Subgroups — C₂²×#91357 #14478762

Order 1920.

3760 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6668	C₂	1	2	yes	maximal, complemented
5	C₂²×#6665	C₂	1	2	yes	maximal
6	C₂×#91357	C₂	1	24	yes	maximal, direct product factor
7	C₂²×#4018	—	3	1	no	maximal, Hall
8	C₂⁴×C₃×D₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂³×Dic₁₅	C₂²	1	2	yes	—
11	C₂³×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×D₅	C₄	1	16	yes	complemented
14	C₂²×C₃×C₄×D₅	C₂²	1	12	yes	—
15	C₂×C₄×Dic₁₅	C₂²	1	12	yes	complemented
16	C₂×C₄×C₅×Dic₃	C₂²	1	12	yes	complemented
17	C₂×#6668	C₂²	1	24	yes	complemented
18	C₂×#6665	C₂²	1	24	yes	—
19	#91357	C₂²	1	64	yes	direct product factor
20	C₂²×#926	—	5	1	no	maximal, Hall
21	C₂³×C₄×D₅	D₃	1	1	yes	characteristic
22	C₂²×C₄²×C₅	—	3	1	no	—
23	C₂²×#685	—	3	2	no	—
24	C₂²×#683	—	3	2	no	—
25	C₂²×C₄×Dic₅	—	3	1	no	—
26	C₂×#4018	—	3	24	no	—
27	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
28	C₂³×C₃×D₅	—	2	8	no	—
29	C₂³×C₃×D₅	C₂×C₄	1	8	yes	—
30	C₂³×C₃×D₅	C₂³	1	6	yes	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
33	C₂²×Dic₁₅	—	2	8	no	—
34	C₂²×Dic₁₅	C₂³	1	12	yes	—
35	C₂²×C₅×Dic₃	—	2	8	no	—
36	C₂²×C₅×Dic₃	C₂³	1	12	yes	—
37	C₂²×C₃×Dic₅	C₂×C₄	1	8	yes	—
38	C₂²×C₃×Dic₅	C₂³	1	6	yes	—
39	C₂×C₃×C₄×D₅	C₂×C₄	1	96	yes	complemented
40	C₂×C₃×C₄×D₅	C₂³	1	16	yes	—
41	C₄×Dic₁₅	C₂³	1	16	yes	complemented
42	C₄×C₅×Dic₃	C₂³	1	16	yes	complemented
43	#6668	C₂³	1	32	yes	complemented
44	#6665	C₂³	1	32	yes	—
45	C₂⁴×C₃×C₄	—	5	1	no	—
46	C₂²×#217	—	5	2	no	—
47	C₂×#926	—	5	24	no	—
48	C₂²×C₄⋊Dic₃	—	5	2	no	—
49	C₂²×C₄×Dic₃	—	5	1	no	—
50	C₂²×C₄×Dic₃	D₅	1	1	yes	characteristic, complemented
51	C₂⁴×D₅	C₂×D₃	1	1	yes	characteristic
52	C₂³×C₄×C₅	—	3	2	no	—
53	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
54	C₂³×Dic₅	—	3	2	no	—
55	C₂³×Dic₅	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₄×D₅	Dic₃	1	16	yes	complemented
57	C₂²×C₄×D₅	C₂×D₃	1	12	yes	—
58	C₂×C₄²×C₅	—	3	12	no	—
59	C₂×#685	—	3	24	no	—
60	C₂×#683	—	3	24	no	—
61	C₂×C₄×Dic₅	—	3	12	no	—
62	#4018	—	3	64	no	—
63	C₂²×#117	—	15	1	no	Sylow
64	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
65	C₂³×C₃×C₅	#41	1	8	yes	—
66	C₂³×C₃×C₅	C₂⁴	1	3	yes	—
67	C₂²×C₃×D₅	—	2	32	no	—
68	C₂²×C₃×D₅	—	2	24	no	—
69	C₂²×C₃×D₅	C₂²×C₄	1	24	yes	—
70	C₂²×C₃×D₅	C₂⁴	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	24	yes	—
72	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
73	C₂×Dic₁₅	—	2	24	no	—
74	C₂×Dic₁₅	C₂⁴	1	8	yes	—
