Subgroups — C₂×#10598203 #11989713

Order 1280.

658 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	#10598203	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₅×M₁₆	C₄	1	2	yes	—
5	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂²×#681	C₄	1	4	yes	—
7	C₅×#423	—	2	2	no	—
8	C₅×#423	C₄	1	2	yes	—
9	C₅×#423	C₂²	1	2	yes	—
10	C₂×C₅×C₄⋊C₈	—	2	2	no	—
11	#65213	—	2	2	no	—
12	#65213	—	2	2	no	—
13	#65213	C₄	1	4	yes	—
14	#65213	C₂²	1	4	yes	—
15	C₂×C₄⋊#168	—	2	2	no	—
16	C₂×C₄⋊#168	—	2	2	no	—
17	C₂×#1309	—	5	1	no	maximal, Sylow
18	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×M₁₆	—	2	2	no	—
21	C₂×C₅×M₁₆	—	4	4	no	—
22	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₅×C₈	—	2	2	no	—
24	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×#681	—	2	4	no	—
34	C₂×#681	—	4	8	no	—
35	C₂×#681	C₂×C₄	1	8	yes	—
36	C₂²×#168	—	2	4	no	—
37	C₅×C₄⋊C₈	—	2	4	no	—
38	C₅×C₄⋊C₈	—	2	4	no	—
39	C₄⋊#168	—	2	4	no	—
40	C₄⋊#168	—	2	4	no	—
41	C₄⋊#168	—	2	4	no	—
42	C₄⋊#168	—	2	4	no	—
43	C₂×#423	—	5	2	no	—
44	C₂×#423	D₅	1	1	yes	characteristic
45	#1309	—	5	4	no	—
46	C₂⁴×C₅	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₅	—	2	2	no	—
48	C₂²×C₄×C₅	—	2	4	no	—
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	—	2	2	no	—
51	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₄²	1	2	yes	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₅×M₁₆	—	4	4	no	—
64	C₅×M₁₆	—	4	4	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	4	4	no	—
67	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
73	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅	C₂×D₄	1	2	yes	—
76	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
77	#681	—	4	8	no	—
78	#681	—	4	8	no	—
79	C₂×#168	—	2	8	no	—
80	C₂×#168	—	4	8	no	—
81	C₂²×M₁₆	—	5	4	no	—
82	C₂²×M₁₆	Dic₅	1	2	yes	—
83	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
84	#423	—	2	2	no	—
85	#423	—	5	4	no	—
86	#423	—	5	4	no	—
87	#423	—	10	2	no	—
88	#423	—	10	2	no	—
89	#423	Dic₅	1	2	yes	—
90	#423	C₂×D₅	1	2	yes	—
91	C₂×C₄⋊C₈	—	2	2	no	—
92	C₂×C₄⋊C₈	—	10	2	no	—
93	C₂×C₄⋊C₈	—	10	2	no	—
94	C₂³×C₅	—	2	4	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	#95	1	1	yes	characteristic
97	C₂³×C₅	#95	1	1	yes	characteristic
98	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	4	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	1	no	—
108	C₂×C₄×C₅	—	2	1	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	1	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	#95	1	1	yes	characteristic
114	C₂×C₄×C₅	#95	1	2	yes	—
115	C₂×C₄×C₅	#95	1	2	yes	—
116	C₂×C₄×C₅	#95	1	1	yes	characteristic
117	C₂×C₄×C₅	#100	1	4	yes	—
118	C₂×C₄×C₅	#100	1	2	yes	—
119	C₂×C₄×C₅	#101	1	4	yes	—
120	C₂×C₄×C₅	#101	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₅×C₈	—	4	4	no	—
126	#168	—	4	8	no	—
127	C₂³×C₄	C₄×D₅	1	2	yes	—
128	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	—	4	4	no	—
131	C₂×M₁₆	—	5	8	no	—
132	C₂×M₁₆	—	10	4	no	—
133	C₂×M₁₆	—	20	8	no	—
134	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	10	4	no	—
137	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
138	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
139	C₂×C₄²	C₄×D₅	1	2	yes	—
140	C₂×C₄²	D₂₀	1	1	yes	characteristic
141	C₂×C₄²	D₂₀	1	1	yes	characteristic
142	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
143	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
