Subgroups — C₄₇×#11198 #26050896

Order 12032.

1042 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#407	C₂	1	2	yes	maximal
2	C₂×C₄₇×#380	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#343	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₄₇	C₂²	1	4	yes	—
6	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
7	C₂²×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
8	C₄₇×#407	—	2	8	no	—
9	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
10	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
11	C₄₇×#380	—	2	2	no	—
12	C₄₇×#380	—	2	2	no	—
13	C₂×#95×C₄₇	C₄	1	2	yes	—
14	C₄₇×#343	—	2	8	no	—
15	C₂⁵×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₄₇	—	2	4	no	—
17	C₂³×C₄×C₄₇	—	2	4	no	—
18	C₂³×C₄×C₄₇	—	2	4	no	—
19	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂³×C₄×C₄₇	D₄	1	2	yes	—
22	C₂³×C₄×C₄₇	D₄	1	2	yes	—
23	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₄₇	—	2	4	no	—
25	C₂×M₁₆×C₄₇	—	2	4	no	—
26	C₂×M₁₆×C₄₇	—	2	4	no	—
27	C₂×M₁₆×C₄₇	—	2	4	no	—
28	C₂×M₁₆×C₄₇	—	2	4	no	—
29	C₂×M₁₆×C₄₇	D₄	1	8	yes	—
30	C₂²×C₈×C₄₇	—	2	4	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	4	4	no	—
38	C₂²⋊C₈×C₄₇	—	2	8	no	—
39	C₂²⋊C₈×C₄₇	—	2	8	no	—
40	#95×C₄₇	—	2	4	no	—
41	C₂⁴×C₄₇	—	2	2	no	—
42	C₂⁴×C₄₇	—	2	2	no	—
43	C₂⁴×C₄₇	—	2	2	no	—
44	C₂⁴×C₄₇	—	2	4	no	—
45	C₂⁴×C₄₇	—	4	2	no	—
46	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	2	no	—
50	C₂²×C₄×C₄₇	—	2	4	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	4	no	—
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	4	no	—
60	C₂²×C₄×C₄₇	—	2	8	no	—
61	C₂²×C₄×C₄₇	—	4	4	no	—
62	C₂²×C₄×C₄₇	—	4	4	no	—
63	C₂²×C₄×C₄₇	—	4	4	no	—
64	C₂²×C₄×C₄₇	—	4	4	no	—
65	C₂²×C₄×C₄₇	—	4	4	no	—
66	C₂²×C₄×C₄₇	—	4	4	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	4	yes	—
71	C₂²×C₄×C₄₇	#41	1	4	yes	—
72	M₁₆×C₄₇	—	2	8	no	—
73	M₁₆×C₄₇	—	2	8	no	—
74	M₁₆×C₄₇	—	4	4	no	—
75	M₁₆×C₄₇	—	4	4	no	—
76	C₂×C₈×C₄₇	—	2	8	no	—
77	C₂×C₈×C₄₇	—	4	4	no	—
78	C₂×C₈×C₄₇	—	4	4	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₄⋊C₄×C₄₇	—	4	4	no	—
81	C₄⋊C₄×C₄₇	—	4	2	no	—
82	C₂²⋊C₄×C₄₇	—	4	4	no	—
83	C₂²⋊C₄×C₄₇	—	4	4	no	—
84	C₂³×C₄₇	—	2	1	no	—
85	C₂³×C₄₇	—	2	1	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	—	2	8	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	1	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	4	8	no	—
97	C₂³×C₄₇	—	4	8	no	—
98	C₂³×C₄₇	—	4	4	no	—
99	C₂³×C₄₇	—	4	4	no	—
100	C₂³×C₄₇	#100	1	2	yes	—
101	C₂³×C₄₇	#101	1	2	yes	—
102	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₄₇	#117	1	2	yes	—
104	C₂³×C₄₇	C₄×D₄	1	4	yes	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	4	4	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	4	no	—
112	C₂×C₄×C₄₇	—	4	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	8	8	no	—
117	C₂×C₄×C₄₇	#120	1	2	yes	—
118	C₂×C₄×C₄₇	#121	1	4	yes	—
119	C₂×C₄×C₄₇	#121	1	4	yes	—
120	C₂×C₄×C₄₇	#123	1	4	yes	—
121	C₂×C₄×C₄₇	#123	1	4	yes	—
122	C₂×C₄×C₄₇	#123	1	2	yes	—
123	C₂×C₄×C₄₇	#124	1	4	yes	—
124	C₂×C₄×C₄₇	#126	1	4	yes	—
125	C₈×C₄₇	—	4	8	no	—
126	#11198	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	2	2	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	2	no	—
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	—	2	4	no	—
134	C₂²×C₄₇	—	2	4	no	—
135	C₂²×C₄₇	—	4	16	no	—
136	C₂²×C₄₇	—	4	1	no	—
137	C₂²×C₄₇	—	4	1	no	—
138	C₂²×C₄₇	—	4	2	no	—
139	C₂²×C₄₇	—	4	2	no	—
140	C₂²×C₄₇	—	4	1	no	—
141	C₂²×C₄₇	—	4	1	no	—
142	C₂²×C₄₇	#381	1	1	yes	characteristic
143	C₂²×C₄₇	#386	1	2	yes	—
144	C₂²×C₄₇	#388	1	4	yes	—
145	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
147	C₂²×C₄₇	#413	1	2	yes	—
148	C₄×C₄₇	—	4	2	no	—
149	C₄×C₄₇	—	8	4	no	—
150	C₄×C₄₇	#394	1	4	yes	—
151	C₄×C₄₇	#397	1	4	yes	—
