Subgroups — C₄₇×#21911 #26061609

Order 12032.

874 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#2297	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#1924	C₂	1	4	yes	maximal, complemented
3	C₄₇×#1923	C₂	1	4	yes	maximal, complemented
4	C₄₇×#1797	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1780	C₂	1	2	yes	maximal
6	C₄₇×#1750	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×#97×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#391	C₄	1	2	yes	—
18	C₄₇×#390	C₄	1	2	yes	—
19	C₄₇×#386	C₄	1	2	yes	—
20	C₄₇×#386	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₄₇×#336	C₂²	1	4	yes	—
23	C₄₇×#336	C₂²	1	2	yes	—
24	C₄₇×#336	C₂²	1	2	yes	—
25	C₂×#41×C₄₇	—	2	4	no	—
26	C₂×#41×C₄₇	—	4	2	no	—
27	C₂×#41×C₄₇	—	4	2	no	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
37	#117×C₄₇	—	4	2	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₇	—	4	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
46	C₄⋊C₈×C₄₇	—	4	4	no	—
47	C₂²⋊C₈×C₄₇	—	2	8	no	—
48	C₂²⋊C₈×C₄₇	—	4	2	no	—
49	C₂²⋊C₈×C₄₇	—	4	2	no	—
50	C₂²⋊C₈×C₄₇	—	4	4	no	—
51	#97×C₄₇	—	4	2	no	—
52	C₄×C₈×C₄₇	—	4	2	no	—
53	#95×C₄₇	C₂×C₄	1	4	yes	—
54	#95×C₄₇	C₂×C₄	1	4	yes	—
55	C₂⁴×C₄₇	C₂²×C₄	1	2	yes	—
56	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	#41×C₄₇	—	4	16	no	—
58	C₂×Q₈×C₄₇	—	2	2	no	—
59	C₂×Q₈×C₄₇	—	4	1	no	—
60	C₂×Q₈×C₄₇	—	4	1	no	—
61	C₂×D₄×C₄₇	—	2	4	no	—
62	C₂×D₄×C₄₇	—	2	2	no	—
63	C₂×D₄×C₄₇	—	4	1	no	—
64	C₂×D₄×C₄₇	—	4	1	no	—
65	C₂×D₄×C₄₇	—	4	2	no	—
66	C₂×D₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	—	2	2	no	—
68	C₂²×C₄×C₄₇	—	2	2	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	—	4	1	no	—
75	C₂²×C₄×C₄₇	—	4	1	no	—
76	C₂²×C₄×C₄₇	—	4	2	no	—
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₄₇	#41	1	4	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₄₇	—	2	4	no	—
85	C₂×C₈×C₄₇	—	2	4	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	4	no	—
89	C₂×C₈×C₄₇	—	2	4	no	—
90	C₄⋊C₄×C₄₇	—	4	2	no	—
91	C₂²⋊C₄×C₄₇	—	2	8	no	—
92	C₂²⋊C₄×C₄₇	—	4	4	no	—
93	C₂²⋊C₄×C₄₇	—	4	2	no	—
94	C₄²×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	4	2	no	—
98	C₂³×C₄₇	—	2	1	no	—
99	C₂³×C₄₇	—	2	1	no	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	—	2	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	—	2	4	no	—
107	C₂³×C₄₇	#117	1	2	yes	—
108	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
109	Q₈×C₄₇	—	4	4	no	—
110	D₄×C₄₇	—	4	4	no	—
111	D₄×C₄₇	—	4	8	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	8	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	4	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	—	2	4	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	4	4	no	—
127	C₂×C₄×C₄₇	—	4	4	no	—
128	C₂×C₄×C₄₇	—	4	4	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	#117	1	2	yes	—
131	C₂×C₄×C₄₇	#131	1	4	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
134	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
135	C₈×C₄₇	—	4	4	no	—
136	C₈×C₄₇	—	4	4	no	—
137	#21911	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	4	1	no	—
141	C₂²×C₄₇	—	4	1	no	—
142	C₂²×C₄₇	—	4	2	no	—
143	C₂²×C₄₇	—	4	2	no	—
144	C₂²×C₄₇	—	4	4	no	—
145	C₂²×C₄₇	—	4	2	no	—
146	C₂²×C₄₇	—	4	2	no	—
147	C₂²×C₄₇	—	4	4	no	—
148	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
149	C₂²×C₄₇	#518	1	1	yes	characteristic
150	C₂²×C₄₇	#518	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#131	1	2	yes	—
152	C₂²×C₄₇	#568	1	2	yes	—
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	—	2	4	no	—
155	C₄×C₄₇	—	4	2	no	—
156	C₄×C₄₇	—	4	2	no	—
157	C₄×C₄₇	—	4	4	no	—
158	C₄×C₄₇	#538	1	4	yes	—
159	#2297	C₂×C₄₇	1	1	yes	characteristic
160	#1924	C₂×C₄₇	1	4	yes	complemented
161	#1923	C₂×C₄₇	1	4	yes	complemented
162	#1797	C₂×C₄₇	1	2	yes	complemented
163	#1780	C₂×C₄₇	1	2	yes	—
164	#1750	C₂×C₄₇	1	2	yes	complemented
165	C₂×C₄₇	—	4	2	no	—
166	C₂×C₄₇	—	8	2	no	—
