Subgroups — C₂²×#65414 #12962993

Order 1280.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 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	C₂³×#681	C₂	1	2	yes	maximal
3	C₂×#65414	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#3951	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₂²×C₅×M₁₆	C₄	1	4	yes	—
7	C₂²×C₅×M₁₆	C₄	1	12	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	12	yes	—
9	C₂³×C₅×C₈	C₂²	1	2	yes	—
10	C₂²×#681	—	2	2	no	—
11	C₂²×#681	—	2	12	no	—
12	C₂²×#681	C₂²	1	4	yes	—
13	C₂²×#681	C₂²	1	24	yes	—
14	C₂³×#168	—	2	2	no	—
15	#65414	C₂²	1	64	yes	direct product factor
16	C₂×#3951	C₂²	1	48	yes	complemented
17	C₂²×#430	—	5	1	no	maximal, Sylow
18	C₂⁵×C₅	Q₈	1	1	yes	characteristic
19	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₅	C₂×C₄	1	6	yes	—
21	C₂³×C₄×C₅	D₄	1	2	yes	—
22	C₂³×C₄×C₅	D₄	1	6	yes	—
23	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	Q₈	1	6	yes	—
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×C₄×C₅	C₂³	1	6	yes	—
27	C₂×C₅×M₁₆	C₂×C₄	1	48	yes	—
28	C₂×C₅×M₁₆	C₂×C₄	1	48	yes	—
29	C₂×C₅×M₁₆	C₂³	1	16	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	12	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	12	yes	—
33	C₂×#681	—	2	24	no	—
34	C₂×#681	—	2	12	no	—
35	C₂×#681	—	2	48	no	—
36	C₂×#681	C₂³	1	24	yes	—
37	C₂×#681	C₂³	1	32	yes	—
38	C₂²×#168	—	2	24	no	—
39	C₂²×#168	—	2	4	no	—
40	#3951	C₂³	1	64	yes	complemented
41	C₂³×M₁₆	—	5	2	no	—
42	C₂³×M₁₆	D₅	1	1	yes	characteristic
43	C₂×#430	—	5	24	no	—
44	C₂²×#108	—	5	4	no	—
45	C₂⁴×C₅	—	4	4	no	—
46	C₂⁴×C₅	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₅	C₄⋊C₄	1	6	yes	—
48	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
49	C₂⁴×C₅	C₂×Q₈	1	6	yes	—
50	C₂²×C₄×C₅	C₄⋊C₄	1	12	yes	—
51	C₂²×C₄×C₅	C₄⋊C₄	1	12	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	2	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	12	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×Q₈	1	3	yes	—
64	C₂²×C₄×C₅	C₂×Q₈	1	6	yes	—
65	C₂²×C₄×C₅	C₂×Q₈	1	12	yes	—
66	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
67	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
68	C₅×M₁₆	C₂²×C₄	1	64	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	24	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	24	yes	—
71	C₂×C₅×C₈	C₂⁴	1	8	yes	—
72	#681	—	2	32	no	—
73	#681	—	2	24	no	—
74	#681	C₂⁴	1	16	yes	—
75	C₂×#168	—	2	24	no	—
76	C₂×#168	—	2	32	no	—
77	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
78	C₂²×M₁₆	—	5	4	no	—
79	C₂²×M₁₆	—	5	24	no	—
80	C₂²×M₁₆	—	10	12	no	—
81	C₂²×M₁₆	—	10	2	no	—
82	C₂²×M₁₆	Dic₅	1	4	yes	—
83	C₂²×M₁₆	Dic₅	1	12	yes	—
84	C₂²×M₁₆	C₂×D₅	1	12	yes	—
85	C₂³×C₈	—	10	2	no	—
86	C₂³×C₈	C₂×D₅	1	2	yes	—
87	#430	—	5	64	no	—
88	C₂×#108	—	5	48	no	—
89	C₂³×C₅	—	2	8	no	—
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	—	4	12	no	—
92	C₂³×C₅	—	4	12	no	—
93	C₂³×C₅	C₂×C₄⋊C₄	1	3	yes	—
94	C₂³×C₅	C₂×C₄⋊C₄	1	12	yes	—
