Subgroups — C₂²×C₄×#676 #12962685

Order 1280.

3950 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#676	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#3947	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#676	C₂	1	24	yes	maximal, direct product factor
7	C₂³×C₄×D₅	C₄	1	2	yes	—
8	C₂³×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₅	C₄	1	2	yes	—
12	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂³×#168	C₂²	1	2	yes	—
14	C₂²×#676	C₄	1	32	yes	direct product factor
15	C₂²×#676	C₂²	1	24	yes	—
16	C₂×C₄²×D₅	C₄	1	12	yes	—
17	C₂×C₄²×D₅	C₂²	1	12	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	12	yes	complemented
19	C₂×#3947	C₂²	1	24	yes	complemented
20	C₂×C₄×#168	C₂²	1	12	yes	complemented
21	C₄×#676	C₂²	1	64	yes	direct product factor
22	C₂²×C₄×M₁₆	—	5	1	no	maximal, Sylow
23	C₂⁴×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×Dic₅	C₂×C₄	1	2	yes	—
27	C₂³×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₅	—	2	16	no	—
29	C₂²×C₄×D₅	C₂×C₄	1	24	yes	—
30	C₂²×C₄×D₅	C₂×C₄	1	6	yes	—
31	C₂²×C₄×D₅	C₂×C₄	1	16	yes	—
32	C₂²×C₄×D₅	C₂³	1	6	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	16	yes	complemented
34	C₂²×C₅×C₈	C₂³	1	12	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	6	yes	—
36	C₂×C₄²×C₅	C₂³	1	6	yes	—
37	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
38	C₂×C₄×Dic₅	C₂×C₄	1	24	yes	—
39	C₂×C₄×Dic₅	C₂³	1	6	yes	—
40	C₂²×#168	C₂×C₄	1	16	yes	complemented
41	C₂²×#168	C₂³	1	12	yes	—
42	C₂×#676	C₂×C₄	1	192	yes	direct product factor
43	C₂×#676	C₂³	1	32	yes	—
44	C₄²×D₅	C₂×C₄	1	48	yes	—
45	C₄²×D₅	C₂³	1	16	yes	—
46	C₄×C₅×C₈	C₂³	1	16	yes	complemented
47	#3947	C₂³	1	32	yes	complemented
48	C₄×#168	C₂³	1	16	yes	complemented
49	C₂³×M₁₆	—	5	2	no	—
50	C₂³×C₄²	—	5	1	no	—
51	C₂×C₄×M₁₆	—	5	24	no	—
52	C₂²×#97	—	5	2	no	—
53	C₂²×C₄×C₈	—	5	1	no	—
54	C₂²×C₄×C₈	D₅	1	1	yes	characteristic, complemented
55	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂³×D₅	—	2	8	no	—
57	C₂³×D₅	C₄²	1	8	yes	—
58	C₂³×D₅	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₅	M₁₆	1	16	yes	complemented
60	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
64	C₂²×Dic₅	—	2	8	no	—
65	C₂²×Dic₅	C₄²	1	8	yes	—
66	C₂²×Dic₅	C₂²×C₄	1	12	yes	—
67	C₂²×Dic₅	C₂²×C₄	1	6	yes	—
68	C₂×C₄×D₅	—	2	96	no	—
69	C₂×C₄×D₅	C₄²	1	48	yes	—
70	C₂×C₄×D₅	C₂²×C₄	1	32	yes	—
71	C₂×C₄×D₅	C₂²×C₄	1	12	yes	—
72	C₂×C₄×D₅	C₂²×C₄	1	48	yes	—
73	C₂×C₄×D₅	C₂⁴	1	4	yes	—
74	C₂×C₅×C₈	C₂²×C₄	1	48	yes	complemented
75	C₂×C₅×C₈	C₂⁴	1	8	yes	—
76	C₄²×C₅	C₂²×C₄	1	12	yes	—
77	C₄²×C₅	C₂⁴	1	4	yes	—
78	C₄×Dic₅	C₂²×C₄	1	12	yes	—
79	C₄×Dic₅	C₂²×C₄	1	32	yes	—
80	C₄×Dic₅	C₂⁴	1	4	yes	—
81	C₂×#168	C₂²×C₄	1	48	yes	complemented
82	C₂×#168	C₂⁴	1	8	yes	—
83	#676	C₂²×C₄	1	128	yes	direct product factor
84	C₂⁴×C₄	—	5	1	no	—
85	C₂⁴×C₄	—	5	2	no	—
86	C₂²×M₁₆	—	5	32	no	—
87	C₂²×M₁₆	—	5	24	no	—
88	C₂³×C₈	—	5	2	no	—
89	C₂³×C₈	C₂×D₅	1	2	yes	—