75	C₂×C₅×Dic₃	—	2	24	no	—
76	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
77	C₂×C₃×Dic₅	C₂²×C₄	1	24	yes	—
78	C₂×C₃×Dic₅	C₂⁴	1	4	yes	—
79	C₃×C₄×D₅	C₂²×C₄	1	64	yes	complemented
80	C₂⁵×C₃	—	5	1	no	—
81	C₂³×C₃×C₄	—	5	1	no	—
82	C₂³×C₃×C₄	—	5	16	no	—
83	C₂³×C₃×C₄	—	5	12	no	—
84	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
85	C₂³×Dic₃	—	5	2	no	—
86	C₂³×Dic₃	C₂×D₅	1	2	yes	—
87	C₂×#217	—	5	24	no	—
88	#926	—	5	64	no	—
89	C₂×C₄⋊Dic₃	—	5	24	no	—
90	C₂×C₄×Dic₃	—	5	12	no	—
91	C₂×C₄×Dic₃	C₂×D₅	1	12	yes	complemented
92	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
93	C₂³×D₅	—	2	8	no	—
94	C₂³×D₅	C₂×Dic₃	1	8	yes	—
95	C₂³×D₅	C₂²×D₃	1	6	yes	—
96	C₂²×C₄×C₅	—	3	12	no	—
97	C₂²×C₄×C₅	—	6	8	no	—
98	C₂²×C₄×C₅	C₂×Dic₃	1	8	yes	—
99	C₂²×C₄×C₅	C₂²×D₃	1	6	yes	—
100	C₂²×Dic₅	—	3	12	no	—
101	C₂²×Dic₅	—	6	8	no	—
102	C₂²×Dic₅	C₂×Dic₃	1	8	yes	—
103	C₂²×Dic₅	C₂²×D₃	1	6	yes	—
104	C₂×C₄×D₅	C₂×Dic₃	1	96	yes	complemented
105	C₂×C₄×D₅	C₂²×D₃	1	16	yes	—
106	C₄²×C₅	—	3	16	no	—
107	#685	—	3	32	no	—
108	#683	—	3	32	no	—
109	C₄×Dic₅	—	3	16	no	—
110	C₂⁴×C₄	—	5	1	no	—
111	C₂×#117	—	15	24	no	—
112	C₂²×C₄⋊C₄	—	15	2	no	—
113	C₂²×C₂²⋊C₄	—	15	2	no	—
114	C₂²×C₄²	—	3	1	no	—
115	C₂²×C₄²	—	15	1	no	—
116	C₂²×C₃×C₅	#117	1	16	yes	complemented
117	C₂²×C₃×C₅	C₂³×C₄	1	6	yes	—
118	C₂²×C₃×C₅	C₂×#41	1	12	yes	—
119	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
120	C₂×C₃×D₅	—	2	48	no	—
121	C₂×C₃×D₅	—	2	8	no	—
122	C₂×C₃×D₅	C₂³×C₄	1	8	yes	—
123	C₃×C₄×C₅	C₂³×C₄	1	8	yes	—
124	Dic₁₅	—	2	8	no	—
125	C₃×Dic₅	C₂³×C₄	1	8	yes	—
126	C₅×Dic₃	—	2	8	no	—
127	C₂⁴×C₃	—	5	8	no	—
128	C₂⁴×C₃	—	5	6	no	—
129	C₂⁴×C₃	—	10	8	no	—
130	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
131	C₂²×C₃×C₄	—	5	8	no	—
132	C₂²×C₃×C₄	—	5	6	no	—
133	C₂²×C₃×C₄	—	5	96	no	—
134	C₂²×C₃×C₄	—	5	16	no	—
135	C₂²×C₃×C₄	C₄×D₅	1	8	yes	—
136	C₂²×C₃×C₄	C₂²×D₅	1	6	yes	—
137	C₂²×Dic₃	—	2	8	no	—
138	C₂²×Dic₃	—	5	12	no	—
139	C₂²×Dic₃	—	10	8	no	—
140	C₂²×Dic₃	C₂²×D₅	1	12	yes	—
141	#217	—	5	32	no	—
142	C₄⋊Dic₃	—	5	32	no	—
143	C₄×Dic₃	—	5	16	no	—
144	C₄×Dic₃	C₂²×D₅	1	16	yes	complemented
145	C₂³×C₅	#235	1	8	yes	—
146	C₂³×C₅	C₂²×Dic₃	1	4	yes	—
147	C₂³×C₅	C₂³×D₃	1	3	yes	—
148	C₂²×D₅	—	2	32	no	—
149	C₂²×D₅	—	2	24	no	—
150	C₂²×D₅	C₂²×Dic₃	1	24	yes	—
151	C₂²×D₅	C₂³×D₃	1	4	yes	—
152	C₂×C₄×C₅	—	3	8	no	—
153	C₂×C₄×C₅	—	6	24	no	—
154	C₂×C₄×C₅	C₂²×Dic₃	1	24	yes	—
155	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
