Subgroups — #108×#170 #13017367

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	M₁₆×#170	C₂	1	2	yes	maximal
2	#10604694	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#170	C₂	1	1	yes	maximal, characteristic
4	Dic₅×#108	C₂	1	1	yes	maximal, characteristic
5	#10598787	C₂	1	2	yes	maximal
6	#65748	C₂²	1	2	yes	—
7	C₂×C₄×#170	C₂²	1	1	yes	characteristic
8	C₂×C₄×#170	C₂²	1	1	yes	characteristic
9	C₂×C₅×#108	C₄	1	1	yes	characteristic
10	M₁₆×Dic₅	C₂²	1	2	yes	—
11	C₂×#3951	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
13	#64882	C₄	1	2	yes	—
14	#64881	C₄	1	2	yes	—
15	#64879	—	2	2	no	—
16	C₈×#170	—	2	2	no	—
17	C₈×#170	C₄	1	2	yes	—
18	C₈×#170	C₄	1	2	yes	—
19	#64778	C₄	1	2	yes	—
20	C₈×#108	—	5	1	no	maximal, Sylow
21	C₂²×#170	—	2	1	no	—
22	C₂²×#170	Q₈	1	2	yes	—
23	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
26	C₂×#681	C₂×C₄	1	2	yes	—
27	C₂²×#168	C₂×C₄	1	1	yes	characteristic
28	C₂²⋊#170	—	2	2	no	—
29	C₄⋊#170	C₂×C₄	1	4	yes	—
30	C₄×#170	—	2	2	no	—
31	C₄×#170	—	2	1	no	—
32	C₄×#170	C₂×C₄	1	2	yes	—
33	C₄×#170	C₂×C₄	1	2	yes	—
34	C₄×#170	D₄	1	1	yes	characteristic
35	C₄×#170	D₄	1	1	yes	characteristic
36	C₅×#108	C₈	1	4	yes	complemented
37	#3951	C₈	1	4	yes	complemented
38	#3947	—	2	2	no	—
39	C₈×Dic₅	—	2	2	no	—
40	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₄×#108	—	5	1	no	—
45	#1564	—	5	1	no	—
46	C₈×M₁₆	—	5	2	no	—
47	C₂×C₈²	—	5	1	no	—
48	#1278	—	5	2	no	—
49	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
51	C₂²×Dic₅	#41	1	1	yes	characteristic
52	C₂×#170	—	2	4	no	—
53	C₂×#170	—	2	2	no	—
54	C₂×#170	—	2	4	no	—
55	C₂×#170	—	2	4	no	—
56	C₂×#170	C₄⋊C₄	1	2	yes	—
57	C₂×#170	C₄⋊C₄	1	2	yes	—
58	C₅×M₁₆	—	2	2	no	—
59	C₅×M₁₆	C₂×C₈	1	4	yes	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
62	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
64	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
65	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
68	C₄×Dic₅	#41	1	1	yes	characteristic
69	#681	—	2	2	no	—
70	#681	C₂×C₈	1	4	yes	—
71	C₂×#168	—	2	2	no	—
72	C₂×#168	—	2	1	no	—
73	C₂×#168	—	2	1	no	—
74	C₂×#168	C₄²	1	1	yes	characteristic
75	C₂×#168	C₄²	1	1	yes	characteristic
76	#431	—	5	2	no	—
77	C₂×#108	—	5	1	no	—
78	C₂×#108	AGL(1,𝔽₅)	1	1	yes	characteristic
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	2	no	—
88	C₈²	—	5	2	no	—
89	C₈²	—	10	2	no	—
90	C₂³×C₅	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
95	C₂×C₄×C₅	#131	1	1	yes	characteristic
96	C₂×C₄×C₅	#131	1	1	yes	characteristic
97	C₂×Dic₅	—	2	2	no	—
98	C₂×Dic₅	—	2	2	no	—
99	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₅	#117	1	1	yes	characteristic
101	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
102	C₂×Dic₅	C₄×Q₈	1	1	yes	characteristic
103	#170	—	2	4	no	—
104	#170	—	2	2	no	—
105	#170	—	4	4	no	—
106	#170	#108	1	4	yes	direct product factor
107	C₅×C₈	—	2	4	no	—
108	C₅×C₈	C₄×C₈	1	2	yes	—
109	C₅×C₈	C₄×C₈	1	2	yes	—
110	#168	—	2	1	no	—
111	#168	—	2	1	no	—
112	#168	—	2	4	no	—
113	C₂×M₁₆	—	5	2	no	—
114	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
115	C₂²×C₈	—	5	2	no	—
116	C₂²×C₈	—	5	1	no	—
117	C₂²×C₈	—	10	1	no	—
118	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₄²	—	5	1	no	—