90	C₂²×C₄²	—	5	2	no	—
91	C₂²×C₄²	—	5	1	no	—
92	C₂²×C₄²	—	5	12	no	—
93	C₂²×C₄²	—	5	12	no	—
94	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₄×M₁₆	—	5	64	no	—
96	C₂×#97	—	5	24	no	—
97	C₂×C₄×C₈	—	5	12	no	—
98	C₂×C₄×C₈	C₂×D₅	1	12	yes	complemented
99	C₂³×C₅	C₂×C₄²	1	4	yes	—
100	C₂³×C₅	C₂×M₁₆	1	8	yes	—
101	C₂³×C₅	C₂³×C₄	1	3	yes	—
102	C₂²×D₅	—	2	24	no	—
103	C₂²×D₅	—	2	32	no	—
104	C₂²×D₅	C₂×C₄²	1	24	yes	—
105	C₂²×D₅	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₅	C₂×C₄²	1	12	yes	—
107	C₂×C₄×C₅	C₂×M₁₆	1	48	yes	complemented
108	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₅	C₂³×C₄	1	12	yes	—
110	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
111	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
112	C₂×Dic₅	—	2	24	no	—
113	C₂×Dic₅	C₂×C₄²	1	24	yes	—
114	C₂×Dic₅	C₂³×C₄	1	8	yes	—
115	C₂×Dic₅	C₂³×C₄	1	4	yes	—
116	C₄×D₅	—	2	64	no	—
117	C₄×D₅	C₂×C₄²	1	48	yes	—
118	C₄×D₅	C₂³×C₄	1	16	yes	—
119	C₅×C₈	C₂³×C₄	1	16	yes	complemented
120	#168	C₂³×C₄	1	16	yes	complemented
121	C₂⁵	—	5	1	no	—
122	C₂³×C₄	—	5	1	no	—
123	C₂³×C₄	—	5	16	no	—
124	C₂³×C₄	—	5	6	no	—
125	C₂³×C₄	—	5	6	no	—
126	C₂³×C₄	—	5	2	no	—
127	C₂³×C₄	—	5	24	no	—
128	C₂³×C₄	—	10	16	no	—
129	C₂³×C₄	C₄×D₅	1	2	yes	—
130	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
131	C₂×M₁₆	—	5	192	no	—
132	C₂×M₁₆	—	5	32	no	—
133	C₂²×C₈	—	5	16	no	—
134	C₂²×C₈	—	5	12	no	—
135	C₂²×C₈	C₄×D₅	1	16	yes	complemented
136	C₂²×C₈	C₂²×D₅	1	12	yes	—
137	C₂×C₄²	—	5	24	no	—
138	C₂×C₄²	—	5	6	no	—
139	C₂×C₄²	—	5	6	no	—
140	C₂×C₄²	—	5	48	no	—
141	C₂×C₄²	—	5	16	no	—
142	C₂×C₄²	C₄×D₅	1	6	yes	—
143	C₂×C₄²	C₂²×D₅	1	6	yes	—
144	#97	—	5	32	no	—
145	C₄×C₈	—	5	16	no	—
146	C₄×C₈	C₂²×D₅	1	16	yes	complemented
147	C₂²×C₅	C₄×M₁₆	1	16	yes	complemented
148	C₂²×C₅	C₂²×C₄²	1	6	yes	—
149	C₂²×C₅	C₂²×M₁₆	1	12	yes	—
150	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
151	C₂×D₅	—	2	48	no	—
152	C₂×D₅	—	2	8	no	—
153	C₂×D₅	C₂²×C₄²	1	8	yes	—
154	C₄×C₅	C₂²×C₄²	1	6	yes	—
155	C₄×C₅	C₂²×M₁₆	1	16	yes	complemented
156	C₄×C₅	C₂⁴×C₄	1	2	yes	—
157	Dic₅	—	2	8	no	—
158	Dic₅	C₂²×C₄²	1	8	yes	—
159	C₂⁴	—	5	6	no	—
160	C₂⁴	—	5	8	no	—
161	C₂⁴	—	10	8	no	—
162	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂²×C₄	—	5	6	no	—
164	C₂²×C₄	—	5	8	no	—
165	C₂²×C₄	—	5	12	no	—
166	C₂²×C₄	—	5	48	no	—
167	C₂²×C₄	—	5	4	no	—
168	C₂²×C₄	—	5	48	no	—
169	C₂²×C₄	—	5	12	no	—
170	C₂²×C₄	—	5	32	no	—
171	C₂²×C₄	—	10	8	no	—
172	C₂²×C₄	—	10	96	no	—
173	C₂²×C₄	#676	1	16	yes	direct product factor
174	C₂²×C₄	C₂×C₄×D₅	1	8	yes	—
175	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
176	C₂²×C₄	C₂×C₄×D₅	1	12	yes	—
177	C₂²×C₄	C₂³×D₅	1	3	yes	—
178	M₁₆	—	5	128	no	—
179	C₂×C₈	—	5	48	no	—
180	C₂×C₈	—	5	8	no	—
181	C₂×C₈	C₂×C₄×D₅	1	48	yes	complemented
182	C₂×C₈	C₂³×D₅	1	8	yes	—
183	C₄²	—	5	32	no	—
184	C₄²	—	5	12	no	—
185	C₄²	—	5	4	no	—
186	C₄²	C₂×C₄×D₅	1	12	yes	—
187	C₄²	C₂³×D₅	1	4	yes	—
