Subgroups — C₂×#11752459 #14319806

Order 1920.

6928 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6763	C₂	1	6	yes	maximal, complemented
3	#11752459	C₂	1	8	yes	maximal, direct product factor
4	C₂×#65800	—	3	1	no	maximal, Hall
5	C₂⁵×C₃×C₅	C₄	1	1	yes	characteristic, complemented
6	C₂⁴×C₃×D₅	—	2	24	no	—
7	C₂⁴×C₃×D₅	C₄	1	4	yes	complemented
8	C₂⁴×C₃×D₅	C₄	1	3	yes	complemented
9	C₂⁴×C₃×D₅	C₂²	1	4	yes	—
10	C₂⁴×C₃×D₅	C₂²	1	3	yes	—
11	C₂³×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	complemented
12	C₂×#6763	—	2	24	no	—
13	C₂×#6763	C₂²	1	24	yes	complemented
14	C₂×#5247	—	5	1	no	maximal, Hall
15	C₂⁵×D₅	D₃	1	1	yes	characteristic
16	C₂²×C₂²⋊AGL(1,𝔽₅)	—	3	6	no	—
17	#65800	—	3	8	no	—
18	C₂⁴×C₃×C₅	—	2	12	no	—
19	C₂⁴×C₃×C₅	C₂×C₄	1	3	yes	—
20	C₂⁴×C₃×C₅	C₂×C₄	1	4	yes	complemented
21	C₂³×C₃×D₅	—	2	128	no	—
22	C₂³×C₃×D₅	—	2	12	no	—
23	C₂³×C₃×D₅	—	2	24	no	—
24	C₂³×C₃×D₅	—	2	48	no	—
25	C₂³×C₃×D₅	—	2	48	no	—
26	C₂³×C₃×D₅	—	2	24	no	—
27	C₂³×C₃×D₅	C₂×C₄	1	3	yes	—
28	C₂³×C₃×D₅	C₂×C₄	1	6	yes	—
29	C₂³×C₃×D₅	C₂×C₄	1	12	yes	complemented
30	C₂³×C₃×D₅	D₄	1	24	yes	—
31	C₂³×C₃×D₅	C₂³	1	6	yes	—
32	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
33	C₂²×ΓL(1,𝔽₁₆)	—	2	24	no	—
34	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	8	yes	complemented
35	#6763	—	2	48	no	—
36	C₂⁶×C₃	—	5	1	no	—
37	C₂²×#217	—	5	6	no	—
38	#5247	—	5	8	no	—
39	C₂⁵×C₅	Dic₃	1	1	yes	characteristic, complemented
40	C₂⁴×D₅	—	2	24	no	—
41	C₂⁴×D₅	Dic₃	1	4	yes	complemented
42	C₂⁴×D₅	Dic₃	1	3	yes	complemented
43	C₂⁴×D₅	C₂×D₃	1	4	yes	—
44	C₂⁴×D₅	C₂×D₃	1	3	yes	—
45	C₂³×AGL(1,𝔽₅)	—	3	4	no	—
46	C₂×C₂²⋊AGL(1,𝔽₅)	—	3	24	no	—
47	C₂×C₂²⋊AGL(1,𝔽₅)	—	6	24	no	—
48	C₂×#379	—	15	1	no	Sylow
49	C₂³×C₃×C₅	—	2	32	no	—
50	C₂³×C₃×C₅	—	2	12	no	—
51	C₂³×C₃×C₅	—	2	24	no	—
52	C₂³×C₃×C₅	C₂²⋊C₄	1	12	yes	complemented
53	C₂³×C₃×C₅	C₂²×C₄	1	6	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₃×D₅	—	2	96	no	—
56	C₂²×C₃×D₅	—	2	128	no	—
57	C₂²×C₃×D₅	—	2	128	no	—
58	C₂²×C₃×D₅	—	2	12	no	—
59	C₂²×C₃×D₅	—	2	24	no	—
60	C₂²×C₃×D₅	—	2	24	no	—
61	C₂²×C₃×D₅	—	2	48	no	—
62	C₂²×C₃×D₅	—	2	96	no	—
63	C₂²×C₃×D₅	—	2	24	no	—
64	C₂²×C₃×D₅	C₂²⋊C₄	1	24	yes	complemented
65	C₂²×C₃×D₅	C₂²⋊C₄	1	12	yes	complemented
66	C₂²×C₃×D₅	C₂²×C₄	1	6	yes	—