152	C₂×#407	C₂×C₄₇	1	2	yes	—
153	C₂×#380	C₂×C₄₇	1	1	yes	characteristic
154	C₂×#343	C₂×C₄₇	1	4	yes	complemented
155	C₂×C₄₇	—	2	4	no	—
156	C₂×C₄₇	—	4	2	no	—
157	C₂×C₄₇	—	4	2	no	—
158	C₂×C₄₇	#1440	1	1	yes	characteristic
159	C₂×C₄₇	#1821	1	1	yes	characteristic
160	C₂×C₄₇	#1822	1	1	yes	characteristic
161	C₂×C₄₇	#1900	1	2	yes	—
162	C₂×C₄₇	#1901	1	2	yes	—
163	C₂⁴×C₄	C₂²×C₄₇	1	1	yes	characteristic
164	C₂²×M₁₆	C₂²×C₄₇	1	4	yes	—
165	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
166	C₂²×C₂²⋊C₄	C₄×C₄₇	1	2	yes	—
167	#407	—	2	8	no	—
168	C₂×C₂²⋊C₈	—	2	2	no	—
169	C₂×C₂²⋊C₈	—	2	2	no	—
170	#380	—	2	2	no	—
171	#380	—	2	2	no	—
172	C₂×#95	C₄×C₄₇	1	2	yes	—
173	#343	—	2	8	no	—
174	C₄₇	#11198	1	1	yes	characteristic, direct product factor, Sylow
175	C₂⁵	C₂×C₄×C₄₇	1	1	yes	characteristic
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	—	2	4	no	—
179	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
180	C₂³×C₄	C₂×C₄×C₄₇	1	4	yes	—
181	C₂³×C₄	D₄×C₄₇	1	2	yes	—
182	C₂³×C₄	D₄×C₄₇	1	2	yes	—
183	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	4	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	D₄×C₄₇	1	8	yes	—
190	C₂²×C₈	—	2	4	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	4	4	no	—
198	C₂²⋊C₈	—	2	8	no	—
199	C₂²⋊C₈	—	2	8	no	—
200	#95	—	2	4	no	—
201	C₂⁴	—	2	2	no	—
202	C₂⁴	—	2	2	no	—
203	C₂⁴	—	2	2	no	—
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	4	2	no	—
206	C₂⁴	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
207	C₂⁴	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
208	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₄₇	1	4	yes	—
229	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
230	C₂²×C₄	#41×C₄₇	1	4	yes	—
231	C₂²×C₄	#41×C₄₇	1	4	yes	—
232	M₁₆	—	2	8	no	—
233	M₁₆	—	2	8	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	—	4	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	4	4	no	—
238	C₂×C₈	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	8	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	4	8	no	—
257	C₂³	—	4	8	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	#100×C₄₇	1	2	yes	—
261	C₂³	#101×C₄₇	1	2	yes	—
262	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
263	C₂³	#117×C₄₇	1	2	yes	—
264	C₂³	C₄×D₄×C₄₇	1	4	yes	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	8	no	—
277	C₂×C₄	#120×C₄₇	1	2	yes	—
278	C₂×C₄	#121×C₄₇	1	4	yes	—
279	C₂×C₄	#121×C₄₇	1	4	yes	—
280	C₂×C₄	#123×C₄₇	1	4	yes	—
281	C₂×C₄	#123×C₄₇	1	4	yes	—
282	C₂×C₄	#123×C₄₇	1	2	yes	—
283	C₂×C₄	#124×C₄₇	1	4	yes	—
284	C₂×C₄	#126×C₄₇	1	4	yes	—
285	C₈	—	4	8	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	16	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₄₇×#381	1	1	yes	characteristic
302	C₂²	C₄₇×#386	1	2	yes	—
303	C₂²	C₄₇×#388	1	4	yes	—
304	C₂²	C₂×#100×C₄₇	1	1	yes	characteristic
305	C₂²	C₂×#101×C₄₇	1	1	yes	characteristic
306	C₂²	C₄₇×#413	1	2	yes	—
307	C₄	—	4	2	no	—
308	C₄	—	8	4	no	—
309	C₄	C₄₇×#394	1	4	yes	—
310	C₄	C₄₇×#397	1	4	yes	—
311	C₂	—	2	4	no	—
312	C₂	—	4	2	no	—
313	C₂	—	4	2	no	—
314	C₂	C₄₇×#1440	1	1	yes	characteristic
315	C₂	C₄₇×#1821	1	1	yes	characteristic
316	C₂	C₄₇×#1822	1	1	yes	characteristic
317	C₂	C₄₇×#1900	1	2	yes	—
318	C₂	C₄₇×#1901	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₄₇
Frattini, cosocle	C₂³×C₄	C₂³×C₄₇
prime radical	C₂⁵×C₄₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂⁴×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄₇×#343	C₂	semidirect
126. #11198	C₄₇	direct (N × H)
154. C₂×#343	C₂×C₄₇	semidirect
174. C₄₇	#11198	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×C₄₇×#407	2
C₂×C₄₇×#380	2
C₂×C₄₇×#343	2
#11198	47

Subgroups — C47×#11198 #26050896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#11198 #26050896