Subgroups — C₂²×#91829 #14479963

Order 1920.

6000 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#1220	C₂	1	4	yes	maximal, complemented
3	C₂³×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91829	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#4086	—	3	1	no	maximal, Hall
7	C₂⁵×C₃×C₅	C₂²	1	2	yes	—
8	C₂⁴×D₁₅	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₄	—	2	8	no	—
10	C₂²×C₃×C₅×D₄	C₂²	1	12	yes	complemented
11	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂²×#1220	—	2	32	no	—
13	C₂²×#1220	C₂²	1	48	yes	complemented
14	C₂³×Dic₁₅	C₂²	1	2	yes	—
15	C₂²×D₆₀	—	2	8	no	—
16	C₂²×D₆₀	C₂²	1	12	yes	complemented
17	#91829	C₂²	1	64	yes	direct product factor
18	C₂×C₄×Dic₁₅	C₂²	1	12	yes	complemented
19	C₂²×#940	—	5	1	no	maximal, Hall
20	C₂³×C₅×D₄	D₃	1	1	yes	characteristic, complemented
21	C₂³×C₂²⋊D₅	—	3	4	no	—
22	C₂³×D₂₀	—	3	1	no	—
23	C₂×#4086	—	3	24	no	—
24	C₂²×C₄×Dic₅	—	3	1	no	—
25	C₂⁴×C₃×C₅	—	2	16	no	—
26	C₂⁴×C₃×C₅	—	2	8	no	—
27	C₂⁴×C₃×C₅	C₂³	1	12	yes	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂³×D₁₅	—	2	8	no	—
30	C₂³×D₁₅	—	2	16	no	—
31	C₂³×D₁₅	C₂³	1	12	yes	—
32	C₂×C₃×C₅×D₄	—	2	48	no	—
33	C₂×C₃×C₅×D₄	C₂³	1	16	yes	complemented
34	C₂²×C₃×C₄×C₅	D₄	1	8	yes	complemented
35	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
36	C₂×#1220	—	2	192	no	—
37	C₂×#1220	C₂³	1	64	yes	complemented
38	C₂²×Dic₁₅	D₄	1	16	yes	complemented
39	C₂²×Dic₁₅	C₂³	1	12	yes	—
40	C₂×D₆₀	—	2	48	no	—
41	C₂×D₆₀	C₂³	1	16	yes	complemented
42	C₄×Dic₁₅	C₂³	1	16	yes	complemented
43	C₂³×C₃×D₄	D₅	1	1	yes	characteristic, complemented
44	C₂³×#67	—	5	4	no	—
45	C₂³×D₁₂	—	5	1	no	—
46	C₂×#940	—	5	24	no	—
47	C₂²×C₄×Dic₃	—	5	1	no	—
48	C₂⁵×C₅	C₂×D₃	1	2	yes	—
49	C₂⁴×D₅	—	3	2	no	—
50	C₂²×C₅×D₄	—	2	8	no	—
51	C₂²×C₅×D₄	C₂×D₃	1	12	yes	complemented
52	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₂²⋊D₅	—	3	48	no	—
54	C₂²×C₂²⋊D₅	—	6	32	no	—
55	C₂³×Dic₅	—	3	2	no	—
56	C₂²×D₂₀	—	3	12	no	—
57	C₂²×D₂₀	—	6	8	no	—
58	#4086	—	3	64	no	—
59	C₂×C₄×Dic₅	—	3	12	no	—
60	C₂²×#127	—	15	1	no	Sylow
61	C₂³×C₃×C₅	—	2	48	no	—
62	C₂³×C₃×C₅	—	2	24	no	—
63	C₂³×C₃×C₅	—	4	32	no	—
64	C₂³×C₃×C₅	C₂×D₄	1	8	yes	—
65	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
66	C₂³×C₃×C₅	C₂⁴	1	8	yes	—
67	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
68	C₂²×D₁₅	—	2	24	no	—
69	C₂²×D₁₅	—	2	48	no	—
70	C₂²×D₁₅	—	4	32	no	—
71	C₂²×D₁₅	C₂⁴	1	8	yes	—
72	C₃×C₅×D₄	—	2	32	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	24	yes	complemented
74	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
75	#1220	—	2	128	no	—
76	C₂×Dic₁₅	C₂×D₄	1	48	yes	complemented
77	C₂×Dic₁₅	C₂⁴	1	8	yes	—
78	D₆₀	—	2	32	no	—
79	C₂⁵×C₃	C₂×D₅	1	2	yes	—
80	C₂⁴×D₃	—	5	2	no	—
