Subgroups — C₄₇×#21617 #26061315

Order 12032.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#478	C₂	1	1	yes	maximal, characteristic
2	C₄×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1989	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1949	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#1920	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1918	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1864	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1862	C₂	1	1	yes	maximal, characteristic
10	C₄₇×#1804	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1776	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#1776	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₇×#1762	C₂	1	1	yes	maximal, characteristic
15	C₄₇×#1762	C₂	1	1	yes	maximal, characteristic
16	C₂×#125×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₄₇	—	2	1	no	—
21	C₄₇×#478	C₄	1	8	yes	complemented
22	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#389	C₂²	1	1	yes	characteristic
35	C₄₇×#389	C₂²	1	1	yes	characteristic
36	C₄₇×#387	—	2	1	no	—
37	C₄₇×#387	—	2	1	no	—
38	C₄₇×#382	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
40	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
41	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
47	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
48	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
49	C₄³×C₄₇	C₂²	1	1	yes	characteristic
50	C₄₇×#340	C₂²	1	1	yes	characteristic
51	C₄₇×#340	C₂²	1	1	yes	characteristic
52	C₄₇×#340	C₂²	1	1	yes	characteristic
53	C₄₇×#340	C₂²	1	1	yes	characteristic
54	C₄₇×#340	C₂²	1	1	yes	characteristic
55	C₄₇×#340	C₂²	1	1	yes	characteristic
56	C₄₇×#340	C₂²	1	1	yes	characteristic
57	C₄₇×#340	C₂²	1	1	yes	characteristic
58	C₄₇×#336	C₂²	1	1	yes	characteristic
59	C₄₇×#336	C₂²	1	1	yes	characteristic
60	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₄₇	—	2	1	no	—
62	C₂³×C₄×C₄₇	—	2	1	no	—
63	C₂³×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	#125×C₄₇	—	2	2	no	—
69	#125×C₄₇	C₂×C₄	1	4	yes	complemented
70	C₄×D₄×C₄₇	—	2	2	no	—
71	C₄×D₄×C₄₇	—	2	2	no	—
72	C₄×D₄×C₄₇	—	2	2	no	—
73	C₄×D₄×C₄₇	—	2	2	no	—
74	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
75	C₄×D₄×C₄₇	Q₈	1	4	yes	—
76	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
90	C₂×C₄²×C₄₇	—	2	1	no	—
91	C₂×C₄²×C₄₇	—	2	1	no	—
92	C₂×C₄²×C₄₇	—	2	1	no	—
93	C₂×C₄²×C₄₇	—	2	1	no	—
94	C₂×C₄²×C₄₇	—	2	1	no	—
95	C₂×C₄²×C₄₇	—	4	4	no	—
96	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
103	#107×C₄₇	—	2	2	no	—
104	#107×C₄₇	C₂×C₄	1	4	yes	complemented
105	#106×C₄₇	—	2	2	no	—
106	#106×C₄₇	C₂×C₄	1	4	yes	complemented
107	C₄⋊C₈×C₄₇	—	2	2	no	—
108	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	complemented
109	#102×C₄₇	—	2	2	no	—
110	#102×C₄₇	—	2	2	no	—
111	#102×C₄₇	C₂×C₄	1	4	yes	—
112	#102×C₄₇	C₂×C₄	1	4	yes	—
113	#97×C₄₇	—	2	2	no	—
114	#97×C₄₇	—	2	2	no	—
115	#95×C₄₇	—	2	1	no	—
116	#95×C₄₇	—	2	1	no	—