67	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×D₅	C₂×D₄	1	24	yes	—
69	C₂²×C₃×D₅	C₂×D₄	1	12	yes	—
70	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
71	C₂×ΓL(1,𝔽₁₆)	—	2	24	no	—
72	C₂×ΓL(1,𝔽₁₆)	—	4	16	no	—
73	C₂⁵×C₃	—	5	3	no	—
74	C₂⁵×C₃	—	5	3	no	—
75	C₂⁵×C₃	—	5	4	no	—
76	C₂⁵×C₃	—	5	4	no	—
77	C₂⁵×C₃	—	10	24	no	—
78	C₂⁵×C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
79	C₂³×Dic₃	—	5	4	no	—
80	C₂×#217	—	5	24	no	—
81	C₂×#217	—	10	24	no	—
82	C₂⁴×C₅	—	2	12	no	—
83	C₂⁴×C₅	C₂×Dic₃	1	3	yes	—
84	C₂⁴×C₅	C₂×Dic₃	1	4	yes	complemented
85	C₂³×D₅	—	2	128	no	—
86	C₂³×D₅	—	2	12	no	—
87	C₂³×D₅	—	2	24	no	—
88	C₂³×D₅	—	2	48	no	—
89	C₂³×D₅	—	2	48	no	—
90	C₂³×D₅	—	2	24	no	—
91	C₂³×D₅	C₂×Dic₃	1	6	yes	—
92	C₂³×D₅	C₂×Dic₃	1	3	yes	—
93	C₂³×D₅	C₂×Dic₃	1	12	yes	complemented
94	C₂³×D₅	#67	1	24	yes	—
95	C₂³×D₅	C₂²×D₃	1	6	yes	—
96	C₂³×D₅	C₂²×D₃	1	1	yes	characteristic
97	C₂²×AGL(1,𝔽₅)	—	3	8	no	—
98	C₂²×AGL(1,𝔽₅)	—	6	24	no	—
99	C₂²⋊AGL(1,𝔽₅)	—	6	48	no	—
100	C₂⁶	—	5	1	no	—
101	C₂²×C₂²⋊C₄	—	15	6	no	—
102	#379	—	15	8	no	—
103	C₂²×C₃×C₅	—	2	32	no	—
104	C₂²×C₃×C₅	—	2	12	no	—
105	C₂²×C₃×C₅	—	2	24	no	—
106	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
107	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	12	yes	complemented
108	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₃×D₅	—	2	96	no	—
110	C₂×C₃×D₅	—	2	12	no	—
111	C₂×C₃×D₅	—	2	24	no	—
112	C₂×C₃×D₅	—	2	48	no	—
113	C₂×C₃×D₅	—	2	48	no	—
114	C₂×C₃×D₅	C₂×C₂²⋊C₄	1	6	yes	—
115	C₂×C₃×D₅	C₂×C₂²⋊C₄	1	12	yes	complemented
116	C₂×C₃×D₅	#120	1	16	yes	—
117	C₂×C₃×D₅	C₂²×D₄	1	6	yes	—
118	ΓL(1,𝔽₁₆)	—	4	8	no	—
119	C₂⁴×C₃	—	2	12	no	—
120	C₂⁴×C₃	—	5	1	no	—
121	C₂⁴×C₃	—	5	24	no	—
122	C₂⁴×C₃	—	5	6	no	—
123	C₂⁴×C₃	—	5	6	no	—
124	C₂⁴×C₃	—	5	3	no	—
125	C₂⁴×C₃	—	5	12	no	—
126	C₂⁴×C₃	—	10	24	no	—
127	C₂⁴×C₃	—	10	24	no	—
128	C₂⁴×C₃	—	10	48	no	—
129	C₂⁴×C₃	—	10	48	no	—
130	C₂⁴×C₃	—	10	128	no	—
131	C₂⁴×C₃	—	10	12	no	—
132	C₂⁴×C₃	C₂×AGL(1,𝔽₅)	1	3	yes	—
133	C₂⁴×C₃	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
134	C₂²×Dic₃	—	5	8	no	—
135	C₂²×Dic₃	—	10	24	no	—
136	#217	—	10	48	no	—
137	C₂³×C₅	—	2	32	no	—
138	C₂³×C₅	—	2	12	no	—
139	C₂³×C₅	—	2	24	no	—
140	C₂³×C₅	#217	1	12	yes	complemented
