Subgroups — C₂×#10624208 #13089385

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	C₂²×C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#1220	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#67153	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#67150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#10624208	C₂	1	8	yes	maximal, direct product factor
8	C₂×Dic₃×D₁₅	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#6819	—	3	1	no	maximal
10	C₂×#6780	—	3	1	no	maximal
11	C₂³×C₃²×C₅	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×C₃⋊Dic₁₅	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊Dic₁₅	C₂²	1	1	yes	characteristic
16	C₂×C₃⋊Dic₁₅	C₂²	1	4	yes	complemented
17	C₃×#1220	C₂²	1	4	yes	complemented
18	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
19	C₃×C₅×#67	C₂²	1	4	yes	complemented
20	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
21	#67153	C₂²	1	4	yes	complemented
22	#67150	C₂²	1	4	yes	complemented
23	D₃×Dic₁₅	C₂²	1	4	yes	complemented
24	Dic₃×D₁₅	C₂²	1	4	yes	complemented
25	C₂×#3805	—	5	1	no	maximal, Hall
26	C₂×C₃×C₅×D₄	—	3	1	no	—
27	C₂×#1220	—	3	1	no	—
28	C₂×#1220	D₃	1	1	yes	characteristic, complemented
29	C₂²×Dic₁₅	—	3	1	no	—
30	C₂²×Dic₁₅	—	3	1	no	—
31	C₂²×Dic₁₅	—	3	1	no	—
32	C₂²×Dic₁₅	—	6	1	no	—
33	#6819	—	3	8	no	—
34	C₂×C₄×D₁₅	—	3	1	no	—
35	C₂×Dic₃₀	—	3	1	no	—
36	C₂×C₅×#67	D₃	1	1	yes	characteristic, complemented
37	C₂×C₃×C₂²⋊D₅	—	3	1	no	—
38	C₂×#1204	—	3	1	no	—
39	C₂×#1201	—	3	1	no	—
40	C₂×D₃×Dic₅	—	3	1	no	—
41	#6780	—	3	8	no	—
42	C₂×D₅×Dic₃	—	3	1	no	—
43	C₂²×C₃²×C₅	—	2	2	no	—
44	C₂²×C₃²×C₅	C₂³	1	2	yes	—
45	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
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	C₃⋊Dic₁₅	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×#711	—	9	1	no	Hall
55	C₂²×C₃⋊Dic₃	—	5	1	no	—
56	C₂×C₃×#67	—	5	1	no	—
57	C₂×C₃×#67	D₅	1	1	yes	characteristic, complemented
58	C₂×#627	—	5	1	no	—
59	C₂×#625	—	5	1	no	—
60	#3805	—	5	8	no	—
61	C₂×D₃×Dic₃	—	5	1	no	—
62	C₂×D₃×Dic₃	—	5	1	no	—
63	C₂³×C₃×C₅	—	2	1	no	—
64	C₂³×C₃×C₅	—	3	1	no	—
65	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
66	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
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	C₃×C₅×D₄	—	3	4	no	—
71	C₂×C₃×C₄×C₅	—	3	1	no	—
72	#1220	—	3	4	no	—
73	#1220	C₂×D₃	1	4	yes	complemented
74	C₂×Dic₁₅	—	3	1	no	—
75	C₂×Dic₁₅	—	3	1	no	—
76	C₂×Dic₁₅	—	3	1	no	—
77	C₂×Dic₁₅	—	3	1	no	—
78	C₂×Dic₁₅	—	3	4	no	—
79	C₂×Dic₁₅	—	3	4	no	—
80	C₂×Dic₁₅	—	3	4	no	—
81	C₂×Dic₁₅	—	6	1	no	—
82	C₂×Dic₁₅	—	6	1	no	—
83	C₂×Dic₁₅	—	6	4	no	—
84	C₂×Dic₁₅	C₂×D₃	1	1	yes	characteristic
85	C₄×D₁₅	—	3	4	no	—
86	Dic₃₀	—	3	4	no	—
