Subgroups — C₂×#11747687 #14413319

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#91347	C₂	1	2	yes	maximal
3	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3949	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90989	C₂	1	2	yes	maximal, complemented
6	#11747687	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65149	—	3	1	no	maximal, Hall
8	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₅	C₄	1	8	yes	complemented
12	C₂×C₈×D₁₅	C₂²	1	4	yes	—
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂×#6667	—	2	2	no	—
16	C₂×Dic₃×Dic₅	—	2	2	no	—
17	#91347	—	2	4	no	—
18	#91347	C₂²	1	8	yes	—
19	C₅×#817	C₂²	1	4	yes	complemented
20	C₃×#3949	C₂²	1	4	yes	complemented
21	#90989	C₂²	1	8	yes	complemented
22	C₂×#4860	—	5	1	no	maximal, Hall
23	C₂²×C₈×D₅	—	3	1	no	—
24	C₂×#4038	—	3	2	no	—
25	C₂×C₅×#106	—	3	1	no	—
26	C₂×#3949	D₃	1	1	yes	characteristic, complemented
27	C₂×#3939	—	3	2	no	—
28	#65149	—	3	8	no	—
29	C₂³×D₁₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₅	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₅	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₅	D₄	1	2	yes	—
34	C₂×C₄×D₁₅	Q₈	1	4	yes	—
35	C₂×C₄×D₁₅	C₂³	1	2	yes	—
36	C₂²×C₅×Dic₃	—	2	2	no	—
37	C₂²×C₃×Dic₅	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
40	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
41	C₂×C₃⋊#168	C₂³	1	2	yes	—
42	C₈×D₁₅	C₂×C₄	1	16	yes	complemented
43	C₅×C₄⋊Dic₃	—	2	2	no	—
44	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
45	C₃×C₄⋊Dic₅	—	2	2	no	—
46	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
47	#6667	—	2	8	no	—
48	Dic₃×Dic₅	—	2	8	no	—
49	C₂²×D₃×C₈	—	5	1	no	—
50	C₂×#892	—	5	2	no	—
51	C₂×C₃×#106	—	5	1	no	—
52	C₂×#817	D₅	1	1	yes	characteristic, complemented
53	C₂×#807	—	5	2	no	—
54	#4860	—	5	8	no	—
55	C₂²×C₄×D₅	—	3	1	no	—
56	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×C₄⋊C₄	—	3	2	no	—
58	C₂×#685	—	6	2	no	—
59	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
60	C₂×C₄×Dic₅	—	6	2	no	—
61	C₂²×#168	—	3	1	no	—
62	C₂×C₈×D₅	—	3	8	no	—
63	C₂×C₈×D₅	—	3	4	no	—
64	#4038	—	3	8	no	—
65	#4038	—	6	4	no	—
66	C₅×#106	—	3	4	no	—
67	#3949	C₂×D₃	1	4	yes	complemented
68	#3939	—	3	8	no	—
69	C₂×#427	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₁₅	—	2	4	no	—
72	C₂²×D₁₅	C₄⋊C₄	1	4	yes	—
73	C₂²×D₁₅	C₂×D₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₁₅	C₄⋊C₄	1	4	yes	—
79	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
80	C₄×D₁₅	C₄⋊C₄	1	8	yes	—
81	C₄×D₁₅	C₂²×C₄	1	4	yes	—
82	C₄×D₁₅	C₂×Q₈	1	4	yes	—
83	C₂×C₅×Dic₃	—	2	4	no	—
84	C₂×C₅×Dic₃	—	4	4	no	—
85	C₂×C₃×Dic₅	—	2	4	no	—
86	C₂×C₃×Dic₅	—	4	4	no	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
88	C₃⋊#168	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×D₃	—	5	1	no	—
90	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
91	C₂×C₃×C₄⋊C₄	—	5	2	no	—
92	C₂×(C₂×D₃)⋊C₄	—	10	2	no	—
93	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
94	C₂×C₄×Dic₃	—	10	2	no	—
95	C₂²×C₃⋊C₈	—	5	1	no	—
96	C₂×D₃×C₈	—	5	8	no	—
97	C₂×D₃×C₈	—	5	4	no	—
98	#892	—	5	8	no	—
99	#892	—	10	4	no	—
100	C₃×#106	—	5	4	no	—
101	#817	C₂×D₅	1	4	yes	complemented
102	#807	—	5	8	no	—
103	C₂³×D₅	—	3	1	no	—
104	C₂²×C₄×C₅	—	6	2	no	—
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	—	2	2	no	—
107	C₂²×Dic₅	—	3	1	no	—
108	C₂×C₄×D₅	—	3	4	no	—
109	C₂×C₄×D₅	—	3	2	no	—
110	C₂×C₄×D₅	—	3	4	no	—
111	C₂×C₄×D₅	—	3	2	no	—
112	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
113	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
