Subgroups — C₂×#20245477 #26370700

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093970	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689364	C₂	1	1	yes	maximal, characteristic
4	#20245477	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15688948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688927	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15688926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#467	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	C₂×#13094003	—	2	1	no	—
11	C₂×#13093970	—	2	4	no	—
12	C₂×#13093970	C₂²	1	4	yes	complemented
13	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
16	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#15689364	—	2	2	no	—
18	#15689364	C₂²	1	4	yes	—
19	C₂×#13093958	C₂²	1	1	yes	characteristic
20	C₂×#13093949	—	2	1	no	—
21	#15688948	C₂²	1	4	yes	complemented
22	#15688927	C₂²	1	4	yes	complemented
23	#15688926	C₂²	1	4	yes	complemented
24	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
27	C₄₇×#467	—	2	2	no	—
28	C₄₇×#467	C₂²	1	4	yes	—
29	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#103×C₄₇	—	2	1	no	—
31	C₂²×Dic₉₄	—	2	1	no	—
32	C₂×Dic₁₈₈	—	2	2	no	—
33	C₂×Dic₁₈₈	—	2	2	no	—
34	C₂×Dic₁₈₈	—	2	1	no	—
35	C₂×Dic₁₈₈	C₂³	1	2	yes	—
36	#13094003	—	2	4	no	—
37	#13094003	—	4	2	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×Dic₄₇⋊C₄	—	4	1	no	—
40	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
41	#13093970	—	2	8	no	—
42	C₂×C₄₇⋊C₁₆	—	2	2	no	—
43	C₂×C₄₇⋊C₁₆	—	2	8	no	—
44	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
45	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
46	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
47	#13093958	—	2	1	no	—
48	#13093958	C₂³	1	2	yes	—
49	#13093949	—	2	4	no	—
50	C₂²×Q₈×C₄₇	—	2	1	no	—
51	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₄₇	—	2	2	no	—
53	C₂×Q₁₆×C₄₇	—	2	1	no	—
54	C₂×Q₁₆×C₄₇	—	2	2	no	—
55	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₄₇	—	2	2	no	—
57	C₂²×C₈×C₄₇	—	2	2	no	—
58	C₂²×C₈×C₄₇	D₄	1	2	yes	—
59	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²⋊Q₈×C₄₇	—	2	4	no	—
63	C₂²⋊Q₈×C₄₇	—	4	2	no	—
64	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
66	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
67	#107×C₄₇	—	2	1	no	—
68	#107×C₄₇	C₂³	1	2	yes	—
69	#103×C₄₇	—	2	4	no	—
70	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
71	C₂×Q₈×C₄₇	—	2	2	no	—
72	C₂×Q₈×C₄₇	—	4	1	no	—
73	C₂×Q₈×C₄₇	—	4	2	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	2	2	no	—
77	C₂²×C₄×C₄₇	—	4	1	no	—
78	C₂²×C₄×C₄₇	—	4	1	no	—
79	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₄₇	D₈	1	2	yes	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	—	4	1	no	—
84	C₂²×Dic₄₇	—	4	1	no	—
85	C₂×Dic₉₄	—	2	2	no	—
86	C₂×Dic₉₄	—	4	2	no	—
87	C₂×Dic₉₄	—	4	2	no	—
88	C₂×Dic₉₄	—	4	1	no	—
89	Q₁₆×C₄₇	—	2	2	no	—
90	Q₁₆×C₄₇	—	2	2	no	—
91	Q₁₆×C₄₇	—	4	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	4	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₄₇	#41	1	2	yes	—
100	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄⋊C₄×C₄₇	—	4	1	no	—
103	C₄⋊C₄×C₄₇	—	4	4	no	—
104	C₂²⋊C₄×C₄₇	—	4	4	no	—
105	C₂²⋊Dic₄₇	—	4	4	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	4	1	no	—