81	C₂²×C₃×D₄	—	2	8	no	—
82	C₂²×C₃×D₄	C₂×D₅	1	12	yes	complemented
83	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
84	C₂²×#67	—	5	48	no	—
85	C₂²×#67	—	10	32	no	—
86	C₂³×Dic₃	—	5	2	no	—
87	C₂²×D₁₂	—	5	12	no	—
88	C₂²×D₁₂	—	10	8	no	—
89	#940	—	5	64	no	—
90	C₂×C₄×Dic₃	—	5	12	no	—
91	C₂⁴×C₅	—	2	16	no	—
92	C₂⁴×C₅	—	2	8	no	—
93	C₂⁴×C₅	C₂²×D₃	1	12	yes	—
94	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
95	C₂³×D₅	—	3	12	no	—
96	C₂³×D₅	—	6	16	no	—
97	C₂³×D₅	—	6	8	no	—
98	C₂×C₅×D₄	—	2	48	no	—
99	C₂×C₅×D₄	C₂²×D₃	1	16	yes	complemented
100	C₂²×C₄×C₅	#67	1	8	yes	complemented
101	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
102	C₂×C₂²⋊D₅	—	3	64	no	—
103	C₂×C₂²⋊D₅	—	6	192	no	—
104	C₂²×Dic₅	—	3	16	no	—
105	C₂²×Dic₅	—	3	12	no	—
106	C₂×D₂₀	—	3	16	no	—
107	C₂×D₂₀	—	6	48	no	—
108	C₄×Dic₅	—	3	16	no	—
109	C₂³×D₄	—	15	4	no	—
110	C₂³×D₄	—	15	1	no	—
111	C₂³×D₄	D₁₅	1	1	yes	characteristic, complemented
112	C₂×#127	—	15	24	no	—
113	C₂²×C₄²	—	15	1	no	—
114	C₂²×C₃×C₅	—	2	16	no	—
115	C₂²×C₃×C₅	—	2	8	no	—
116	C₂²×C₃×C₅	—	4	48	no	—
117	C₂²×C₃×C₅	#127	1	16	yes	complemented
118	C₂²×C₃×C₅	C₂²×D₄	1	12	yes	—
119	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
120	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
121	C₂×D₁₅	—	2	8	no	—
122	C₂×D₁₅	—	2	16	no	—
123	C₂×D₁₅	—	4	48	no	—
124	C₃×C₄×C₅	C₂²×D₄	1	8	yes	complemented
125	Dic₁₅	C₂²×D₄	1	16	yes	complemented
126	C₂⁴×C₃	—	2	16	no	—
127	C₂⁴×C₃	—	2	8	no	—
128	C₂⁴×C₃	C₂²×D₅	1	12	yes	—
129	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
130	C₂³×D₃	—	5	12	no	—
131	C₂³×D₃	—	10	16	no	—
132	C₂³×D₃	—	10	8	no	—
133	C₂×C₃×D₄	—	2	48	no	—
134	C₂×C₃×D₄	C₂²×D₅	1	16	yes	complemented
135	C₂²×C₃×C₄	C₂²⋊D₅	1	8	yes	complemented
136	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
137	C₂×#67	—	5	64	no	—
138	C₂×#67	—	10	192	no	—
139	C₂²×Dic₃	—	5	16	no	—
140	C₂²×Dic₃	—	5	12	no	—
141	C₂×D₁₂	—	5	16	no	—
142	C₂×D₁₂	—	10	48	no	—
143	C₄×Dic₃	—	5	16	no	—
144	C₂³×C₅	—	2	48	no	—
145	C₂³×C₅	—	2	24	no	—
146	C₂³×C₅	—	4	32	no	—
147	C₂³×C₅	D₃×D₄	1	8	yes	—
148	C₂³×C₅	C₂×#67	1	4	yes	—
149	C₂³×C₅	C₂³×D₃	1	8	yes	—
150	C₂³×C₅	C₂³×D₃	1	3	yes	—
151	C₂²×D₅	—	3	8	no	—
152	C₂²×D₅	—	6	48	no	—
153	C₂²×D₅	—	6	24	no	—
154	C₂²×D₅	—	12	32	no	—
155	C₅×D₄	—	2	32	no	—
156	C₂×C₄×C₅	C₂×#67	1	24	yes	complemented
157	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
158	C₂²⋊D₅	—	6	128	no	—
159	C₂×Dic₅	—	3	48	no	—
160	C₂×Dic₅	—	3	8	no	—
161	D₂₀	—	6	32	no	—
162	C₂⁵	—	15	2	no	—
163	C₂⁵	C₂×D₁₅	1	2	yes	—
164	C₂²×D₄	—	2	8	no	—
165	C₂²×D₄	—	15	48	no	—
166	C₂²×D₄	—	15	12	no	—
167	C₂²×D₄	—	30	32	no	—
168	C₂²×D₄	—	30	8	no	—
169	C₂²×D₄	C₂×D₁₅	1	12	yes	complemented
170	C₂³×C₄	—	15	2	no	—
171	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
