Subgroups — C₂×#11746316 #14314171

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 316 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×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746316	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64715	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	—	2	1	no	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×D₁₂₀	—	2	1	no	—
10	C₂×D₁₂₀	—	2	1	no	—
11	C₂×D₁₂₀	—	2	1	no	—
12	C₂×D₁₂₀	C₄	1	2	yes	—
13	C₂×D₁₂₀	C₄	1	2	yes	—
14	C₂×D₁₂₀	C₂²	1	2	yes	—
15	C₃×C₅×#108	—	2	1	no	—
16	C₃×C₅×#108	C₂²	1	2	yes	complemented
17	#91096	—	2	1	no	—
18	#91096	C₂²	1	2	yes	complemented
19	C₂×C₃⋊#672	—	2	1	no	—
20	C₂×#4495	—	5	1	no	maximal, Hall
21	C₂²×D₄₀	—	3	1	no	—
22	C₂×C₅×#108	D₃	1	1	yes	characteristic, complemented
23	C₂×#3944	—	3	1	no	—
24	#64715	—	3	4	no	—
25	C₂³×D₁₅	—	4	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×D₆₀	—	2	2	no	—
28	C₂×D₆₀	—	2	2	no	—
29	C₂×D₆₀	—	2	2	no	—
30	C₂×D₆₀	—	4	1	no	—
31	C₂×D₆₀	—	4	1	no	—
32	C₂×D₆₀	—	4	1	no	—
33	C₃×C₅×M₁₆	—	2	2	no	—
34	C₃×C₅×M₁₆	—	4	1	no	—
35	C₂×C₃×C₅×C₈	—	4	1	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	D₁₂₀	—	2	2	no	—
43	D₁₂₀	—	2	2	no	—
44	D₁₂₀	—	2	2	no	—
45	D₁₂₀	C₂×C₄	1	4	yes	—
46	C₃⋊#672	—	2	2	no	—
47	C₂²×D₂₄	—	5	1	no	—
48	C₂×C₃×#108	D₅	1	1	yes	characteristic, complemented
49	C₂×#812	—	5	1	no	—
50	#4495	—	5	4	no	—
51	C₂²×D₂₀	—	6	1	no	—
52	C₂×C₅×M₁₆	—	2	1	no	—
53	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×D₄₀	—	3	2	no	—
55	C₂×D₄₀	—	3	2	no	—
56	C₂×D₄₀	—	3	2	no	—
57	C₂×D₄₀	—	6	1	no	—
58	C₂×D₄₀	—	6	1	no	—
59	C₂×D₄₀	—	6	1	no	—
60	C₅×#108	—	2	1	no	—
61	C₅×#108	C₂×D₃	1	2	yes	complemented
62	#3944	—	3	2	no	—
63	#3944	—	6	1	no	—
64	C₂×#672	—	6	1	no	—
65	C₂×#361	—	15	1	no	Sylow
66	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
67	C₂²×D₁₅	—	4	2	no	—
68	C₂²×D₁₅	—	4	2	no	—
69	C₂²×D₁₅	—	4	2	no	—
70	C₂²×D₁₅	—	8	4	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	D₆₀	—	2	4	no	—
78	D₆₀	—	4	2	no	—
79	D₆₀	—	4	2	no	—
80	D₆₀	—	4	2	no	—
81	C₃×C₅×C₈	—	4	2	no	—
82	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₁₂	—	10	1	no	—
87	C₂×C₃×M₁₆	—	2	1	no	—
88	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
89	C₂×D₂₄	—	5	2	no	—
90	C₂×D₂₄	—	5	2	no	—
91	C₂×D₂₄	—	5	2	no	—
92	C₂×D₂₄	—	10	1	no	—
93	C₂×D₂₄	—	10	1	no	—
94	C₂×D₂₄	—	10	1	no	—
95	C₃×#108	—	2	1	no	—
96	C₃×#108	C₂×D₅	1	2	yes	complemented
97	#812	—	5	2	no	—
98	#812	—	10	1	no	—
99	C₂×C₃⋊C₁₆	—	10	1	no	—
100	C₂³×D₅	—	12	1	no	—
101	C₂²×C₄×C₅	#67	1	1	yes	characteristic
102	C₂×D₂₀	—	6	2	no	—
103	C₂×D₂₀	—	6	2	no	—
104	C₂×D₂₀	—	6	2	no	—
105	C₂×D₂₀	—	12	1	no	—
106	C₂×D₂₀	—	12	1	no	—
107	C₂×D₂₀	—	12	1	no	—
108	C₅×M₁₆	—	2	2	no	—
109	C₅×M₁₆	—	4	1	no	—
110	C₂×C₅×C₈	—	4	1	no	—
111	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
112	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
113	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
114	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
115	C₂×C₅×C₈	#67	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
117	D₄₀	—	3	4	no	—
118	D₄₀	—	6	2	no	—
119	D₄₀	—	6	2	no	—
120	D₄₀	—	6	2	no	—
121	#672	—	6	2	no	—
122	C₂²×D₈	—	15	1	no	—
123	C₂×M₃₂	—	15	1	no	—
124	C₂×#108	D₁₅	1	1	yes	characteristic, complemented
125	#361	—	15	4	no	—
126	C₂²×C₃×C₅	—	2	2	no	—
127	C₂²×C₃×C₅	#102	1	1	yes	characteristic