108	Dic₄₇⋊C₄	—	4	4	no	—
109	Dic₁₈₈	—	2	2	no	—
110	Dic₁₈₈	—	2	2	no	—
111	Dic₁₈₈	—	4	2	no	—
112	C₄₇⋊C₁₆	—	2	2	no	—
113	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
114	C₂³×C₄₇	—	2	2	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	Q₃₂	1	4	yes	complemented
117	C₂³×C₄₇	C₂×D₈	1	2	yes	—
118	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₄₇	—	4	2	no	—
120	Q₈×C₄₇	—	4	2	no	—
121	Q₈×C₄₇	—	8	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	8	2	no	—
129	C₂×C₄×C₄₇	—	8	2	no	—
130	C₂×C₄×C₄₇	#121	1	2	yes	—
131	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#135	1	2	yes	—
134	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	8	2	no	—
138	C₂×Dic₄₇	—	8	2	no	—
139	Dic₉₄	—	4	2	no	—
140	Dic₉₄	—	4	2	no	—
141	Dic₉₄	—	8	2	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	#121	1	2	yes	—
144	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×#2216	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	8	no	—
149	C₂²×C₄₇	—	2	2	no	—
150	C₂²×C₄₇	#466	1	2	yes	—
151	C₂²×C₄₇	C₂×Q₃₂	1	2	yes	—
152	C₂²×C₄₇	C₂×Q₃₂	1	4	yes	complemented
153	C₂²×C₄₇	#508	1	2	yes	—
154	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	—	8	2	no	—
158	C₄×C₄₇	#466	1	2	yes	—
159	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
160	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	—	8	2	no	—
163	C₂²×Q₃₂	—	47	1	no	—
164	C₂³×C₁₆	—	47	1	no	—
165	C₂×#467	—	47	1	no	—
166	C₂×#467	D₄₇	1	1	yes	characteristic
167	#2216	—	47	8	no	—
168	C₂×#366	—	47	1	no	—
169	C₂×#358	—	47	1	no	—
170	C₂×#358	—	47	1	no	—
171	C₂×C₄₇	—	2	4	no	—
172	C₂×C₄₇	#2216	1	4	yes	complemented
173	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
174	C₂×C₄₇	C₂²×Q₃₂	1	1	yes	characteristic
175	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
176	C₂²×Q₁₆	—	47	1	no	—
177	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
178	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
179	C₂×C₂²⋊Q₈	—	2	1	no	—
180	C₂×C₂²⋊Q₈	—	94	1	no	—
181	C₂×Q₃₂	—	47	4	no	—
182	C₂×Q₃₂	—	94	4	no	—
183	C₂²×C₁₆	—	47	1	no	—
184	C₂²×C₁₆	—	47	1	no	—
185	C₂²×C₁₆	—	47	4	no	—
186	C₂²×C₁₆	—	94	4	no	—
187	#467	—	2	2	no	—
188	#467	—	47	4	no	—
189	#467	—	94	2	no	—
190	#467	C₂×D₄₇	1	4	yes	—
191	C₂×#107	—	47	1	no	—
192	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
193	C₂×#103	—	2	1	no	—
194	C₂×#103	—	94	1	no	—
195	#366	—	47	4	no	—
196	#358	—	47	4	no	—
197	#358	—	47	4	no	—
198	C₄₇	C₂×#2216	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	94	1	no	—
201	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	—	47	2	no	—
206	C₂×Q₁₆	—	94	2	no	—
207	C₂×Q₁₆	—	94	2	no	—
208	C₂×Q₁₆	—	94	1	no	—
209	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
214	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
215	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
216	C₂²⋊Q₈	—	2	4	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₂²⋊Q₈	—	94	4	no	—
219	C₂²⋊Q₈	—	188	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	4	1	no	—
222	C₂×C₄⋊C₄	—	94	1	no	—
223	C₂×C₄⋊C₄	—	188	1	no	—
224	C₂×C₂²⋊C₄	—	4	1	no	—
225	C₂×C₂²⋊C₄	—	188	1	no	—
226	Q₃₂	—	94	8	no	—
227	C₂×C₁₆	—	47	4	no	—
228	C₂×C₁₆	—	47	2	no	—
229	C₂×C₁₆	—	47	2	no	—