95	C₂³×C₅	C₂×C₄⋊C₄	1	12	yes	—
96	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₅	C₂²×Q₈	1	3	yes	—
98	C₂³×C₅	C₂²×Q₈	1	4	yes	—
99	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
100	C₂×C₄×C₅	C₂×C₄⋊C₄	1	24	yes	—
101	C₂×C₄×C₅	C₂×C₄⋊C₄	1	24	yes	—
102	C₂×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
103	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
104	C₂×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
105	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
106	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
107	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
108	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
111	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
112	C₂×C₄×C₅	C₂²×Q₈	1	3	yes	—
113	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
114	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
115	C₅×C₈	C₂³×C₄	1	16	yes	—
116	#168	—	2	16	no	—
117	C₂⁵	Dic₁₀	1	1	yes	characteristic
118	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
119	C₂³×C₄	Dic₁₀	1	6	yes	—
120	C₂³×C₄	D₂₀	1	6	yes	—
121	C₂³×C₄	D₂₀	1	2	yes	—
122	C₂³×C₄	C₂×Dic₅	1	2	yes	—
123	C₂³×C₄	C₂×Dic₅	1	6	yes	—
124	C₂³×C₄	C₂²×D₅	1	6	yes	—
125	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
126	C₂×M₁₆	—	5	24	no	—
127	C₂×M₁₆	—	5	32	no	—
128	C₂×M₁₆	—	10	24	no	—
129	C₂×M₁₆	—	10	12	no	—
130	C₂×M₁₆	—	10	48	no	—
131	C₂×M₁₆	C₂×Dic₅	1	48	yes	—
132	C₂×M₁₆	C₂×Dic₅	1	48	yes	—
133	C₂×M₁₆	C₂²×D₅	1	16	yes	—
134	C₂²×C₈	—	10	24	no	—
135	C₂²×C₈	—	10	4	no	—
136	C₂²×C₈	C₂×Dic₅	1	4	yes	—
137	C₂²×C₈	C₂×Dic₅	1	12	yes	—
138	C₂²×C₈	C₂²×D₅	1	12	yes	—
139	#108	—	5	64	no	—
140	C₂²×C₅	—	2	24	no	—
141	C₂²×C₅	—	2	6	no	—
142	C₂²×C₅	—	2	6	no	—
143	C₂²×C₅	—	4	12	no	—
144	C₂²×C₅	—	4	4	no	—
145	C₂²×C₅	#430	1	4	yes	complemented
146	C₂²×C₅	C₂²×C₄⋊C₄	1	3	yes	—
147	C₂²×C₅	C₂²×C₄⋊C₄	1	8	yes	—
148	C₂²×C₅	C₂²×C₄⋊C₄	1	3	yes	—
149	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
150	C₄×C₅	C₂²×C₄⋊C₄	1	3	yes	—
151	C₄×C₅	C₂²×C₄⋊C₄	1	3	yes	—
152	C₄×C₅	C₂²×C₄⋊C₄	1	8	yes	—
153	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
154	C₄×C₅	C₂³×D₄	1	1	yes	characteristic
155	C₂⁴	—	4	4	no	—
156	C₂⁴	C₄⋊Dic₅	1	2	yes	—
157	C₂⁴	C₄⋊Dic₅	1	6	yes	—
158	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
159	C₂⁴	C₂×Dic₁₀	1	6	yes	—
160	C₂²×C₄	C₄⋊Dic₅	1	6	yes	—
161	C₂²×C₄	C₄⋊Dic₅	1	12	yes	—
162	C₂²×C₄	C₄⋊Dic₅	1	24	yes	—
163	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
164	C₂²×C₄	C₄⋊Dic₅	1	12	yes	—
165	C₂²×C₄	C₂×Dic₁₀	1	6	yes	—
166	C₂²×C₄	C₂×Dic₁₀	1	12	yes	—
167	C₂²×C₄	C₂×Dic₁₀	1	3	yes	—
168	C₂²×C₄	C₂×D₂₀	1	12	yes	—
169	C₂²×C₄	C₂×D₂₀	1	3	yes	—
170	C₂²×C₄	C₂×D₂₀	1	12	yes	—
171	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
172	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₅	1	3	yes	—
174	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
175	C₂²×C₄	C₂²×Dic₅	1	12	yes	—
176	C₂²×C₄	C₂³×D₅	1	3	yes	—
