Subgroups — C₄₇×#14296 #26053994

Order 12032.

1262 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#383	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₇×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1444	C₂	1	2	yes	maximal
5	C₄₇×#1442	C₂	1	2	yes	maximal
6	C₄₇×#1441	C₂	1	4	yes	maximal
7	C₄₇×#1436	C₂	1	4	yes	maximal
8	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	2	yes	—
12	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#383	—	2	4	no	—
14	C₄₇×#382	C₄	1	8	yes	complemented
15	C₄×C₄⋊C₄×C₄₇	—	2	4	no	—
16	C₂×#95×C₄₇	C₂²	1	2	yes	—
17	C₂×#95×C₄₇	C₂²	1	2	yes	—
18	C₂×#95×C₄₇	C₂²	1	2	yes	—
19	C₂×#95×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#95×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×#95×C₄₇	C₂²	1	4	yes	—
22	C₂×#95×C₄₇	C₂²	1	4	yes	—
23	C₂×#95×C₄₇	C₂²	1	4	yes	—
24	C₂×#95×C₄₇	C₂²	1	2	yes	—
25	C₂×#95×C₄₇	C₂²	1	2	yes	—
26	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	4	yes	—
30	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
37	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
38	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₄₇	—	2	4	no	—
41	C₂×C₄²×C₄₇	—	4	2	no	—
42	C₂×C₄²×C₄₇	—	4	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	D₄	1	4	yes	—
45	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
46	#95×C₄₇	—	2	4	no	—
47	#95×C₄₇	—	2	4	no	—
48	#95×C₄₇	—	2	2	no	—
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	—	2	4	no	—
51	#95×C₄₇	—	2	8	no	—
52	#95×C₄₇	—	4	4	no	—
53	#95×C₄₇	—	4	4	no	—
54	#95×C₄₇	C₂×C₄	1	8	yes	—
55	#95×C₄₇	C₂×C₄	1	8	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	—	2	4	no	—
64	C₂²×C₄×C₄₇	—	2	4	no	—
65	C₂²×C₄×C₄₇	—	2	4	no	—
66	C₂²×C₄×C₄₇	—	2	4	no	—
67	C₂²×C₄×C₄₇	—	2	4	no	—
68	C₂²×C₄×C₄₇	—	2	4	no	—
69	C₂²×C₄×C₄₇	—	2	4	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	—	2	4	no	—
72	C₂²×C₄×C₄₇	—	2	4	no	—
73	C₂²×C₄×C₄₇	—	2	4	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	—	2	2	no	—
76	C₂²×C₄×C₄₇	—	2	4	no	—
77	C₂²×C₄×C₄₇	—	2	4	no	—
78	C₂²×C₄×C₄₇	—	2	2	no	—
79	C₂²×C₄×C₄₇	—	2	4	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	—	2	2	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₄₇	#41	1	8	yes	—
95	C₄⋊C₄×C₄₇	—	2	16	no	—
96	C₄⋊C₄×C₄₇	—	4	8	no	—
97	C₄⋊C₄×C₄₇	—	4	16	no	—
98	C₄²×C₄₇	—	4	8	no	—
99	C₄²×C₄₇	C₄⋊C₄	1	16	yes	complemented
100	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
102	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂³×C₄₇	C₂×#41	1	4	yes	—
104	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
105	C₂³×C₄₇	H(5,𝔽₂)	1	2	yes	—
106	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
107	C₂³×C₄₇	2₋¹⁺⁴	1	2	yes	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	2	4	no	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	2	2	no	—
116	C₂×C₄×C₄₇	—	2	4	no	—
117	C₂×C₄×C₄₇	—	2	4	no	—
118	C₂×C₄×C₄₇	—	2	4	no	—
119	C₂×C₄×C₄₇	—	2	8	no	—
