Subgroups — C₃×C₈×#700 #14280378

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65680	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#700	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#170⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#170	C₂	1	2	yes	maximal, complemented
6	C₈×#700	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#700	C₄	1	2	yes	—
8	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4003	C₄	1	2	yes	—
12	C₃×#4002	C₄	1	2	yes	—
13	C₃×C₄⋊#170	C₄	1	2	yes	—
14	C₃×C₄×#170	C₂²	1	2	yes	—
15	C₃×C₄×#170	C₂²	1	2	yes	—
16	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
17	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
18	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
19	C₃×C₄×#168	C₄	1	1	yes	characteristic
20	C₃×C₈×M₁₆	—	5	1	no	maximal, Hall
21	#65680	C₂×C₃	1	1	yes	characteristic
22	C₄×#700	C₂×C₃	1	1	yes	characteristic
23	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
24	#170⋊C₈	C₂×C₃	1	2	yes	complemented
25	C₈×#170	C₂×C₃	1	2	yes	complemented
26	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×#170	C₂×C₄	1	4	yes	—
30	C₂×C₃×#170	C₂×C₄	1	4	yes	—
31	C₃×#700	C₈	1	8	yes	direct product factor
32	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
40	C₃×C₈×D₅	—	2	2	no	—
41	C₃×C₈×D₅	—	2	2	no	—
42	C₃×#431	—	5	1	no	—
43	C₃×C₄×M₁₆	—	5	1	no	—
44	C₂×C₃×C₄×C₈	—	5	1	no	—
45	C₃×#322	—	5	2	no	—
46	C₃×C₈²	—	5	2	no	—
47	C₂×#700	C₃×C₄	1	2	yes	—
48	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
50	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
51	#4003	C₃×C₄	1	2	yes	—
52	#4002	C₃×C₄	1	2	yes	—
53	C₄⋊#170	C₃×C₄	1	2	yes	—
54	C₄×#170	C₂²×C₃	1	2	yes	—
55	C₄×#170	C₂²×C₃	1	2	yes	—
56	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
57	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
58	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
59	C₄×#168	C₃×C₄	1	1	yes	characteristic
60	C₈×M₁₆	—	5	1	no	Sylow
61	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
66	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄×D₅	—	2	1	no	—
69	C₃×C₄×D₅	—	2	1	no	—
70	C₃×C₄×D₅	—	2	2	no	—
71	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
72	C₃×#170	—	2	4	no	—
73	C₃×#170	C₂×C₈	1	8	yes	complemented
74	C₃×C₅×C₈	—	2	1	no	—
75	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
76	C₃×#168	—	2	1	no	—
77	C₃×#168	M₁₆	1	2	yes	complemented
78	C₂×C₃×M₁₆	—	5	2	no	—
79	C₂²×C₃×C₈	—	5	1	no	—
80	C₂²×C₃×C₈	—	5	1	no	—
81	C₂×C₃×C₄²	—	5	1	no	—
82	C₃×C₄⋊C₈	—	5	2	no	—
83	C₃×C₂²⋊C₈	—	5	2	no	—
84	C₃×#97	—	5	2	no	—
85	C₃×C₄×C₈	—	5	2	no	—
86	C₃×C₄×C₈	—	5	2	no	—
87	C₃×C₄×C₈	—	5	1	no	—
88	C₃×C₄×C₈	—	5	1	no	—
89	C₃×C₄×C₈	—	5	1	no	—
90	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
91	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×#170	C₂×C₃×C₄	1	4	yes	—
95	C₂×#170	C₂×C₃×C₄	1	4	yes	—
96	#700	C₃×C₈	1	8	yes	direct product factor
97	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
100	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
102	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
103	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
105	C₈×D₅	—	2	2	no	—
106	C₈×D₅	—	2	2	no	—
107	#431	—	5	1	no	—
108	C₄×M₁₆	—	5	1	no	—
109	C₂×C₄×C₈	—	5	1	no	—
110	#322	—	5	2	no	—
111	C₈²	—	5	2	no	—
112	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	—	2	1	no	—
115	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
116	C₃×C₄×C₅	—	2	1	no	—
117	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
118	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
119	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
120	C₃×Dic₅	—	2	1	no	—
121	C₃×Dic₅	C₂²×C₈	1	2	yes	—
122	C₃×Dic₅	#131	1	2	yes	—
123	C₂²×C₃×C₄	—	5	1	no	—
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₃×M₁₆	—	5	8	no	—
127	C₂×C₃×C₈	—	5	4	no	—
128	C₂×C₃×C₈	—	5	1	no	—
129	C₂×C₃×C₈	—	5	4	no	—
130	C₂×C₃×C₈	—	5	1	no	—
131	C₂×C₃×C₈	—	10	2	no	—
132	C₂×C₃×C₈	—	10	2	no	—
133	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₃×C₄²	—	5	1	no	—
136	C₃×C₄²	—	5	1	no	—
137	C₃×C₄²	—	5	1	no	—
138	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₂²×D₅	C₃×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
144	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₄×D₅	—	2	1	no	—
147	C₄×D₅	—	2	1	no	—
148	C₄×D₅	—	2	2	no	—
149	C₄×D₅	C₂×C₃×C₈	1	4	yes	—
