Subgroups — C₂×#11746384 #14314174

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×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746384	C₂	1	4	yes	maximal, direct product factor
5	C₂×#64783	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×D₄	—	2	1	no	—
7	C₂×C₃×C₅×D₈	—	2	1	no	—
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₂×C₃×C₅×D₈	C₄	1	2	yes	—
11	C₂×C₃×C₅×D₈	C₄	1	2	yes	—
12	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6710	—	2	1	no	—
15	#91103	—	2	1	no	—
16	#91103	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×#4563	—	5	1	no	maximal, Hall
21	C₂²×C₅×D₈	D₃	1	1	yes	characteristic
22	C₂×#3951	—	3	1	no	—
23	C₂×#3944	—	3	1	no	—
24	#64783	—	3	4	no	—
25	C₂⁴×C₃×C₅	—	4	1	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂×C₃×C₅×D₄	—	4	1	no	—
31	C₂×C₃×C₅×D₄	—	4	1	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₃×C₅×D₈	—	2	2	no	—
34	C₃×C₅×D₈	—	2	2	no	—
35	C₃×C₅×D₈	—	2	2	no	—
36	C₃×C₅×D₈	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	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	D₄	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
43	#6710	—	2	2	no	—
44	#6710	—	4	1	no	—
45	C₂×C₃⋊#168	—	4	1	no	—
46	C₃⋊#672	—	2	2	no	—
47	C₂²×C₃×D₈	D₅	1	1	yes	characteristic
48	C₂×#819	—	5	1	no	—
49	C₂×#812	—	5	1	no	—
50	#4563	—	5	4	no	—
51	C₂²×C₅×D₄	—	2	1	no	—
52	C₂×C₅×D₈	—	2	1	no	—
53	C₂×C₅×D₈	—	2	1	no	—
54	C₂×C₅×D₈	—	2	1	no	—
55	C₂×C₅×D₈	Dic₃	1	2	yes	—
56	C₂×C₅×D₈	Dic₃	1	2	yes	—
57	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
58	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂×#681	—	6	1	no	—
60	#3951	—	3	2	no	—
61	#3951	—	6	1	no	—
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	—	4	2	no	—
67	C₂³×C₃×C₅	—	4	2	no	—
68	C₂³×C₃×C₅	—	4	2	no	—
69	C₂³×C₃×C₅	—	8	4	no	—
70	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
71	C₃×C₅×D₄	—	2	4	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₃×C₅×D₄	—	4	2	no	—
74	C₃×C₅×D₄	—	4	2	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
85	C₃⋊#168	—	4	2	no	—
86	C₂²×C₃×D₄	—	2	1	no	—
87	C₂×C₃×D₈	—	2	1	no	—
88	C₂×C₃×D₈	—	2	1	no	—
89	C₂×C₃×D₈	—	2	1	no	—
90	C₂×C₃×D₈	Dic₅	1	2	yes	—
91	C₂×C₃×D₈	Dic₅	1	2	yes	—
92	C₂×C₃×D₈	C₂×D₅	1	2	yes	—
93	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#208	—	10	1	no	—
95	#819	—	5	2	no	—
96	#819	—	10	1	no	—
97	#812	—	5	2	no	—
98	#812	—	10	1	no	—
99	C₂×C₃⋊C₁₆	—	10	1	no	—
100	C₂⁴×C₅	—	4	1	no	—
101	C₂×C₅×D₄	—	2	2	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	—	4	1	no	—
105	C₂×C₅×D₄	—	4	1	no	—
106	C₂×C₅×D₄	—	4	1	no	—
107	C₂²×C₄×C₅	#67	1	1	yes	characteristic
108	C₅×D₈	—	2	2	no	—
109	C₅×D₈	—	2	2	no	—
110	C₅×D₈	—	2	2	no	—
111	C₅×D₈	C₂×Dic₃	1	4	yes	—
112	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₂×C₅×C₈	#67	1	1	yes	characteristic
115	C₂×C₅×C₈	#67	1	1	yes	characteristic
116	C₂×C₅×C₈	#67	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
118	#681	—	6	2	no	—
119	#681	—	12	1	no	—
120	C₂×#168	—	12	1	no	—
121	#672	—	6	2	no	—
122	C₂²×D₈	D₁₅	1	1	yes	characteristic
123	C₂×M₃₂	—	15	1	no	—
124	C₂×#108	—	15	1	no	—
125	#361	—	15	4	no	—
126	C₂²×C₃×C₅	—	2	2	no	—
