Subgroups — C₄₇×#23744 #26063442

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2296	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2101	C₂	1	4	yes	maximal, complemented
4	C₄₇×#2100	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2099	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1735	C₂	1	2	yes	maximal
7	C₄₇×#1735	C₂	1	2	yes	maximal
8	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
10	C₂×#117×C₄₇	C₂²	1	2	yes	—
11	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#389	C₂²	1	2	yes	—
15	C₄₇×#389	C₂²	1	2	yes	—
16	C₄₇×#385	—	2	2	no	—
17	C₄₇×#384	C₂²	1	1	yes	characteristic
18	C₄₇×#384	C₂²	1	1	yes	characteristic
19	C₄₇×#382	C₂²	1	2	yes	—
20	C₄₇×#380	—	2	2	no	—
21	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₄×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₄₇×#340	C₂²	1	4	yes	—
24	C₄₇×#340	C₂²	1	4	yes	—
25	C₄₇×#340	C₂²	1	4	yes	—
26	C₄₇×#340	C₂²	1	4	yes	—
27	C₄₇×#336	C₂²	1	2	yes	—
28	C₄₇×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₄₇	D₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	—	2	4	no	—
32	C₂²×C₈×C₄₇	—	2	4	no	—
33	C₂²×C₈×C₄₇	—	2	4	no	—
34	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
35	#117×C₄₇	—	2	2	no	—
36	#117×C₄₇	—	2	4	no	—
37	#117×C₄₇	—	2	8	no	—
38	#117×C₄₇	Q₈	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	—	2	1	no	—
51	C₂×C₄²×C₄₇	—	2	4	no	—
52	C₂×C₄²×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	D₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#95×C₄₇	—	2	4	no	—
58	#95×C₄₇	—	2	2	no	—
59	#95×C₄₇	—	2	2	no	—
60	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	2	2	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	—	4	2	no	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	4	yes	—
74	C₂²×C₄×C₄₇	#41	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	4	8	no	—
80	C₄⋊C₄×C₄₇	—	2	2	no	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	4	4	no	—
85	C₄⋊C₄×C₄₇	—	4	4	no	—
86	C₄⋊C₄×C₄₇	—	4	4	no	—
87	C₄⋊C₄×C₄₇	—	4	4	no	—
88	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
89	C₂²⋊C₄×C₄₇	—	2	4	no	—
90	C₂²⋊C₄×C₄₇	—	4	2	no	—
91	C₂²⋊C₄×C₄₇	—	4	4	no	—
92	C₄²×C₄₇	—	2	4	no	—
93	C₄²×C₄₇	—	2	4	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	4	2	no	—
97	C₄²×C₄₇	—	4	4	no	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₄₇	#123	1	2	yes	—
102	C₂³×C₄₇	#124	1	2	yes	—
103	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	8	no	—
111	C₂×C₄×C₄₇	—	4	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	4	no	—
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#124	1	2	yes	—
123	C₂×C₄×C₄₇	#125	1	8	yes	—
124	C₂×C₄×C₄₇	#126	1	8	yes	—
125	C₂×C₄×C₄₇	#135	1	4	yes	—
126	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
128	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₈×C₄₇	—	4	4	no	—
131	#23744	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₄₇	—	2	4	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	—	2	2	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	#400	1	4	yes	—
137	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
138	C₂²×C₄₇	C₂×#123	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×#135	1	2	yes	—
141	C₂²×C₄₇	#575	1	2	yes	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	4	4	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	—	8	2	no	—
146	C₄×C₄₇	—	8	2	no	—
147	C₄×C₄₇	—	8	2	no	—
148	C₄×C₄₇	#400	1	4	yes	—
149	C₄×C₄₇	C₂×#125	1	2	yes	—
150	C₄×C₄₇	C₂×#126	1	2	yes	—
151	#2296	C₂×C₄₇	1	1	yes	characteristic
152	#2285	C₂×C₄₇	1	1	yes	characteristic
153	#2101	C₂×C₄₇	1	4	yes	complemented