120	C₂×C₄×C₄₇	—	2	8	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	—	2	8	no	—
123	C₂×C₄×C₄₇	—	2	8	no	—
124	C₂×C₄×C₄₇	—	2	8	no	—
125	C₂×C₄×C₄₇	—	4	4	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	—	4	4	no	—
131	C₂×C₄×C₄₇	—	4	4	no	—
132	C₂×C₄×C₄₇	—	4	4	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	—	4	4	no	—
140	C₂×C₄×C₄₇	—	4	4	no	—
141	C₂×C₄×C₄₇	—	4	4	no	—
142	C₂×C₄×C₄₇	—	4	4	no	—
143	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
144	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
145	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
146	#14296	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄₇	C₄×#41	1	2	yes	—
149	C₂²×C₄₇	#518	1	1	yes	characteristic
150	C₂²×C₄₇	#519	1	1	yes	characteristic
151	C₂²×C₄₇	#520	1	2	yes	—
152	C₂²×C₄₇	#525	1	2	yes	—
153	C₂²×C₄₇	#533	1	2	yes	—
154	C₂²×C₄₇	#535	1	1	yes	characteristic
155	C₂²×C₄₇	#536	1	1	yes	characteristic
156	C₂²×C₄₇	#537	1	2	yes	—
157	C₂²×C₄₇	#538	1	4	yes	—
158	C₂²×C₄₇	#539	1	4	yes	—
159	C₂²×C₄₇	#541	1	4	yes	—
160	C₂²×C₄₇	#542	1	4	yes	—
161	C₂²×C₄₇	#544	1	2	yes	—
162	C₂²×C₄₇	#544	1	2	yes	—
163	C₄×C₄₇	—	2	4	no	—
164	C₄×C₄₇	—	2	4	no	—
165	C₄×C₄₇	—	2	4	no	—
166	C₄×C₄₇	—	4	8	no	—
167	C₄×C₄₇	—	4	8	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₄×C₄₇	—	8	2	no	—
170	C₂×#383	C₂×C₄₇	1	1	yes	characteristic
171	C₂×#382	C₂×C₄₇	1	1	yes	characteristic
172	C₂×C₄×C₄⋊C₄	C₂×C₄₇	1	1	yes	characteristic
173	#1444	C₂×C₄₇	1	2	yes	—
174	#1442	C₂×C₄₇	1	2	yes	—
175	#1441	C₂×C₄₇	1	4	yes	—
176	#1436	C₂×C₄₇	1	4	yes	—
177	C₂×C₄₇	#2296	1	1	yes	characteristic
178	C₂×C₄₇	#2316	1	1	yes	characteristic
179	C₂×C₄₇	#2317	1	1	yes	characteristic
180	C₂×C₄₇	#2336	1	2	yes	—
181	C₂×C₄₇	#2337	1	2	yes	—
182	C₂×C₄₇	#2636	1	2	yes	—
183	C₂×C₄₇	#2637	1	2	yes	—
184	C₂×C₄₇	#2666	1	4	yes	—
185	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₄₇	1	4	yes	—
187	C₂²×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
188	C₂²×C₄²	C₂²×C₄₇	1	2	yes	—
189	C₂²×C₄²	C₂²×C₄₇	1	1	yes	characteristic
190	#383	—	2	4	no	—
191	#382	C₄×C₄₇	1	8	yes	complemented
192	C₄×C₄⋊C₄	—	2	4	no	—
193	C₂×#95	C₂²×C₄₇	1	2	yes	—
194	C₂×#95	C₂²×C₄₇	1	4	yes	—
195	C₂×#95	C₂²×C₄₇	1	2	yes	—
196	C₂×#95	C₂²×C₄₇	1	2	yes	—
197	C₂×#95	C₂²×C₄₇	1	1	yes	characteristic
198	C₂×#95	C₂²×C₄₇	1	1	yes	characteristic
199	C₂×#95	C₂²×C₄₇	1	4	yes	—
200	C₂×#95	C₂²×C₄₇	1	4	yes	—
201	C₂×#95	C₂²×C₄₇	1	2	yes	—
202	C₂×#95	C₂²×C₄₇	1	2	yes	—
203	C₄₇	#14296	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
205	C₂³×C₄	C₂³×C₄₇	1	2	yes	—
206	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
207	C₂³×C₄	C₂³×C₄₇	1	4	yes	—
208	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
209	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
210	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	8	no	—
213	C₂×C₄⋊C₄	—	2	8	no	—
214	C₂×C₄⋊C₄	—	4	4	no	—
215	C₂×C₄⋊C₄	—	4	4	no	—
