Subgroups — C₄₇×#10825 #26050523

Order 12032.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#344	C₂	1	2	yes	maximal
3	C₂×C₄₇×#339	C₂	1	2	yes	maximal
4	C₂×C₄₇×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×#117×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₄₇	—	2	4	no	—
12	C₂×#97×C₄₇	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	complemented
14	C₄₇×#344	—	2	4	no	—
15	C₄₇×#339	C₄	1	8	yes	complemented
16	C₄₇×#338	—	2	2	no	—
17	C₄₇×H(3,ℤ/4)	—	2	2	no	—
18	C₂³×C₄×C₄₇	—	2	1	no	—
19	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	—	2	2	no	—
21	C₂×M₁₆×C₄₇	—	2	1	no	—
22	C₂×M₁₆×C₄₇	—	2	1	no	—
23	C₂×M₁₆×C₄₇	—	2	2	no	—
24	C₂×M₁₆×C₄₇	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂×M₁₆×C₄₇	—	2	2	no	—
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂×M₁₆×C₄₇	—	2	2	no	—
30	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
31	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
32	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
33	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
34	C₂²×C₈×C₄₇	—	2	2	no	—
35	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
37	#117×C₄₇	—	2	2	no	—
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	2	no	—
40	#117×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	—	2	2	no	—
44	C₂×C₄²×C₄₇	—	2	1	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
49	#97×C₄₇	—	2	4	no	—
50	C₄×C₈×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	#41	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	1	no	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	2	4	no	—
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	M₁₆×C₄₇	—	2	4	no	—
73	M₁₆×C₄₇	—	4	2	no	—
74	M₁₆×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	2	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₄₇	—	2	4	no	—
85	C₄⋊C₄×C₄₇	—	4	2	no	—
86	C₂²⋊C₄×C₄₇	—	2	4	no	—
87	C₂²⋊C₄×C₄₇	—	4	2	no	—
88	C₄²×C₄₇	—	2	1	no	—
89	C₄²×C₄₇	—	2	1	no	—
90	C₄²×C₄₇	—	2	4	no	—
91	C₄²×C₄₇	—	2	4	no	—
92	C₄²×C₄₇	—	4	2	no	—
93	C₄²×C₄₇	C₄²	1	2	yes	—
94	C₄²×C₄₇	C₄²	1	2	yes	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	#117	1	1	yes	characteristic
97	C₂³×C₄₇	C₄×D₄	1	2	yes	—
98	C₂³×C₄₇	#124	1	1	yes	characteristic
99	C₂³×C₄₇	#125	1	1	yes	characteristic
100	C₂³×C₄₇	#126	1	2	yes	—
101	C₂×C₄×C₄₇	—	2	4	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	4	8	no	—
105	C₂×C₄×C₄₇	—	4	2	no	—
106	C₂×C₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	8	4	no	—
108	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#117	1	2	yes	—
114	C₂×C₄×C₄₇	#117	1	2	yes	—
115	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#121	1	2	yes	—
120	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₇	#123	1	2	yes	—
122	C₂×C₄×C₄₇	#123	1	2	yes	—
123	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#126	1	2	yes	—
126	C₈×C₄₇	—	2	4	no	—
127	C₈×C₄₇	—	2	4	no	—
128	C₈×C₄₇	—	4	4	no	—
129	#10825	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	4	2	no	—
133	C₂²×C₄₇	—	4	1	no	—
134	C₂²×C₄₇	—	4	1	no	—
135	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₇	#382	1	1	yes	characteristic
137	C₂²×C₄₇	#383	1	1	yes	characteristic
138	C₂²×C₄₇	#387	1	2	yes	—
139	C₂²×C₄₇	#388	1	2	yes	—
140	C₄×C₄₇	—	2	4	no	—
141	C₄×C₄₇	—	4	2	no	—
142	C₄×C₄₇	—	8	4	no	—
143	C₄×C₄₇	#376	1	1	yes	characteristic
144	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
145	C₄×C₄₇	#380	1	1	yes	characteristic
146	C₄×C₄₇	#381	1	1	yes	characteristic
147	C₄×C₄₇	#387	1	2	yes	—
148	C₄×C₄₇	#388	1	2	yes	—
149	C₂×C₄×M₁₆	C₂×C₄₇	1	1	yes	characteristic
150	C₂×#344	C₂×C₄₇	1	2	yes	—
151	C₂×#339	C₂×C₄₇	1	2	yes	—
152	C₂×#338	C₂×C₄₇	1	1	yes	characteristic
153	C₂×H(3,ℤ/4)	C₂×C₄₇	1	1	yes	characteristic
154	C₂×C₄₇	—	2	1	no	—
