Subgroups — C₂×#11751253 #14416482

Order 1920.

1216 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×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Dic₁₅	C₂	1	2	yes	maximal, complemented
3	C₂×#91116	C₂	1	2	yes	maximal, complemented
4	#11751253	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65441	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂³×Dic₁₅	—	2	2	no	—
10	C₂×C₃×C₅×D₈	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6719	—	2	2	no	—
14	D₄×Dic₁₅	—	2	4	no	—
15	D₄×Dic₁₅	C₂²	1	8	yes	complemented
16	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
18	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
19	#91116	C₂²	1	8	yes	complemented
20	#91101	C₂²	1	4	yes	complemented
21	#91099	C₂²	1	4	yes	complemented
22	C₂×#5152	—	5	1	no	maximal, Hall
23	C₂²×C₅×D₈	D₃	1	1	yes	characteristic
24	C₂×D₄×Dic₅	—	3	2	no	—
25	C₂×#3964	—	3	2	no	—
26	#65441	—	3	8	no	—
27	C₂×#3949	—	3	1	no	—
28	C₂×#3947	—	3	1	no	—
29	C₂⁴×C₃×C₅	—	2	2	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	4	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	complemented
33	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×Dic₁₅	—	2	2	no	—
36	C₂²×Dic₁₅	—	2	8	no	—
37	C₂²×Dic₁₅	—	4	8	no	—
38	C₂²×Dic₁₅	D₄	1	2	yes	—
39	C₃×C₅×D₈	C₂×C₄	1	16	yes	complemented
40	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	#6719	—	2	8	no	—
43	C₄⋊Dic₁₅	—	2	2	no	—
44	C₄⋊Dic₁₅	C₂³	1	4	yes	—
45	C₄×Dic₁₅	D₄	1	2	yes	—
46	C₄×Dic₁₅	C₂³	1	2	yes	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₂×C₃⋊#168	C₂³	1	2	yes	—
49	C₂²×C₃×D₈	D₅	1	1	yes	characteristic
50	C₂×D₄×Dic₃	—	5	2	no	—
51	C₂×#832	—	5	2	no	—
52	#5152	—	5	8	no	—
53	C₂×#817	—	5	1	no	—
54	C₂×#815	—	5	1	no	—
55	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
56	C₂³×Dic₅	—	6	2	no	—
57	C₂×C₅×D₈	Dic₃	1	8	yes	complemented
58	C₂×C₅×D₈	C₂×D₃	1	4	yes	—
59	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	C₂×C₂²⋊Dic₅	—	6	2	no	—
61	D₄×Dic₅	—	3	8	no	—
62	D₄×Dic₅	—	6	4	no	—
63	C₂×C₄⋊Dic₅	—	3	2	no	—
64	C₂×C₄×Dic₅	—	3	1	no	—
65	C₂²×#168	—	3	1	no	—
66	#3964	—	3	8	no	—
67	#3949	—	3	4	no	—
68	#3947	—	3	4	no	—
69	C₂×#442	—	15	1	no	Sylow
70	C₂³×C₃×C₅	—	2	8	no	—
71	C₂³×C₃×C₅	—	2	4	no	—
72	C₂³×C₃×C₅	—	4	4	no	—
73	C₂³×C₃×C₅	—	4	4	no	—
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₅×D₄	—	2	8	no	—
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	C₂²×C₄	1	8	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₅	—	2	4	no	—
83	C₂×Dic₁₅	—	2	4	no	—
84	C₂×Dic₁₅	—	4	16	no	—
85	C₂×Dic₁₅	—	4	4	no	—
86	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
88	C₃⋊#168	—	2	2	no	—
89	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
90	C₂³×Dic₃	—	10	2	no	—
91	C₂×C₃×D₈	Dic₅	1	8	yes	complemented