154	#2100	C₂×C₄₇	1	2	yes	complemented
155	#2099	C₂×C₄₇	1	2	yes	complemented
156	#1735	C₂×C₄₇	1	2	yes	—
157	#1735	C₂×C₄₇	1	2	yes	—
158	C₂×#336	C₂×C₄₇	1	1	yes	characteristic
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	C₂×#400	1	1	yes	characteristic
161	C₂×C₄₇	#3076	1	2	yes	—
162	C₂×C₄₇	#3091	1	2	yes	—
163	C₂×C₄₇	#3131	1	1	yes	characteristic
164	C₂×C₄₇	#3135	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₄₇	1	2	yes	—
166	C₂×#117	C₂²×C₄₇	1	2	yes	—
167	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
168	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
170	#389	C₂²×C₄₇	1	2	yes	—
171	#389	C₂²×C₄₇	1	2	yes	—
172	#385	—	2	2	no	—
173	#384	C₂²×C₄₇	1	1	yes	characteristic
174	#384	C₂²×C₄₇	1	1	yes	characteristic
175	#382	C₂²×C₄₇	1	2	yes	—
176	#380	—	2	2	no	—
177	C₄×C₄⋊C₄	—	2	2	no	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	#340	C₂²×C₄₇	1	4	yes	—
180	#340	C₂²×C₄₇	1	4	yes	—
181	#340	C₂²×C₄₇	1	4	yes	—
182	#340	C₂²×C₄₇	1	4	yes	—
183	#336	C₂²×C₄₇	1	2	yes	—
184	#336	C₂²×C₄₇	1	2	yes	—
185	C₄₇	#23744	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³×C₄	D₄×C₄₇	1	2	yes	—
187	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
192	#117	—	2	2	no	—
193	#117	—	2	4	no	—
194	#117	—	2	8	no	—
195	#117	Q₈×C₄₇	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	D₄×C₄₇	1	2	yes	—
212	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
214	#95	—	2	4	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
229	C₂²×C₄	#41×C₄₇	1	2	yes	—
230	C₂²×C₄	#41×C₄₇	1	4	yes	—
231	C₂²×C₄	#41×C₄₇	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	8	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	C₂×Q₈×C₄₇	1	4	yes	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	4	4	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂²⋊Q₈×C₄₇	1	2	yes	—
258	C₂³	#123×C₄₇	1	2	yes	—
259	C₂³	#124×C₄₇	1	2	yes	—
260	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
278	C₂×C₄	#123×C₄₇	1	2	yes	—
279	C₂×C₄	#124×C₄₇	1	2	yes	—
280	C₂×C₄	#125×C₄₇	1	8	yes	—
281	C₂×C₄	#126×C₄₇	1	8	yes	—
282	C₂×C₄	#135×C₄₇	1	4	yes	—
283	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
284	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
285	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
286	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
287	C₈	—	4	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	C₄₇×#400	1	4	yes	—
293	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
294	C₂²	C₂×#123×C₄₇	1	1	yes	characteristic
295	C₂²	C₂×#124×C₄₇	1	1	yes	characteristic
296	C₂²	C₂×#135×C₄₇	1	2	yes	—
297	C₂²	C₄₇×#575	1	2	yes	—
298	C₄	—	2	4	no	—
299	C₄	—	4	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₄₇×#400	1	4	yes	—
305	C₄	C₂×#125×C₄₇	1	2	yes	—
306	C₄	C₂×#126×C₄₇	1	2	yes	—
307	C₂	—	2	4	no	—
308	C₂	C₂×C₄₇×#400	1	1	yes	characteristic
309	C₂	C₄₇×#3076	1	2	yes	—
310	C₂	C₄₇×#3091	1	2	yes	—
311	C₂	C₄₇×#3131	1	1	yes	characteristic
312	C₂	C₄₇×#3135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₇ ◁ C₂²×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄₇×#2101	C₂	semidirect
4. C₄₇×#2100	C₂	semidirect
5. C₄₇×#2099	C₂	semidirect
131. #23744	C₄₇	direct (N × H)
153. #2101	C₂×C₄₇	semidirect
154. #2100	C₂×C₄₇	semidirect
155. #2099	C₂×C₄₇	semidirect
185. C₄₇	#23744	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₇×#2296	2
C₄₇×#2285	2
C₄₇×#2101	2
C₄₇×#2100	2
C₄₇×#2099	2
C₄₇×#1735	2
C₂×C₄₇×#336	2
#23744	47

Subgroups — C47×#23744 #26063442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23744 #26063442