Subgroups — C₂×C₄×#3951 #12097396

Order 1280.

1086 conjugacy classes of subgroups.

The subgroup lattice has 254 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×#3951	C₂	1	2	yes	maximal
3	C₂×C₄×#681	C₂	1	2	yes	maximal
4	C₂×#64772	C₂	1	2	yes	maximal
5	C₄×#3951	C₂	1	8	yes	maximal, direct product factor
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×#681	C₂²	1	4	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
13	C₂×#3951	C₄	1	16	yes	direct product factor
14	C₂×#3951	C₂²	1	8	yes	—
15	C₄×#681	—	2	4	no	—
16	C₄×#681	C₂²	1	8	yes	—
17	C₂×#3935	—	2	2	no	—
18	C₂×C₄×#168	—	2	2	no	—
19	#64772	C₂²	1	8	yes	—
20	C₂×C₄×#108	—	5	1	no	maximal, Sylow
21	C₂³×C₄×C₅	Q₈	1	2	yes	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂²×C₅×C₈	—	2	4	no	—
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	2	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	Q₈	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	2	yes	—
38	C₂×#681	—	2	4	no	—
39	C₂×#681	—	2	8	no	—
40	C₂×#681	C₂×C₄	1	16	yes	—
41	C₂×#681	C₂³	1	8	yes	—
42	C₂²×#168	—	2	4	no	—
43	C₄×C₅×C₈	C₂×C₄	1	16	yes	—
44	#3951	C₂×C₄	1	32	yes	direct product factor
45	#3935	—	2	8	no	—
46	C₄×#168	—	2	8	no	—
47	C₂²×#108	—	5	2	no	—
48	C₂×C₄×M₁₆	—	5	2	no	—
49	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
50	C₄×#108	—	5	8	no	—
51	C₂×#341	—	5	2	no	—
52	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₅	—	2	8	no	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
63	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	#41	1	4	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	4	no	—
69	C₂×C₅×C₈	C₄²	1	8	yes	—
70	C₂×C₅×C₈	C₄²	1	8	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
73	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
74	C₄²×C₅	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅	C₄⋊C₄	1	4	yes	—
76	C₄²×C₅	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅	C₂²×C₄	1	4	yes	—
78	C₄²×C₅	C₂×D₄	1	2	yes	—
79	C₄²×C₅	C₂×D₄	1	2	yes	—
80	#681	—	2	16	no	—
81	#681	—	2	8	no	—
82	#681	C₂²×C₄	1	16	yes	—
83	C₂×#168	—	2	16	no	—
84	C₂×#168	—	2	8	no	—
85	C₂²×M₁₆	—	5	4	no	—
86	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
88	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
89	C₂×#108	—	5	16	no	—
90	C₂×#108	—	5	8	no	—
91	C₄×M₁₆	—	5	8	no	—
92	C₄×M₁₆	—	10	4	no	—
93	C₂×#97	—	10	2	no	—
94	C₂×C₄×C₈	—	10	2	no	—
95	C₂×C₄×C₈	Dic₅	1	4	yes	—
96	C₂×C₄×C₈	Dic₅	1	4	yes	—
97	C₂×C₄×C₈	C₂×D₅	1	4	yes	—
98	#341	—	5	8	no	—
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₅	C₄×Q₈	1	4	yes	—
103	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	8	no	—
106	C₂×C₄×C₅	—	2	8	no	—
107	C₂×C₄×C₅	#108	1	8	yes	complemented
108	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₅	#117	1	2	yes	—
115	C₂×C₄×C₅	#117	1	2	yes	—
116	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
118	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×#41	1	2	yes	—
123	C₅×C₈	—	2	4	no	—
124	C₅×C₈	C₂×C₄²	1	8	yes	—
125	#168	—	2	16	no	—
126	C₂³×C₄	Dic₁₀	1	2	yes	—
127	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₂×M₁₆	—	5	16	no	—
129	C₂×M₁₆	—	5	8	no	—
130	C₂×M₁₆	—	10	4	no	—
131	C₂×M₁₆	—	10	8	no	—
132	C₂²×C₈	—	2	4	no	—
133	C₂²×C₈	—	10	4	no	—
134	C₂²×C₈	C₄×D₅	1	8	yes	—
135	C₂²×C₈	C₂×Dic₅	1	2	yes	—
136	C₂²×C₈	C₂×Dic₅	1	2	yes	—
137	C₂²×C₈	C₂×Dic₅	1	2	yes	—
138	C₂²×C₈	C₂×Dic₅	1	2	yes	—
139	C₂²×C₈	C₂²×D₅	1	2	yes	—
140	C₂²×C₈	C₂²×D₅	1	2	yes	—
141	C₂×C₄²	Dic₁₀	1	2	yes	—
142	C₂×C₄²	D₂₀	1	1	yes	characteristic
143	C₂×C₄²	D₂₀	1	2	yes	—
144	C₂×C₄²	D₂₀	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₅	1	2	yes	—
146	C₂×C₄²	C₂×Dic₅	1	2	yes	—
147	C₂×C₄²	C₂²×D₅	1	2	yes	—
148	#108	—	5	32	no	—
149	#97	—	10	8	no	—
150	C₄×C₈	—	10	8	no	—
151	C₄×C₈	C₂×Dic₅	1	16	yes	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	2	no	—
154	C₂²×C₅	—	2	8	no	—
