Subgroups — C₂×#11756353 #14495256

Order 1920.

2816 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91842	C₂	1	2	yes	maximal, complemented
3	C₂×#91826	C₂	1	3	yes	maximal, complemented
4	C₂×#91825	C₂	1	3	yes	maximal, complemented
5	#11756353	C₂	1	16	yes	maximal, direct product factor
6	C₂×#91774	C₂	1	6	yes	maximal, complemented
7	C₂×#65928	—	3	1	no	maximal, Hall
8	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic, complemented
9	C₂⁴×D₁₅	C₂²	1	1	yes	characteristic, complemented
10	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
11	C₂²×#1220	C₂²	1	6	yes	—
12	C₂²×Dic₃₀	C₂²	1	3	yes	complemented
13	C₃×C₅×#120	C₂²	1	8	yes	complemented
14	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	complemented
15	#91842	C₂²	1	16	yes	complemented
16	C₂×#6719	C₂²	1	3	yes	complemented
17	C₂×#6719	C₂²	1	6	yes	complemented
18	#91826	C₂²	1	24	yes	complemented
19	#91825	C₂²	1	24	yes	complemented
20	C₂×#6714	C₂²	1	6	yes	complemented
21	C₂×C₄×Dic₁₅	C₂²	1	3	yes	complemented
22	#91774	C₂²	1	48	yes	complemented
23	C₂×#5594	—	5	1	no	maximal, Hall
24	C₂×C₅×#120	D₃	1	1	yes	characteristic, complemented
25	C₂×#4099	—	3	2	no	—
26	C₂×#4083	—	3	3	no	—
27	C₂×#4082	—	3	3	no	—
28	#65928	—	3	16	no	—
29	C₂×#4031	—	3	6	no	—
30	C₂⁴×C₃×C₅	—	4	6	no	—
31	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
33	C₂⁴×C₃×C₅	C₂³	1	4	yes	complemented
34	C₂³×D₁₅	—	4	6	no	—
35	C₂³×D₁₅	C₂³	1	2	yes	—
36	C₂³×D₁₅	C₂³	1	4	yes	complemented
37	C₂×C₃×C₅×D₄	—	4	6	no	—
38	C₂×C₃×C₅×D₄	C₂³	1	12	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
40	C₂×#1220	—	4	12	no	—
41	C₂×#1220	C₂³	1	24	yes	—
42	C₂²×Dic₁₅	C₂³	1	6	yes	—
43	C₂×Dic₃₀	—	4	6	no	—
44	C₂×Dic₃₀	C₂³	1	12	yes	complemented
45	C₃×C₅×C₂²⋊C₄	C₂³	1	12	yes	complemented
46	#6719	C₂³	1	24	yes	complemented
47	#6719	C₂³	1	12	yes	complemented
48	#6714	C₂³	1	24	yes	complemented
49	C₄×Dic₁₅	C₂³	1	12	yes	complemented
50	C₂×C₃×#120	D₅	1	1	yes	characteristic, complemented
51	C₂×#953	—	5	2	no	—
52	C₂×#937	—	5	3	no	—
53	C₂×#936	—	5	3	no	—
54	#5594	—	5	16	no	—
55	C₂×#885	—	5	6	no	—
56	C₂⁵×C₅	C₂×D₃	1	1	yes	characteristic, complemented
57	C₂⁴×D₅	—	3	1	no	—
58	C₂²×C₅×D₄	C₂×D₃	1	3	yes	—
59	C₂²×C₂²⋊D₅	—	3	6	no	—
60	C₂²×Dic₁₀	—	3	3	no	—
61	C₅×#120	C₂×D₃	1	8	yes	complemented
62	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	3	yes	complemented
