Subgroups — C₂×#10598048 #12009014

Order 1280.

422 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64977	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64681	C₂	1	1	yes	maximal, characteristic, complemented
4	#10598048	C₂	1	4	yes	maximal, direct product factor
5	C₂²×#676	—	2	1	no	—
6	C₂×C₄×D₂₀	C₄	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3952	—	2	1	no	—
10	C₂²×#672	—	2	1	no	—
11	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
12	#64977	—	2	1	no	—
13	#64977	—	2	1	no	—
14	#64977	C₄	1	2	yes	—
15	#64977	C₂²	1	2	yes	—
16	C₅×C₄⋊C₁₆	—	2	1	no	—
17	C₅×C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	#64681	—	2	1	no	—
19	#64681	C₂²	1	2	yes	complemented
20	C₂×#1330	—	5	1	no	maximal, Sylow
21	C₂²×D₂₀	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₅	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×#685	—	2	1	no	—
27	C₂×C₄⋊Dic₅	C₈	1	1	yes	characteristic
28	C₂²×#168	—	2	1	no	—
29	C₂×#676	—	2	2	no	—
30	C₂×#676	—	2	2	no	—
31	C₂×#676	—	4	2	no	—
32	C₂×#676	—	4	2	no	—
33	C₄×D₂₀	—	2	1	no	—
34	C₄×D₂₀	—	2	1	no	—
35	C₄×D₂₀	C₈	1	2	yes	—
36	C₄×D₂₀	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₁₆	—	2	2	no	—
38	C₂×C₅×C₁₆	—	4	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
43	#3952	—	2	2	no	—
44	#3952	—	2	2	no	—
45	C₂×#672	—	2	4	no	—
46	C₂×#672	—	2	2	no	—
47	C₄⋊#168	—	2	1	no	—
48	C₄⋊#168	C₂×C₄	1	2	yes	—
49	C₂×#436	—	5	1	no	—
50	C₂×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
51	C₂×#346	—	5	1	no	—
52	#1330	—	5	4	no	—
53	C₂³×D₅	—	2	1	no	—
54	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
57	C₂²×Dic₅	—	2	1	no	—
58	C₂×D₂₀	—	2	1	no	—
59	C₂×D₂₀	—	4	2	no	—
60	C₂×D₂₀	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₅	—	2	2	no	—
62	C₂×C₄×D₅	—	2	2	no	—
63	C₂×C₄×D₅	—	4	2	no	—
64	C₂×C₄×D₅	—	4	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₅×C₈	D₈	1	2	yes	—
69	C₂×C₅×C₈	SD₁₆	1	2	yes	—
70	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₅	M₁₆	1	1	yes	characteristic
75	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
76	#685	—	2	2	no	—
77	#685	—	2	2	no	—
78	C₄⋊Dic₅	—	2	1	no	—
79	C₄⋊Dic₅	C₂×C₈	1	2	yes	—
80	C₂×#168	—	2	2	no	—
81	C₂×#168	—	4	2	no	—
82	#676	—	4	4	no	—
83	#676	—	4	2	no	—
84	#676	—	4	2	no	—
85	C₅×C₁₆	—	4	2	no	—
86	#672	—	2	4	no	—
87	C₂²×M₁₆	—	10	1	no	—
88	C₂×C₄×D₄	—	5	1	no	—
89	C₂²×C₁₆	—	2	1	no	—
90	C₂²×C₁₆	—	10	1	no	—
91	#436	—	5	2	no	—
92	#436	—	5	2	no	—
93	#436	—	10	1	no	—
94	#436	—	10	1	no	—
95	C₂×C₄⋊C₈	—	5	1	no	—
96	C₂×C₂²⋊C₈	—	10	1	no	—
97	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
98	C₄⋊C₁₆	—	2	1	no	—
99	C₄⋊C₁₆	C₂×D₅	1	2	yes	complemented