128	C₂²×C₃×C₅	#102	1	1	yes	characteristic
129	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×D₁₅	—	4	4	no	—
131	C₂×D₁₅	—	8	4	no	—
132	C₂×D₁₅	—	8	4	no	—
133	C₂×D₁₅	—	8	4	no	—
134	C₃×C₄×C₅	#102	1	1	yes	characteristic
135	C₃×C₄×C₅	#102	1	1	yes	characteristic
136	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
138	C₂³×D₃	—	20	1	no	—
139	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
140	C₂×D₁₂	—	10	2	no	—
141	C₂×D₁₂	—	10	2	no	—
142	C₂×D₁₂	—	10	2	no	—
143	C₂×D₁₂	—	20	1	no	—
144	C₂×D₁₂	—	20	1	no	—
145	C₂×D₁₂	—	20	1	no	—
146	C₃×M₁₆	—	2	2	no	—
147	C₃×M₁₆	—	4	1	no	—
148	C₂×C₃×C₈	—	4	1	no	—
149	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
150	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
151	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
152	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
153	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
154	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
155	D₂₄	—	5	4	no	—
156	D₂₄	—	10	2	no	—
157	D₂₄	—	10	2	no	—
158	D₂₄	—	10	2	no	—
159	C₃⋊C₁₆	—	10	2	no	—
160	C₂³×C₅	#215	1	1	yes	characteristic
161	C₂²×D₅	—	12	2	no	—
162	C₂²×D₅	—	12	2	no	—
163	C₂²×D₅	—	12	2	no	—
164	C₂²×D₅	—	24	4	no	—
165	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	#213	1	1	yes	characteristic
168	C₂×C₄×C₅	#213	1	1	yes	characteristic
169	C₂×C₄×C₅	#215	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
171	D₂₀	—	6	4	no	—
172	D₂₀	—	12	2	no	—
173	D₂₀	—	12	2	no	—
174	D₂₀	—	12	2	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
179	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
180	C₂²×D₄	—	30	1	no	—
181	C₂×D₈	—	15	2	no	—
182	C₂×D₈	—	15	2	no	—
183	C₂×D₈	—	15	2	no	—
184	C₂×D₈	—	30	1	no	—
185	C₂×D₈	—	30	1	no	—
186	C₂×D₈	—	30	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
189	M₃₂	—	15	2	no	—
190	M₃₂	—	30	1	no	—
191	C₂×C₁₆	—	30	1	no	—
192	#108	—	2	1	no	—
193	#108	C₂×D₁₅	1	2	yes	complemented
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	#361	1	2	yes	complemented
197	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
198	D₁₅	—	8	4	no	—
199	C₂³×C₃	#688	1	1	yes	characteristic
200	C₂²×D₃	—	20	2	no	—
201	C₂²×D₃	—	20	2	no	—
202	C₂²×D₃	—	20	2	no	—
203	C₂²×D₃	—	40	4	no	—
204	C₂×C₃×C₄	#685	1	1	yes	characteristic
205	C₂×C₃×C₄	#685	1	1	yes	characteristic
206	C₂×C₃×C₄	#686	1	1	yes	characteristic
207	C₂×C₃×C₄	#686	1	1	yes	characteristic
208	C₂×C₃×C₄	#688	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	D₁₂	—	10	4	no	—
211	D₁₂	—	20	2	no	—
212	D₁₂	—	20	2	no	—
213	D₁₂	—	20	2	no	—
214	C₃×C₈	—	4	2	no	—
215	C₃×C₈	#685	1	1	yes	characteristic
216	C₃×C₈	#685	1	1	yes	characteristic
217	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
218	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	#809	1	1	yes	characteristic
221	C₂²×C₅	#809	1	1	yes	characteristic
222	C₂²×C₅	C₂×#215	1	1	yes	characteristic
223	C₂×D₅	—	12	4	no	—
224	C₂×D₅	—	24	4	no	—
225	C₂×D₅	—	24	4	no	—
226	C₂×D₅	—	24	4	no	—
227	C₄×C₅	#809	1	1	yes	characteristic
228	C₄×C₅	#809	1	1	yes	characteristic
229	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
230	C₄×C₅	C₂×#213	1	1	yes	characteristic
231	C₂⁴	—	60	1	no	—
232	C₂×D₄	—	30	2	no	—
233	C₂×D₄	—	30	2	no	—
234	C₂×D₄	—	30	2	no	—
235	C₂×D₄	—	60	1	no	—
236	C₂×D₄	—	60	1	no	—
237	C₂×D₄	—	60	1	no	—
238	C₂²×C₄	#1220	1	1	yes	characteristic
