Subgroups — C₂×#10624199 #13089354

Order 1440.

710 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₄×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#67153	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#67152	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#67150	C₂	1	1	yes	maximal, characteristic, complemented
8	#10624199	C₂	1	8	yes	maximal, direct product factor
9	C₂×#6817	—	3	1	no	maximal
10	C₂×#6771	—	3	1	no	maximal
11	C₂²×C₃⋊D₁₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃×D₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊Dic₁₅	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊D₁₅	C₂²	1	4	yes	complemented
17	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
18	C₃×C₄×D₁₅	C₂²	1	4	yes	complemented
19	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×D₃	C₂²	1	4	yes	complemented
21	#67153	C₂²	1	4	yes	complemented
22	#67152	C₂²	1	4	yes	complemented
23	#67151	C₂²	1	4	yes	complemented
24	#67150	C₂²	1	4	yes	complemented
25	C₂×#3798	—	5	1	no	maximal, Hall
26	C₂²×C₃×C₄×C₅	—	3	1	no	—
27	C₂×#1220	—	3	1	no	—
28	C₂×#1220	—	3	1	no	—
29	#6817	—	3	8	no	—
30	C₂×D₆₀	—	3	1	no	—
31	C₂×C₄×D₁₅	—	3	1	no	—
32	C₂×C₄×D₁₅	—	3	1	no	—
33	C₂×C₄×D₁₅	—	6	1	no	—
34	C₂×C₄×D₁₅	D₃	1	1	yes	characteristic, complemented
35	C₂×Dic₃₀	—	3	1	no	—
36	C₂×C₄×C₅×D₃	D₃	1	1	yes	characteristic, complemented
37	C₂×C₃×C₄×D₅	—	3	1	no	—
38	C₂×#1204	—	3	1	no	—
39	C₂×#1203	—	3	1	no	—
40	C₂×#1202	—	3	1	no	—
41	C₂×#1201	—	3	1	no	—
42	#6771	—	3	8	no	—
43	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₃⋊D₁₅	—	2	2	no	—
45	C₂×C₃⋊D₁₅	C₂³	1	2	yes	—
46	C₂×C₃×D₁₅	—	2	2	no	—
47	C₂×C₃×D₁₅	C₂³	1	2	yes	—
48	C₂×C₃×C₅×D₃	—	2	2	no	—
49	C₂×C₃×C₅×D₃	C₂³	1	2	yes	—
50	C₃²×C₄×C₅	C₂³	1	2	yes	—
51	C₃⋊Dic₁₅	C₂³	1	2	yes	—
52	C₃×Dic₁₅	C₂³	1	2	yes	—
53	C₃×C₅×Dic₃	C₂³	1	2	yes	—
54	C₂×#709	—	9	1	no	Hall
55	C₂×C₄×C₃⋊D₃	—	5	1	no	—
56	C₂×C₃×C₄×D₃	—	5	1	no	—
57	C₂×C₃×C₄×D₃	D₅	1	1	yes	characteristic, complemented
58	C₂×#627	—	5	1	no	—
59	C₂×#626	—	5	1	no	—
60	C₂×#626	—	5	1	no	—
61	C₂×#625	—	5	1	no	—
62	#3798	—	5	8	no	—
63	C₂³×C₃×C₅	—	3	1	no	—
64	C₂²×D₁₅	—	3	1	no	—
65	C₂²×D₁₅	—	3	1	no	—
66	C₂²×D₁₅	—	6	1	no	—
67	C₂²×D₁₅	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₅×D₃	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×D₅	—	3	1	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	3	1	no	—