117	#95×C₄₇	—	2	1	no	—
118	#95×C₄₇	—	2	1	no	—
119	#95×C₄₇	—	2	1	no	—
120	#95×C₄₇	—	2	1	no	—
121	#95×C₄₇	—	2	1	no	—
122	#95×C₄₇	—	2	1	no	—
123	#95×C₄₇	—	2	1	no	—
124	C₂⁴×C₄₇	—	2	1	no	—
125	C₂×D₄×C₄₇	—	2	1	no	—
126	C₂×D₄×C₄₇	—	2	1	no	—
127	C₂×D₄×C₄₇	—	2	1	no	—
128	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₄₇	C₂×Q₈	1	2	yes	—
130	C₂×D₄×C₄₇	#41	1	2	yes	—
131	C₂²×C₄×C₄₇	—	2	4	no	—
132	C₂²×C₄×C₄₇	—	2	1	no	—
133	C₂²×C₄×C₄₇	—	2	1	no	—
134	C₂²×C₄×C₄₇	—	2	1	no	—
135	C₂²×C₄×C₄₇	—	2	1	no	—
136	C₂²×C₄×C₄₇	—	2	1	no	—
137	C₂²×C₄×C₄₇	—	2	4	no	—
138	C₂²×C₄×C₄₇	—	2	4	no	—
139	C₂²×C₄×C₄₇	—	2	1	no	—
140	C₂²×C₄×C₄₇	—	2	1	no	—
141	C₂²×C₄×C₄₇	—	4	2	no	—
142	C₂²×C₄×C₄₇	—	4	2	no	—
143	C₂²×C₄×C₄₇	—	4	2	no	—
144	C₂²×C₄×C₄₇	—	4	2	no	—
145	C₂²×C₄×C₄₇	—	4	2	no	—
146	C₂²×C₄×C₄₇	—	4	2	no	—
147	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
154	C₂×C₈×C₄₇	—	2	1	no	—
155	C₂×C₈×C₄₇	—	2	1	no	—
156	C₂×C₈×C₄₇	—	2	1	no	—
157	C₂×C₈×C₄₇	—	2	1	no	—
158	C₂×C₈×C₄₇	—	2	1	no	—
159	C₂×C₈×C₄₇	—	2	1	no	—
160	C₂×C₈×C₄₇	—	2	1	no	—
161	C₂×C₈×C₄₇	—	2	1	no	—
162	C₂×C₈×C₄₇	—	2	1	no	—
163	C₂×C₈×C₄₇	—	2	1	no	—
164	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
165	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₄₇	—	2	4	no	—
167	C₄⋊C₄×C₄₇	—	2	4	no	—
168	C₄⋊C₄×C₄₇	—	2	1	no	—
169	C₄⋊C₄×C₄₇	—	2	1	no	—
170	C₄⋊C₄×C₄₇	—	2	1	no	—
171	C₄⋊C₄×C₄₇	—	2	1	no	—
172	C₄⋊C₄×C₄₇	—	2	1	no	—
173	C₄⋊C₄×C₄₇	—	2	1	no	—
174	C₄⋊C₄×C₄₇	—	2	1	no	—
175	C₄⋊C₄×C₄₇	—	2	1	no	—
176	C₄⋊C₄×C₄₇	—	2	1	no	—
177	C₄⋊C₄×C₄₇	—	4	2	no	—
178	C₄⋊C₄×C₄₇	—	4	2	no	—
179	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
180	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
181	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₄₇	C₂×Q₈	1	2	yes	—
183	C₄⋊C₄×C₄₇	#41	1	2	yes	—
184	C₂²⋊C₄×C₄₇	—	2	4	no	—
185	C₂²⋊C₄×C₄₇	—	2	4	no	—
186	C₂²⋊C₄×C₄₇	—	2	4	no	—
187	C₄²×C₄₇	—	2	2	no	—
188	C₄²×C₄₇	—	2	1	no	—
189	C₄²×C₄₇	—	2	1	no	—
190	C₄²×C₄₇	—	4	1	no	—
191	C₄²×C₄₇	—	4	1	no	—
192	C₄²×C₄₇	—	4	1	no	—
193	C₄²×C₄₇	—	4	2	no	—
194	C₄²×C₄₇	—	4	2	no	—
195	C₄²×C₄₇	—	4	2	no	—
196	C₄²×C₄₇	—	4	2	no	—
197	C₄²×C₄₇	—	4	2	no	—
198	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
199	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
200	C₄²×C₄₇	#41	1	2	yes	—
201	C₄²×C₄₇	#41	1	2	yes	—
202	C₂³×C₄₇	—	2	2	no	—
203	C₂³×C₄₇	—	2	2	no	—
204	C₂³×C₄₇	—	2	2	no	—
205	C₂³×C₄₇	—	4	1	no	—
206	C₂³×C₄₇	—	4	1	no	—
207	C₂³×C₄₇	—	4	1	no	—
208	C₂³×C₄₇	—	4	1	no	—
209	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
210	D₄×C₄₇	—	2	2	no	—
211	D₄×C₄₇	—	2	2	no	—
212	D₄×C₄₇	—	2	2	no	—
213	D₄×C₄₇	C₄×Q₈	1	4	yes	—
214	C₂×C₄×C₄₇	—	2	1	no	—
215	C₂×C₄×C₄₇	—	2	1	no	—
216	C₂×C₄×C₄₇	—	2	1	no	—
217	C₂×C₄×C₄₇	—	2	1	no	—
218	C₂×C₄×C₄₇	—	2	1	no	—
219	C₂×C₄×C₄₇	—	2	1	no	—
220	C₂×C₄×C₄₇	—	2	1	no	—
221	C₂×C₄×C₄₇	—	2	1	no	—
222	C₂×C₄×C₄₇	—	2	1	no	—
223	C₂×C₄×C₄₇	—	2	1	no	—
224	C₂×C₄×C₄₇	—	2	2	no	—
225	C₂×C₄×C₄₇	—	2	2	no	—
226	C₂×C₄×C₄₇	—	2	2	no	—
227	C₂×C₄×C₄₇	—	2	2	no	—