156	C₂×Dic₅	—	3	8	no	—
157	C₂×Dic₅	—	6	24	no	—
158	C₂×Dic₅	C₂²×Dic₃	1	24	yes	—
159	C₂×Dic₅	C₂³×D₃	1	4	yes	—
160	C₄×D₅	C₂²×Dic₃	1	64	yes	complemented
161	C₂⁵	—	5	1	no	—
162	C₂³×C₄	—	3	2	no	—
163	C₂³×C₄	—	5	1	no	—
164	C₂³×C₄	—	5	16	no	—
165	C₂³×C₄	—	5	12	no	—
166	C₂³×C₄	—	15	2	no	—
167	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
168	#117	—	15	64	no	—
169	C₂×C₄⋊C₄	—	15	24	no	—
170	C₂×C₂²⋊C₄	—	15	24	no	—
171	C₂×C₄²	—	3	12	no	—
172	C₂×C₄²	—	15	12	no	—
173	C₂×C₃×C₅	C₂×#117	1	12	yes	complemented
174	C₂×C₃×C₅	C₂⁴×C₄	1	1	yes	characteristic
175	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
176	C₃×D₅	—	2	8	no	—
177	C₂³×C₃	—	5	24	no	—
178	C₂³×C₃	—	5	4	no	—
179	C₂³×C₃	—	10	24	no	—
180	C₂³×C₃	—	10	32	no	—
181	C₂³×C₃	C₂×C₄×D₅	1	4	yes	—
182	C₂³×C₃	#709	1	8	yes	—
183	C₂³×C₃	C₂³×D₅	1	3	yes	—
184	C₂×C₃×C₄	—	5	24	no	—
185	C₂×C₃×C₄	—	5	4	no	—
186	C₂×C₃×C₄	—	5	64	no	—
187	C₂×C₃×C₄	C₂×C₄×D₅	1	24	yes	—
188	C₂×C₃×C₄	C₂³×D₅	1	4	yes	—
189	C₂×Dic₃	—	2	24	no	—
190	C₂×Dic₃	—	5	8	no	—
191	C₂×Dic₃	—	10	24	no	—
192	C₂×Dic₃	C₂³×D₅	1	8	yes	—
193	C₂²×C₅	#926	1	16	yes	complemented
194	C₂²×C₅	C₂×#235	1	12	yes	—
195	C₂²×C₅	C₂³×Dic₃	1	6	yes	—
196	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
197	C₂×D₅	—	2	8	no	—
198	C₂×D₅	—	2	48	no	—
199	C₂×D₅	C₂³×Dic₃	1	8	yes	—
200	C₄×C₅	—	6	8	no	—
201	C₄×C₅	C₂³×Dic₃	1	8	yes	—
202	Dic₅	—	6	8	no	—
203	Dic₅	C₂³×Dic₃	1	8	yes	—
204	C₂⁴	—	5	8	no	—
205	C₂⁴	—	5	6	no	—
206	C₂⁴	—	10	8	no	—
207	C₂⁴	C₂×D₃×D₅	1	1	yes	characteristic
208	C₂²×C₄	—	3	12	no	—
209	C₂²×C₄	—	5	8	no	—
210	C₂²×C₄	—	5	6	no	—
211	C₂²×C₄	—	5	96	no	—
212	C₂²×C₄	—	5	16	no	—
213	C₂²×C₄	—	6	8	no	—
214	C₂²×C₄	—	15	12	no	—
215	C₂²×C₄	—	30	8	no	—
216	C₂²×C₄	D₅×Dic₃	1	8	yes	—
217	C₂²×C₄	C₂×D₃×D₅	1	6	yes	—
218	C₄⋊C₄	—	15	32	no	—
219	C₂²⋊C₄	—	15	32	no	—
220	C₄²	—	3	16	no	—
221	C₄²	—	15	16	no	—
222	C₃×C₅	C₂²×#117	1	1	yes	characteristic, complemented, Hall
223	C₂²×C₃	—	5	8	no	—
224	C₂²×C₃	—	10	48	no	—
225	C₂²×C₃	—	10	8	no	—
226	C₂²×C₃	#4018	1	16	yes	complemented
227	C₂²×C₃	C₂²×C₄×D₅	1	6	yes	—
228	C₂²×C₃	C₂×#709	1	12	yes	—
229	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
230	C₃×C₄	—	5	8	no	—
231	C₃×C₄	C₂²×C₄×D₅	1	8	yes	—
232	Dic₃	—	2	8	no	—
233	Dic₃	—	10	8	no	—
234	C₂×C₅	C₂×#926	1	12	yes	complemented
235	C₂×C₅	C₂²×#235	1	2	yes	—
236	C₂×C₅	C₂⁴×Dic₃	1	1	yes	characteristic
237	D₅	—	2	8	no	—
238	C₂³	—	5	24	no	—
239	C₂³	—	5	4	no	—
240	C₂³	—	10	24	no	—
241	C₂³	—	10	32	no	—
242	C₂³	#6771	1	8	yes	—