72	C₂×C₃×C₄×C₅	—	3	4	no	—
73	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
75	#1220	—	3	4	no	—
76	#1220	—	3	4	no	—
77	C₂×Dic₁₅	—	3	1	no	—
78	C₂×Dic₁₅	—	3	1	no	—
79	C₂×Dic₁₅	—	6	1	no	—
80	C₂×Dic₁₅	C₂×D₃	1	1	yes	characteristic
81	D₆₀	—	3	4	no	—
82	C₄×D₁₅	—	3	4	no	—
83	C₄×D₁₅	—	3	4	no	—
84	C₄×D₁₅	—	6	4	no	—
85	C₄×D₁₅	C₂×D₃	1	4	yes	complemented
86	Dic₃₀	—	3	4	no	—
87	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
88	C₄×C₅×D₃	C₂×D₃	1	4	yes	complemented
89	C₂×C₃×Dic₅	—	3	1	no	—
90	C₃×C₄×D₅	—	3	4	no	—
91	#1204	—	3	4	no	—
92	#1203	—	3	4	no	—
93	#1202	—	3	4	no	—
94	#1201	—	3	4	no	—
95	C₂×#235	—	15	1	no	—
96	C₂×#235	—	15	1	no	—
97	C₂×C₃²×C₅	#41	1	2	yes	complemented
98	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
99	C₃⋊D₁₅	—	2	2	no	—
100	C₃×D₁₅	—	2	2	no	—
101	C₃×C₅×D₃	—	2	2	no	—
102	C₂²×C₄×C₅	—	3	1	no	—
103	C₂×C₂²⋊D₅	—	9	1	no	—
104	C₂×C₂²⋊D₅	—	9	1	no	—
105	#709	—	9	8	no	—
106	C₂×D₂₀	—	9	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₄×D₅	—	9	1	no	—
109	C₂×Dic₁₀	—	9	1	no	—
110	C₂²×C₃⋊D₃	—	5	1	no	—
111	C₂²×C₃×D₃	—	5	1	no	—
112	C₂²×C₃×D₃	C₂×D₅	1	1	yes	characteristic
113	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
114	C₂×C₃⋊Dic₃	—	5	1	no	—
115	C₄×C₃⋊D₃	—	5	4	no	—
116	C₂×C₃×Dic₃	—	5	1	no	—
117	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
118	C₃×C₄×D₃	—	5	4	no	—
119	C₃×C₄×D₃	C₂×D₅	1	4	yes	complemented
120	#627	—	5	4	no	—
121	#626	—	5	4	no	—
122	#626	—	5	4	no	—
123	#625	—	5	4	no	—
124	C₂²×C₃×C₅	—	2	1	no	—
125	C₂²×C₃×C₅	—	3	2	no	—
126	C₂²×C₃×C₅	—	6	2	no	—
127	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂×D₁₅	—	2	2	no	—
130	C₂×D₁₅	—	3	2	no	—
131	C₂×D₁₅	—	3	2	no	—
132	C₂×D₁₅	—	6	2	no	—
133	C₂×D₁₅	—	6	4	no	—
134	C₂×D₁₅	—	6	2	no	—
135	C₂×D₁₅	—	6	2	no	—
136	C₂×D₁₅	C₂²×D₃	1	2	yes	—
137	C₂×C₅×D₃	—	2	2	no	—
138	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
139	C₂×C₃×D₅	—	3	2	no	—
140	C₂×C₃×D₅	—	6	2	no	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	—	3	2	no	—
143	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
144	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
145	Dic₁₅	—	3	2	no	—
146	Dic₁₅	—	3	2	no	—
147	Dic₁₅	—	6	2	no	—
148	Dic₁₅	C₂²×D₃	1	2	yes	—
149	C₃×Dic₅	—	3	2	no	—
150	C₅×Dic₃	C₂²×D₃	1	2	yes	—
151	C₂²×C₃×C₄	—	3	1	no	—
