Subgroups — C₂×C₄×#3944 #12251931

Order 1280.

918 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#3944	C₂	1	2	yes	maximal
3	C₄×#3944	C₂	1	8	yes	maximal, direct product factor
4	C₂×#64681	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#672	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3944	C₄	1	16	yes	direct product factor
16	C₂×#3944	C₂²	1	8	yes	—
17	C₂²×#672	C₂²	1	4	yes	—
18	#64681	C₂²	1	8	yes	complemented
19	C₄×#672	C₂²	1	8	yes	complemented
20	C₂×C₄×M₃₂	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	C₈	1	2	yes	—
22	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	C₈	1	8	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	2	yes	—
32	C₂×C₄²×C₅	C₈	1	4	yes	—
33	C₂×C₄²×C₅	C₈	1	2	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₅×C₈	C₂³	1	2	yes	—
42	C₄×C₅×C₈	C₂³	1	2	yes	—
43	#3944	C₂×C₄	1	32	yes	direct product factor
44	C₂×#672	C₂×C₄	1	16	yes	complemented
45	C₂×#672	C₂³	1	8	yes	—
46	C₂²×M₃₂	—	5	2	no	—
47	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
48	C₄×M₃₂	—	5	8	no	—
49	C₂×#346	—	5	2	no	—
50	C₂×C₄×C₁₆	—	5	2	no	—
51	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅	—	2	8	no	—
53	C₂²×C₄×C₅	C₄²	1	4	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅	C₂×C₈	1	8	yes	—
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
59	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₄²	1	8	yes	—
63	C₂×C₅×C₈	C₄²	1	4	yes	—
64	C₂×C₅×C₈	C₂×C₈	1	16	yes	—
65	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
66	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
67	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
69	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
74	C₄²×C₅	C₂×C₈	1	2	yes	—
75	C₄²×C₅	C₂×C₈	1	8	yes	—
76	C₄²×C₅	C₂×C₈	1	2	yes	—
77	C₄²×C₅	C₂²×C₄	1	2	yes	—
78	C₄²×C₅	C₂²×C₄	1	2	yes	—
79	#672	C₂²×C₄	1	16	yes	complemented
80	C₂³×C₈	Dic₅	1	1	yes	characteristic
81	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂²×C₄²	Dic₅	1	1	yes	characteristic
83	C₂×M₃₂	—	5	16	no	—
84	C₂×M₃₂	—	5	8	no	—
85	C₂²×C₁₆	—	5	4	no	—
86	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
87	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
88	C₂×C₄×C₈	Dic₅	1	4	yes	—
89	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₅	1	4	yes	—
91	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
92	#346	—	5	8	no	—
93	C₄×C₁₆	—	5	8	no	—
94	C₂³×C₅	—	2	4	no	—
95	C₂³×C₅	C₄×C₈	1	4	yes	—
96	C₂³×C₅	C₂²×C₈	1	2	yes	—
97	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	16	no	—
100	C₂×C₄×C₅	C₄×C₈	1	8	yes	—
101	C₂×C₄×C₅	C₄×C₈	1	4	yes	—
102	C₂×C₄×C₅	M₃₂	1	8	yes	complemented
103	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
104	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
111	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
112	C₅×C₈	C₂×C₄²	1	2	yes	—
113	C₅×C₈	C₂×C₄²	1	4	yes	—
114	C₅×C₈	C₂²×C₈	1	4	yes	—
115	C₅×C₈	C₂²×C₈	1	4	yes	—
116	C₅×C₈	C₂³×C₄	1	2	yes	—
117	C₂³×C₄	#168	1	2	yes	—
118	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
119	C₂²×C₈	#168	1	8	yes	—
120	C₂²×C₈	C₄×D₅	1	8	yes	—
121	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
122	C₂²×C₈	C₂×Dic₅	1	4	yes	—
123	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
124	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₅	1	2	yes	—
126	C₂²×C₈	C₂²×D₅	1	2	yes	—
127	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₂×C₄²	#168	1	4	yes	—
129	C₂×C₄²	#168	1	2	yes	—
130	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₅	1	4	yes	—
132	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
133	M₃₂	—	5	32	no	—
134	C₂×C₁₆	—	5	16	no	—
135	C₂×C₁₆	—	5	8	no	—
136	C₄×C₈	C₂×Dic₅	1	2	yes	—
137	C₄×C₈	C₂×Dic₅	1	4	yes	—
138	C₄×C₈	C₂×Dic₅	1	4	yes	—