92	C₂×C₃×D₈	C₂×D₅	1	4	yes	—
93	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#217	—	10	2	no	—
95	D₄×Dic₃	—	5	8	no	—
96	D₄×Dic₃	—	10	4	no	—
97	C₂×C₄⋊Dic₃	—	5	2	no	—
98	C₂×C₄×Dic₃	—	5	1	no	—
99	C₂²×C₃⋊C₈	—	5	1	no	—
100	#832	—	5	8	no	—
101	#817	—	5	4	no	—
102	#815	—	5	4	no	—
103	C₂⁴×C₅	—	2	2	no	—
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	4	no	—
106	C₂×C₅×D₄	C₂×Dic₃	1	8	yes	complemented
107	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₂²×Dic₅	—	3	2	no	—
110	C₂²×Dic₅	—	6	8	no	—
111	C₂²×Dic₅	—	6	2	no	—
112	C₂²×Dic₅	—	12	8	no	—
113	C₅×D₈	C₂×Dic₃	1	16	yes	complemented
114	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
115	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
116	C₂²⋊Dic₅	—	6	8	no	—
117	C₄⋊Dic₅	—	3	4	no	—
118	C₄⋊Dic₅	—	6	2	no	—
119	C₄×Dic₅	—	3	2	no	—
120	C₄×Dic₅	—	3	2	no	—
121	C₂×#168	—	3	2	no	—
122	C₂×#168	—	6	2	no	—
123	C₂²×D₈	D₁₅	1	1	yes	characteristic
124	C₂×C₄×D₄	—	15	2	no	—
125	#442	—	15	8	no	—
126	C₂×#106	—	15	1	no	—
127	C₂×#102	—	15	2	no	—
128	C₂×#97	—	15	1	no	—
129	C₂²×C₃×C₅	—	2	8	no	—
130	C₂²×C₃×C₅	—	4	4	no	—
131	C₂²×C₃×C₅	—	4	4	no	—
132	C₂²×C₃×C₅	—	4	16	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
134	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
135	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
139	Dic₁₅	—	2	4	no	—
140	Dic₁₅	—	4	4	no	—
141	C₂⁴×C₃	—	2	2	no	—
142	C₂×C₃×D₄	—	2	2	no	—
143	C₂×C₃×D₄	—	2	4	no	—
144	C₂×C₃×D₄	C₂×Dic₅	1	8	yes	complemented
145	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	5	2	no	—
148	C₂²×Dic₃	—	10	8	no	—
149	C₂²×Dic₃	—	10	2	no	—
150	C₂²×Dic₃	—	20	8	no	—
151	C₃×D₈	C₂×Dic₅	1	16	yes	complemented
152	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
153	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
154	#217	—	10	8	no	—
155	C₄⋊Dic₃	—	5	4	no	—
156	C₄⋊Dic₃	—	10	2	no	—
157	C₄×Dic₃	—	5	2	no	—
158	C₄×Dic₃	—	5	2	no	—
159	C₂×C₃⋊C₈	—	5	2	no	—
160	C₂×C₃⋊C₈	—	10	2	no	—
161	C₂³×C₅	—	2	8	no	—
162	C₂³×C₅	—	2	4	no	—
163	C₂³×C₅	—	4	4	no	—
164	C₂³×C₅	—	4	4	no	—
165	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
166	C₅×D₄	—	2	8	no	—
167	C₅×D₄	—	2	4	no	—
168	C₅×D₄	C₂²×Dic₃	1	8	yes	complemented
169	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	#237	1	2	yes	—
171	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
172	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
173	C₂×Dic₅	—	3	4	no	—
174	C₂×Dic₅	—	6	4	no	—
175	C₂×Dic₅	—	6	4	no	—
176	C₂×Dic₅	—	12	16	no	—
177	C₂×Dic₅	—	12	4	no	—
178	C₅×C₈	C₂²×Dic₃	1	4	yes	complemented
179	#168	—	6	2	no	—
180	C₂²×D₄	C₂×D₁₅	1	2	yes	—
181	C₂³×C₄	—	30	2	no	—
182	C₂×D₈	Dic₁₅	1	8	yes	complemented
183	C₂×D₈	C₂×D₁₅	1	4	yes	—
184	C₂²×C₈	—	15	1	no	—
185	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