141	C₂³×C₅	C₂²×Dic₃	1	6	yes	—
142	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
143	C₂²×D₅	—	2	96	no	—
144	C₂²×D₅	—	2	128	no	—
145	C₂²×D₅	—	2	128	no	—
146	C₂²×D₅	—	2	12	no	—
147	C₂²×D₅	—	2	24	no	—
148	C₂²×D₅	—	2	24	no	—
149	C₂²×D₅	—	2	48	no	—
150	C₂²×D₅	—	2	96	no	—
151	C₂²×D₅	—	2	24	no	—
152	C₂²×D₅	#217	1	24	yes	complemented
153	C₂²×D₅	#217	1	12	yes	complemented
154	C₂²×D₅	C₂²×Dic₃	1	6	yes	—
155	C₂²×D₅	C₂²×Dic₃	1	1	yes	characteristic
156	C₂²×D₅	C₂×#67	1	24	yes	—
157	C₂²×D₅	C₂×#67	1	12	yes	—
158	C₂²×D₅	C₂³×D₃	1	1	yes	characteristic
159	C₂×AGL(1,𝔽₅)	—	6	24	no	—
160	C₂×AGL(1,𝔽₅)	—	12	16	no	—
161	C₂⁵	—	5	3	no	—
162	C₂⁵	—	5	3	no	—
163	C₂⁵	—	5	4	no	—
164	C₂⁵	—	5	4	no	—
165	C₂⁵	—	10	24	no	—
166	C₂⁵	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
167	C₂³×C₄	—	15	4	no	—
168	C₂×C₂²⋊C₄	—	15	24	no	—
169	C₂×C₂²⋊C₄	—	30	24	no	—
170	C₂×C₃×C₅	—	2	12	no	—
171	C₂×C₃×C₅	#379	1	4	yes	complemented
172	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	3	yes	—
173	C₃×D₅	—	2	12	no	—
174	C₃×D₅	#379	1	4	yes	complemented
175	C₃×D₅	C₂×#120	1	4	yes	—
176	C₂³×C₃	—	2	32	no	—
177	C₂³×C₃	—	2	12	no	—
178	C₂³×C₃	—	2	24	no	—
179	C₂³×C₃	—	5	12	no	—
180	C₂³×C₃	—	5	1	no	—
181	C₂³×C₃	—	5	1	no	—
182	C₂³×C₃	—	5	12	no	—
183	C₂³×C₃	—	5	24	no	—
184	C₂³×C₃	—	5	24	no	—
185	C₂³×C₃	—	5	6	no	—
186	C₂³×C₃	—	10	96	no	—
187	C₂³×C₃	—	10	24	no	—
188	C₂³×C₃	—	10	24	no	—
189	C₂³×C₃	—	10	48	no	—
190	C₂³×C₃	—	10	128	no	—
191	C₂³×C₃	—	10	128	no	—
192	C₂³×C₃	—	10	96	no	—
193	C₂³×C₃	—	10	12	no	—
194	C₂³×C₃	—	10	24	no	—
195	C₂³×C₃	C₂²⋊AGL(1,𝔽₅)	1	12	yes	complemented
196	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	6	yes	—
197	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₂×Dic₃	—	10	24	no	—
199	C₂×Dic₃	—	20	16	no	—
200	C₂²×C₅	—	2	32	no	—
201	C₂²×C₅	—	2	24	no	—
202	C₂²×C₅	—	2	12	no	—
203	C₂²×C₅	C₂×#217	1	6	yes	—
204	C₂²×C₅	C₂×#217	1	12	yes	complemented
205	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
206	C₂×D₅	—	2	96	no	—
207	C₂×D₅	—	2	24	no	—
208	C₂×D₅	—	2	12	no	—
209	C₂×D₅	—	2	48	no	—
210	C₂×D₅	—	2	48	no	—
211	C₂×D₅	C₂×#217	1	6	yes	—
212	C₂×D₅	C₂×#217	1	12	yes	complemented
213	C₂×D₅	#953	1	16	yes	—
214	C₂×D₅	C₂²×#67	1	6	yes	—
215	AGL(1,𝔽₅)	—	12	8	no	—
216	C₂⁴	—	2	12	no	—