167	C₂×C₄₇	#2312	1	1	yes	characteristic
168	C₂×C₄₇	#2925	1	2	yes	—
169	C₂×C₄₇	#2977	1	1	yes	characteristic
170	C₂×C₄₇	#2978	1	1	yes	characteristic
171	C₂×C₄₇	#2981	1	2	yes	—
172	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
173	C₂×#117	C₂²×C₄₇	1	1	yes	characteristic
174	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₄₇	1	4	yes	—
176	C₂×C₂²⋊C₈	C₂²×C₄₇	1	4	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
178	C₂×C₂²⋊C₈	C₂²×C₄₇	1	4	yes	—
179	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
180	C₂×#97	C₂²×C₄₇	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	—
182	#391	C₄×C₄₇	1	2	yes	—
183	#390	C₄×C₄₇	1	2	yes	—
184	#386	C₄×C₄₇	1	2	yes	—
185	#386	C₄×C₄₇	1	2	yes	—
186	C₄×C₂²⋊C₄	C₂²×C₄₇	1	4	yes	—
187	#336	C₂²×C₄₇	1	2	yes	—
188	#336	C₂²×C₄₇	1	2	yes	—
189	#336	C₂²×C₄₇	1	4	yes	—
190	C₄₇	#21911	1	1	yes	characteristic, direct product factor, Sylow
191	C₂×#41	—	2	4	no	—
192	C₂×#41	—	4	2	no	—
193	C₂×#41	—	4	2	no	—
194	C₂²×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
195	C₂²×D₄	C₂×C₄×C₄₇	1	2	yes	—
196	C₂²×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
197	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
198	C₂³×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
200	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
202	C₂²×C₈	C₂³×C₄₇	1	4	yes	—
203	#117	—	4	2	no	—
204	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
205	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×C₄×C₄₇	1	4	yes	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂×C₄×C₄₇	1	2	yes	—
210	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
211	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
212	C₄⋊C₈	—	4	4	no	—
213	C₂²⋊C₈	—	2	8	no	—
214	C₂²⋊C₈	—	4	4	no	—
215	C₂²⋊C₈	—	4	2	no	—
216	C₂²⋊C₈	—	4	2	no	—
217	#97	—	4	2	no	—
218	C₄×C₈	—	4	2	no	—
219	#95	C₂×C₄×C₄₇	1	4	yes	—
220	#95	C₂×C₄×C₄₇	1	4	yes	—
221	C₂⁴	C₂²×C₄×C₄₇	1	2	yes	—
222	C₂⁴	C₂²×C₄×C₄₇	1	1	yes	characteristic
223	#41	—	4	16	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₄₇	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
248	C₂²×C₄	#41×C₄₇	1	4	yes	—
249	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	4	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	4	no	—
272	C₂³	—	2	4	no	—
273	C₂³	#117×C₄₇	1	2	yes	—
274	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
275	Q₈	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	8	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	#117×C₄₇	1	2	yes	—
297	C₂×C₄	#131×C₄₇	1	4	yes	—
298	C₂×C₄	C₂×#41×C₄₇	1	2	yes	—
299	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
300	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
301	C₈	—	4	4	no	—
302	C₈	—	4	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	4	no	—
313	C₂²	C₂×#117×C₄₇	1	1	yes	characteristic
314	C₂²	C₄₇×#518	1	1	yes	characteristic
315	C₂²	C₄₇×#518	1	1	yes	characteristic
316	C₂²	C₂×#131×C₄₇	1	2	yes	—
317	C₂²	C₄₇×#568	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	4	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	4	no	—
323	C₄	C₄₇×#538	1	4	yes	—
324	C₂	—	4	2	no	—
325	C₂	—	8	2	no	—
326	C₂	C₄₇×#2312	1	1	yes	characteristic
327	C₂	C₄₇×#2925	1	2	yes	—
328	C₂	C₄₇×#2977	1	1	yes	characteristic
329	C₂	C₄₇×#2978	1	1	yes	characteristic
330	C₂	C₄₇×#2981	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₄₇×#1924	C₂	semidirect
3. C₄₇×#1923	C₂	semidirect
4. C₄₇×#1797	C₂	semidirect
6. C₄₇×#1750	C₂	semidirect
137. #21911	C₄₇	direct (N × H)
160. #1924	C₂×C₄₇	semidirect
161. #1923	C₂×C₄₇	semidirect
162. #1797	C₂×C₄₇	semidirect
164. #1750	C₂×C₄₇	semidirect
190. C₄₇	#21911	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×#2297	2
C₄₇×#1924	2
C₄₇×#1923	2
C₄₇×#1797	2
C₄₇×#1780	2
C₄₇×#1750	2
#21911	47

Subgroups — C47×#21911 #26061609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#21911 #26061609