yes	—
227	C₈	C₂²×C₅×D₄	1	2	yes	—
228	C₅	C₂²×#349	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	2	6	no	—
230	C₂²	—	2	6	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	4	4	no	—
233	C₂²	—	4	12	no	—
234	C₂²	—	4	12	no	—
235	C₂²	C₅×#349	1	4	yes	direct product factor
236	C₂²	C₂×C₅×C₂²⋊C₈	1	3	yes	—
237	C₂²	C₂×C₅×C₂²⋊C₈	1	3	yes	—
238	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	C₂×C₅×C₂²⋊C₈	1	3	yes	—
241	C₄	C₂×C₅×C₂²⋊C₈	1	3	yes	—
242	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
243	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
244	C₂	—	2	3	no	—
245	C₂	—	2	1	no	—
246	C₂	—	4	4	no	—
247	C₂	C₂×C₅×#349	1	6	yes	direct product factor
248	C₂	C₂²×C₅×C₂²⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂⁵×C₅	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×M₃₂	C₂	semidirect
3. C₂×C₅×#349	C₂	direct (N × H)
12. C₂×C₅×M₃₂	C₂²	semidirect
14. C₅×#349	C₂²	direct (N × H)
15. C₂²×#349	C₅	direct (N × H)
39. C₅×M₃₂	C₂³	semidirect
42. C₂²×M₃₂	C₂×C₅	semidirect
43. C₂×#349	C₂×C₅	direct (N × H)
87. C₂×M₃₂	C₂²×C₅	semidirect
89. #349	C₂²×C₅	direct (N × H)
142. M₃₂	C₂³×C₅	semidirect
150. C₂²×C₅	#349	direct (N × H)
197. C₂×C₅	C₂×#349	direct (N × H)
228. C₅	C₂²×#349	direct (N × H)
235. C₂²	C₅×#349	direct (N × H)
247. C₂	C₂×C₅×#349	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
C₂²×C₅×M₃₂	2
C₂×C₅×#349	2
C₂²×#349	5

Subgroups — C22×C5×#349 #11956055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#349 #11956055