186	C₄×D₄	—	15	8	no	—
187	C₄×D₄	—	30	4	no	—
188	C₂×C₄⋊C₄	—	15	2	no	—
189	C₂×C₂²⋊C₄	—	30	2	no	—
190	C₂×C₄²	—	15	1	no	—
191	#106	—	15	4	no	—
192	#102	—	15	8	no	—
193	#97	—	15	4	no	—
194	C₂×C₃×C₅	—	4	8	no	—
195	C₂×C₃×C₅	#442	1	4	yes	complemented
196	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
198	C₂³×C₃	—	2	4	no	—
199	C₂³×C₃	—	2	8	no	—
200	C₂³×C₃	—	4	4	no	—
201	C₂³×C₃	—	4	4	no	—
202	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
203	C₃×D₄	—	2	4	no	—
204	C₃×D₄	—	2	8	no	—
205	C₃×D₄	C₂²×Dic₅	1	8	yes	complemented
206	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
207	C₂×C₃×C₄	#711	1	2	yes	—
208	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
209	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
210	C₂×Dic₃	—	5	4	no	—
211	C₂×Dic₃	—	10	4	no	—
212	C₂×Dic₃	—	10	4	no	—
213	C₂×Dic₃	—	20	16	no	—
214	C₂×Dic₃	—	20	4	no	—
215	C₃×C₈	C₂²×Dic₅	1	4	yes	complemented
216	C₃⋊C₈	—	10	2	no	—
217	C₂²×C₅	—	2	8	no	—
218	C₂²×C₅	—	4	16	no	—
219	C₂²×C₅	—	4	4	no	—
220	C₂²×C₅	—	4	4	no	—
221	C₂²×C₅	#911	1	4	yes	—
222	C₂²×C₅	D₄×Dic₃	1	2	yes	—
223	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
224	C₄×C₅	D₄×Dic₃	1	2	yes	—
225	C₄×C₅	C₂×#237	1	1	yes	characteristic
226	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
227	Dic₅	—	6	4	no	—
228	Dic₅	—	12	4	no	—
229	C₂⁴	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	C₂×Dic₁₅	1	8	yes	complemented
233	C₂×D₄	C₂²×D₁₅	1	4	yes	—
234	C₂²×C₄	—	15	2	no	—
235	C₂²×C₄	—	30	8	no	—
236	C₂²×C₄	—	30	2	no	—
237	C₂²×C₄	—	60	8	no	—
238	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
239	D₈	C₂×Dic₁₅	1	16	yes	complemented
240	C₂×C₈	—	15	2	no	—
241	C₂×C₈	—	30	2	no	—
242	C₂×C₈	C₂×Dic₁₅	1	4	yes	complemented
243	C₂×C₈	C₂²×D₁₅	1	2	yes	—
244	C₄⋊C₄	—	15	4	no	—
245	C₄⋊C₄	—	30	2	no	—
246	C₂²⋊C₄	—	30	8	no	—
247	C₄²	—	15	2	no	—
248	C₄²	—	15	2	no	—
249	C₃×C₅	C₂×#442	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	8	no	—
251	C₂²×C₃	—	4	4	no	—
252	C₂²×C₃	—	4	4	no	—
253	C₂²×C₃	—	4	16	no	—
254	C₂²×C₃	#4057	1	4	yes	—
255	C₂²×C₃	D₄×Dic₅	1	2	yes	—
256	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
257	C₃×C₄	D₄×Dic₅	1	2	yes	—
258	C₃×C₄	C₂×#711	1	1	yes	characteristic
259	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
260	Dic₃	—	10	4	no	—
261	Dic₃	—	20	4	no	—
262	C₂×C₅	—	4	8	no	—
263	C₂×C₅	#5152	1	4	yes	complemented
264	C₂×C₅	C₂×#911	1	2	yes	—
265	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
266	C₂³	—	2	8	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	4	no	—
270	C₂³	D₄×D₁₅	1	1	yes	characteristic
271	D₄	—	2	8	no	—
272	D₄	—	2	4	no	—
273	D₄	C₂²×Dic₁₅	1	8	yes	complemented
274	C₂×C₄	—	15	4	no	—
275	C₂×C₄	—	30	4	no	—
276	C₂×C₄	—	30	4	no	—
277	C₂×C₄	—	60	16	no	—