87	C₅×#67	C₂×D₃	1	4	yes	complemented
88	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
89	C₃×C₂²⋊D₅	—	3	4	no	—
90	C₂×C₃×Dic₅	—	3	1	no	—
91	#1204	—	3	4	no	—
92	#1201	—	3	4	no	—
93	D₃×Dic₅	—	3	4	no	—
94	D₅×Dic₃	—	3	4	no	—
95	C₂×#237	—	15	1	no	—
96	C₂×#237	—	15	1	no	—
97	C₂×C₃²×C₅	—	2	2	no	—
98	C₂×C₃²×C₅	#41	1	2	yes	complemented
99	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
100	C₃×D₁₅	—	2	2	no	—
101	C₃×C₅×D₃	—	2	2	no	—
102	C₂×C₅×D₄	—	3	1	no	—
103	C₂×C₂²⋊D₅	—	3	1	no	—
104	C₂×C₂²⋊D₅	—	9	1	no	—
105	C₂²×Dic₅	—	9	1	no	—
106	C₂²×Dic₅	—	9	1	no	—
107	#711	—	9	8	no	—
108	C₂×C₄×D₅	—	9	1	no	—
109	C₂×Dic₁₀	—	9	1	no	—
110	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
111	C₂²×C₃×D₃	—	5	1	no	—
112	C₂²×C₃×D₃	C₂×D₅	1	1	yes	characteristic
113	C₂×C₃⋊Dic₃	—	5	1	no	—
114	C₂×C₃⋊Dic₃	—	5	1	no	—
115	C₂×C₃⋊Dic₃	—	5	4	no	—
116	C₃×#67	—	5	4	no	—
117	C₃×#67	C₂×D₅	1	4	yes	complemented
118	C₂×C₃×Dic₃	—	5	1	no	—
119	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
120	#627	—	5	4	no	—
121	#625	—	5	4	no	—
122	D₃×Dic₃	—	5	4	no	—
123	D₃×Dic₃	—	5	4	no	—
124	C₂²×C₃×C₅	—	2	4	no	—
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	—	2	2	no	—
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	—	3	2	no	—
130	C₂²×C₃×C₅	—	6	2	no	—
131	C₂²×C₃×C₅	C₂²×D₃	1	2	yes	—
132	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₂²×C₃×C₅	C₂²×D₃	1	2	yes	—
134	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₂×D₁₅	—	2	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	—	3	2	no	—
142	Dic₁₅	—	3	2	no	—
143	Dic₁₅	—	3	2	no	—
144	Dic₁₅	—	3	2	no	—
145	Dic₁₅	—	3	2	no	—
146	Dic₁₅	—	6	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	C₂×C₃×D₄	—	3	1	no	—
152	C₂×C₃×D₄	—	15	1	no	—
153	C₂×#67	—	5	1	no	—
154	C₂×#67	—	15	1	no	—
155	C₂×#67	—	15	1	no	—
156	C₂×#67	D₁₅	1	1	yes	characteristic, complemented
157	C₂²×Dic₃	—	15	1	no	—
158	C₂²×Dic₃	—	15	1	no	—
159	C₂²×Dic₃	—	15	1	no	—
160	C₂²×Dic₃	—	15	1	no	—
161	C₂²×Dic₃	—	30	1	no	—
162	#237	—	15	8	no	—
163	#237	—	15	8	no	—
164	C₂×C₄×D₃	—	15	1	no	—
165	C₂×C₄×D₃	—	15	1	no	—
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	C₂³×C₅	D₃²	1	1	yes	characteristic
171	C₂²×D₅	—	3	1	no	—
172	C₅×D₄	—	3	4	no	—
173	C₂×C₄×C₅	—	3	1	no	—
174	C₂²⋊D₅	—	3	4	no	—
175	C₂²⋊D₅	—	9	4	no	—
176	C₂×Dic₅	—	3	1	no	—
177	C₂×Dic₅	—	9	4	no	—
178	C₂×Dic₅	—	9	1	no	—
179	C₂×Dic₅	—	9	1	no	—
180	C₂×Dic₅	—	9	4	no	—
181	C₄×D₅	—	9	4	no	—
182	Dic₁₀	—	9	4	no	—
183	C₂²×C₃²	—	2	2	no	—
184	C₂²×C₃²	C₂²×D₅	1	2	yes	—
185	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
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	C₃⋊Dic₃	—	5	2	no	—
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	4	no	—