230	C₂×C₁₆	—	94	8	no	—
231	C₂×C₁₆	—	94	2	no	—
232	#107	—	2	1	no	—
233	#107	—	47	2	no	—
234	#107	—	94	1	no	—
235	#107	C₂²×D₄₇	1	2	yes	—
236	#103	—	2	4	no	—
237	#103	—	94	4	no	—
238	C₂⁴	#10625334	1	1	yes	characteristic
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	94	2	no	—
244	C₂×Q₈	—	188	2	no	—
245	C₂×Q₈	—	188	2	no	—
246	C₂×Q₈	—	188	1	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	188	1	no	—
252	C₂²×C₄	—	188	1	no	—
253	C₂²×C₄	#10625334	1	2	yes	—
254	C₂²×C₄	#10625334	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	4	2	no	—
260	Q₁₆	—	94	2	no	—
261	Q₁₆	—	94	2	no	—
262	Q₁₆	—	188	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	D₄×D₄₇	1	2	yes	—
268	C₂×C₈	#10625352	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	94	2	no	—
276	C₄⋊C₄	—	188	4	no	—
277	C₄⋊C₄	—	188	1	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	188	4	no	—
280	C₁₆	—	47	4	no	—
281	C₁₆	—	94	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	#13093970	1	4	yes	complemented
285	C₂³	C₂×#10625334	1	2	yes	—
286	C₂³	C₂×#10625334	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	376	2	no	—
291	Q₈	—	188	2	no	—
292	Q₈	—	188	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	376	2	no	—
302	C₂×C₄	—	376	2	no	—
303	C₂×C₄	—	188	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	#13093996	1	2	yes	—
306	C₂×C₄	C₂×#10625334	1	2	yes	—
307	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
308	C₂×C₄	#13094044	1	2	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	#13093996	1	2	yes	—
312	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
313	C₈	C₂×#10625352	1	1	yes	characteristic
314	C₂²	—	2	2	no	—
315	C₂²	—	2	8	no	—
316	C₂²	—	2	2	no	—
317	C₂²	#15689357	1	2	yes	—
318	C₂²	C₂×#13093970	1	2	yes	—
319	C₂²	C₂×#13093970	1	4	yes	complemented
320	C₂²	#15689553	1	2	yes	—
321	C₂²	C₂²×#10625334	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	376	2	no	—
326	C₄	—	376	2	no	—
327	C₄	#15689357	1	2	yes	—
328	C₄	C₂×#13093996	1	1	yes	characteristic
329	C₄	C₂×#13094044	1	1	yes	characteristic
330	C₂	—	2	4	no	—
331	C₂	#20245477	1	4	yes	direct product factor
332	C₂	C₂×#15689357	1	1	yes	characteristic
333	C₂	C₂²×#13093970	1	1	yes	characteristic
334	C₂	C₂×#15689553	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#467	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
hypercenter	C₂×#467	D₄₇
hypocenter	C₄₇	C₂×#2216
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093970	C₂	semidirect
4. #20245477	C₂	direct (N × H)
5. C₂×#15688948	C₂	semidirect
6. C₂×#15688927	C₂	semidirect
7. C₂×#15688926	C₂	semidirect
12. C₂×#13093970	C₂²	semidirect
21. #15688948	C₂²	semidirect
22. #15688927	C₂²	semidirect
23. #15688926	C₂²	semidirect
116. C₂³×C₄₇	Q₃₂	semidirect
152. C₂²×C₄₇	C₂×Q₃₂	semidirect
172. C₂×C₄₇	#2216	semidirect
198. C₄₇	C₂×#2216	semidirect
284. C₂³	#13093970	semidirect
319. C₂²	C₂×#13093970	semidirect
331. C₂	#20245477	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×#13093970	2
C₂³×C₄₇⋊C₁₆	2
C₂×#15689364	2
#20245477	2
C₂×#15688948	2
C₂×#15688927	2
C₂×#15688926	2
C₂×C₄₇×#467	2
C₂×#2216	47

Subgroups — C2×#20245477 #26370700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245477 #26370700