228	C₂×C₄×C₄₇	—	2	1	no	—
229	C₂×C₄×C₄₇	—	2	2	no	—
230	C₂×C₄×C₄₇	—	2	1	no	—
231	C₂×C₄×C₄₇	—	2	1	no	—
232	C₂×C₄×C₄₇	—	4	8	no	—
233	C₂×C₄×C₄₇	—	4	8	no	—
234	C₂×C₄×C₄₇	—	4	8	no	—
235	C₂×C₄×C₄₇	—	4	1	no	—
236	C₂×C₄×C₄₇	—	4	1	no	—
237	C₂×C₄×C₄₇	—	4	1	no	—
238	C₂×C₄×C₄₇	—	4	1	no	—
239	C₂×C₄×C₄₇	—	4	1	no	—
240	C₂×C₄×C₄₇	—	4	1	no	—
241	C₂×C₄×C₄₇	—	4	1	no	—
242	C₂×C₄×C₄₇	—	4	1	no	—
243	C₂×C₄×C₄₇	—	4	1	no	—
244	C₂×C₄×C₄₇	—	4	1	no	—
245	C₂×C₄×C₄₇	—	4	1	no	—
246	C₂×C₄×C₄₇	—	4	1	no	—
247	C₂×C₄×C₄₇	—	4	1	no	—
248	C₂×C₄×C₄₇	—	4	1	no	—
249	C₂×C₄×C₄₇	—	4	1	no	—
250	C₂×C₄×C₄₇	—	4	1	no	—
251	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
256	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
257	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
258	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
261	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
262	C₈×C₄₇	—	2	2	no	—
263	C₈×C₄₇	—	2	2	no	—
264	C₈×C₄₇	—	4	1	no	—
265	C₈×C₄₇	—	4	1	no	—
266	#21617	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
267	C₂²×C₄₇	—	2	4	no	—
268	C₂²×C₄₇	—	4	2	no	—
269	C₂²×C₄₇	—	4	2	no	—
270	C₂²×C₄₇	—	4	2	no	—
271	C₂²×C₄₇	—	4	2	no	—
272	C₂²×C₄₇	—	4	2	no	—
273	C₂²×C₄₇	—	4	2	no	—
274	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
275	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂²×C₄₇	#525	1	1	yes	characteristic
277	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂²×C₄₇	#575	1	1	yes	characteristic
279	C₂²×C₄₇	#575	1	1	yes	characteristic
280	C₂²×C₄₇	#575	1	1	yes	characteristic
281	C₄×C₄₇	—	2	2	no	—
282	C₄×C₄₇	—	2	2	no	—
283	C₄×C₄₇	—	2	2	no	—
284	C₄×C₄₇	—	2	2	no	—
285	C₄×C₄₇	—	2	2	no	—
286	C₄×C₄₇	—	4	2	no	—
287	C₄×C₄₇	—	4	1	no	—
288	C₄×C₄₇	—	4	2	no	—
289	C₄×C₄₇	—	4	2	no	—
290	C₄×C₄₇	—	4	2	no	—
291	C₄×C₄₇	—	4	2	no	—
292	C₄×C₄₇	—	8	1	no	—
293	C₄×C₄₇	—	8	1	no	—
294	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
295	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
296	C₄×C₄₇	#529	1	1	yes	characteristic
297	C₄×C₄₇	#533	1	1	yes	characteristic
298	C₂×#478	C₂×C₄₇	1	1	yes	characteristic
299	C₄×#125	C₂×C₄₇	1	1	yes	characteristic, complemented
300	C₄²×D₄	C₂×C₄₇	1	1	yes	characteristic
301	#1989	C₂×C₄₇	1	1	yes	characteristic, complemented
302	#1949	C₂×C₄₇	1	1	yes	characteristic, complemented
303	#1920	C₂×C₄₇	1	1	yes	characteristic, complemented
304	#1918	C₂×C₄₇	1	1	yes	characteristic, complemented
305	#1864	C₂×C₄₇	1	1	yes	characteristic
306	#1862	C₂×C₄₇	1	1	yes	characteristic
307	#1804	C₂×C₄₇	1	1	yes	characteristic
308	#1776	C₂×C₄₇	1	1	yes	characteristic, complemented
309	#1776	C₂×C₄₇	1	1	yes	characteristic, complemented
310	#1767	C₂×C₄₇	1	1	yes	characteristic, complemented
311	#1762	C₂×C₄₇	1	1	yes	characteristic
312	#1762	C₂×C₄₇	1	1	yes	characteristic
313	C₂×C₄₇	—	4	4	no	—
314	C₂×C₄₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic
315	C₂×C₄₇	#2943	1	1	yes	characteristic
316	C₂×C₄₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