139	C₄×C₈	C₂×Dic₅	1	2	yes	—
140	C₄×C₈	C₂²×D₅	1	2	yes	—
141	C₄×C₈	C₂²×D₅	1	2	yes	—
142	C₂²×C₅	—	2	8	no	—
143	C₂²×C₅	—	2	4	no	—
144	C₂²×C₅	C₂×C₄×C₈	1	4	yes	—
145	C₂²×C₅	C₂×C₄×C₈	1	2	yes	—
146	C₂²×C₅	C₂×M₃₂	1	4	yes	—
147	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₅	—	2	4	no	—
149	C₄×C₅	C₂×C₄×C₈	1	2	yes	—
150	C₄×C₅	C₂×C₄×C₈	1	4	yes	—
151	C₄×C₅	C₂×M₃₂	1	8	yes	complemented
152	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
153	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
154	C₂⁴	C₂×#168	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	C₂×#168	1	2	yes	—
157	C₂²×C₄	C₂×#168	1	8	yes	—
158	C₂²×C₄	C₂×#168	1	2	yes	—
159	C₂²×C₄	C₂×#168	1	1	yes	characteristic
160	C₂²×C₄	C₂×#168	1	4	yes	—
161	C₂²×C₄	C₂×#168	1	2	yes	—
162	C₂²×C₄	C₄×Dic₅	1	4	yes	—
163	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
165	C₂×C₈	C₂×#168	1	16	yes	—
166	C₂×C₈	C₂×#168	1	4	yes	—
167	C₂×C₈	C₂×#168	1	4	yes	—
168	C₂×C₈	C₄×Dic₅	1	8	yes	—
169	C₂×C₈	C₄×Dic₅	1	4	yes	—
170	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
171	C₂×C₈	C₂×C₄×D₅	1	8	yes	—
172	C₂×C₈	C₂²×Dic₅	1	2	yes	—
173	C₂×C₈	C₂²×Dic₅	1	2	yes	—
174	C₂×C₈	C₂²×Dic₅	1	2	yes	—
175	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
176	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
177	C₄²	C₂×#168	1	2	yes	—
178	C₄²	C₂×#168	1	8	yes	—
179	C₄²	C₂×#168	1	2	yes	—
180	C₄²	C₂²×Dic₅	1	2	yes	—
181	C₄²	C₂²×Dic₅	1	2	yes	—
182	C₁₆	—	5	16	no	—
183	C₂×C₅	—	2	4	no	—
184	C₂×C₅	C₄×M₃₂	1	4	yes	complemented
185	C₂×C₅	C₂²×C₄×C₈	1	1	yes	characteristic
186	C₂×C₅	C₂²×M₃₂	1	2	yes	—
187	C₂³	—	2	4	no	—
188	C₂³	C₄×#168	1	4	yes	—
189	C₂³	C₂²×#168	1	2	yes	—
190	C₂³	C₂²×#168	1	1	yes	characteristic
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	C₄×#168	1	8	yes	—
194	C₂×C₄	C₄×#168	1	4	yes	—
195	C₂×C₄	#3944	1	8	yes	direct product factor
196	C₂×C₄	C₂²×#168	1	4	yes	—
197	C₂×C₄	C₂²×#168	1	4	yes	—
198	C₂×C₄	C₂²×#168	1	2	yes	—
199	C₂×C₄	C₂²×#168	1	4	yes	—
200	C₂×C₄	C₂²×#168	1	1	yes	characteristic
201	C₂×C₄	C₂²×#168	1	2	yes	—
202	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
203	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
204	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
205	C₈	C₂²×#168	1	4	yes	—
206	C₈	C₂²×#168	1	4	yes	—
207	C₈	C₂×C₄×Dic₅	1	2	yes	—
208	C₈	C₂×C₄×Dic₅	1	4	yes	—
209	C₈	C₂²×C₄×D₅	1	2	yes	—
210	C₅	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	8	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₂×C₄×#168	1	4	yes	—
214	C₂²	C₂×C₄×#168	1	2	yes	—
215	C₂²	C₂×#3944	1	4	yes	—
216	C₂²	C₂³×#168	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	C₂×C₄×#168	1	2	yes	—
219	C₄	C₂×C₄×#168	1	4	yes	—
220	C₄	C₂×#3944	1	8	yes	direct product factor
221	C₄	C₂³×#168	1	1	yes	characteristic
222	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₄×#3944	1	4	yes	direct product factor
225	C₂	C₂²×C₄×#168	1	1	yes	characteristic
226	C₂	C₂²×#3944	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₅
commutator	C₂×C₅	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#3944	C₂	direct (N × H)
4. C₂×#64681	C₂	semidirect
5. C₂×C₄×#672	C₂	semidirect
15. C₂×#3944	C₄	direct (N × H)
18. #64681	C₂²	semidirect
19. C₄×#672	C₂²	semidirect
43. #3944	C₂×C₄	direct (N × H)
44. C₂×#672	C₂×C₄	semidirect
79. #672	C₂²×C₄	semidirect
102. C₂×C₄×C₅	M₃₂	semidirect
151. C₄×C₅	C₂×M₃₂	semidirect
184. C₂×C₅	C₄×M₃₂	semidirect
195. C₂×C₄	#3944	direct (N × H)
210. C₅	C₂×C₄×M₃₂	semidirect
220. C₄	C₂×#3944	direct (N × H)
224. C₂	C₄×#3944	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂²×#3944	2
C₄×#3944	2
C₂×#64681	2
C₂×C₄×#672	2
C₂×C₄×M₃₂	5

Subgroups — C2×C4×#3944 #12251931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#3944 #12251931