188	C₂×C₅	C₂×C₄×M₁₆	1	12	yes	complemented
189	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
190	C₂×C₅	C₂³×M₁₆	1	2	yes	—
191	D₅	—	2	8	no	—
192	C₂³	—	5	24	no	—
193	C₂³	—	5	4	no	—
194	C₂³	—	10	24	no	—
195	C₂³	—	10	32	no	—
196	C₂³	C₄²×D₅	1	4	yes	—
197	C₂³	C₂×#676	1	8	yes	—
198	C₂³	C₂²×C₄×D₅	1	3	yes	—
199	C₂×C₄	—	5	24	no	—
200	C₂×C₄	—	5	4	no	—
201	C₂×C₄	—	5	48	no	—
202	C₂×C₄	—	5	16	no	—
203	C₂×C₄	—	5	8	no	—
204	C₂×C₄	—	10	24	no	—
205	C₂×C₄	—	10	64	no	—
206	C₂×C₄	C₄²×D₅	1	12	yes	—
207	C₂×C₄	C₂×#676	1	48	yes	direct product factor
208	C₂×C₄	C₂²×C₄×D₅	1	3	yes	—
209	C₂×C₄	C₂²×C₄×D₅	1	12	yes	—
210	C₂×C₄	C₂²×C₄×D₅	1	8	yes	—
211	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
212	C₈	—	5	16	no	—
213	C₈	C₂²×C₄×D₅	1	16	yes	complemented
214	C₅	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
215	C₂²	—	5	8	no	—
216	C₂²	—	10	48	no	—
217	C₂²	—	10	8	no	—
218	C₂²	C₄×#676	1	16	yes	direct product factor
219	C₂²	C₂×C₄²×D₅	1	6	yes	—
220	C₂²	C₂²×#676	1	12	yes	—
221	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
222	C₄	—	5	8	no	—
223	C₄	—	10	8	no	—
224	C₄	C₂×C₄²×D₅	1	6	yes	—
225	C₄	C₂²×#676	1	16	yes	direct product factor
226	C₄	C₂³×C₄×D₅	1	2	yes	—
227	C₂	—	10	8	no	—
228	C₂	C₂×C₄×#676	1	12	yes	direct product factor
229	C₂	C₂²×C₄²×D₅	1	1	yes	characteristic
230	C₂	C₂³×#676	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅
commutator	C₂×C₅	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁴×D₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₅×C₈	C₂	semidirect
4. C₂²×#3947	C₂	semidirect
5. C₂²×C₄×#168	C₂	semidirect
6. C₂×C₄×#676	C₂	direct (N × H)
14. C₂²×#676	C₄	direct (N × H)
18. C₂×C₄×C₅×C₈	C₂²	semidirect
19. C₂×#3947	C₂²	semidirect
20. C₂×C₄×#168	C₂²	semidirect
21. C₄×#676	C₂²	direct (N × H)
33. C₂²×C₅×C₈	C₂×C₄	semidirect
40. C₂²×#168	C₂×C₄	semidirect
42. C₂×#676	C₂×C₄	direct (N × H)
46. C₄×C₅×C₈	C₂³	semidirect
47. #3947	C₂³	semidirect
48. C₄×#168	C₂³	semidirect
54. C₂²×C₄×C₈	D₅	semidirect
59. C₂²×C₄×C₅	M₁₆	semidirect
74. C₂×C₅×C₈	C₂²×C₄	semidirect
81. C₂×#168	C₂²×C₄	semidirect
83. #676	C₂²×C₄	direct (N × H)
98. C₂×C₄×C₈	C₂×D₅	semidirect
107. C₂×C₄×C₅	C₂×M₁₆	semidirect
119. C₅×C₈	C₂³×C₄	semidirect
120. #168	C₂³×C₄	semidirect
135. C₂²×C₈	C₄×D₅	semidirect
146. C₄×C₈	C₂²×D₅	semidirect
147. C₂²×C₅	C₄×M₁₆	semidirect
155. C₄×C₅	C₂²×M₁₆	semidirect
173. C₂²×C₄	#676	direct (N × H)
181. C₂×C₈	C₂×C₄×D₅	semidirect
188. C₂×C₅	C₂×C₄×M₁₆	semidirect
207. C₂×C₄	C₂×#676	direct (N × H)
213. C₈	C₂²×C₄×D₅	semidirect
214. C₅	C₂²×C₄×M₁₆	semidirect
218. C₂²	C₄×#676	direct (N × H)
225. C₄	C₂²×#676	direct (N × H)
228. C₂	C₂×C₄×#676	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×#676	2
C₂²×C₄²×D₅	2
C₂²×C₄×C₅×C₈	2
C₂²×#3947	2
C₂²×C₄×#168	2
C₂×C₄×#676	2
C₂²×C₄×M₁₆	5

Subgroups — C22×C4×#676 #12962685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#676 #12962685