196	C₂×C₃×C₅	—	2	2	no	—
197	C₂×C₃×C₅	—	2	2	no	—
198	C₂×C₃×C₅	—	2	1	no	—
199	C₂×C₃×C₅	—	2	2	no	—
200	C₂×C₃×C₅	—	6	2	no	—
201	C₂×C₃×C₅	#237	1	2	yes	complemented
202	C₂×C₃×C₅	#237	1	2	yes	complemented
203	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂³×D₃	1	1	yes	characteristic
205	D₁₅	—	2	2	no	—
206	C₃×D₅	—	6	2	no	—
207	C₅×D₃	—	2	2	no	—
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	—	3	1	no	—
210	C₂³×C₃	—	15	1	no	—
211	C₂³×C₃	D₃×D₅	1	1	yes	characteristic
212	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
213	C₂²×D₃	—	5	1	no	—
214	C₂²×D₃	C₂×D₁₅	1	1	yes	characteristic
215	C₃×D₄	—	3	4	no	—
216	C₃×D₄	—	15	4	no	—
217	C₂×C₃×C₄	—	3	1	no	—
218	C₂×C₃×C₄	—	15	1	no	—
219	#67	—	5	4	no	—
220	#67	—	15	4	no	—
221	#67	—	15	4	no	—
222	#67	C₂×D₁₅	1	4	yes	complemented
223	C₂×Dic₃	—	5	1	no	—
224	C₂×Dic₃	—	15	4	no	—
225	C₂×Dic₃	—	15	1	no	—
226	C₂×Dic₃	—	15	1	no	—
227	C₂×Dic₃	—	15	4	no	—
228	C₂×Dic₃	—	15	1	no	—
229	C₂×Dic₃	—	15	1	no	—
230	C₂×Dic₃	—	15	4	no	—
231	C₂×Dic₃	—	15	4	no	—
232	C₂×Dic₃	—	30	4	no	—
233	C₂×Dic₃	—	30	1	no	—
234	C₂×Dic₃	—	30	1	no	—
235	C₂×Dic₃	C₂×D₁₅	1	1	yes	characteristic
236	C₄×D₃	—	15	4	no	—
237	C₄×D₃	—	15	4	no	—
238	Dic₆	—	15	4	no	—
239	Dic₆	—	15	4	no	—
240	C₂²×C₅	—	2	2	no	—
241	C₂²×C₅	—	3	2	no	—
242	C₂²×C₅	—	6	2	no	—
243	C₂²×C₅	C₂×D₃²	1	2	yes	—
244	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
245	C₂×D₅	—	3	2	no	—
246	C₂×D₅	—	6	2	no	—
247	C₄×C₅	—	3	2	no	—
248	Dic₅	—	3	2	no	—
249	Dic₅	—	9	2	no	—
250	Dic₅	—	9	2	no	—
251	C₂×C₃²	—	2	2	no	—
252	C₂×C₃²	#711	1	2	yes	complemented
253	C₂×C₃²	C₂³×D₅	1	1	yes	characteristic
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	—	3	1	no	—
259	C₂×D₄	—	15	1	no	—
260	C₂×D₄	—	45	1	no	—
261	C₂²×C₄	—	45	1	no	—
262	C₂²×C₄	—	45	1	no	—
263	C₂²×C₄	—	45	1	no	—
264	C₃×C₅	—	2	1	no	—
265	C₃×C₅	C₂×#237	1	1	yes	characteristic, complemented
266	C₃×C₅	C₂×#237	1	1	yes	characteristic, complemented
267	C₂²×C₃	—	2	4	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	3	2	no	—
273	C₂²×C₃	—	6	2	no	—
274	C₂²×C₃	—	15	2	no	—
275	C₂²×C₃	—	30	2	no	—
276	C₂²×C₃	C₂×D₃×D₅	1	2	yes	—
277	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
278	C₂²×C₃	C₂²×D₁₅	1	2	yes	—
279	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
280	C₂×D₃	—	2	2	no	—
281	C₂×D₃	—	5	2	no	—
282	C₂×D₃	—	10	2	no	—
283	C₂×D₃	C₂²×D₁₅	1	2	yes	—
284	C₃×C₄	—	3	2	no	—
285	C₃×C₄	—	15	2	no	—
286	Dic₃	—	5	2	no	—
287	Dic₃	—	15	2	no	—
288	Dic₃	—	15	2	no	—
289	Dic₃	—	15	2	no	—
290	Dic₃	—	15	2	no	—
291	Dic₃	—	30	2	no	—
292	Dic₃	—	30	2	no	—
293	Dic₃	C₂²×D₁₅	1	2	yes	—
294	C₂×C₅	—	2	2	no	—
295	C₂×C₅	—	6	2	no	—
296	C₂×C₅	#3805	1	2	yes	complemented