127	C₂²×C₃×C₅	—	4	4	no	—
128	C₂²×C₃×C₅	—	8	4	no	—
129	C₂²×C₃×C₅	—	8	4	no	—
130	C₂²×C₃×C₅	—	8	4	no	—
131	C₂²×C₃×C₅	#102	1	1	yes	characteristic
132	C₂²×C₃×C₅	#102	1	1	yes	characteristic
133	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
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	C₂⁴×C₃	—	4	1	no	—
139	C₂×C₃×D₄	—	2	2	no	—
140	C₂×C₃×D₄	—	2	2	no	—
141	C₂×C₃×D₄	—	2	2	no	—
142	C₂×C₃×D₄	—	4	1	no	—
143	C₂×C₃×D₄	—	4	1	no	—
144	C₂×C₃×D₄	—	4	1	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
146	C₃×D₈	—	2	2	no	—
147	C₃×D₈	—	2	2	no	—
148	C₃×D₈	—	2	2	no	—
149	C₃×D₈	C₂×Dic₅	1	4	yes	—
150	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
151	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
152	C₂×C₃×C₈	C₂²⋊D₅	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	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
156	#208	—	10	2	no	—
157	#208	—	20	1	no	—
158	C₂×C₃⋊C₈	—	20	1	no	—
159	C₃⋊C₁₆	—	10	2	no	—
160	C₂³×C₅	—	4	2	no	—
161	C₂³×C₅	—	4	2	no	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	—	8	4	no	—
164	C₂³×C₅	#214	1	1	yes	characteristic
165	C₅×D₄	—	2	4	no	—
166	C₅×D₄	—	4	2	no	—
167	C₅×D₄	—	4	2	no	—
168	C₅×D₄	—	4	2	no	—
169	C₂×C₄×C₅	#213	1	1	yes	characteristic
170	C₂×C₄×C₅	#213	1	1	yes	characteristic
171	C₂×C₄×C₅	#214	1	1	yes	characteristic
172	C₂×C₄×C₅	#217	1	1	yes	characteristic
173	C₂×C₄×C₅	#217	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
175	C₅×C₈	#217	1	1	yes	characteristic
176	C₅×C₈	#217	1	1	yes	characteristic
177	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
178	C₅×C₈	C₂×#67	1	1	yes	characteristic
179	#168	—	12	2	no	—
180	C₂²×D₄	—	2	1	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	1	no	—
183	C₂×D₈	—	2	1	no	—
184	C₂×D₈	Dic₁₅	1	2	yes	—
185	C₂×D₈	Dic₁₅	1	2	yes	—
186	C₂×D₈	C₂×D₁₅	1	2	yes	—
187	C₂×M₁₆	—	30	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	—	15	2	no	—
193	#108	—	30	1	no	—
194	C₂×C₃×C₅	—	2	1	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	8	4	no	—
197	C₂×C₃×C₅	#361	1	2	yes	complemented
198	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
199	C₂³×C₃	—	4	2	no	—
200	C₂³×C₃	—	4	2	no	—
201	C₂³×C₃	—	4	2	no	—
202	C₂³×C₃	—	8	4	no	—
203	C₂³×C₃	#687	1	1	yes	characteristic
204	C₃×D₄	—	2	4	no	—
205	C₃×D₄	—	4	2	no	—
206	C₃×D₄	—	4	2	no	—
207	C₃×D₄	—	4	2	no	—
208	C₂×C₃×C₄	#686	1	1	yes	characteristic
209	C₂×C₃×C₄	#686	1	1	yes	characteristic
210	C₂×C₃×C₄	#687	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
215	C₃×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
216	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
217	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
218	C₃⋊C₈	—	20	2	no	—
219	C₂²×C₅	—	2	2	no	—
220	C₂²×C₅	—	4	4	no	—
221	C₂²×C₅	—	8	4	no	—
222	C₂²×C₅	—	8	4	no	—
223	C₂²×C₅	—	8	4	no	—
224	C₂²×C₅	#832	1	1	yes	characteristic
225	C₂²×C₅	#832	1	1	yes	characteristic
226	C₂²×C₅	C₂×#214	1	1	yes	characteristic
227	C₄×C₅	#832	1	1	yes	characteristic
228	C₄×C₅	#832	1	1	yes	characteristic
229	C₄×C₅	C₂×#213	1	1	yes	characteristic
230	C₄×C₅	C₂×#217	1	1	yes	characteristic
231	C₂⁴	—	4	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂²×C₄	#1220	1	1	yes	characteristic
