Subgroups — C₂×#11746312 #14313781

Order 1920.

544 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	C₂²×D₁₂₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746312	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64711	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
9	C₂×D₁₂₀	—	2	1	no	—
10	C₂×D₁₂₀	—	2	2	no	—
11	C₂×D₁₂₀	C₄	1	4	yes	complemented
12	C₂×D₁₂₀	C₂²	1	2	yes	—
13	C₃×C₅×#107	—	2	1	no	—
14	C₃×C₅×#107	C₂²	1	2	yes	complemented
15	C₂×C₃⋊#672	—	2	2	no	—
16	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
17	C₂×#4491	—	5	1	no	maximal, Hall
18	C₂²×D₄₀	—	3	1	no	—
19	C₂×C₅×#107	D₃	1	1	yes	characteristic, complemented
20	C₂²×#672	—	3	1	no	—
21	#64711	—	3	4	no	—
22	C₂³×D₁₅	—	4	1	no	—
23	C₂²×C₃×C₄×C₅	—	4	1	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×D₆₀	—	2	4	no	—
26	C₂×D₆₀	—	2	2	no	—
27	C₂×D₆₀	—	4	1	no	—
28	C₂×D₆₀	—	4	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	4	1	no	—
35	D₁₂₀	—	2	4	no	—
36	D₁₂₀	—	2	2	no	—
37	D₁₂₀	C₂×C₄	1	4	yes	complemented
38	C₃⋊#672	—	2	2	no	—
39	C₂²×D₂₄	—	5	1	no	—
40	C₂×C₃×#107	D₅	1	1	yes	characteristic, complemented
41	C₂²×C₃⋊C₁₆	—	5	1	no	—
42	#4491	—	5	4	no	—
43	C₂²×D₂₀	—	6	1	no	—
44	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×D₄₀	—	3	4	no	—
47	C₂×D₄₀	—	3	2	no	—
48	C₂×D₄₀	—	6	2	no	—
49	C₂×D₄₀	—	6	1	no	—
50	C₅×#107	—	2	1	no	—
51	C₅×#107	C₂×D₃	1	2	yes	complemented
52	C₂×#672	—	3	2	no	—
53	C₂×#672	—	6	2	no	—
54	C₂×#357	—	15	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂²×D₁₅	—	4	4	no	—
57	C₂²×D₁₅	—	4	2	no	—
58	C₂²×D₁₅	—	8	2	no	—
59	C₂²×D₁₅	—	8	2	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	8	2	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	D₆₀	—	2	4	no	—
67	D₆₀	—	4	4	no	—
68	D₆₀	—	4	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₂	—	10	1	no	—
73	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂×C₃×C₄⋊C₄	—	2	1	no	—
75	C₂×D₂₄	—	5	4	no	—
76	C₂×D₂₄	—	5	2	no	—
77	C₂×D₂₄	—	10	2	no	—
78	C₂×D₂₄	—	10	1	no	—
79	C₃×#107	—	2	1	no	—
80	C₃×#107	C₂×D₅	1	2	yes	complemented
81	C₂×C₃⋊C₁₆	—	5	2	no	—
82	C₂×C₃⋊C₁₆	—	10	2	no	—
83	C₂³×D₅	—	12	1	no	—
84	C₂²×C₄×C₅	—	4	1	no	—
85	C₂²×C₄×C₅	#67	1	1	yes	characteristic
86	C₂×D₂₀	—	6	4	no	—
87	C₂×D₂₀	—	6	2	no	—
88	C₂×D₂₀	—	12	1	no	—
89	C₂×D₂₀	—	12	2	no	—
90	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
91	C₂×C₅×C₈	D₁₂	1	2	yes	—
92	C₂×C₅×C₈	#67	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	D₄₀	—	3	4	no	—
97	D₄₀	—	6	4	no	—
98	D₄₀	—	6	2	no	—
99	#672	—	6	2	no	—
100	C₂²×D₈	—	15	1	no	—
101	C₂²×C₁₆	—	15	1	no	—
102	C₂×#107	D₁₅	1	1	yes	characteristic, complemented
103	#357	—	15	4	no	—
104	C₂²×C₃×C₅	#102	1	2	yes	—
105	C₂²×C₃×C₅	D₁₆	1	2	yes	—
106	C₂²×C₃×C₅	SD₃₂	1	2	yes	—
107	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
108	C₂×D₁₅	—	4	4	no	—
109	C₂×D₁₅	—	8	8	no	—
110	C₂×D₁₅	—	8	2	no	—
111	C₂×D₁₅	—	8	2	no	—
112	C₃×C₄×C₅	—	8	2	no	—
113	C₃×C₄×C₅	#102	1	2	yes	—