297	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
298	D₅	—	6	2	no	—
299	C₃²	C₂×#711	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	3	1	no	—
301	C₂³	—	15	1	no	—
302	C₂³	D₃×D₁₅	1	1	yes	characteristic
303	Q₈	—	45	4	no	—
304	D₄	—	3	4	no	—
305	D₄	—	15	4	no	—
306	D₄	—	45	4	no	—
307	C₂×C₄	—	3	1	no	—
308	C₂×C₄	—	15	1	no	—
309	C₂×C₄	—	45	4	no	—
310	C₂×C₄	—	45	4	no	—
311	C₂×C₄	—	45	4	no	—
312	C₂×C₄	—	45	1	no	—
313	C₂×C₄	—	45	1	no	—
314	C₂×C₃	—	2	4	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	6	2	no	—
320	C₂×C₃	—	30	2	no	—
321	C₂×C₃	#6780	1	2	yes	complemented
322	C₂×C₃	#6819	1	2	yes	complemented
323	C₂×C₃	C₂²×D₃×D₅	1	1	yes	characteristic
324	C₂×C₃	C₂³×D₁₅	1	1	yes	characteristic
325	D₃	—	2	2	no	—
326	D₃	—	10	2	no	—
327	C₅	C₂×#3805	1	1	yes	characteristic, complemented, Sylow
328	C₂²	—	2	2	no	—
329	C₂²	—	3	2	no	—
330	C₂²	—	6	2	no	—
331	C₂²	—	15	2	no	—
332	C₂²	—	30	2	no	—
333	C₂²	C₂×D₃×D₁₅	1	2	yes	—
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	—	45	2	no	—
339	C₃	—	2	1	no	—
340	C₃	C₂×#6780	1	1	yes	characteristic, complemented
341	C₃	C₂×#6819	1	1	yes	characteristic, complemented
342	C₂	—	2	2	no	—
343	C₂	—	6	2	no	—
344	C₂	—	30	2	no	—
345	C₂	#10624208	1	2	yes	direct product factor
346	C₂	C₂²×D₃×D₁₅	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃⋊Dic₁₅	C₂	semidirect
2. C₂×C₃×#1220	C₂	semidirect
3. C₂×C₃×C₅×#67	C₂	semidirect
4. C₂×#67153	C₂	semidirect
5. C₂×#67150	C₂	semidirect
6. C₂×D₃×Dic₁₅	C₂	semidirect
7. #10624208	C₂	direct (N × H)
8. C₂×Dic₃×D₁₅	C₂	semidirect
16. C₂×C₃⋊Dic₁₅	C₂²	semidirect
17. C₃×#1220	C₂²	semidirect
19. C₃×C₅×#67	C₂²	semidirect
21. #67153	C₂²	semidirect
22. #67150	C₂²	semidirect
23. D₃×Dic₁₅	C₂²	semidirect
24. Dic₃×D₁₅	C₂²	semidirect
28. C₂×#1220	D₃	semidirect
36. C₂×C₅×#67	D₃	semidirect
57. C₂×C₃×#67	D₅	semidirect
73. #1220	C₂×D₃	semidirect
87. C₅×#67	C₂×D₃	semidirect
98. C₂×C₃²×C₅	#41	semidirect
117. C₃×#67	C₂×D₅	semidirect
156. C₂×#67	D₁₅	semidirect
168. C₃²×C₅	C₂×#41	semidirect
201. C₂×C₃×C₅	#237	semidirect
202. C₂×C₃×C₅	#237	semidirect
222. #67	C₂×D₁₅	semidirect
252. C₂×C₃²	#711	semidirect
265. C₃×C₅	C₂×#237	semidirect
266. C₃×C₅	C₂×#237	semidirect
296. C₂×C₅	#3805	semidirect
299. C₃²	C₂×#711	semidirect
321. C₂×C₃	#6780	semidirect
322. C₂×C₃	#6819	semidirect
327. C₅	C₂×#3805	semidirect
340. C₃	C₂×#6780	semidirect
341. C₃	C₂×#6819	semidirect
345. C₂	#10624208	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊Dic₁₅	2
C₂×C₃×#1220	2
C₂×C₃×C₅×#67	2
C₂×#67153	2
C₂×#67150	2
C₂×D₃×Dic₁₅	2
#10624208	2
C₂×Dic₃×D₁₅	2
C₂×#6819	3
C₂×#6780	3
C₂×#3805	5

Subgroups — C2×#10624208 #13089385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10624208 #13089385