152	C₂²×C₃×C₄	—	15	1	no	—
153	C₂×#67	—	15	1	no	—
154	C₂×#67	—	15	1	no	—
155	C₂×#67	—	15	1	no	—
156	C₂×#67	—	15	1	no	—
157	#235	—	15	8	no	—
158	#235	—	15	8	no	—
159	C₂×D₁₂	—	15	1	no	—
160	C₂×D₁₂	—	15	1	no	—
161	C₂×C₄×D₃	—	5	1	no	—
162	C₂×C₄×D₃	—	15	1	no	—
163	C₂×C₄×D₃	—	15	1	no	—
164	C₂×C₄×D₃	—	30	1	no	—
165	C₂×C₄×D₃	D₁₅	1	1	yes	characteristic, complemented
166	C₂×Dic₆	—	15	1	no	—
167	C₂×Dic₆	—	15	1	no	—
168	C₃²×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
169	C₂³×C₅	—	3	1	no	—
170	C₂²×D₅	—	3	1	no	—
171	C₂²×D₅	—	9	1	no	—
172	C₂×C₄×C₅	—	3	1	no	—
173	C₂×C₄×C₅	—	3	4	no	—
174	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
175	C₂²⋊D₅	—	9	4	no	—
176	C₂²⋊D₅	—	9	4	no	—
177	C₂×Dic₅	—	3	1	no	—
178	C₂×Dic₅	—	9	1	no	—
179	D₂₀	—	9	4	no	—
180	C₄×D₅	—	3	4	no	—
181	C₄×D₅	—	9	4	no	—
182	Dic₁₀	—	9	4	no	—
183	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₃⋊D₃	—	5	2	no	—
185	C₂×C₃⋊D₃	—	10	2	no	—
186	C₂×C₃×D₃	—	2	2	no	—
187	C₂×C₃×D₃	—	5	2	no	—
188	C₂×C₃×D₃	—	10	2	no	—
189	C₂×C₃×D₃	C₂²×D₅	1	2	yes	—
190	C₃²×C₄	C₂²×D₅	1	2	yes	—
191	C₃⋊Dic₃	—	5	2	no	—
192	C₃×Dic₃	—	5	2	no	—
193	C₃×Dic₃	C₂²×D₅	1	2	yes	—
194	C₂×#41	—	45	1	no	Sylow
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	2	2	no	—
197	C₂×C₃×C₅	—	6	2	no	—
198	C₂×C₃×C₅	#235	1	2	yes	complemented
199	C₂×C₃×C₅	#235	1	2	yes	complemented
200	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
201	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
202	D₁₅	—	2	2	no	—
203	D₁₅	—	6	4	no	—
204	D₁₅	—	6	2	no	—
205	D₁₅	—	6	2	no	—
206	C₃×D₅	—	6	2	no	—
207	C₅×D₃	—	2	2	no	—
208	C₂³×C₃	—	3	1	no	—
209	C₂³×C₃	—	15	1	no	—
210	C₂²×D₃	—	5	1	no	—
211	C₂²×D₃	—	15	1	no	—
212	C₂²×D₃	—	15	1	no	—
213	C₂²×D₃	—	30	1	no	—
214	C₂²×D₃	C₂×D₁₅	1	1	yes	characteristic
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	3	1	no	—
217	C₂×C₃×C₄	—	3	4	no	—
218	C₂×C₃×C₄	—	15	1	no	—
219	C₂×C₃×C₄	—	15	4	no	—
220	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
222	#67	—	15	4	no	—
223	#67	—	15	4	no	—
224	#67	—	15	4	no	—
225	#67	—	15	4	no	—
226	C₂×Dic₃	—	5	1	no	—
227	C₂×Dic₃	—	15	1	no	—
228	C₂×Dic₃	—	15	1	no	—
229	C₂×Dic₃	—	30	1	no	—
230	C₂×Dic₃	C₂×D₁₅	1	1	yes	characteristic
231	D₁₂	—	15	4	no	—
232	D₁₂	—	15	4	no	—
233	C₄×D₃	—	5	4	no	—
234	C₄×D₃	—	15	4	no	—
235	C₄×D₃	—	15	4	no	—
236	C₄×D₃	—	30	4	no	—