114	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂³×D₃	—	20	1	no	—
117	C₂²×C₃×C₄	—	4	1	no	—
118	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
119	C₂×D₁₂	—	10	4	no	—
120	C₂×D₁₂	—	10	2	no	—
121	C₂×D₁₂	—	20	1	no	—
122	C₂×D₁₂	—	20	2	no	—
123	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
124	C₂×C₃×C₈	D₂₀	1	2	yes	—
125	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	—	4	1	no	—
129	D₂₄	—	5	4	no	—
130	D₂₄	—	10	4	no	—
131	D₂₄	—	10	2	no	—
132	C₃⋊C₁₆	—	10	2	no	—
133	C₂³×C₅	#213	1	1	yes	characteristic
134	C₂²×D₅	—	12	4	no	—
135	C₂²×D₅	—	12	2	no	—
136	C₂²×D₅	—	24	2	no	—
137	C₂²×D₅	—	24	2	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	—	8	2	no	—
140	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
141	C₂×C₄×C₅	#213	1	1	yes	characteristic
142	C₂×C₄×C₅	#215	1	2	yes	—
143	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
144	D₂₀	—	6	4	no	—
145	D₂₀	—	12	4	no	—
146	D₂₀	—	12	2	no	—
147	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
148	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
149	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
150	C₂²×D₄	—	30	1	no	—
151	C₂×D₈	—	15	4	no	—
152	C₂×D₈	—	15	2	no	—
153	C₂×D₈	—	30	2	no	—
154	C₂×D₈	—	30	1	no	—
155	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₁₆	—	15	2	no	—
158	C₂×C₁₆	—	30	2	no	—
159	#107	—	2	1	no	—
160	#107	C₂×D₁₅	1	2	yes	complemented
161	C₂×C₃×C₅	#357	1	4	yes	complemented
162	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
165	D₁₅	—	8	4	no	—
166	C₂³×C₃	#686	1	1	yes	characteristic
167	C₂²×D₃	—	20	4	no	—
168	C₂²×D₃	—	20	2	no	—
169	C₂²×D₃	—	40	2	no	—
170	C₂²×D₃	—	40	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	8	2	no	—
173	C₂×C₃×C₄	#685	1	2	yes	—
174	C₂×C₃×C₄	#686	1	1	yes	characteristic
175	C₂×C₃×C₄	#688	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
177	D₁₂	—	10	4	no	—
178	D₁₂	—	20	4	no	—
179	D₁₂	—	20	2	no	—
180	C₃×C₈	#685	1	2	yes	—
181	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
182	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
183	C₂²×C₅	#809	1	2	yes	—
184	C₂²×C₅	#826	1	2	yes	—
185	C₂²×C₅	#828	1	2	yes	—
186	C₂²×C₅	C₂×#213	1	1	yes	characteristic
187	C₂×D₅	—	12	4	no	—
188	C₂×D₅	—	24	2	no	—
189	C₂×D₅	—	24	2	no	—
190	C₂×D₅	—	24	8	no	—
191	C₄×C₅	—	8	2	no	—
192	C₄×C₅	#809	1	2	yes	—
193	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₄×C₅	C₂×#215	1	1	yes	characteristic
195	C₂⁴	—	60	1	no	—
196	C₂×D₄	—	30	4	no	—
197	C₂×D₄	—	30	2	no	—
198	C₂×D₄	—	60	1	no	—
199	C₂×D₄	—	60	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	#1220	1	1	yes	characteristic
202	D₈	—	15	4	no	—
203	D₈	—	30	4	no	—
204	D₈	—	30	2	no	—
205	C₂×C₈	C₄×D₁₅	1	2	yes	—
206	C₂×C₈	D₆₀	1	2	yes	—
207	C₂×C₈	#1220	1	1	yes	characteristic
208	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₁₆	—	30	2	no	—
212	C₃×C₅	C₂×#357	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	#3941	1	2	yes	—