63	#4099	—	3	16	no	—
64	C₂×C₂²⋊Dic₅	—	3	6	no	—
65	C₂×C₂²⋊Dic₅	—	3	3	no	—
66	#4083	—	3	24	no	—
67	#4082	—	3	24	no	—
68	C₂×#685	—	3	6	no	—
69	C₂×C₄×Dic₅	—	3	3	no	—
70	#4031	—	3	48	no	—
71	C₂×H(3,𝔽₄)	—	15	1	no	Sylow
72	C₂³×C₃×C₅	—	2	12	no	—
73	C₂³×C₃×C₅	—	2	6	no	—
74	C₂³×C₃×C₅	—	2	6	no	—
75	C₂³×C₃×C₅	—	4	12	no	—
76	C₂³×C₃×C₅	—	4	16	no	—
77	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
78	C₂³×C₃×C₅	C₂⁴	1	6	yes	—
79	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
80	C₂²×D₁₅	—	2	12	no	—
81	C₂²×D₁₅	—	4	12	no	—
82	C₂²×D₁₅	—	4	16	no	—
83	C₂²×D₁₅	C₂⁴	1	4	yes	—
84	C₃×C₅×D₄	—	4	12	no	—
85	C₂×C₃×C₄×C₅	—	2	6	no	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
87	#1220	—	4	24	no	—
88	C₂×Dic₁₅	—	2	12	no	—
89	C₂×Dic₁₅	C₂⁴	1	12	yes	—
90	Dic₃₀	—	4	12	no	—
91	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic, complemented
92	C₂⁴×D₃	—	5	1	no	—
93	C₂²×C₃×D₄	C₂×D₅	1	3	yes	—
94	C₂²×#67	—	5	6	no	—
95	C₂²×Dic₆	—	5	3	no	—
96	C₃×#120	C₂×D₅	1	8	yes	complemented
97	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	3	yes	complemented
98	#953	—	5	16	no	—
99	C₂×#217	—	5	6	no	—
100	C₂×#217	—	5	3	no	—
101	#937	—	5	24	no	—
102	#936	—	5	24	no	—
103	C₂×(C₂×D₃)⋊C₄	—	5	6	no	—
104	C₂×C₄×Dic₃	—	5	3	no	—
105	#885	—	5	48	no	—
106	C₂⁴×C₅	—	4	6	no	—
107	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₂⁴×C₅	C₂²×D₃	1	3	yes	—
109	C₂⁴×C₅	C₂²×D₃	1	4	yes	complemented
110	C₂³×D₅	—	3	2	no	—
111	C₂³×D₅	—	3	4	no	—
112	C₂³×D₅	—	12	6	no	—
113	C₂×C₅×D₄	—	4	6	no	—
114	C₂×C₅×D₄	C₂²×D₃	1	12	yes	—
115	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
116	C₂×C₂²⋊D₅	—	3	24	no	—
117	C₂×C₂²⋊D₅	—	12	12	no	—
118	C₂²×Dic₅	—	3	6	no	—
119	C₂×Dic₁₀	—	3	12	no	—
120	C₂×Dic₁₀	—	12	6	no	—
121	C₅×C₂²⋊C₄	C₂²×D₃	1	12	yes	complemented
122	C₂²⋊Dic₅	—	3	24	no	—
123	C₂²⋊Dic₅	—	3	12	no	—
124	#685	—	3	24	no	—
125	C₄×Dic₅	—	3	12	no	—
126	H(3,𝔽₄)	—	15	16	no	—
127	C₂×#124	—	15	3	no	—
128	C₂×#124	—	15	6	no	—
129	C₂×#120	—	15	2	no	—
130	C₂×#120	—	15	3	no	—
131	C₂×#120	D₁₅	1	1	yes	characteristic, complemented
132	C₂²×C₃×C₅	—	2	12	no	—
133	C₂²×C₃×C₅	—	2	6	no	—
134	C₂²×C₃×C₅	—	2	6	no	—
135	C₂²×C₃×C₅	—	4	16	no	—
136	C₂²×C₃×C₅	—	4	4	no	—
137	C₂²×C₃×C₅	—	4	12	no	—
138	C₂²×C₃×C₅	H(5,𝔽₂)	1	6	yes	—
139	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
140	C₂×D₁₅	—	2	12	no	—