172	#127	—	15	64	no	—
173	C₂×C₄²	—	15	12	no	—
174	C₂×C₃×C₅	—	4	8	no	—
175	C₂×C₃×C₅	C₂×#127	1	12	yes	complemented
176	C₂×C₃×C₅	C₂³×D₄	1	2	yes	—
177	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
178	D₁₅	—	4	8	no	—
179	C₂³×C₃	—	2	48	no	—
180	C₂³×C₃	—	2	24	no	—
181	C₂³×C₃	—	4	32	no	—
182	C₂³×C₃	D₄×D₅	1	8	yes	—
183	C₂³×C₃	C₂×C₂²⋊D₅	1	4	yes	—
184	C₂³×C₃	C₂³×D₅	1	8	yes	—
185	C₂³×C₃	C₂³×D₅	1	3	yes	—
186	C₂²×D₃	—	5	8	no	—
187	C₂²×D₃	—	10	48	no	—
188	C₂²×D₃	—	10	24	no	—
189	C₂²×D₃	—	20	32	no	—
190	C₃×D₄	—	2	32	no	—
191	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	24	yes	complemented
192	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
193	#67	—	10	128	no	—
194	C₂×Dic₃	—	5	48	no	—
195	C₂×Dic₃	—	5	8	no	—
196	D₁₂	—	10	32	no	—
197	C₂²×C₅	—	2	16	no	—
198	C₂²×C₅	—	2	8	no	—
199	C₂²×C₅	—	4	48	no	—
200	C₂²×C₅	#940	1	16	yes	complemented
201	C₂²×C₅	C₂×D₃×D₄	1	12	yes	—
202	C₂²×C₅	C₂²×#67	1	6	yes	—
203	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
204	C₂×D₅	—	6	16	no	—
205	C₂×D₅	—	6	8	no	—
206	C₂×D₅	—	12	48	no	—
207	C₄×C₅	C₂²×#67	1	8	yes	complemented
208	Dic₅	—	3	16	no	—
209	C₂⁴	—	2	16	no	—
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	15	12	no	—
212	C₂⁴	—	30	16	no	—
213	C₂⁴	—	30	8	no	—
214	C₂⁴	C₂²×D₁₅	1	12	yes	—
215	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
216	C₂×D₄	—	2	48	no	—
217	C₂×D₄	—	15	64	no	—
218	C₂×D₄	—	15	16	no	—
219	C₂×D₄	—	30	192	no	—
220	C₂×D₄	—	30	48	no	—
221	C₂×D₄	C₂²×D₁₅	1	16	yes	complemented
222	C₂²×C₄	—	15	16	no	—
223	C₂²×C₄	—	15	12	no	—
224	C₂²×C₄	#1220	1	8	yes	complemented
225	C₂²×C₄	C₂²×D₁₅	1	6	yes	—
226	C₄²	—	15	16	no	—
227	C₃×C₅	C₂²×#127	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	16	no	—
229	C₂²×C₃	—	2	8	no	—
230	C₂²×C₃	—	4	48	no	—
231	C₂²×C₃	#4086	1	16	yes	complemented
232	C₂²×C₃	C₂×D₄×D₅	1	12	yes	—
233	C₂²×C₃	C₂²×C₂²⋊D₅	1	6	yes	—
234	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
235	C₂×D₃	—	10	16	no	—
236	C₂×D₃	—	10	8	no	—
237	C₂×D₃	—	20	48	no	—
238	C₃×C₄	C₂²×C₂²⋊D₅	1	8	yes	complemented
239	Dic₃	—	5	16	no	—
240	C₂×C₅	—	4	8	no	—
241	C₂×C₅	C₂×#940	1	12	yes	complemented
242	C₂×C₅	C₂²×D₃×D₄	1	2	yes	—
243	C₂×C₅	C₂³×#67	1	1	yes	characteristic
244	D₅	—	12	8	no	—
245	C₂³	—	2	48	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	4	32	no	—
248	C₂³	—	15	8	no	—
249	C₂³	—	30	48	no	—
250	C₂³	—	30	24	no	—
251	C₂³	—	60	32	no	—
252	C₂³	D₄×D₁₅	1	8	yes	—
253	C₂³	C₂×#1220	1	4	yes	—
254	C₂³	C₂³×D₁₅	1	8	yes	—
255	C₂³	C₂³×D₁₅	1	3	yes	—
256	D₄	—	2	32	no	—
257	D₄	—	30	128	no	—
258	D₄	—	30	32	no	—
259	C₂×C₄	—	15	48	no	—
260	C₂×C₄	—	15	8	no	—
261	C₂×C₄	C₂×#1220	1	24	yes	complemented
262	C₂×C₄	C₂³×D₁₅	1	4	yes	—
263	C₂×C₃	—	4	8	no	—