317	C₂×C₄₇	#3077	1	1	yes	characteristic
318	C₂×C₄₇	#3080	1	1	yes	characteristic
319	C₂×C₄₇	#3111	1	1	yes	characteristic
320	C₂×C₄₇	#3114	1	1	yes	characteristic
321	C₂×#125	C₂²×C₄₇	1	1	yes	characteristic
322	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
323	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
324	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
325	C₂²×C₄²	—	2	1	no	—
326	#478	C₄×C₄₇	1	8	yes	complemented
327	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
328	C₂×#107	C₂²×C₄₇	1	1	yes	characteristic
329	C₂×#106	C₂²×C₄₇	1	1	yes	characteristic
330	C₂×#106	C₂²×C₄₇	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
333	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic
334	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic
335	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic
336	C₂×#102	C₂²×C₄₇	1	1	yes	characteristic
337	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
338	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
339	#389	C₂²×C₄₇	1	1	yes	characteristic
340	#389	C₂²×C₄₇	1	1	yes	characteristic
341	#387	—	2	1	no	—
342	#387	—	2	1	no	—
343	#382	C₂²×C₄₇	1	1	yes	characteristic
344	C₄×C₄⋊C₄	—	2	1	no	—
345	C₄×C₄⋊C₄	—	2	1	no	—
346	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
347	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
348	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
350	C₄×C₄⋊C₄	C₂²×C₄₇	1	1	yes	characteristic
351	C₄×C₂²⋊C₄	—	2	1	no	—
352	C₄×C₂²⋊C₄	—	2	1	no	—
353	C₄×C₂²⋊C₄	—	2	1	no	—
354	C₄³	C₂²×C₄₇	1	1	yes	characteristic
355	#340	C₂²×C₄₇	1	1	yes	characteristic
356	#340	C₂²×C₄₇	1	1	yes	characteristic
357	#340	C₂²×C₄₇	1	1	yes	characteristic
358	#340	C₂²×C₄₇	1	1	yes	characteristic
359	#340	C₂²×C₄₇	1	1	yes	characteristic
360	#340	C₂²×C₄₇	1	1	yes	characteristic
361	#340	C₂²×C₄₇	1	1	yes	characteristic
362	#340	C₂²×C₄₇	1	1	yes	characteristic
363	#336	C₂²×C₄₇	1	1	yes	characteristic
364	#336	C₂²×C₄₇	1	1	yes	characteristic
365	C₄₇	#21617	1	1	yes	characteristic, direct product factor, Sylow
366	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	—	2	1	no	—
369	C₂³×C₄	—	2	1	no	—
370	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
374	#125	—	2	2	no	—
375	#125	C₂×C₄×C₄₇	1	4	yes	complemented
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	C₂×C₄×C₄₇	1	4	yes	—
381	C₄×D₄	Q₈×C₄₇	1	4	yes	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	2	1	no	—
398	C₂×C₄²	—	2	1	no	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	—	4	4	no	—
402	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
403	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
404	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
405	C₂×C₄²	D₄×C₄₇	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
409	#107	—	2	2	no	—
410	#107	C₂×C₄×C₄₇	1	4	yes	complemented
411	#106	—	2	2	no	—
412	#106	C₂×C₄×C₄₇	1	4	yes	complemented
413	C₄⋊C₈	—	2	2	no	—
414	C₄⋊C₈	C₂×C₄×C₄₇	1	4	yes	complemented