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	2	2	no	—
242	D₈	C₂×Dic₁₅	1	4	yes	—
243	M₁₆	—	30	2	no	—
244	M₁₆	—	60	1	no	—
245	C₂×C₈	—	60	1	no	—
246	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
247	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
248	C₂×C₈	#1220	1	1	yes	characteristic
249	C₂×C₈	#1220	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	—	4	4	no	—
256	C₂²×C₃	—	8	4	no	—
257	C₂²×C₃	—	8	4	no	—
258	C₂²×C₃	—	8	4	no	—
259	C₂²×C₃	#3964	1	1	yes	characteristic
260	C₂²×C₃	#3964	1	1	yes	characteristic
261	C₂²×C₃	C₂×#687	1	1	yes	characteristic
262	C₃×C₄	#3964	1	1	yes	characteristic
263	C₃×C₄	#3964	1	1	yes	characteristic
264	C₃×C₄	C₂×#686	1	1	yes	characteristic
265	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
266	C₂×C₅	—	2	1	no	—
267	C₂×C₅	—	2	1	no	—
268	C₂×C₅	—	8	4	no	—
269	C₂×C₅	#4563	1	2	yes	complemented
270	C₂×C₅	C₂×#832	1	1	yes	characteristic
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	8	4	no	—
275	C₂³	#6716	1	1	yes	characteristic
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	#6715	1	1	yes	characteristic
281	C₂×C₄	#6715	1	1	yes	characteristic
282	C₂×C₄	#6716	1	1	yes	characteristic
283	C₂×C₄	#6719	1	1	yes	characteristic
284	C₂×C₄	#6719	1	1	yes	characteristic
285	C₂×C₄	C₂×#1220	1	1	yes	characteristic
286	C₈	—	60	2	no	—
287	C₈	#6719	1	1	yes	characteristic
288	C₈	#6719	1	1	yes	characteristic
289	C₈	C₂²×Dic₁₅	1	1	yes	characteristic
290	C₈	C₂×#1220	1	1	yes	characteristic
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	8	4	no	—
294	C₂×C₃	#64783	1	2	yes	complemented
295	C₂×C₃	C₂×#3964	1	1	yes	characteristic
296	C₅	C₂×#4563	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	8	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	#91116	1	1	yes	characteristic
303	C₂²	#91116	1	1	yes	characteristic
304	C₂²	C₂×#6716	1	1	yes	characteristic
305	C₄	#91116	1	1	yes	characteristic
306	C₄	#91116	1	1	yes	characteristic
307	C₄	C₂×#6715	1	1	yes	characteristic
308	C₄	C₂×#6719	1	1	yes	characteristic
309	C₃	C₂×#64783	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	8	4	no	—
313	C₂	#11746384	1	2	yes	direct product factor
314	C₂	C₂×#91116	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#91116
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₅
Fitting, prime radical	C₂²×C₃×C₅×D₈	C₂
socle	C₂²×C₃×C₅	#102
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×D₈	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 ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91103	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. #11746384	C₂	direct (N × H)
16. #91103	C₂²	semidirect
18. #91096	C₂²	semidirect
197. C₂×C₃×C₅	#361	semidirect
253. C₃×C₅	C₂×#361	semidirect
269. C₂×C₅	#4563	semidirect
294. C₂×C₃	#64783	semidirect
296. C₅	C₂×#4563	semidirect
309. C₃	C₂×#64783	semidirect
313. C₂	#11746384	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×C₃×C₅×D₈	2
C₂×#91103	2
C₂×#91096	2
#11746384	2
C₂×#64783	3
C₂×#4563	5

Subgroups — C2×#11746384 #14314174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11746384 #14314174