100	#346	—	5	2	no	—
101	#346	—	10	1	no	—
102	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
103	C₂²×D₅	—	2	2	no	—
104	C₂²×D₅	—	4	2	no	—
105	C₂²×D₅	—	4	2	no	—
106	C₂²×D₅	—	4	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₅	#102	1	2	yes	—
113	C₂×C₄×C₅	#104	1	2	yes	—
114	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₂×Dic₅	—	2	2	no	—
118	C₂×Dic₅	—	4	2	no	—
119	D₂₀	—	4	2	no	—
120	D₂₀	—	4	2	no	—
121	C₄×D₅	—	4	4	no	—
122	C₄×D₅	—	4	2	no	—
123	C₄×D₅	—	4	2	no	—
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	—	2	1	no	—
126	C₅×C₈	#102	1	2	yes	—
127	C₅×C₈	C₂×D₈	1	1	yes	characteristic
128	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
129	#168	—	4	2	no	—
130	C₂²×D₄	—	5	1	no	—
131	C₂³×C₄	—	10	1	no	—
132	C₂×M₁₆	—	10	2	no	—
133	C₂×M₁₆	—	10	2	no	—
134	C₂×M₁₆	—	20	2	no	—
135	C₂×M₁₆	—	20	2	no	—
136	C₂²×C₈	—	10	1	no	—
137	C₂²×C₈	D₂₀	1	1	yes	characteristic
138	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
139	C₄×D₄	—	5	2	no	—
140	C₄×D₄	—	5	2	no	—
141	C₄×D₄	—	10	1	no	—
142	C₄×D₄	—	10	1	no	—
143	C₂×C₄⋊C₄	—	5	1	no	—
144	C₂×C₂²⋊C₄	—	10	1	no	—
145	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	4	2	no	—
148	C₂×C₁₆	—	10	4	no	—
149	C₂×C₁₆	—	10	2	no	—
150	C₄⋊C₈	—	5	2	no	—
151	C₄⋊C₈	—	10	1	no	—
152	C₂²⋊C₈	—	10	2	no	—
153	C₂²⋊C₈	—	10	2	no	—
154	C₄×C₈	C₄×D₅	1	1	yes	characteristic
155	C₄×C₈	D₂₀	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
157	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
158	C₂²×C₅	D₄⋊C₈	1	2	yes	—
159	C₂²×C₅	#349	1	2	yes	—
160	C₂²×C₅	#350	1	2	yes	—
161	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×D₅	—	4	2	no	—
163	C₂×D₅	—	4	2	no	—
164	C₂×D₅	—	4	2	no	—
165	C₂×D₅	—	8	4	no	—
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	2	1	no	—
168	C₄×C₅	—	2	1	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	D₄⋊C₈	1	2	yes	—
171	C₄×C₅	C₂×#102	1	1	yes	characteristic
172	C₄×C₅	C₂×#104	1	1	yes	characteristic
173	Dic₅	—	4	2	no	—
174	C₂⁴	—	10	1	no	—
175	C₂×D₄	—	5	2	no	—
176	C₂×D₄	—	10	1	no	—
177	C₂×D₄	—	20	2	no	—
178	C₂²×C₄	—	10	1	no	—
179	C₂²×C₄	—	10	2	no	—
180	C₂²×C₄	—	10	2	no	—
181	C₂²×C₄	—	20	2	no	—
182	C₂²×C₄	—	20	2	no	—
183	C₂²×C₄	C₈×D₅	1	1	yes	characteristic
184	C₂²×C₄	#676	1	1	yes	characteristic
185	C₂²×C₄	#685	1	1	yes	characteristic
186	M₁₆	—	20	4	no	—
187	M₁₆	—	20	2	no	—
188	M₁₆	—	20	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	10	2	no	—
191	C₂×C₈	—	20	2	no	—
192	C₂×C₈	#685	1	1	yes	characteristic
193	C₂×C₈	#685	1	1	yes	characteristic
194	C₂×C₈	#686	1	2	yes	—
195	C₂×C₈	#688	1	2	yes	—
196	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
198	C₄⋊C₄	—	5	2	no	—