150	#170	—	2	4	no	—
151	#170	C₂×C₃×C₈	1	8	yes	complemented
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	C₃×M₁₆	1	2	yes	complemented
154	#168	—	2	1	no	—
155	#168	C₃×M₁₆	1	2	yes	complemented
156	C₂×M₁₆	—	5	2	no	—
157	C₂²×C₈	—	5	1	no	—
158	C₂²×C₈	—	5	1	no	—
159	C₂×C₄²	—	5	1	no	—
160	C₄⋊C₈	—	5	2	no	—
161	C₂²⋊C₈	—	5	2	no	—
162	#97	—	5	2	no	—
163	C₄×C₈	—	5	2	no	—
164	C₄×C₈	—	5	2	no	—
165	C₄×C₈	—	5	1	no	—
166	C₄×C₈	—	5	1	no	—
167	C₄×C₈	—	5	1	no	—
168	C₄×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
169	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
170	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
171	C₂×C₃×C₅	#405	1	1	yes	characteristic
172	C₃×D₅	—	2	2	no	—
173	C₂³×C₃	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	4	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	10	2	no	—
179	C₂×C₃×C₄	—	10	1	no	—
180	C₂×C₃×C₄	—	10	1	no	—
181	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₃×C₈	—	2	1	no	—
185	C₃×C₈	—	5	8	no	—
186	C₃×C₈	—	5	2	no	—
187	C₃×C₈	—	10	4	no	—
188	C₃×C₈	—	10	1	no	—
189	C₃×C₈	#700	1	2	yes	direct product factor
190	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
191	C₂×D₅	—	2	1	no	—
192	C₂×D₅	—	2	1	no	—
193	C₂×D₅	C₃×C₄×C₈	1	2	yes	—
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
196	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
197	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
198	Dic₅	—	2	1	no	—
199	Dic₅	C₂²×C₃×C₈	1	2	yes	—
200	Dic₅	C₃×#131	1	2	yes	—
201	C₂²×C₄	—	5	1	no	—
202	C₂²×C₄	—	5	1	no	—
203	C₂²×C₄	—	5	1	no	—
204	M₁₆	—	5	8	no	—
205	C₂×C₈	—	5	4	no	—
206	C₂×C₈	—	5	1	no	—
207	C₂×C₈	—	5	4	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	10	2	no	—
210	C₂×C₈	—	10	2	no	—
211	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₄²	—	5	1	no	—
214	C₄²	—	5	1	no	—
215	C₄²	—	5	1	no	—
216	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
218	C₂²×C₃	—	5	2	no	—
219	C₂²×C₃	—	10	1	no	—
220	C₂²×C₃	—	10	1	no	—
221	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	5	2	no	—
224	C₃×C₄	—	5	2	no	—
225	C₃×C₄	—	10	1	no	—
226	C₃×C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
227	C₃×C₄	C₂×#700	1	1	yes	characteristic
228	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
229	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
230	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
231	C₂×C₅	C₃×#405	1	1	yes	characteristic
232	D₅	—	2	2	no	—
233	C₂³	—	5	1	no	—
234	C₂×C₄	—	5	1	no	—
235	C₂×C₄	—	5	1	no	—
236	C₂×C₄	—	5	4	no	—
237	C₂×C₄	—	5	1	no	—
238	C₂×C₄	—	10	2	no	—
239	C₂×C₄	—	10	1	no	—
240	C₂×C₄	—	10	1	no	—
241	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	5	8	no	—
246	C₈	—	5	2	no	—
247	C₈	—	10	4	no	—
248	C₈	—	10	1	no	—
249	C₈	C₃×#700	1	2	yes	direct product factor
250	C₂×C₃	—	10	2	no	—
251	C₂×C₃	C₄×#700	1	1	yes	characteristic
252	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₂×C₃	#65718	1	1	yes	characteristic
254	C₅	C₃×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
255	C₂²	—	5	2	no	—
256	C₂²	—	10	1	no	—
257	C₂²	—	10	1	no	—
258	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	5	2	no	—
261	C₄	—	5	2	no	—
262	C₄	—	10	1	no	—
263	C₄	C₃×C₈×AGL(1,𝔽₅)	1	2	yes	—
264	C₄	C₂×C₃×#700	1	1	yes	characteristic
265	C₄	C₃×D₅⋊M₁₆	1	1	yes	characteristic
266	C₃	C₈×#700	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	10	2	no	—
268	C₂	C₃×C₄×#700	1	1	yes	characteristic
269	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₂	C₃×#65718	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₈×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#170⋊C₈	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. C₈×#700	C₃	direct (N × H)
24. #170⋊C₈	C₂×C₃	semidirect
25. C₈×#170	C₂×C₃	semidirect
31. C₃×#700	C₈	direct (N × H)
73. C₃×#170	C₂×C₈	semidirect
75. C₃×C₅×C₈	M₁₆	semidirect
77. C₃×#168	M₁₆	semidirect
96. #700	C₃×C₈	direct (N × H)
151. #170	C₂×C₃×C₈	semidirect
153. C₅×C₈	C₃×M₁₆	semidirect
155. #168	C₃×M₁₆	semidirect
189. C₃×C₈	#700	direct (N × H)
217. C₃×C₅	C₈×M₁₆	semidirect
249. C₈	C₃×#700	direct (N × H)
254. C₅	C₃×C₈×M₁₆	semidirect
266. C₃	C₈×#700	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#65680	2
C₃×C₄×#700	2
C₃×C₄×C₈×D₅	2
C₃×#170⋊C₈	2
C₃×C₈×#170	2
C₈×#700	3
C₃×C₈×M₁₆	5

Subgroups — C3×C8×#700 #14280378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×#700 #14280378