114	C₅×C₄⋊C₄	—	3	4	no	—
115	C₅×C₄⋊C₄	—	6	2	no	—
116	#685	—	6	8	no	—
117	C₄⋊Dic₅	—	2	2	no	—
118	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
119	C₄×Dic₅	—	6	8	no	—
120	C₂×#168	—	3	4	no	—
121	C₂×#168	—	3	2	no	—
122	C₈×D₅	—	3	16	no	—
123	C₂³×C₈	—	15	1	no	—
124	C₂×#117	—	15	2	no	—
125	#427	—	15	8	no	—
126	C₂×#107	—	15	2	no	—
127	C₂×#106	—	3	1	no	—
128	C₂×#106	—	5	1	no	—
129	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
130	C₂²×C₃×C₅	#135	1	4	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×D₁₅	—	2	4	no	—
133	C₂×D₁₅	—	2	8	no	—
134	C₂×D₁₅	C₂×C₄⋊C₄	1	4	yes	—
135	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
138	Dic₁₅	C₂×C₄⋊C₄	1	4	yes	—
139	C₃×Dic₅	—	4	4	no	—
140	C₅×Dic₃	—	4	4	no	—
141	C₂³×D₃	—	5	1	no	—
142	C₂²×C₃×C₄	—	10	2	no	—
143	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₂²×Dic₃	—	2	2	no	—
145	C₂²×Dic₃	—	5	1	no	—
146	C₂×C₄×D₃	—	5	4	no	—
147	C₂×C₄×D₃	—	5	2	no	—
148	C₂×C₄×D₃	—	5	4	no	—
149	C₂×C₄×D₃	—	5	2	no	—
150	C₂×C₃×C₈	C₄×D₅	1	4	yes	—
151	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
152	C₃×C₄⋊C₄	—	5	4	no	—
153	C₃×C₄⋊C₄	—	10	2	no	—
154	(C₂×D₃)⋊C₄	—	10	8	no	—
155	C₄⋊Dic₃	—	2	2	no	—
156	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
157	C₄×Dic₃	—	10	8	no	—
158	C₂×C₃⋊C₈	—	5	4	no	—
159	C₂×C₃⋊C₈	—	5	2	no	—
160	D₃×C₈	—	5	16	no	—
161	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
162	C₂²×D₅	—	3	4	no	—
163	C₂²×D₅	—	3	2	no	—
164	C₂²×D₅	—	6	4	no	—
165	C₂×C₄×C₅	—	6	4	no	—
166	C₂×C₄×C₅	—	12	4	no	—
167	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
168	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
170	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
171	C₂×Dic₅	—	2	4	no	—
172	C₂×Dic₅	—	3	4	no	—
173	C₂×Dic₅	—	3	2	no	—
174	C₂×Dic₅	—	4	4	no	—
175	C₄×D₅	—	3	8	no	—
176	C₄×D₅	—	3	4	no	—
177	C₄×D₅	—	3	4	no	—
178	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
179	#168	—	3	4	no	—
180	C₂³×C₄	—	15	1	no	—
181	C₂²×C₈	—	15	1	no	—
182	C₂²×C₈	—	15	8	no	—
183	C₂²×C₈	—	15	4	no	—
184	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
185	#117	—	15	8	no	—
186	#117	—	30	4	no	—
187	C₂×C₄⋊C₄	—	3	2	no	—
188	C₂×C₄⋊C₄	—	5	2	no	—
189	C₂×C₂²⋊C₄	—	30	2	no	—
190	C₂×C₄²	—	30	2	no	—
191	#107	—	15	8	no	—
192	#106	—	3	4	no	—
193	#106	—	5	4	no	—
194	C₂×C₃×C₅	#427	1	4	yes	complemented
195	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×#135	1	2	yes	—
197	D₁₅	—	2	4	no	—
198	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
199	C₂²×D₃	—	5	2	no	—
200	C₂²×D₃	—	5	4	no	—
201	C₂²×D₃	—	10	4	no	—
202	C₂×C₃×C₄	—	10	4	no	—
203	C₂×C₃×C₄	—	20	4	no	—
204	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
205	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
206	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
207	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
208	C₂×Dic₃	—	2	4	no	—
209	C₂×Dic₃	—	4	4	no	—
210	C₂×Dic₃	—	5	2	no	—
211	C₂×Dic₃	—	5	4	no	—
212	C₄×D₃	—	5	4	no	—
213	C₄×D₃	—	5	4	no	—
214	C₄×D₃	—	5	8	no	—
215	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
216	C₃⋊C₈	—	5	4	no	—
217	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
218	C₂²×C₅	#916	1	4	yes	—
219	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
220	C₂×D₅	—	3	4	no	—
221	C₂×D₅	—	6	4	no	—
222	C₂×D₅	—	6	8	no	—
223	C₄×C₅	—	12	4	no	—
224	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
225	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
226	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