155	C₂×C₄₇	—	2	2	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	#1434	1	1	yes	characteristic
159	C₂×C₄₇	#1841	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
162	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
163	C₂×#117	C₂²×C₄₇	1	2	yes	—
164	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
165	C₄×M₁₆	—	2	4	no	—
166	C₂×#97	C₄×C₄₇	1	2	yes	complemented
167	C₂×C₄×C₈	C₄×C₄₇	1	2	yes	complemented
168	#344	—	2	4	no	—
169	#339	C₄×C₄₇	1	8	yes	complemented
170	#338	—	2	2	no	—
171	H(3,ℤ/4)	—	2	2	no	—
172	C₄₇	#10825	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³×C₄	—	2	1	no	—
174	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	C₂×C₄×C₄₇	1	4	yes	—
186	C₂×M₁₆	D₄×C₄₇	1	4	yes	—
187	C₂×M₁₆	D₄×C₄₇	1	2	yes	—
188	C₂×M₁₆	Q₈×C₄₇	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
191	C₂²×C₈	C₂×C₄×C₄₇	1	2	yes	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	C₂×C₄×C₄₇	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
202	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
204	#97	—	2	4	no	—
205	C₄×C₈	—	2	4	no	—
206	C₂⁴	#41×C₄₇	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₈×C₄₇	1	1	yes	characteristic
217	C₂²×C₄	#41×C₄₇	1	2	yes	—
218	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
219	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
220	C₂²×C₄	#41×C₄₇	1	2	yes	—
221	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂²⋊C₄×C₄₇	1	4	yes	—
238	C₂×C₈	C₄⋊C₄×C₄₇	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₄²×C₄₇	1	2	yes	—
249	C₄²	C₄²×C₄₇	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	#117×C₄₇	1	1	yes	characteristic
252	C₂³	C₄×D₄×C₄₇	1	2	yes	—
253	C₂³	#124×C₄₇	1	1	yes	characteristic
254	C₂³	#125×C₄₇	1	1	yes	characteristic
255	C₂³	#126×C₄₇	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂×C₄²×C₄₇	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	1	yes	characteristic
266	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
267	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
268	C₂×C₄	#117×C₄₇	1	2	yes	—
269	C₂×C₄	#117×C₄₇	1	2	yes	—
270	C₂×C₄	#117×C₄₇	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
273	C₂×C₄	C₄×Q₈×C₄₇	1	2	yes	—
274	C₂×C₄	#121×C₄₇	1	2	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
276	C₂×C₄	#123×C₄₇	1	2	yes	—
277	C₂×C₄	#123×C₄₇	1	2	yes	—
278	C₂×C₄	#124×C₄₇	1	1	yes	characteristic
279	C₂×C₄	#125×C₄₇	1	1	yes	characteristic
280	C₂×C₄	#126×C₄₇	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
290	C₂²	C₄₇×#382	1	1	yes	characteristic
291	C₂²	C₄₇×#383	1	1	yes	characteristic
292	C₂²	C₄₇×#387	1	2	yes	—
293	C₂²	C₄₇×#388	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₄₇×#376	1	1	yes	characteristic
298	C₄	C₄×C₄⋊C₄×C₄₇	1	1	yes	characteristic
299	C₄	C₄₇×#380	1	1	yes	characteristic
300	C₄	C₄₇×#381	1	1	yes	characteristic
301	C₄	C₄₇×#387	1	2	yes	—
302	C₄	C₄₇×#388	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	C₄₇×#1434	1	1	yes	characteristic
308	C₂	C₄₇×#1841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₇	#117
commutator	C₂×C₄	C₂×C₄²×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
socle	C₂²×C₄₇	#383
prime radical	C₂⁴×C₄₇	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂²×M₁₆×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×#97×C₄₇	C₄	semidirect
13. C₂×C₄×C₈×C₄₇	C₄	semidirect
15. C₄₇×#339	C₄	semidirect
129. #10825	C₄₇	direct (N × H)
166. C₂×#97	C₄×C₄₇	semidirect
167. C₂×C₄×C₈	C₄×C₄₇	semidirect
169. #339	C₄×C₄₇	semidirect
172. C₄₇	#10825	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×M₁₆×C₄₇	2
C₂×C₄₇×#344	2
C₂×C₄₇×#339	2
C₂×C₄₇×#338	2
C₂×C₄₇×H(3,ℤ/4)	2
#10825	47

Subgroups — C47×#10825 #26050523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#10825 #26050523