141	C₂×D₁₅	—	4	16	no	—
142	C₂×D₁₅	—	4	8	no	—
143	C₃×C₄×C₅	—	2	6	no	—
144	Dic₁₅	—	2	12	no	—
145	C₂⁴×C₃	—	4	6	no	—
146	C₂⁴×C₃	C₂²×D₅	1	3	yes	—
147	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
148	C₂⁴×C₃	C₂²×D₅	1	4	yes	complemented
149	C₂³×D₃	—	5	2	no	—
150	C₂³×D₃	—	5	4	no	—
151	C₂³×D₃	—	20	6	no	—
152	C₂×C₃×D₄	—	4	6	no	—
153	C₂×C₃×D₄	C₂²×D₅	1	12	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
155	C₂×#67	—	5	24	no	—
156	C₂×#67	—	20	12	no	—
157	C₂²×Dic₃	—	5	6	no	—
158	C₂×Dic₆	—	5	12	no	—
159	C₂×Dic₆	—	20	6	no	—
160	C₃×C₂²⋊C₄	C₂²×D₅	1	12	yes	complemented
161	#217	—	5	24	no	—
162	#217	—	5	12	no	—
163	(C₂×D₃)⋊C₄	—	5	24	no	—
164	C₄×Dic₃	—	5	12	no	—
165	C₂³×C₅	—	2	12	no	—
166	C₂³×C₅	—	2	6	no	—
167	C₂³×C₅	—	2	6	no	—
168	C₂³×C₅	—	4	12	no	—
169	C₂³×C₅	—	4	16	no	—
170	C₂³×C₅	C₂³×D₃	1	2	yes	—
171	C₂³×C₅	C₂³×D₃	1	6	yes	—
172	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
173	C₂²×D₅	—	3	4	no	—
174	C₂²×D₅	—	6	12	no	—
175	C₂²×D₅	—	12	12	no	—
176	C₂²×D₅	—	12	16	no	—
177	C₅×D₄	—	4	12	no	—
178	C₂×C₄×C₅	—	2	6	no	—
179	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
180	C₂²⋊D₅	—	12	24	no	—
181	C₂×Dic₅	—	3	12	no	—
182	C₂×Dic₅	—	6	12	no	—
183	Dic₁₀	—	12	12	no	—
184	C₂⁵	—	15	1	no	—
185	C₂⁵	C₂×D₁₅	1	1	yes	characteristic, complemented
186	C₂²×Q₈	—	15	3	no	—
187	C₂²×D₄	—	15	6	no	—
188	C₂²×D₄	C₂×D₁₅	1	3	yes	—
189	#124	—	15	24	no	—
190	#124	—	15	48	no	—
191	#120	—	15	16	no	—
192	#120	—	15	24	no	—
193	#120	C₂×D₁₅	1	8	yes	complemented
194	C₂×C₂²⋊C₄	—	15	3	no	—
195	C₂×C₂²⋊C₄	—	15	6	no	—
196	C₂×C₂²⋊C₄	—	15	6	no	—
197	C₂×C₂²⋊C₄	C₂×D₁₅	1	3	yes	complemented
198	C₂×C₄²	—	15	3	no	—
199	C₂×C₃×C₅	—	4	2	no	—
200	C₂×C₃×C₅	—	4	6	no	—
201	C₂×C₃×C₅	H(3,𝔽₄)	1	4	yes	complemented
202	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	3	yes	—
203	D₁₅	—	4	4	no	—
204	C₂³×C₃	—	2	12	no	—
205	C₂³×C₃	—	2	6	no	—
206	C₂³×C₃	—	2	6	no	—
207	C₂³×C₃	—	4	12	no	—
208	C₂³×C₃	—	4	16	no	—
209	C₂³×C₃	C₂³×D₅	1	2	yes	—
210	C₂³×C₃	C₂³×D₅	1	6	yes	—
211	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
212	C₂²×D₃	—	5	4	no	—
213	C₂²×D₃	—	10	12	no	—
214	C₂²×D₃	—	20	12	no	—
215	C₂²×D₃	—	20	16	no	—
216	C₃×D₄	—	4	12	no	—
217	C₂×C₃×C₄	—	2	6	no	—
218	C₂×C₃×C₄	C₂³×D₅	1	6	yes	—
219	#67	—	20	24	no	—
220	C₂×Dic₃	—	5	12	no	—
221	C₂×Dic₃	—	10	12	no	—
222	Dic₆	—	20	12	no	—