217	C₂⁴	—	5	1	no	—
218	C₂⁴	—	5	24	no	—
219	C₂⁴	—	5	6	no	—
220	C₂⁴	—	5	6	no	—
221	C₂⁴	—	5	3	no	—
222	C₂⁴	—	5	12	no	—
223	C₂⁴	—	10	24	no	—
224	C₂⁴	—	10	24	no	—
225	C₂⁴	—	10	48	no	—
226	C₂⁴	—	10	48	no	—
227	C₂⁴	—	10	128	no	—
228	C₂⁴	—	10	12	no	—
229	C₂⁴	C₂×ΓL(1,𝔽₁₆)	1	3	yes	—
230	C₂⁴	C₂×ΓL(1,𝔽₁₆)	1	4	yes	complemented
231	C₂²×C₄	—	15	8	no	—
232	C₂²×C₄	—	30	24	no	—
233	C₂²⋊C₄	—	30	48	no	—
234	C₃×C₅	C₂×#379	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	2	32	no	—
236	C₂²×C₃	—	2	24	no	—
237	C₂²×C₃	—	2	12	no	—
238	C₂²×C₃	—	5	16	no	—
239	C₂²×C₃	—	5	6	no	—
240	C₂²×C₃	—	5	6	no	—
241	C₂²×C₃	—	5	12	no	—
242	C₂²×C₃	—	10	48	no	—
243	C₂²×C₃	—	10	48	no	—
244	C₂²×C₃	—	10	96	no	—
245	C₂²×C₃	—	10	24	no	—
246	C₂²×C₃	—	10	12	no	—
247	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	6	yes	—
248	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	12	yes	complemented
249	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
250	Dic₃	—	20	8	no	—
251	C₂×C₅	—	2	12	no	—
252	C₂×C₅	#5247	1	4	yes	complemented
253	C₂×C₅	C₂²×#217	1	3	yes	—
254	D₅	—	2	12	no	—
255	D₅	#5247	1	4	yes	complemented
256	D₅	C₂×#953	1	4	yes	—
257	C₂³	—	2	32	no	—
258	C₂³	—	2	12	no	—
259	C₂³	—	2	24	no	—
260	C₂³	—	5	12	no	—
261	C₂³	—	5	1	no	—
262	C₂³	—	5	1	no	—
263	C₂³	—	5	12	no	—
264	C₂³	—	5	24	no	—
265	C₂³	—	5	24	no	—
266	C₂³	—	5	6	no	—
267	C₂³	—	10	96	no	—
268	C₂³	—	10	24	no	—
269	C₂³	—	10	24	no	—
270	C₂³	—	10	48	no	—
271	C₂³	—	10	128	no	—
272	C₂³	—	10	128	no	—
273	C₂³	—	10	96	no	—
274	C₂³	—	10	12	no	—
275	C₂³	—	10	24	no	—
276	C₂³	#6763	1	12	yes	complemented
277	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	6	yes	—
278	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
279	C₂×C₄	—	30	24	no	—
280	C₂×C₄	—	60	16	no	—
281	C₂×C₃	—	2	12	no	—
282	C₂×C₃	—	5	4	no	—
283	C₂×C₃	—	5	4	no	—
284	C₂×C₃	—	10	12	no	—
285	C₂×C₃	#65800	1	4	yes	complemented
286	C₂×C₃	C₂²×C₂²⋊AGL(1,𝔽₅)	1	3	yes	—
287	C₅	C₂×#5247	1	1	yes	characteristic, complemented, Sylow
288	C₂²	—	2	32	no	—
289	C₂²	—	2	24	no	—
290	C₂²	—	2	12	no	—
291	C₂²	—	5	16	no	—
292	C₂²	—	5	6	no	—
293	C₂²	—	5	6	no	—
294	C₂²	—	5	12	no	—
295	C₂²	—	10	48	no	—
296	C₂²	—	10	48	no	—
297	C₂²	—	10	96	no	—
298	C₂²	—	10	24	no	—
299	C₂²	—	10	12	no	—