199	C₄⋊C₄	—	10	1	no	—
200	C₂²⋊C₄	—	10	2	no	—
201	C₂²⋊C₄	—	10	2	no	—
202	C₄²	C₈×D₅	1	1	yes	characteristic
203	C₄²	#676	1	1	yes	characteristic
204	C₄²	#685	1	1	yes	characteristic
205	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
206	C₁₆	—	4	2	no	—
207	C₁₆	—	10	4	no	—
208	C₂×C₅	#1330	1	4	yes	complemented
209	C₂×C₅	C₂×D₄⋊C₈	1	1	yes	characteristic
210	C₂×C₅	C₂×#349	1	1	yes	characteristic
211	C₂×C₅	C₂×#350	1	1	yes	characteristic
212	D₅	—	8	2	no	—
213	C₂³	—	10	2	no	—
214	C₂³	—	20	2	no	—
215	C₂³	—	20	2	no	—
216	C₂³	—	20	2	no	—
217	C₂³	#3952	1	1	yes	characteristic
218	D₄	—	20	2	no	—
219	D₄	—	20	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	10	2	no	—
223	C₂×C₄	—	20	2	no	—
224	C₂×C₄	—	20	2	no	—
225	C₂×C₄	—	20	4	no	—
226	C₂×C₄	—	20	2	no	—
227	C₂×C₄	#3941	1	2	yes	—
228	C₂×C₄	#3952	1	1	yes	characteristic
229	C₂×C₄	#3952	1	1	yes	characteristic
230	C₂×C₄	#3952	1	1	yes	characteristic
231	C₂×C₄	#3957	1	2	yes	—
232	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
233	C₂×C₄	C₂×#676	1	1	yes	characteristic
234	C₂×C₄	C₂×#685	1	1	yes	characteristic
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	20	2	no	—
238	C₈	#3941	1	2	yes	—
239	C₈	C₂×#686	1	1	yes	characteristic
240	C₈	C₂×#688	1	1	yes	characteristic
241	C₅	C₂×#1330	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	20	2	no	—
243	C₂²	—	20	2	no	—
244	C₂²	—	20	2	no	—
245	C₂²	—	40	4	no	—
246	C₂²	#64703	1	2	yes	—
247	C₂²	#64728	1	2	yes	—
248	C₂²	#64734	1	2	yes	—
249	C₂²	C₂×#3952	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	20	2	no	—
255	C₄	#64703	1	2	yes	—
256	C₄	C₂×#3941	1	1	yes	characteristic
257	C₄	C₂×#3957	1	1	yes	characteristic
258	C₂	—	40	2	no	—
259	C₂	#10598048	1	4	yes	direct product factor
260	C₂	C₂×#64703	1	1	yes	characteristic
261	C₂	C₂×#64728	1	1	yes	characteristic
262	C₂	C₂×#64734	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#685
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×C₄⋊C₁₆	C₂
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂²×D₂₀	C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅
hypocenter	C₅	C₂×#1330
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
3. C₂×#64681	C₂	semidirect
4. #10598048	C₂	direct (N × H)
17. C₅×C₄⋊C₁₆	C₂²	semidirect
19. #64681	C₂²	semidirect
50. C₂×C₄⋊C₁₆	D₅	semidirect
99. C₄⋊C₁₆	C₂×D₅	semidirect
208. C₂×C₅	#1330	semidirect
241. C₅	C₂×#1330	semidirect
259. C₂	#10598048	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×#64977	2
C₂×C₅×C₄⋊C₁₆	2
C₂×#64681	2
#10598048	2
C₂×#1330	5

Subgroups — C2×#10598048 #12009014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598048 #12009014