Subgroups — C₂×#10600093 #12252527

Order 1280.

662 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×#65196	C₂	1	2	yes	maximal, complemented
2	C₂×#64989	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#64735	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600093	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4050	C₄	1	1	yes	characteristic
8	C₂×#4047	C₂²	1	2	yes	—
9	C₂×C₅×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3951	C₄	1	1	yes	characteristic
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₂²×#672	C₂²	1	1	yes	characteristic
14	C₂×#3937	C₄	1	2	yes	—
15	#65196	—	2	4	no	—
16	#65196	C₂²	1	8	yes	complemented
17	C₂×#3930	—	2	2	no	—
18	C₂×D₅×C₁₆	—	2	2	no	—
19	#64989	C₄	1	4	yes	—
20	#64989	C₂²	1	4	yes	—
21	C₅×#364	C₂²	1	4	yes	complemented
22	#64735	C₂²	1	8	yes	complemented
23	C₄×#672	C₂²	1	4	yes	complemented
24	C₂×#2170	—	5	1	no	maximal, Sylow
25	C₂×#709	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×#681	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₀	C₈	1	1	yes	characteristic
31	#4050	C₈	1	4	yes	—
32	#4050	C₂×C₄	1	4	yes	—
33	C₂×D₄₀	C₈	1	1	yes	characteristic
34	C₂×#677	C₈	1	2	yes	—
35	#4047	—	2	2	no	—
36	#4047	—	2	2	no	—
37	#4047	C₂×C₄	1	4	yes	—
38	#4047	C₂³	1	4	yes	—
39	C₂×#676	—	2	2	no	—
40	C₂×C₈×D₅	—	2	2	no	—
41	C₅×M₃₂	—	2	2	no	—
42	C₅×M₃₂	C₂³	1	4	yes	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	C₂³	1	2	yes	—
46	#3951	C₂×C₄	1	4	yes	—
47	C₂×#672	—	2	1	no	—
48	C₂×#672	—	2	1	no	—
49	C₂×#672	D₄	1	1	yes	characteristic
50	C₂×#672	D₄	1	1	yes	characteristic
51	C₂×#672	D₄	1	2	yes	—
52	C₂×#672	C₂³	1	2	yes	—
53	C₂×#672	C₂³	1	2	yes	—
54	#3937	C₂×C₄	1	8	yes	—
55	#3930	—	2	8	no	—
56	D₅×C₁₆	—	2	8	no	—
57	C₂×#504	—	5	2	no	—
58	C₂×#443	—	5	1	no	—
59	#2170	—	5	8	no	—
60	C₂×#364	D₅	1	1	yes	characteristic, complemented
61	C₂×#350	—	5	2	no	—
62	C₂×C₄×C₁₆	—	5	1	no	—
63	C₂²×C₄×C₅	—	2	1	no	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₅	—	2	2	no	—
66	#709	—	2	2	no	—
67	#709	—	2	2	no	—
68	#709	C₂×C₈	1	4	yes	—
69	#709	C₂²×C₄	1	4	yes	—
70	C₂×D₂₀	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₅	—	2	2	no	—
72	C₂×Dic₁₀	C₂×C₈	1	2	yes	—
73	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
74	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
76	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	#41	1	1	yes	characteristic
81	C₂×C₅×C₈	#41	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
83	C₄²×C₅	—	2	1	no	—
84	C₄²×C₅	C₂²×C₄	1	2	yes	—
85	#681	—	2	2	no	—
86	#681	C₂²×C₄	1	4	yes	—
87	C₂×#168	—	2	2	no	—
88	Dic₂₀	C₂×C₈	1	4	yes	—
89	D₄₀	C₂×C₈	1	4	yes	—
90	#677	C₂×C₈	1	8	yes	—
91	#676	—	2	4	no	—
92	#676	—	2	4	no	—
93	C₈×D₅	—	2	4	no	—
94	C₈×D₅	—	2	4	no	—
95	C₅×C₁₆	—	2	4	no	—
96	#672	—	2	2	no	—
97	#672	C₂×D₄	1	2	yes	—
98	#672	C₂×D₄	1	2	yes	—