264	C₂×C₃	C₂×#4086	1	12	yes	complemented
265	C₂×C₃	C₂²×D₄×D₅	1	2	yes	—
266	C₂×C₃	C₂³×C₂²⋊D₅	1	1	yes	characteristic
267	D₃	—	20	8	no	—
268	C₅	C₂²×#940	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	16	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	4	48	no	—
272	C₂²	—	30	16	no	—
273	C₂²	—	30	8	no	—
274	C₂²	—	60	48	no	—
275	C₂²	#91829	1	16	yes	direct product factor
276	C₂²	C₂×D₄×D₁₅	1	12	yes	—
277	C₂²	C₂²×#1220	1	6	yes	—
278	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
279	C₄	—	15	16	no	—
280	C₄	C₂²×#1220	1	8	yes	complemented
281	C₃	C₂²×#4086	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	4	8	no	—
283	C₂	—	60	8	no	—
284	C₂	C₂×#91829	1	12	yes	direct product factor
285	C₂	C₂²×D₄×D₁₅	1	2	yes	—
286	C₂	C₂³×#1220	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂³×C₃×C₅×D₄	C₂
socle	C₂⁴×C₃×C₅	C₂³
hypercenter	C₂³×D₄	D₁₅
hypocenter	C₃×C₅	C₂²×#127
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₄	C₂	semidirect
2. C₂³×#1220	C₂	semidirect
3. C₂³×D₆₀	C₂	semidirect
4. C₂×#91829	C₂	direct (N × H)
5. C₂²×C₄×Dic₁₅	C₂	semidirect
10. C₂²×C₃×C₅×D₄	C₂²	semidirect
13. C₂²×#1220	C₂²	semidirect
16. C₂²×D₆₀	C₂²	semidirect
17. #91829	C₂²	direct (N × H)
18. C₂×C₄×Dic₁₅	C₂²	semidirect
20. C₂³×C₅×D₄	D₃	semidirect
33. C₂×C₃×C₅×D₄	C₂³	semidirect
34. C₂²×C₃×C₄×C₅	D₄	semidirect
37. C₂×#1220	C₂³	semidirect
38. C₂²×Dic₁₅	D₄	semidirect
41. C₂×D₆₀	C₂³	semidirect
42. C₄×Dic₁₅	C₂³	semidirect
43. C₂³×C₃×D₄	D₅	semidirect
51. C₂²×C₅×D₄	C₂×D₃	semidirect
73. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
76. C₂×Dic₁₅	C₂×D₄	semidirect
82. C₂²×C₃×D₄	C₂×D₅	semidirect
99. C₂×C₅×D₄	C₂²×D₃	semidirect
100. C₂²×C₄×C₅	#67	semidirect
111. C₂³×D₄	D₁₅	semidirect
117. C₂²×C₃×C₅	#127	semidirect
124. C₃×C₄×C₅	C₂²×D₄	semidirect
125. Dic₁₅	C₂²×D₄	semidirect
134. C₂×C₃×D₄	C₂²×D₅	semidirect
135. C₂²×C₃×C₄	C₂²⋊D₅	semidirect
156. C₂×C₄×C₅	C₂×#67	semidirect
169. C₂²×D₄	C₂×D₁₅	semidirect
175. C₂×C₃×C₅	C₂×#127	semidirect
191. C₂×C₃×C₄	C₂×C₂²⋊D₅	semidirect
200. C₂²×C₅	#940	semidirect
207. C₄×C₅	C₂²×#67	semidirect
221. C₂×D₄	C₂²×D₁₅	semidirect
224. C₂²×C₄	#1220	semidirect
227. C₃×C₅	C₂²×#127	semidirect
231. C₂²×C₃	#4086	semidirect
238. C₃×C₄	C₂²×C₂²⋊D₅	semidirect
241. C₂×C₅	C₂×#940	semidirect
261. C₂×C₄	C₂×#1220	semidirect
264. C₂×C₃	C₂×#4086	semidirect
268. C₅	C₂²×#940	semidirect
275. C₂²	#91829	direct (N × H)
280. C₄	C₂²×#1220	semidirect
281. C₃	C₂²×#4086	semidirect
284. C₂	C₂×#91829	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₅×D₄	2
C₂³×#1220	2
C₂³×D₆₀	2
C₂×#91829	2
C₂²×C₄×Dic₁₅	2
C₂²×#4086	3
C₂²×#940	5

Subgroups — C22×#91829 #14479963

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#91829 #14479963