278	C₂×C₄	—	60	4	no	—
279	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
280	C₂×C₄	#6819	1	2	yes	—
281	C₂×C₄	C₂²×Dic₁₅	1	2	yes	—
282	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
283	C₈	—	30	2	no	—
284	C₈	C₂²×Dic₁₅	1	4	yes	complemented
285	C₂×C₃	—	4	8	no	—
286	C₂×C₃	#65441	1	4	yes	complemented
287	C₂×C₃	C₂×#4057	1	2	yes	—
288	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
289	C₅	C₂×#5152	1	1	yes	characteristic, complemented, Sylow
290	C₂²	—	2	8	no	—
291	C₂²	—	4	16	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	4	4	no	—
294	C₂²	#91800	1	4	yes	—
295	C₂²	D₄×Dic₁₅	1	2	yes	—
296	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
297	C₄	—	30	4	no	—
298	C₄	—	60	4	no	—
299	C₄	D₄×Dic₁₅	1	2	yes	—
300	C₄	C₂×#6819	1	1	yes	characteristic
301	C₄	C₂³×Dic₁₅	1	1	yes	characteristic
302	C₃	C₂×#65441	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	4	8	no	—
304	C₂	#11751253	1	4	yes	direct product factor
305	C₂	C₂×#91800	1	2	yes	—
306	C₂	C₂×D₄×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting, prime radical	C₂²×C₃×C₅×D₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×D₈	D₁₅
hypocenter	C₃×C₅	C₂×#442
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
2. C₂×D₄×Dic₁₅	C₂	semidirect
3. C₂×#91116	C₂	semidirect
4. #11751253	C₂	direct (N × H)
5. C₂×#91101	C₂	semidirect
6. C₂×#91099	C₂	semidirect
10. C₂×C₃×C₅×D₈	C₄	semidirect
15. D₄×Dic₁₅	C₂²	semidirect
19. #91116	C₂²	semidirect
20. #91101	C₂²	semidirect
21. #91099	C₂²	semidirect
32. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
39. C₃×C₅×D₈	C₂×C₄	semidirect
40. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
57. C₂×C₅×D₈	Dic₃	semidirect
77. C₃×C₅×D₄	C₂²×C₄	semidirect
87. C₃×C₅×C₈	C₂²×C₄	semidirect
91. C₂×C₃×D₈	Dic₅	semidirect
106. C₂×C₅×D₄	C₂×Dic₃	semidirect
113. C₅×D₈	C₂×Dic₃	semidirect
114. C₂×C₅×C₈	C₂×Dic₃	semidirect
144. C₂×C₃×D₄	C₂×Dic₅	semidirect
151. C₃×D₈	C₂×Dic₅	semidirect
152. C₂×C₃×C₈	C₂×Dic₅	semidirect
168. C₅×D₄	C₂²×Dic₃	semidirect
178. C₅×C₈	C₂²×Dic₃	semidirect
182. C₂×D₈	Dic₁₅	semidirect
195. C₂×C₃×C₅	#442	semidirect
205. C₃×D₄	C₂²×Dic₅	semidirect
215. C₃×C₈	C₂²×Dic₅	semidirect
232. C₂×D₄	C₂×Dic₁₅	semidirect
239. D₈	C₂×Dic₁₅	semidirect
242. C₂×C₈	C₂×Dic₁₅	semidirect
249. C₃×C₅	C₂×#442	semidirect
263. C₂×C₅	#5152	semidirect
273. D₄	C₂²×Dic₁₅	semidirect
284. C₈	C₂²×Dic₁₅	semidirect
286. C₂×C₃	#65441	semidirect
289. C₅	C₂×#5152	semidirect
302. C₃	C₂×#65441	semidirect
304. C₂	#11751253	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×D₄×Dic₁₅	2
C₂×#91116	2
#11751253	2
C₂×#91101	2
C₂×#91099	2
C₂×#65441	3
C₂×#5152	5

Subgroups — C2×#11751253 #14416482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751253 #14416482