Subgroups — C₈×#3996 #13017369

Order 1280.

374 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#3996	C₂	1	1	yes	maximal, characteristic
2	#10603815	C₂	1	1	yes	maximal, characteristic
3	C₈×#700	C₂	1	2	yes	maximal
4	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
5	#10598531	C₂	1	2	yes	maximal
6	C₂×#3996	C₄	1	2	yes	—
7	#65680	C₂²	1	2	yes	—
8	C₄×#700	C₂²	1	2	yes	—
9	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
12	#64896	C₄	1	2	yes	—
13	#64883	C₄	1	2	yes	—
14	#64882	C₄	1	2	yes	—
15	#170⋊C₈	—	2	2	no	—
16	C₈×#170	—	2	2	no	—
17	C₅×C₈²	C₄	1	1	yes	characteristic
18	C₈×#168	C₄	1	1	yes	characteristic
19	C₈×#168	C₄	1	1	yes	characteristic
20	C₈×#168	C₄	1	1	yes	characteristic
21	C₈×#108	—	5	1	no	maximal, Sylow
22	C₂×#700	C₂×C₄	1	4	yes	—
23	C₂×C₈×D₅	—	2	1	no	—
24	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
27	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
28	C₄²×D₅	C₂³	1	1	yes	characteristic
29	#4003	—	2	2	no	—
30	#4002	—	2	2	no	—
31	C₄⋊#170	C₂×C₄	1	4	yes	—
32	C₄×#170	—	2	2	no	—
33	C₄×#170	—	2	2	no	—
34	#3996	C₈	1	8	yes	direct product factor
35	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₅	—	2	1	no	—
39	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₅	D₄	1	1	yes	characteristic
42	C₈×Dic₅	D₄	1	1	yes	characteristic
43	C₄×#168	C₂×C₄	1	1	yes	characteristic
44	C₄×#168	C₂×C₄	1	1	yes	characteristic
45	C₄×#168	C₂×C₄	1	1	yes	characteristic
46	C₄×#108	—	5	1	no	—
47	#1564	—	5	1	no	—
48	C₈×M₁₆	—	5	2	no	—
49	C₂×C₈²	—	5	1	no	—
50	#1278	—	5	2	no	—
51	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×D₅	#41	1	1	yes	characteristic
54	C₂×#170	—	2	4	no	—
55	C₂×#170	—	2	4	no	—
56	#700	—	2	4	no	—
57	#700	C₂×C₈	1	8	yes	—
58	C₂×C₅×C₈	—	2	1	no	—
59	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
60	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
61	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
66	C₄×Dic₅	#41	1	1	yes	characteristic
67	C₂×#168	—	2	1	no	—
68	C₂×#168	C₄²	1	1	yes	characteristic
69	C₂×#168	C₄²	1	1	yes	characteristic
70	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
71	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
72	C₈×D₅	—	2	4	no	—
73	C₈×D₅	—	2	2	no	—
74	C₈×D₅	—	2	2	no	—
75	C₈×D₅	—	2	2	no	—
76	C₈×D₅	C₂×C₈	1	4	yes	—
77	#431	—	5	2	no	—
78	C₂×#108	—	5	2	no	—
79	C₄×M₁₆	—	5	2	no	—
80	C₂×C₄×C₈	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₂×C₄×C₈	—	5	1	no	—
83	#341	—	5	2	no	—
84	#335	—	5	2	no	—
85	#334	—	5	2	no	—
86	#322	—	10	2	no	—
87	C₈²	—	5	1	no	—
88	C₈²	—	5	1	no	—
89	C₈²	—	5	1	no	—
90	C₈²	—	10	2	no	—
91	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₂²×D₅	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	#117	1	1	yes	characteristic
96	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
97	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
98	C₂×Dic₅	C₄×Q₈	1	1	yes	characteristic
99	C₄×D₅	—	2	1	no	—
100	C₄×D₅	—	2	1	no	—
101	C₄×D₅	—	2	2	no	—
102	C₄×D₅	C₂²×C₈	1	2	yes	—
103	C₄×D₅	#131	1	2	yes	—
104	#170	—	2	8	no	—
105	#170	—	4	4	no	—
106	C₅×C₈	—	2	2	no	—
107	C₅×C₈	—	2	1	no	—
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	C₄×C₈	1	2	yes	—
110	C₅×C₈	#108	1	2	yes	complemented
111	#168	—	2	2	no	—
112	#168	—	2	1	no	—
113	#168	—	2	1	no	—