214	C₂²×C₃	#3958	1	2	yes	—
215	C₂²×C₃	#3960	1	2	yes	—
216	C₂²×C₃	C₂×#686	1	1	yes	characteristic
217	C₂×D₃	—	20	4	no	—
218	C₂×D₃	—	40	2	no	—
219	C₂×D₃	—	40	2	no	—
220	C₂×D₃	—	40	8	no	—
221	C₃×C₄	—	8	2	no	—
222	C₃×C₄	#3941	1	2	yes	—
223	C₃×C₄	C₂×#685	1	1	yes	characteristic
224	C₃×C₄	C₂×#688	1	1	yes	characteristic
225	C₂×C₅	#4491	1	4	yes	complemented
226	C₂×C₅	C₂×#809	1	1	yes	characteristic
227	C₂×C₅	C₂×#826	1	1	yes	characteristic
228	C₂×C₅	C₂×#828	1	1	yes	characteristic
229	D₅	—	24	4	no	—
230	C₂³	—	60	2	no	—
231	C₂³	—	60	4	no	—
232	C₂³	—	120	2	no	—
233	C₂³	—	120	2	no	—
234	C₂³	#6715	1	1	yes	characteristic
235	D₄	—	30	4	no	—
236	D₄	—	60	2	no	—
237	D₄	—	60	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	#6714	1	2	yes	—
241	C₂×C₄	#6715	1	1	yes	characteristic
242	C₂×C₄	#6717	1	2	yes	—
243	C₂×C₄	C₂×#1220	1	1	yes	characteristic
244	C₈	#6714	1	2	yes	—
245	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
246	C₈	C₂×D₆₀	1	1	yes	characteristic
247	C₂×C₃	#64711	1	4	yes	complemented
248	C₂×C₃	C₂×#3941	1	1	yes	characteristic
249	C₂×C₃	C₂×#3958	1	1	yes	characteristic
250	C₂×C₃	C₂×#3960	1	1	yes	characteristic
251	D₃	—	40	4	no	—
252	C₅	C₂×#4491	1	1	yes	characteristic, complemented, Sylow
253	C₂²	—	60	4	no	—
254	C₂²	—	120	8	no	—
255	C₂²	—	120	2	no	—
256	C₂²	—	120	2	no	—
257	C₂²	#91093	1	2	yes	—
258	C₂²	#91110	1	2	yes	—
259	C₂²	#91112	1	2	yes	—
260	C₂²	C₂×#6715	1	1	yes	characteristic
261	C₄	—	8	2	no	—
262	C₄	#91093	1	2	yes	—
263	C₄	C₂×#6714	1	1	yes	characteristic
264	C₄	C₂×#6717	1	1	yes	characteristic
265	C₃	C₂×#64711	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	120	4	no	—
267	C₂	#11746312	1	4	yes	direct product factor
268	C₂	C₂×#91093	1	1	yes	characteristic
269	C₂	C₂×#91110	1	1	yes	characteristic
270	C₂	C₂×#91112	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6715
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#107	C₂
socle	C₂³×C₃×C₅	D₈
prime radical	C₂²×D₁₂₀	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×#107	D₁₅
hypocenter	C₃×C₅	C₂×#357
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#107	C₂	semidirect
3. C₂²×C₃⋊#672	C₂	semidirect
4. #11746312	C₂	direct (N × H)
11. C₂×D₁₂₀	C₄	semidirect
14. C₃×C₅×#107	C₂²	semidirect
16. C₂×C₃⋊#672	C₂²	semidirect
19. C₂×C₅×#107	D₃	semidirect
37. D₁₂₀	C₂×C₄	semidirect
40. C₂×C₃×#107	D₅	semidirect
51. C₅×#107	C₂×D₃	semidirect
80. C₃×#107	C₂×D₅	semidirect
102. C₂×#107	D₁₅	semidirect
160. #107	C₂×D₁₅	semidirect
161. C₂×C₃×C₅	#357	semidirect
212. C₃×C₅	C₂×#357	semidirect
225. C₂×C₅	#4491	semidirect
247. C₂×C₃	#64711	semidirect
252. C₅	C₂×#4491	semidirect
265. C₃	C₂×#64711	semidirect
267. C₂	#11746312	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×D₁₂₀	2
C₂×C₃×C₅×#107	2
C₂²×C₃⋊#672	2
#11746312	2
C₂×#64711	3
C₂×#4491	5

Subgroups — C2×#11746312 #14313781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746312 #14313781