237	C₄×D₃	C₂×D₁₅	1	4	yes	complemented
238	Dic₆	—	15	4	no	—
239	Dic₆	—	15	4	no	—
240	C₂²×C₅	—	3	2	no	—
241	C₂²×C₅	—	6	2	no	—
242	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
243	C₂×D₅	—	3	2	no	—
244	C₂×D₅	—	6	2	no	—
245	C₂×D₅	—	9	2	no	—
246	C₂×D₅	—	18	2	no	—
247	C₄×C₅	—	3	2	no	—
248	C₄×C₅	C₂×D₃²	1	2	yes	—
249	Dic₅	—	3	2	no	—
250	Dic₅	—	9	2	no	—
251	C₂×C₃²	#709	1	2	yes	complemented
252	C₂×C₃²	C₂³×D₅	1	1	yes	characteristic
253	C₃⋊D₃	—	10	2	no	—
254	C₃×D₃	—	2	2	no	—
255	C₃×D₃	—	10	2	no	—
256	#41	—	45	8	no	—
257	C₂×Q₈	—	45	1	no	—
258	C₂×D₄	—	45	1	no	—
259	C₂×D₄	—	45	1	no	—
260	C₂×D₄	—	45	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	15	1	no	—
263	C₂²×C₄	—	45	1	no	—
264	C₃×C₅	—	2	1	no	—
265	C₃×C₅	C₂×#235	1	1	yes	characteristic, complemented
266	C₃×C₅	C₂×#235	1	1	yes	characteristic, complemented
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	3	2	no	—
269	C₂²×C₃	—	6	2	no	—
270	C₂²×C₃	—	15	2	no	—
271	C₂²×C₃	—	30	2	no	—
272	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
273	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
274	C₂×D₃	—	2	2	no	—
275	C₂×D₃	—	5	2	no	—
276	C₂×D₃	—	10	2	no	—
277	C₂×D₃	—	15	2	no	—
278	C₂×D₃	—	15	2	no	—
279	C₂×D₃	—	30	2	no	—
280	C₂×D₃	—	30	4	no	—
281	C₂×D₃	—	30	2	no	—
282	C₂×D₃	—	30	2	no	—
283	C₂×D₃	C₂²×D₁₅	1	2	yes	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	3	2	no	—
286	C₃×C₄	—	15	2	no	—
287	C₃×C₄	C₂×D₃×D₅	1	2	yes	—
288	C₃×C₄	C₂²×D₁₅	1	2	yes	—
289	Dic₃	—	5	2	no	—
290	Dic₃	—	15	2	no	—
291	Dic₃	—	15	2	no	—
292	Dic₃	—	30	2	no	—
293	Dic₃	C₂²×D₁₅	1	2	yes	—
294	C₂×C₅	—	6	2	no	—
295	C₂×C₅	#3798	1	2	yes	complemented
296	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
297	D₅	—	6	2	no	—
298	D₅	—	18	2	no	—
299	C₃²	C₂×#709	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	3	1	no	—
301	C₂³	—	15	1	no	—
302	C₂³	—	45	1	no	—
303	Q₈	—	45	4	no	—
304	D₄	—	45	4	no	—
305	D₄	—	45	4	no	—
306	D₄	—	45	4	no	—
307	C₂×C₄	—	3	1	no	—
308	C₂×C₄	—	3	4	no	—
309	C₂×C₄	—	15	1	no	—
310	C₂×C₄	—	15	4	no	—
311	C₂×C₄	—	45	1	no	—
312	C₂×C₄	—	45	4	no	—
313	C₂×C₄	D₃×D₁₅	1	1	yes	characteristic
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	6	2	no	—
317	C₂×C₃	—	30	2	no	—
318	C₂×C₃	#6771	1	2	yes	complemented
319	C₂×C₃	#6817	1	2	yes	complemented
320	C₂×C₃	C₂²×D₃×D₅	1	1	yes	characteristic
321	C₂×C₃	C₂³×D₁₅	1	1	yes	characteristic