114	#168	C₄×C₈	1	2	yes	—
115	#168	#108	1	2	yes	complemented
116	C₂×M₁₆	—	5	4	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	—	5	1	no	—
119	C₂²×C₈	—	5	1	no	—
120	C₂²×C₈	—	5	1	no	—
121	C₂²×C₈	—	10	1	no	—
122	C₂×C₄²	—	5	1	no	—
123	#108	—	5	8	no	—
124	C₄⋊C₈	—	5	4	no	—
125	C₂²⋊C₈	—	10	2	no	—
126	#97	—	10	2	no	—
127	C₄×C₈	—	5	1	no	—
128	C₄×C₈	—	5	1	no	—
129	C₄×C₈	—	5	1	no	—
130	C₄×C₈	—	5	1	no	—
131	C₄×C₈	—	5	1	no	—
132	C₄×C₈	—	5	1	no	—
133	C₄×C₈	—	5	1	no	—
134	C₄×C₈	—	10	1	no	—
135	C₄×C₈	—	10	2	no	—
136	C₄×C₈	—	10	2	no	—
137	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂×D₅	—	2	1	no	—
142	C₂×D₅	—	2	1	no	—
143	C₂×D₅	C₈×Q₈	1	2	yes	—
144	C₄×C₅	—	2	1	no	—
145	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₅	#405	1	1	yes	characteristic
147	C₄×C₅	C₂×#108	1	1	yes	characteristic
148	C₄×C₅	C₂×#108	1	1	yes	characteristic
149	Dic₅	—	2	1	no	—
150	Dic₅	C₈×D₄	1	2	yes	—
151	Dic₅	#443	1	2	yes	—
152	C₂²×C₄	—	5	1	no	—
153	C₂²×C₄	—	5	1	no	—
154	C₂²×C₄	—	5	1	no	—
155	M₁₆	—	5	8	no	—
156	M₁₆	—	10	4	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	4	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	5	1	no	—
163	C₂×C₈	—	10	2	no	—
164	C₂×C₈	—	10	2	no	—
165	C₂×C₈	—	10	4	no	—
166	C₂×C₈	—	10	4	no	—
167	C₂×C₈	—	10	1	no	—
168	C₂×C₈	—	10	2	no	—
169	C₂×C₈	—	10	4	no	—
170	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
174	C₄²	—	5	1	no	—
175	C₄²	—	5	1	no	—
176	C₄²	—	5	1	no	—
177	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₅	#1772	1	1	yes	characteristic
180	C₂×C₅	C₄×#108	1	1	yes	characteristic
181	D₅	—	2	2	no	—
182	C₂³	—	5	1	no	—
183	C₂×C₄	—	5	1	no	—
184	C₂×C₄	—	5	1	no	—
185	C₂×C₄	—	5	2	no	—
186	C₂×C₄	—	5	2	no	—
187	C₂×C₄	—	5	1	no	—
188	C₂×C₄	—	10	2	no	—
189	C₂×C₄	—	10	1	no	—
190	C₂×C₄	—	10	1	no	—
191	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₂×C₄	#4130	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	—	5	2	no	—
198	C₈	—	5	2	no	—
199	C₈	—	10	2	no	—
200	C₈	—	10	8	no	—
201	C₈	—	10	1	no	—
202	C₈	—	10	1	no	—
203	C₈	—	20	4	no	—
204	C₈	C₈×AGL(1,𝔽₅)	1	2	yes	—
205	C₈	#3996	1	2	yes	direct product factor
206	C₅	C₈×#108	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	5	2	no	—
208	C₂²	—	10	1	no	—
209	C₂²	—	10	1	no	—
210	C₂²	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	5	2	no	—
213	C₄	—	5	2	no	—
214	C₄	—	10	1	no	—
215	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₄	#65718	1	1	yes	characteristic
217	C₄	C₂×#3996	1	1	yes	characteristic
218	C₄	D₅⋊#108	1	1	yes	characteristic
219	C₂	—	10	2	no	—
220	C₂	C₈×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
221	C₂	#10603851	1	1	yes	characteristic
222	C₂	C₄×#3996	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	C₄×C₄⋊C₄
prime radical	C₂²×D₅	C₄×Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	C₈×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
34. #3996	C₈	direct (N × H)
110. C₅×C₈	#108	semidirect
115. #168	#108	semidirect
205. C₈	#3996	direct (N × H)
206. C₅	C₈×#108	semidirect

Maximal subgroups

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

subgroup	index
C₄×#3996	2
#10603815	2
C₈×#700	2
C₈²×D₅	2
#10598531	2
C₈×#108	5

Subgroups — C8×#3996 #13017369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#3996 #13017369