243	C₂³	C₂×D₅×Dic₃	1	4	yes	—
244	C₂³	C₂²×D₃×D₅	1	3	yes	—
245	C₂×C₄	—	3	8	no	—
246	C₂×C₄	—	5	24	no	—
247	C₂×C₄	—	5	4	no	—
248	C₂×C₄	—	5	64	no	—
249	C₂×C₄	—	6	24	no	—
250	C₂×C₄	—	15	8	no	—
251	C₂×C₄	—	30	24	no	—
252	C₂×C₄	C₂×D₅×Dic₃	1	24	yes	—
253	C₂×C₄	C₂²×D₃×D₅	1	4	yes	—
254	C₂×C₃	—	10	8	no	—
255	C₂×C₃	C₂×#4018	1	12	yes	complemented
256	C₂×C₃	C₂³×C₄×D₅	1	1	yes	characteristic
257	C₂×C₃	C₂²×#709	1	2	yes	—
258	C₅	C₂²×#926	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	5	8	no	—
260	C₂²	—	10	8	no	—
261	C₂²	—	10	48	no	—
262	C₂²	#91357	1	16	yes	direct product factor
263	C₂²	C₂×#6771	1	12	yes	—
264	C₂²	C₂²×D₅×Dic₃	1	6	yes	—
265	C₂²	C₂³×D₃×D₅	1	1	yes	characteristic
266	C₄	—	5	8	no	—
267	C₄	—	6	8	no	—
268	C₄	—	30	8	no	—
269	C₄	C₂²×D₅×Dic₃	1	8	yes	—
270	C₃	C₂²×#4018	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	10	8	no	—
272	C₂	C₂×#91357	1	12	yes	direct product factor
273	C₂	C₂²×#6771	1	2	yes	—
274	C₂	C₂³×D₅×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃×D₅
commutator	C₂×C₃×C₅	C₂⁴×C₄
Frattini	C₂²	C₂³×D₃×D₅
Fitting	C₂³×C₃×C₄×C₅	C₂²
socle	C₂⁴×C₃×C₅	C₂³
prime radical	C₂⁴×C₃×D₅	C₂²
cosocle	C₂²×C₃×C₅	C₂⁵
hypocenter	C₃×C₅	C₂²×#117
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₁₅	C₂	semidirect
3. C₂²×C₄×C₅×Dic₃	C₂	semidirect
4. C₂²×#6668	C₂	semidirect
6. C₂×#91357	C₂	direct (N × H)
13. C₂²×C₃×C₄×D₅	C₄	semidirect
15. C₂×C₄×Dic₁₅	C₂²	semidirect
16. C₂×C₄×C₅×Dic₃	C₂²	semidirect
17. C₂×#6668	C₂²	semidirect
19. #91357	C₂²	direct (N × H)
39. C₂×C₃×C₄×D₅	C₂×C₄	semidirect
41. C₄×Dic₁₅	C₂³	semidirect
42. C₄×C₅×Dic₃	C₂³	semidirect
43. #6668	C₂³	semidirect
50. C₂²×C₄×Dic₃	D₅	semidirect
56. C₂²×C₄×D₅	Dic₃	semidirect
79. C₃×C₄×D₅	C₂²×C₄	semidirect
91. C₂×C₄×Dic₃	C₂×D₅	semidirect
104. C₂×C₄×D₅	C₂×Dic₃	semidirect
116. C₂²×C₃×C₅	#117	semidirect
144. C₄×Dic₃	C₂²×D₅	semidirect
160. C₄×D₅	C₂²×Dic₃	semidirect
173. C₂×C₃×C₅	C₂×#117	semidirect
193. C₂²×C₅	#926	semidirect
222. C₃×C₅	C₂²×#117	semidirect
226. C₂²×C₃	#4018	semidirect
234. C₂×C₅	C₂×#926	semidirect
255. C₂×C₃	C₂×#4018	semidirect
258. C₅	C₂²×#926	semidirect
262. C₂²	#91357	direct (N × H)
270. C₃	C₂²×#4018	semidirect
272. C₂	C₂×#91357	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×C₄×D₅	2
C₂²×C₄×Dic₁₅	2
C₂²×C₄×C₅×Dic₃	2
C₂²×#6668	2
C₂²×#6665	2
C₂×#91357	2
C₂²×#4018	3
C₂²×#926	5

Subgroups — C22×#91357 #14478762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#91357 #14478762