155	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
156	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
157	C₂²×C₅	C₂×#108	1	4	yes	—
158	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₅	C₂×#108	1	8	yes	complemented
164	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
165	C₄×C₅	C₂×#117	1	1	yes	characteristic
166	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
167	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
170	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
171	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
172	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
173	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
175	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
177	C₂²×C₄	#709	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
179	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
180	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
181	M₁₆	—	5	16	no	—
182	M₁₆	—	10	8	no	—
183	M₁₆	—	10	16	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	10	8	no	—
188	C₂×C₈	—	10	16	no	—
189	C₂×C₈	C₄×Dic₅	1	8	yes	—
190	C₂×C₈	C₄×Dic₅	1	8	yes	—
191	C₂×C₈	C₂×C₄×D₅	1	8	yes	—
192	C₂×C₈	C₂²×Dic₅	1	4	yes	—
193	C₂×C₈	C₂²×Dic₅	1	4	yes	—
194	C₄²	C₄⋊Dic₅	1	2	yes	—
195	C₄²	C₄⋊Dic₅	1	4	yes	—
196	C₄²	C₄⋊Dic₅	1	2	yes	—
197	C₄²	C₂×D₂₀	1	2	yes	—
198	C₄²	C₂×D₂₀	1	2	yes	—
199	C₄²	C₂²×Dic₅	1	4	yes	—
200	C₂×C₅	—	2	2	no	—
201	C₂×C₅	—	2	2	no	—
202	C₂×C₅	C₄×#108	1	4	yes	complemented
203	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₅	C₂²×#108	1	2	yes	—
205	C₂³	—	2	4	no	—
206	C₂³	C₄×Dic₁₀	1	4	yes	—
207	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
208	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
209	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	#3951	1	8	yes	direct product factor
214	C₂×C₄	C₄×Dic₁₀	1	4	yes	—
215	C₂×C₄	C₄×D₂₀	1	4	yes	—
216	C₂×C₄	C₄×D₂₀	1	4	yes	—
217	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
218	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
219	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
220	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
221	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
223	C₂×C₄	#4072	1	2	yes	—
224	C₂×C₄	#4072	1	2	yes	—
225	C₂×C₄	C₂²×C₄×D₅	1	2	yes	—
226	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
227	C₂×C₄	C₂×#709	1	2	yes	—
228	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	—	10	16	no	—
231	C₈	C₂×C₄×Dic₅	1	8	yes	—
232	C₅	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
233	C₂²	—	2	8	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
237	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
238	C₂²	C₂×#3951	1	4	yes	—
239	C₂²	C₂×C₄×Dic₁₀	1	2	yes	—
240	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
243	C₄	C₄×C₄⋊Dic₅	1	2	yes	—
244	C₄	C₂×#3951	1	8	yes	direct product factor
245	C₄	C₂×C₄×D₂₀	1	2	yes	—
246	C₄	C₂²×C₄×Dic₅	1	1	yes	characteristic
247	C₄	C₂×#4072	1	1	yes	characteristic
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	C₄×#3951	1	4	yes	direct product factor
251	C₂	C₂×C₄×C₄⋊Dic₅	1	1	yes	characteristic
252	C₂	C₂²×#3951	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₀
commutator	C₄×C₅	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂×C₄×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. C₄×#3951	C₂	direct (N × H)
13. C₂×#3951	C₄	direct (N × H)
44. #3951	C₂×C₄	direct (N × H)
107. C₂×C₄×C₅	#108	semidirect
163. C₄×C₅	C₂×#108	semidirect
202. C₂×C₅	C₄×#108	semidirect
213. C₂×C₄	#3951	direct (N × H)
232. C₅	C₂×C₄×#108	semidirect
244. C₄	C₂×#3951	direct (N × H)
250. C₂	C₄×#3951	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×#3951	2
C₂×C₄×#681	2
C₂×#64772	2
C₄×#3951	2
C₂×C₄×#108	5

Subgroups — C2×C4×#3951 #12097396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#3951 #12097396