223	C₂²×C₅	—	2	12	no	—
224	C₂²×C₅	—	2	6	no	—
225	C₂²×C₅	—	2	6	no	—
226	C₂²×C₅	—	4	16	no	—
227	C₂²×C₅	—	4	4	no	—
228	C₂²×C₅	—	4	12	no	—
229	C₂²×C₅	#1004	1	6	yes	—
230	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
231	C₂×D₅	—	6	12	no	—
232	C₂×D₅	—	12	16	no	—
233	C₂×D₅	—	12	8	no	—
234	C₄×C₅	—	2	6	no	—
235	Dic₅	—	6	12	no	—
236	C₂⁴	—	4	6	no	—
237	C₂⁴	—	15	2	no	—
238	C₂⁴	—	15	4	no	—
239	C₂⁴	—	60	6	no	—
240	C₂⁴	C₂²×D₁₅	1	1	yes	characteristic
241	C₂⁴	C₂²×D₁₅	1	3	yes	—
242	C₂⁴	C₂²×D₁₅	1	4	yes	complemented
243	C₂×Q₈	—	15	12	no	—
244	C₂×Q₈	—	60	6	no	—
245	C₂×D₄	—	4	6	no	—
246	C₂×D₄	—	15	24	no	—
247	C₂×D₄	—	60	12	no	—
248	C₂×D₄	C₂²×D₁₅	1	12	yes	—
249	C₂²×C₄	—	15	6	no	—
250	C₂²×C₄	C₂²×D₁₅	1	3	yes	—
251	C₂²⋊C₄	—	15	12	no	—
252	C₂²⋊C₄	—	15	24	no	—
253	C₂²⋊C₄	—	15	24	no	—
254	C₂²⋊C₄	C₂²×D₁₅	1	12	yes	complemented
255	C₄²	—	15	12	no	—
256	C₃×C₅	C₂×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	2	12	no	—
258	C₂²×C₃	—	2	6	no	—
259	C₂²×C₃	—	2	6	no	—
260	C₂²×C₃	—	4	16	no	—
261	C₂²×C₃	—	4	4	no	—
262	C₂²×C₃	—	4	12	no	—
263	C₂²×C₃	#4144	1	6	yes	—
264	C₂²×C₃	C₂⁴×D₅	1	1	yes	characteristic
265	C₂×D₃	—	10	12	no	—
266	C₂×D₃	—	20	16	no	—
267	C₂×D₃	—	20	8	no	—
268	C₃×C₄	—	2	6	no	—
269	Dic₃	—	10	12	no	—
270	C₂×C₅	—	4	2	no	—
271	C₂×C₅	—	4	6	no	—
272	C₂×C₅	#5594	1	4	yes	complemented
273	C₂×C₅	C₂×#1004	1	3	yes	—
274	D₅	—	12	4	no	—
275	C₂³	—	2	6	no	—
276	C₂³	—	2	6	no	—
277	C₂³	—	2	12	no	—
278	C₂³	—	4	12	no	—
279	C₂³	—	4	16	no	—
280	C₂³	—	15	4	no	—
281	C₂³	—	30	12	no	—
282	C₂³	—	60	12	no	—
283	C₂³	—	60	16	no	—
284	C₂³	C₂³×D₁₅	1	2	yes	—
285	C₂³	C₂³×D₁₅	1	6	yes	—
286	C₂³	C₂³×D₁₅	1	1	yes	characteristic
287	Q₈	—	60	12	no	—
288	D₄	—	4	12	no	—
289	D₄	—	60	24	no	—
290	C₂×C₄	—	2	6	no	—
291	C₂×C₄	—	15	12	no	—
292	C₂×C₄	—	30	12	no	—
293	C₂×C₄	C₂³×D₁₅	1	6	yes	—
294	C₂×C₃	—	4	2	no	—
295	C₂×C₃	—	4	6	no	—
296	C₂×C₃	#65928	1	4	yes	complemented
297	C₂×C₃	C₂×#4144	1	3	yes	—
298	D₃	—	20	4	no	—
299	C₅	C₂×#5594	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	2	6	no	—
301	C₂²	—	2	6	no	—
302	C₂²	—	2	12	no	—
303	C₂²	—	4	16	no	—
304	C₂²	—	4	12	no	—
305	C₂²	—	4	4	no	—
306	C₂²	—	30	12	no	—
307	C₂²	—	60	16	no	—
308	C₂²	—	60	8	no	—
309	C₂²	#92095	1	6	yes	—
310	C₂²	C₂⁴×D₁₅	1	1	yes	characteristic
311	C₄	—	2	6	no	—
312	C₄	—	30	12	no	—