239	D₈	—	15	4	no	—
240	D₈	—	30	2	no	—
241	D₈	—	30	2	no	—
242	D₈	—	30	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
247	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
248	C₂×C₈	D₆₀	1	1	yes	characteristic
249	C₂×C₈	D₆₀	1	1	yes	characteristic
250	C₂×C₈	#1220	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
252	C₁₆	—	30	2	no	—
253	C₃×C₅	C₂×#361	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	#3941	1	1	yes	characteristic
256	C₂²×C₃	#3941	1	1	yes	characteristic
257	C₂²×C₃	C₂×#688	1	1	yes	characteristic
258	C₂×D₃	—	20	4	no	—
259	C₂×D₃	—	40	4	no	—
260	C₂×D₃	—	40	4	no	—
261	C₂×D₃	—	40	4	no	—
262	C₃×C₄	#3941	1	1	yes	characteristic
263	C₃×C₄	#3941	1	1	yes	characteristic
264	C₃×C₄	C₂×#685	1	1	yes	characteristic
265	C₃×C₄	C₂×#686	1	1	yes	characteristic
266	C₂×C₅	—	2	1	no	—
267	C₂×C₅	—	2	1	no	—
268	C₂×C₅	#4495	1	2	yes	complemented
269	C₂×C₅	C₂×#809	1	1	yes	characteristic
270	D₅	—	24	4	no	—
271	C₂³	—	60	2	no	—
272	C₂³	—	60	2	no	—
273	C₂³	—	60	2	no	—
274	C₂³	—	120	4	no	—
275	C₂³	#6717	1	1	yes	characteristic
276	D₄	—	30	4	no	—
277	D₄	—	60	2	no	—
278	D₄	—	60	2	no	—
279	D₄	—	60	2	no	—
280	C₂×C₄	#6714	1	1	yes	characteristic
281	C₂×C₄	#6714	1	1	yes	characteristic
282	C₂×C₄	#6715	1	1	yes	characteristic
283	C₂×C₄	#6715	1	1	yes	characteristic
284	C₂×C₄	#6717	1	1	yes	characteristic
285	C₂×C₄	C₂×#1220	1	1	yes	characteristic
286	C₈	—	4	2	no	—
287	C₈	#6714	1	1	yes	characteristic
288	C₈	#6714	1	1	yes	characteristic
289	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
290	C₈	C₂×D₆₀	1	1	yes	characteristic
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	#64715	1	2	yes	complemented
294	C₂×C₃	C₂×#3941	1	1	yes	characteristic
295	D₃	—	40	4	no	—
296	C₅	C₂×#4495	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	60	4	no	—
299	C₂²	—	120	4	no	—
300	C₂²	—	120	4	no	—
301	C₂²	—	120	4	no	—
302	C₂²	#91093	1	1	yes	characteristic
303	C₂²	#91093	1	1	yes	characteristic
304	C₂²	C₂×#6717	1	1	yes	characteristic
305	C₄	#91093	1	1	yes	characteristic
306	C₄	#91093	1	1	yes	characteristic
307	C₄	C₂×#6714	1	1	yes	characteristic
308	C₄	C₂×#6715	1	1	yes	characteristic
309	C₃	C₂×#64715	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	120	4	no	—
313	C₂	#11746316	1	2	yes	direct product factor
314	C₂	C₂×#91093	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#91093
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂²×C₃×C₅	#102
prime radical	C₂²×D₁₂₀	C₂
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂×#108	D₁₅
hypocenter	C₃×C₅	C₂×#361
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#108	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. #11746316	C₂	direct (N × H)
16. C₃×C₅×#108	C₂²	semidirect
18. #91096	C₂²	semidirect
22. C₂×C₅×#108	D₃	semidirect
48. C₂×C₃×#108	D₅	semidirect
61. C₅×#108	C₂×D₃	semidirect
96. C₃×#108	C₂×D₅	semidirect
124. C₂×#108	D₁₅	semidirect
193. #108	C₂×D₁₅	semidirect
196. C₂×C₃×C₅	#361	semidirect
253. C₃×C₅	C₂×#361	semidirect
268. C₂×C₅	#4495	semidirect
293. C₂×C₃	#64715	semidirect
296. C₅	C₂×#4495	semidirect
309. C₃	C₂×#64715	semidirect
313. C₂	#11746316	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×#108	2
C₂×#91096	2
#11746316	2
C₂×#64715	3
C₂×#4495	5

Subgroups — C2×#11746316 #14314171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746316 #14314171