227	Dic₅	—	3	4	no	—
228	Dic₅	—	4	4	no	—
229	C₂⁴	—	15	1	no	—
230	C₂²×C₄	—	6	2	no	—
231	C₂²×C₄	—	10	2	no	—
232	C₂²×C₄	—	15	1	no	—
233	C₂²×C₄	—	15	4	no	—
234	C₂²×C₄	—	15	2	no	—
235	C₂²×C₄	—	15	4	no	—
236	C₂²×C₄	—	15	2	no	—
237	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
238	C₂×C₈	—	15	4	no	—
239	C₂×C₈	—	15	2	no	—
240	C₂×C₈	—	15	16	no	—
241	C₂×C₈	#1200	1	4	yes	—
242	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
243	C₄⋊C₄	—	3	4	no	—
244	C₄⋊C₄	—	5	4	no	—
245	C₄⋊C₄	—	6	2	no	—
246	C₄⋊C₄	—	10	2	no	—
247	C₂²⋊C₄	—	30	8	no	—
248	C₄²	—	30	8	no	—
249	C₃×C₅	C₂×#427	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
251	C₂²×C₃	#4062	1	4	yes	—
252	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
253	C₂×D₃	—	5	4	no	—
254	C₂×D₃	—	10	4	no	—
255	C₂×D₃	—	10	8	no	—
256	C₃×C₄	—	20	4	no	—
257	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
258	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
259	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
260	Dic₃	—	4	4	no	—
261	Dic₃	—	5	4	no	—
262	C₂×C₅	#4860	1	4	yes	complemented
263	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₅	C₂×#916	1	2	yes	—
265	D₅	—	6	4	no	—
266	C₂³	—	15	4	no	—
267	C₂³	—	15	2	no	—
268	C₂³	—	30	4	no	—
269	C₂³	#6777	1	1	yes	characteristic
270	C₂×C₄	—	6	4	no	—
271	C₂×C₄	—	10	4	no	—
272	C₂×C₄	—	12	4	no	—
273	C₂×C₄	—	15	4	no	—
274	C₂×C₄	—	15	2	no	—
275	C₂×C₄	—	15	8	no	—
276	C₂×C₄	—	15	4	no	—
277	C₂×C₄	—	15	4	no	—
278	C₂×C₄	—	20	4	no	—
279	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
280	C₂×C₄	#6777	1	1	yes	characteristic
281	C₂×C₄	C₂×#1200	1	2	yes	—
282	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
283	C₈	—	15	4	no	—
284	C₈	C₂×#1200	1	4	yes	—
285	C₂×C₃	#65149	1	4	yes	complemented
286	C₂×C₃	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
287	C₂×C₃	C₂×#4062	1	2	yes	—
288	D₃	—	10	4	no	—
289	C₅	C₂×#4860	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	15	4	no	—
291	C₂²	—	30	4	no	—
292	C₂²	—	30	8	no	—
293	C₂²	#91276	1	4	yes	—
294	C₂²	#91437	1	2	yes	—
295	C₂²	C₂×#6777	1	1	yes	characteristic
296	C₄	—	12	4	no	—
297	C₄	—	15	4	no	—
298	C₄	—	20	4	no	—
299	C₄	#91437	1	2	yes	—
300	C₄	C₂×D₁₅⋊Q₈	1	1	yes	characteristic
301	C₄	C₂²×#1200	1	1	yes	characteristic
302	C₃	C₂×#65149	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	30	4	no	—
304	C₂	#11747687	1	4	yes	direct product factor
305	C₂	C₂×#91276	1	2	yes	—
306	C₂	C₂×#91437	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#6777
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×D₁₅	D₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#427
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#817	C₂	semidirect
4. C₂×C₃×#3949	C₂	semidirect
5. C₂×#90989	C₂	semidirect
6. #11747687	C₂	direct (N × H)
11. C₂×C₈×D₁₅	C₄	semidirect
19. C₅×#817	C₂²	semidirect
20. C₃×#3949	C₂²	semidirect
21. #90989	C₂²	semidirect
26. C₂×#3949	D₃	semidirect
42. C₈×D₁₅	C₂×C₄	semidirect
52. C₂×#817	D₅	semidirect
67. #3949	C₂×D₃	semidirect
101. #817	C₂×D₅	semidirect
194. C₂×C₃×C₅	#427	semidirect
249. C₃×C₅	C₂×#427	semidirect
262. C₂×C₅	#4860	semidirect
285. C₂×C₃	#65149	semidirect
289. C₅	C₂×#4860	semidirect
302. C₃	C₂×#65149	semidirect
304. C₂	#11747687	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×C₈×D₁₅	2
C₂×#91347	2
C₂×C₅×#817	2
C₂×C₃×#3949	2
C₂×#90989	2
#11747687	2
C₂×#65149	3
C₂×#4860	5

Subgroups — C2×#11747687 #14413319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11747687 #14413319