216	C₂×C₄⋊C₄	—	4	4	no	—
217	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	8	yes	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	—	4	2	no	—
220	C₂×C₄²	—	4	2	no	—
221	C₂×C₄²	C₂×C₄×C₄₇	1	4	yes	—
222	C₂×C₄²	D₄×C₄₇	1	4	yes	—
223	C₂×C₄²	Q₈×C₄₇	1	4	yes	—
224	#95	—	2	4	no	—
225	#95	—	2	4	no	—
226	#95	—	2	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	8	no	—
230	#95	—	4	4	no	—
231	#95	—	4	4	no	—
232	#95	C₂×C₄×C₄₇	1	8	yes	—
233	#95	C₂×C₄×C₄₇	1	8	yes	—
234	C₂⁴	C₂⁴×C₄₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	C₂²×C₄×C₄₇	1	8	yes	—
263	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
264	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
265	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
269	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
270	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
271	C₂²×C₄	C₂×Q₈×C₄₇	1	2	yes	—
272	C₂²×C₄	#41×C₄₇	1	8	yes	—
273	C₄⋊C₄	—	2	16	no	—
274	C₄⋊C₄	—	4	8	no	—
275	C₄⋊C₄	—	4	16	no	—
276	C₄²	—	4	8	no	—
277	C₄²	C₄⋊C₄×C₄₇	1	16	yes	complemented
278	C₂³	C₂³×C₄×C₄₇	1	1	yes	characteristic
279	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
280	C₂³	C₂²×Q₈×C₄₇	1	1	yes	characteristic
281	C₂³	C₂×#41×C₄₇	1	4	yes	—
282	C₂³	H(5,𝔽₂)×C₄₇	1	2	yes	—
283	C₂³	H(5,𝔽₂)×C₄₇	1	2	yes	—
284	C₂³	2₋¹⁺⁴×C₄₇	1	2	yes	—
285	C₂³	2₋¹⁺⁴×C₄₇	1	2	yes	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
322	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
323	C₂×C₄	C₂×C₄⋊C₄×C₄₇	1	4	yes	—
324	C₂²	C₂²×C₄⋊C₄×C₄₇	1	1	yes	characteristic
325	C₂²	C₄×#41×C₄₇	1	2	yes	—
326	C₂²	C₄₇×#518	1	1	yes	characteristic
327	C₂²	C₄₇×#519	1	1	yes	characteristic
328	C₂²	C₄₇×#520	1	2	yes	—
329	C₂²	C₄₇×#525	1	2	yes	—
330	C₂²	C₄₇×#533	1	2	yes	—
331	C₂²	C₄₇×#535	1	1	yes	characteristic
332	C₂²	C₄₇×#536	1	1	yes	characteristic
333	C₂²	C₄₇×#537	1	2	yes	—
334	C₂²	C₄₇×#538	1	4	yes	—
335	C₂²	C₄₇×#539	1	4	yes	—
336	C₂²	C₄₇×#541	1	4	yes	—
337	C₂²	C₄₇×#542	1	4	yes	—
338	C₂²	C₄₇×#544	1	2	yes	—
339	C₂²	C₄₇×#544	1	2	yes	—
340	C₄	—	2	4	no	—
341	C₄	—	2	4	no	—
342	C₄	—	2	4	no	—
343	C₄	—	4	8	no	—
344	C₄	—	4	8	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₂	C₄₇×#2296	1	1	yes	characteristic
348	C₂	C₄₇×#2316	1	1	yes	characteristic
349	C₂	C₄₇×#2317	1	1	yes	characteristic
350	C₂	C₄₇×#2336	1	2	yes	—
351	C₂	C₄₇×#2337	1	2	yes	—
352	C₂	C₄₇×#2636	1	2	yes	—
353	C₂	C₄₇×#2637	1	2	yes	—
354	C₂	C₄₇×#2666	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
14. C₄₇×#382	C₄	semidirect
99. C₄²×C₄₇	C₄⋊C₄	semidirect
146. #14296	C₄₇	direct (N × H)
191. #382	C₄×C₄₇	semidirect
203. C₄₇	#14296	direct (N × H)
277. C₄²	C₄⋊C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#383	2
C₂×C₄₇×#382	2
C₂×C₄×C₄⋊C₄×C₄₇	2
C₄₇×#1444	2
C₄₇×#1442	2
C₄₇×#1441	2
C₄₇×#1436	2
#14296	47

Subgroups — C47×#14296 #26053994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#14296 #26053994