322	D₃	—	2	2	no	—
323	D₃	—	10	2	no	—
324	D₃	—	30	4	no	—
325	D₃	—	30	2	no	—
326	D₃	—	30	2	no	—
327	C₅	C₂×#3798	1	1	yes	characteristic, complemented, Sylow
328	C₂²	—	3	2	no	—
329	C₂²	—	6	2	no	—
330	C₂²	—	15	2	no	—
331	C₂²	—	30	2	no	—
332	C₂²	—	45	2	no	—
333	C₂²	—	90	2	no	—
334	C₂²	C₂×D₃×D₁₅	1	1	yes	characteristic
335	C₄	—	3	2	no	—
336	C₄	—	15	2	no	—
337	C₄	—	45	2	no	—
338	C₄	C₂×D₃×D₁₅	1	2	yes	—
339	C₃	—	2	1	no	—
340	C₃	C₂×#6771	1	1	yes	characteristic, complemented
341	C₃	C₂×#6817	1	1	yes	characteristic, complemented
342	C₂	—	6	2	no	—
343	C₂	—	30	2	no	—
344	C₂	—	90	2	no	—
345	C₂	#10624199	1	2	yes	direct product factor
346	C₂	C₂²×D₃×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₃×D₁₅
commutator, cosocle	C₂×C₃²×C₅	C₂⁴
Frattini	C₂	C₂²×D₃×D₁₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
hypocenter	C₃²×C₅	C₂×#41
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×C₃⋊D₁₅	C₂	semidirect
2. C₂×C₃×C₄×D₁₅	C₂	semidirect
3. C₂×C₃×C₄×C₅×D₃	C₂	semidirect
4. C₂×#67153	C₂	semidirect
5. C₂×#67152	C₂	semidirect
6. C₂×#67151	C₂	semidirect
7. C₂×#67150	C₂	semidirect
8. #10624199	C₂	direct (N × H)
16. C₄×C₃⋊D₁₅	C₂²	semidirect
18. C₃×C₄×D₁₅	C₂²	semidirect
20. C₃×C₄×C₅×D₃	C₂²	semidirect
21. #67153	C₂²	semidirect
22. #67152	C₂²	semidirect
23. #67151	C₂²	semidirect
24. #67150	C₂²	semidirect
34. C₂×C₄×D₁₅	D₃	semidirect
36. C₂×C₄×C₅×D₃	D₃	semidirect
57. C₂×C₃×C₄×D₃	D₅	semidirect
85. C₄×D₁₅	C₂×D₃	semidirect
88. C₄×C₅×D₃	C₂×D₃	semidirect
97. C₂×C₃²×C₅	#41	semidirect
119. C₃×C₄×D₃	C₂×D₅	semidirect
165. C₂×C₄×D₃	D₁₅	semidirect
168. C₃²×C₅	C₂×#41	semidirect
198. C₂×C₃×C₅	#235	semidirect
199. C₂×C₃×C₅	#235	semidirect
237. C₄×D₃	C₂×D₁₅	semidirect
251. C₂×C₃²	#709	semidirect
265. C₃×C₅	C₂×#235	semidirect
266. C₃×C₅	C₂×#235	semidirect
295. C₂×C₅	#3798	semidirect
299. C₃²	C₂×#709	semidirect
318. C₂×C₃	#6771	semidirect
319. C₂×C₃	#6817	semidirect
327. C₅	C₂×#3798	semidirect
340. C₃	C₂×#6771	semidirect
341. C₃	C₂×#6817	semidirect
345. C₂	#10624199	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×C₄×C₃⋊D₁₅	2
C₂×C₃×C₄×D₁₅	2
C₂×C₃×C₄×C₅×D₃	2
C₂×#67153	2
C₂×#67152	2
C₂×#67151	2
C₂×#67150	2
#10624199	2
C₂×#6817	3
C₂×#6771	3
C₂×#3798	5

Subgroups — C2×#10624199 #13089354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624199 #13089354