144	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
145	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	2	4	no	—
148	C₄⋊C₈	—	10	4	no	—
149	C₄⋊C₈	—	10	4	no	—
150	C₄⋊C₈	—	10	4	no	—
151	C₄⋊C₈	—	10	4	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	4	4	no	—
157	C₂²×C₅	#343	1	4	yes	—
158	C₂²×C₅	#343	1	2	yes	—
159	C₂²×C₅	C₂×#95	1	1	yes	characteristic
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	2	2	no	—
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	#343	1	4	yes	—
165	C₄×C₅	#343	1	2	yes	—
166	C₄×C₅	C₂×#100	1	2	yes	—
167	C₄×C₅	C₂×#100	1	1	yes	characteristic
168	C₄×C₅	C₂×#101	1	2	yes	—
169	C₄×C₅	C₂×#101	1	1	yes	characteristic
170	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
176	C₂²×C₄	C₄×Dic₅	1	2	yes	—
177	C₂²×C₄	#683	1	2	yes	—
178	C₂²×C₄	#683	1	2	yes	—
179	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
181	C₂²×C₄	#685	1	2	yes	—
182	C₂²×C₄	#685	1	2	yes	—
183	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
187	M₁₆	—	4	4	no	—
188	M₁₆	—	4	4	no	—
189	M₁₆	—	20	8	no	—
190	M₁₆	—	20	8	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	4	no	—
193	C₂×C₈	—	10	8	no	—
194	C₂×C₈	—	20	8	no	—
195	C₄²	#683	1	2	yes	—
196	C₄²	#683	1	2	yes	—
197	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
198	C₄²	C₄⋊Dic₅	1	1	yes	characteristic
199	C₄²	#685	1	2	yes	—
200	C₄²	#685	1	2	yes	—
201	C₄²	C₂²⋊Dic₅	1	2	yes	—
202	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
203	C₄²	C₂×D₂₀	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
205	C₂×C₅	—	4	1	no	—
206	C₂×C₅	—	4	1	no	—
207	C₂×C₅	#1309	1	4	yes	complemented
208	C₂×C₅	C₂×#343	1	2	yes	—
209	C₂×C₅	C₂×#343	1	1	yes	characteristic
210	C₂³	—	2	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	#3963	1	1	yes	characteristic
213	C₂³	#3963	1	1	yes	characteristic
214	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	#3955	1	4	yes	—
230	C₂×C₄	#3956	1	4	yes	—
231	C₂×C₄	#3963	1	1	yes	characteristic
232	C₂×C₄	#3963	1	2	yes	—
233	C₂×C₄	#3963	1	2	yes	—
234	C₂×C₄	#3963	1	1	yes	characteristic
235	C₂×C₄	#3965	1	2	yes	—
236	C₂×C₄	#3968	1	2	yes	—
237	C₂×C₄	C₂×#683	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
239	C₂×C₄	C₂×#685	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
241	C₈	—	4	4	no	—
242	C₈	—	20	8	no	—
243	C₅	C₂×#1309	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	4	no	—
249	C₂²	#64750	1	2	yes	—
250	C₂²	#64774	1	4	yes	—
251	C₂²	C₂×#3963	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	#64750	1	2	yes	—
257	C₄	#64774	1	4	yes	—
258	C₄	C₂×#3955	1	2	yes	—
259	C₄	C₂×#3956	1	2	yes	—
260	C₄	C₂×#3965	1	1	yes	characteristic
261	C₄	C₂×#3968	1	1	yes	characteristic
262	C₂	—	4	1	no	—
263	C₂	—	4	1	no	—
264	C₂	#10598203	1	4	yes	direct product factor
265	C₂	C₂×#64750	1	1	yes	characteristic
266	C₂	C₂×#64774	1	2	yes	—

	subgroup	G/H
center	C₂³	#3963
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₂×C₅×#423	C₂
socle	C₂³×C₅	#95
prime radical	C₂⁴×C₅	C₄²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₂×#423	D₅
hypocenter	C₅	C₂×#1309
LCS	G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #10598203	C₂	direct (N × H)
207. C₂×C₅	#1309	semidirect
243. C₅	C₂×#1309	semidirect
264. C₂	#10598203	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×#423	2
C₂×#65213	2
#10598203	2
C₂×#1309	5

Subgroups — C2×#10598203 #11989713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598203 #11989713