120	#108	—	5	4	no	—
121	#108	#170	1	4	yes	direct product factor
122	C₄⋊C₈	—	5	4	no	—
123	C₂²⋊C₈	—	10	2	no	—
124	#97	—	10	2	no	—
125	C₄×C₈	—	5	2	no	—
126	C₄×C₈	—	5	1	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	2	no	—
133	C₄×C₈	—	10	2	no	—
134	C₄×C₈	—	10	2	no	—
135	C₄×C₈	—	10	1	no	—
136	C₂²×C₅	—	2	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
140	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
141	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
142	C₄×C₅	#405	1	1	yes	characteristic
143	C₄×C₅	C₈×D₄	1	1	yes	characteristic
144	C₄×C₅	C₈×D₄	1	1	yes	characteristic
145	Dic₅	—	2	1	no	—
146	Dic₅	—	2	1	no	—
147	Dic₅	C₂×#108	1	2	yes	—
148	Dic₅	#443	1	2	yes	—
149	C₂²×C₄	—	5	1	no	—
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	5	4	no	—
154	M₁₆	—	10	2	no	—
155	M₁₆	C₂×#170	1	4	yes	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	5	2	no	—
158	C₂×C₈	—	5	2	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	10	4	no	—
162	C₂×C₈	—	10	2	no	—
163	C₂×C₈	—	10	4	no	—
164	C₂×C₈	—	10	1	no	—
165	C₂×C₈	—	10	2	no	—
166	C₂×C₈	—	10	1	no	—
167	C₂×C₈	—	10	4	no	—
168	C₂×C₈	D₅⋊C₈	1	2	yes	—
169	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₈	C₂×#170	1	2	yes	—
172	C₄²	—	5	1	no	—
173	C₄²	—	5	1	no	—
174	C₄²	—	5	1	no	—
175	C₄²	—	5	1	no	—
176	C₂×C₅	—	2	1	no	—
177	C₂×C₅	—	2	1	no	—
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	C₂³	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂×C₄	—	5	1	no	—
183	C₂×C₄	—	5	1	no	—
184	C₂×C₄	—	5	1	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	10	2	no	—
187	C₂×C₄	—	10	2	no	—
188	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₂×C₄	#4131	1	1	yes	characteristic
191	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₂×C₄	#4133	1	1	yes	characteristic
193	C₂×C₄	C₂²×#170	1	1	yes	characteristic
194	C₈	—	2	4	no	—
195	C₈	—	5	4	no	—
196	C₈	—	10	2	no	—
197	C₈	—	10	4	no	—
198	C₈	—	10	1	no	—
199	C₈	—	10	1	no	—
200	C₈	—	10	4	no	—
201	C₈	—	20	4	no	—
202	C₈	C₄×#170	1	2	yes	—
203	C₈	C₄×#170	1	2	yes	—
204	C₅	C₈×#108	1	1	yes	characteristic, complemented, Sylow
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	#65703	1	1	yes	characteristic
208	C₂²	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₂²	Q₈×#170	1	1	yes	characteristic
210	C₄	—	5	2	no	—
211	C₄	—	5	2	no	—
212	C₄	—	10	1	no	—
213	C₄	—	10	1	no	—
214	C₄	#65701	1	1	yes	characteristic
215	C₄	#65702	1	1	yes	characteristic
216	C₄	C₂×C₄×#170	1	1	yes	characteristic
217	C₄	D₄×#170	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	AGL(1,𝔽₅)×#108	1	1	yes	characteristic
221	C₂	#10604089	1	1	yes	characteristic
222	C₂	C₄⋊C₄×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×#108	C₄
socle	C₂²×C₅	C₄×C₄⋊C₄
prime radical	C₂³×C₅	C₄×Q₈
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	C₂×#108	AGL(1,𝔽₅)
hypocenter	C₅	C₈×#108
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
36. C₅×#108	C₈	semidirect
37. #3951	C₈	semidirect
106. #170	#108	direct (N × H)
121. #108	#170	direct (N × H)
204. 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
M₁₆×#170	2
#10604694	2
C₂×C₈×#170	2
Dic₅×#108	2
#10598787	2
C₈×#108	5