300	C₂²	C₂×#6763	1	6	yes	—
301	C₂²	C₂×#6763	1	12	yes	complemented
302	C₂²	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
303	C₄	—	60	8	no	—
304	C₃	C₂×#65800	1	1	yes	characteristic, complemented, Sylow
305	C₂	—	2	12	no	—
306	C₂	—	5	4	no	—
307	C₂	—	5	4	no	—
308	C₂	—	10	12	no	—
309	C₂	#11752459	1	4	yes	direct product factor
310	C₂	C₂²×#6763	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂²×C₃×C₅	C₂³×C₄
Frattini	C₂²	C₂³×ΓL(1,𝔽₁₆)
Fitting	C₂⁵×C₃×C₅	C₄
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁵×C₃×D₅	C₂
cosocle	C₂²×C₃×D₅	C₂⁴
hypercenter	C₂⁵	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×#379
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂²×#6763	C₂	semidirect
3. #11752459	C₂	direct (N × H)
5. C₂⁵×C₃×C₅	C₄	semidirect
7. C₂⁴×C₃×D₅	C₄	semidirect
8. C₂⁴×C₃×D₅	C₄	semidirect
11. C₂³×ΓL(1,𝔽₁₆)	C₂²	semidirect
13. C₂×#6763	C₂²	semidirect
20. C₂⁴×C₃×C₅	C₂×C₄	semidirect
29. C₂³×C₃×D₅	C₂×C₄	semidirect
34. C₂²×ΓL(1,𝔽₁₆)	C₂³	semidirect
39. C₂⁵×C₅	Dic₃	semidirect
41. C₂⁴×D₅	Dic₃	semidirect
42. C₂⁴×D₅	Dic₃	semidirect
52. C₂³×C₃×C₅	C₂²⋊C₄	semidirect
64. C₂²×C₃×D₅	C₂²⋊C₄	semidirect
65. C₂²×C₃×D₅	C₂²⋊C₄	semidirect
78. C₂⁵×C₃	AGL(1,𝔽₅)	semidirect
84. C₂⁴×C₅	C₂×Dic₃	semidirect
93. C₂³×D₅	C₂×Dic₃	semidirect
107. C₂²×C₃×C₅	C₂×C₂²⋊C₄	semidirect
115. C₂×C₃×D₅	C₂×C₂²⋊C₄	semidirect
133. C₂⁴×C₃	C₂×AGL(1,𝔽₅)	semidirect
140. C₂³×C₅	#217	semidirect
152. C₂²×D₅	#217	semidirect
153. C₂²×D₅	#217	semidirect
166. C₂⁵	ΓL(1,𝔽₁₆)	semidirect
171. C₂×C₃×C₅	#379	semidirect
174. C₃×D₅	#379	semidirect
195. C₂³×C₃	C₂²⋊AGL(1,𝔽₅)	semidirect
204. C₂²×C₅	C₂×#217	semidirect
212. C₂×D₅	C₂×#217	semidirect
230. C₂⁴	C₂×ΓL(1,𝔽₁₆)	semidirect
234. C₃×C₅	C₂×#379	semidirect
248. C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	semidirect
252. C₂×C₅	#5247	semidirect
255. D₅	#5247	semidirect
276. C₂³	#6763	semidirect
285. C₂×C₃	#65800	semidirect
287. C₅	C₂×#5247	semidirect
301. C₂²	C₂×#6763	semidirect
304. C₃	C₂×#65800	semidirect
309. C₂	#11752459	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃×D₅	2
C₂²×#6763	2
#11752459	2
C₂×#65800	3
C₂×#5247	5

Subgroups — C2×#11752459 #14319806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11752459 #14319806