99	C₂×#135	—	5	1	no	—
100	C₂×#131	—	5	2	no	—
101	#504	—	5	8	no	—
102	#504	—	10	4	no	—
103	C₂×M₃₂	—	10	2	no	—
104	C₂×M₃₂	C₂×D₅	1	2	yes	—
105	C₂²×C₁₆	—	5	1	no	—
106	C₂²×C₁₆	—	5	1	no	—
107	C₂²×C₁₆	—	10	2	no	—
108	#443	—	5	4	no	—
109	#443	—	5	4	no	—
110	C₂×#108	—	5	1	no	—
111	C₂×#104	—	5	2	no	—
112	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
113	#364	C₂×D₅	1	4	yes	complemented
114	#350	—	5	8	no	—
115	C₄×C₁₆	—	5	4	no	—
116	C₂³×C₅	#131	1	1	yes	characteristic
117	C₂²×D₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅	#131	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂²⋊D₅	—	2	4	no	—
127	C₂²⋊D₅	—	2	4	no	—
128	C₂×Dic₅	—	2	2	no	—
129	D₂₀	—	2	2	no	—
130	D₂₀	C₂²×C₈	1	4	yes	—
131	C₄×D₅	—	2	4	no	—
132	C₄×D₅	—	2	4	no	—
133	Dic₁₀	—	2	2	no	—
134	Dic₁₀	C₂²×C₈	1	4	yes	—
135	C₅×C₈	C₄×D₄	1	1	yes	characteristic
136	C₅×C₈	C₄×D₄	1	1	yes	characteristic
137	C₅×C₈	C₂²×C₈	1	2	yes	—
138	C₅×C₈	C₂²×C₈	1	2	yes	—
139	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
140	C₅×C₈	C₂×#41	1	1	yes	characteristic
141	#168	—	2	4	no	—
142	C₂×#41	—	5	2	no	—
143	#135	—	5	4	no	—
144	#135	—	5	4	no	—
145	C₂×Q₁₆	—	5	1	no	—
146	C₂×SD₁₆	—	5	2	no	—
147	C₂×D₈	—	5	1	no	—
148	#131	—	5	4	no	—
149	#131	—	5	4	no	—
150	#131	—	10	2	no	—
151	#131	—	10	2	no	—
152	C₂×M₁₆	—	5	2	no	—
153	C₂×M₁₆	—	10	2	no	—
154	C₂²×C₈	—	10	2	no	—
155	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
158	M₃₂	—	2	2	no	—
159	M₃₂	—	10	8	no	—
160	M₃₂	C₂²×D₅	1	4	yes	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	5	2	no	—
163	C₂×C₁₆	—	5	1	no	—
164	C₂×C₁₆	—	5	1	no	—
165	C₂×C₁₆	—	5	2	no	—
166	C₂×C₁₆	—	5	2	no	—
167	C₂×C₁₆	—	10	8	no	—
168	C₂×C₁₆	—	10	1	no	—
169	C₂×C₁₆	—	10	1	no	—
170	#108	—	5	4	no	—
171	#104	—	5	8	no	—
172	C₄×C₈	C₄×D₅	1	2	yes	—
173	C₄×C₈	C₂²×D₅	1	2	yes	—
174	C₂²×C₅	—	2	2	no	—
175	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
176	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
177	C₂²×C₅	C₂×#131	1	1	yes	characteristic
178	C₂×D₅	—	2	4	no	—
179	C₂×D₅	—	4	4	no	—
180	C₄×C₅	—	2	4	no	—
181	C₄×C₅	C₈×D₄	1	1	yes	characteristic
182	C₄×C₅	C₈×D₄	1	1	yes	characteristic
183	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
185	Dic₅	—	2	4	no	—
186	#41	—	5	4	no	—
187	#41	—	5	4	no	—
188	#41	—	10	2	no	—
189	#41	—	10	2	no	—
190	C₂×Q₈	—	5	2	no	—
191	C₂×D₄	—	5	2	no	—
192	C₂×D₄	—	10	2	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	10	2	no	—
195	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
196	Q₁₆	—	5	4	no	—
197	SD₁₆	—	5	8	no	—
198	D₈	—	5	4	no	—
199	M₁₆	—	5	4	no	—
200	M₁₆	—	10	2	no	—
201	M₁₆	—	10	4	no	—
202	M₁₆	—	10	4	no	—
203	C₂×C₈	—	10	2	no	—
204	C₂×C₈	—	10	4	no	—
205	C₂×C₈	—	10	4	no	—
206	C₂×C₈	C₈×D₅	1	1	yes	characteristic
207	C₂×C₈	C₈×D₅	1	1	yes	characteristic