177	C₂²×C₄	C₂³×D₅	1	4	yes	—
178	M₁₆	—	5	16	no	—
179	M₁₆	—	10	32	no	—
180	M₁₆	—	10	24	no	—
181	M₁₆	C₂²×Dic₅	1	64	yes	—
182	C₂×C₈	—	10	24	no	—
183	C₂×C₈	—	10	32	no	—
184	C₂×C₈	C₂²×Dic₅	1	24	yes	—
185	C₂×C₈	C₂²×Dic₅	1	24	yes	—
186	C₂×C₈	C₂³×D₅	1	8	yes	—
187	C₂×C₅	—	2	8	no	—
188	C₂×C₅	—	2	3	no	—
189	C₂×C₅	—	2	1	no	—
190	C₂×C₅	C₂×#430	1	6	yes	complemented
191	C₂×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic
192	C₂³	—	2	8	no	—
193	C₂³	—	2	4	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	4	12	no	—
196	C₂³	C₂×C₄⋊Dic₅	1	12	yes	—
197	C₂³	C₂×C₄⋊Dic₅	1	3	yes	—
198	C₂³	C₂×C₄⋊Dic₅	1	12	yes	—
199	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
200	C₂³	C₂²×Dic₁₀	1	3	yes	—
201	C₂³	C₂²×Dic₁₀	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₅	1	24	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₅	1	6	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₅	1	12	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₅	1	24	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₅	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
209	C₂×C₄	C₂²×Dic₁₀	1	3	yes	—
210	C₂×C₄	C₂²×Dic₁₀	1	4	yes	—
211	C₂×C₄	C₂²×D₂₀	1	3	yes	—
212	C₂×C₄	C₂²×D₂₀	1	8	yes	—
213	C₂×C₄	C₂²×D₂₀	1	3	yes	—
214	C₂×C₄	C₂³×Dic₅	1	3	yes	—
215	C₂×C₄	C₂³×Dic₅	1	3	yes	—
216	C₂×C₄	C₂³×Dic₅	1	8	yes	—
217	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
218	C₈	—	10	16	no	—
219	C₈	C₂³×Dic₅	1	16	yes	—
220	C₅	C₂²×#430	1	1	yes	characteristic, complemented, Sylow
221	C₂²	—	2	6	no	—
222	C₂²	—	2	24	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	4	4	no	—
225	C₂²	—	4	12	no	—
226	C₂²	#65414	1	4	yes	direct product factor
227	C₂²	C₂²×C₄⋊Dic₅	1	8	yes	—
228	C₂²	C₂²×C₄⋊Dic₅	1	3	yes	—
229	C₂²	C₂²×C₄⋊Dic₅	1	3	yes	—
230	C₂²	C₂³×Dic₁₀	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊Dic₅	1	3	yes	—
232	C₄	C₂²×C₄⋊Dic₅	1	8	yes	—
233	C₄	C₂²×C₄⋊Dic₅	1	3	yes	—
234	C₄	C₂³×D₂₀	1	1	yes	characteristic
235	C₄	C₂⁴×Dic₅	1	1	yes	characteristic
236	C₂	—	2	3	no	—
237	C₂	—	2	8	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×#65414	1	6	yes	direct product factor
240	C₂	C₂³×C₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₄×C₅	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂³×C₅×M₁₆	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁵×C₅	Q₈
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂³×M₁₆	D₅
hypocenter	C₅	C₂²×#430
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#65414	C₂	direct (N × H)
4. C₂²×#3951	C₂	semidirect
15. #65414	C₂²	direct (N × H)
16. C₂×#3951	C₂²	semidirect
40. #3951	C₂³	semidirect
145. C₂²×C₅	#430	semidirect
190. C₂×C₅	C₂×#430	semidirect
220. C₅	C₂²×#430	semidirect
226. C₂²	#65414	direct (N × H)
239. C₂	C₂×#65414	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
C₂³×#681	2
C₂×#65414	2
C₂²×#3951	2
C₂²×#430	5

Subgroups — C22×#65414 #12962993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65414 #12962993