313	C₃	C₂×#65928	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	4	6	no	—
315	C₂	—	4	2	no	—
316	C₂	—	60	4	no	—
317	C₂	#11756353	1	4	yes	direct product factor
318	C₂	C₂×#92095	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₅
commutator, cosocle	C₂²×C₃×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₁₅
Fitting	C₂×C₃×C₅×#120	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×#120	D₁₅
hypocenter	C₃×C₅	C₂×H(3,𝔽₄)
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×#120

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#120	C₂	semidirect
2. C₂×#91842	C₂	semidirect
3. C₂×#91826	C₂	semidirect
4. C₂×#91825	C₂	semidirect
5. #11756353	C₂	direct (N × H)
6. C₂×#91774	C₂	semidirect
8. C₂⁵×C₃×C₅	C₂²	semidirect
9. C₂⁴×D₁₅	C₂²	semidirect
12. C₂²×Dic₃₀	C₂²	semidirect
13. C₃×C₅×#120	C₂²	semidirect
14. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
15. #91842	C₂²	semidirect
16. C₂×#6719	C₂²	semidirect
17. C₂×#6719	C₂²	semidirect
18. #91826	C₂²	semidirect
19. #91825	C₂²	semidirect
20. C₂×#6714	C₂²	semidirect
21. C₂×C₄×Dic₁₅	C₂²	semidirect
22. #91774	C₂²	semidirect
24. C₂×C₅×#120	D₃	semidirect
33. C₂⁴×C₃×C₅	C₂³	semidirect
36. C₂³×D₁₅	C₂³	semidirect
44. C₂×Dic₃₀	C₂³	semidirect
45. C₃×C₅×C₂²⋊C₄	C₂³	semidirect
46. #6719	C₂³	semidirect
47. #6719	C₂³	semidirect
48. #6714	C₂³	semidirect
49. C₄×Dic₁₅	C₂³	semidirect
50. C₂×C₃×#120	D₅	semidirect
56. C₂⁵×C₅	C₂×D₃	semidirect
61. C₅×#120	C₂×D₃	semidirect
62. C₂×C₅×C₂²⋊C₄	C₂×D₃	semidirect
91. C₂⁵×C₃	C₂×D₅	semidirect
96. C₃×#120	C₂×D₅	semidirect
97. C₂×C₃×C₂²⋊C₄	C₂×D₅	semidirect
109. C₂⁴×C₅	C₂²×D₃	semidirect
121. C₅×C₂²⋊C₄	C₂²×D₃	semidirect
131. C₂×#120	D₁₅	semidirect
148. C₂⁴×C₃	C₂²×D₅	semidirect
160. C₃×C₂²⋊C₄	C₂²×D₅	semidirect
185. C₂⁵	C₂×D₁₅	semidirect
193. #120	C₂×D₁₅	semidirect
197. C₂×C₂²⋊C₄	C₂×D₁₅	semidirect
201. C₂×C₃×C₅	H(3,𝔽₄)	semidirect
242. C₂⁴	C₂²×D₁₅	semidirect
254. C₂²⋊C₄	C₂²×D₁₅	semidirect
256. C₃×C₅	C₂×H(3,𝔽₄)	semidirect
272. C₂×C₅	#5594	semidirect
296. C₂×C₃	#65928	semidirect
299. C₅	C₂×#5594	semidirect
313. C₃	C₂×#65928	semidirect
317. C₂	#11756353	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#120	2
C₂×#91842	2
C₂×#91826	2
C₂×#91825	2
#11756353	2
C₂×#91774	2
C₂×#65928	3
C₂×#5594	5

Subgroups — C2×#11756353 #14495256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11756353 #14495256