208	C₂×C₈	C₈×D₅	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
211	C₂×C₈	D₄×D₅	1	1	yes	characteristic
212	C₂×C₈	D₄×D₅	1	1	yes	characteristic
213	C₂×C₈	#713	1	1	yes	characteristic
214	C₂×C₈	#713	1	1	yes	characteristic
215	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
216	C₄²	—	2	1	no	—
217	C₄²	C₂×C₄×D₅	1	2	yes	—
218	C₁₆	—	2	4	no	—
219	C₁₆	—	5	2	no	—
220	C₁₆	—	5	2	no	—
221	C₁₆	—	10	2	no	—
222	C₂×C₅	—	2	1	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	#2170	1	2	yes	complemented
225	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
226	D₅	—	4	4	no	—
227	C₂³	—	10	2	no	—
228	C₂³	#4053	1	1	yes	characteristic
229	Q₈	—	5	4	no	—
230	Q₈	—	10	2	no	—
231	D₄	—	5	4	no	—
232	D₄	—	10	4	no	—
233	D₄	—	10	4	no	—
234	D₄	—	10	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	10	2	no	—
238	C₂×C₄	—	10	4	no	—
239	C₂×C₄	—	10	4	no	—
240	C₂×C₄	#4039	1	1	yes	characteristic
241	C₂×C₄	#4039	1	1	yes	characteristic
242	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
244	C₂×C₄	#4053	1	1	yes	characteristic
245	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
246	C₈	—	10	4	no	—
247	C₈	#4039	1	1	yes	characteristic
248	C₈	#4039	1	1	yes	characteristic
249	C₈	C₂×C₈×D₅	1	2	yes	—
250	C₈	C₂×C₈×D₅	1	2	yes	—
251	C₈	C₂×D₄×D₅	1	1	yes	characteristic
252	C₈	C₂×#713	1	1	yes	characteristic
253	C₅	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	2	2	no	—
255	C₂²	—	10	4	no	—
256	C₂²	—	20	4	no	—
257	C₂²	#65123	1	1	yes	characteristic
258	C₂²	#65123	1	1	yes	characteristic
259	C₂²	C₂×#4053	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	10	4	no	—
262	C₄	#65123	1	1	yes	characteristic
263	C₄	#65123	1	1	yes	characteristic
264	C₄	C₂×#4039	1	1	yes	characteristic
265	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	20	4	no	—
269	C₂	#10600093	1	2	yes	direct product factor
270	C₂	C₂×#65123	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅
commutator	C₄×C₅	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#364	C₂
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×#4050	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#364	D₅
hypocenter	C₅	C₂×#2170
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#65196	C₂	semidirect
3. C₂×C₅×#364	C₂	semidirect
4. C₂×#64735	C₂	semidirect
5. C₂×C₄×#672	C₂	semidirect
6. #10600093	C₂	direct (N × H)
16. #65196	C₂²	semidirect
21. C₅×#364	C₂²	semidirect
22. #64735	C₂²	semidirect
23. C₄×#672	C₂²	semidirect
60. C₂×#364	D₅	semidirect
113. #364	C₂×D₅	semidirect
224. C₂×C₅	#2170	semidirect
253. C₅	C₂×#2170	semidirect
269. C₂	#10600093	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×#65196	2
C₂×#64989	2
C₂×C₅×#364	2
C₂×#64735	2
C₂×C₄×#672	2
#10600093	2
C₂×#2170	5

Subgroups — C2×#10600093 #12252527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600093 #12252527