415	#102	—	2	2	no	—
416	#102	—	2	2	no	—
417	#102	C₂×C₄×C₄₇	1	4	yes	—
418	#102	C₂×C₄×C₄₇	1	4	yes	—
419	#97	—	2	2	no	—
420	#97	—	2	2	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	C₂⁴	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	C₂²×C₄×C₄₇	1	2	yes	—
435	C₂×D₄	C₂×Q₈×C₄₇	1	2	yes	—
436	C₂×D₄	#41×C₄₇	1	2	yes	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	C₂²×C₄×C₄₇	1	2	yes	—
471	C₂×C₈	C₂²×C₄×C₄₇	1	2	yes	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	—	4	2	no	—
485	C₄⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
486	C₄⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
487	C₄⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
488	C₄⋊C₄	C₂×Q₈×C₄₇	1	2	yes	—
489	C₄⋊C₄	#41×C₄₇	1	2	yes	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	1	no	—
499	C₄²	—	4	2	no	—
500	C₄²	—	4	2	no	—
501	C₄²	—	4	2	no	—
502	C₄²	—	4	2	no	—
503	C₄²	—	4	2	no	—
504	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
505	C₄²	C₂×Q₈×C₄₇	1	2	yes	—
506	C₄²	#41×C₄₇	1	2	yes	—
507	C₄²	#41×C₄₇	1	2	yes	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	1	no	—
515	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	—	2	2	no	—
519	D₄	C₄×Q₈×C₄₇	1	4	yes	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
558	C₂×C₄	C₄×D₄×C₄₇	1	2	yes	—
559	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
560	C₂×C₄	C₂²⋊Q₈×C₄₇	1	2	yes	—
561	C₂×C₄	AGL(1,ℤ/8)×C₄₇	1	2	yes	—
562	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
563	C₂×C₄	C₂²×Q₈×C₄₇	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
566	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	2	no	—
575	C₂²	—	4	2	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	2	no	—
578	C₂²	—	4	2	no	—
579	C₂²	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
580	C₂²	C₂×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
581	C₂²	C₄₇×#525	1	1	yes	characteristic
582	C₂²	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
583	C₂²	C₄₇×#575	1	1	yes	characteristic
584	C₂²	C₄₇×#575	1	1	yes	characteristic
585	C₂²	C₄₇×#575	1	1	yes	characteristic
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	8	1	no	—
598	C₄	—	8	1	no	—
599	C₄	C₂×C₄×Q₈×C₄₇	1	1	yes	characteristic
600	C₄	C₄×#41×C₄₇	1	1	yes	characteristic
601	C₄	C₄₇×#529	1	1	yes	characteristic
602	C₄	C₄₇×#533	1	1	yes	characteristic
603	C₂	—	4	4	no	—
604	C₂	C₄×C₂²⋊Q₈×C₄₇	1	1	yes	characteristic
605	C₂	C₄₇×#2943	1	1	yes	characteristic
606	C₂	C₄×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
607	C₂	C₄₇×#3077	1	1	yes	characteristic
608	C₂	C₄₇×#3080	1	1	yes	characteristic
609	C₂	C₄₇×#3111	1	1	yes	characteristic
610	C₂	C₄₇×#3114	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂²×D₄×C₄₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₄³×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×#125×C₄₇	C₂	semidirect
4. C₄₇×#1989	C₂	semidirect
5. C₄₇×#1949	C₂	semidirect
6. C₄₇×#1920	C₂	semidirect
7. C₄₇×#1918	C₂	semidirect
11. C₄₇×#1776	C₂	semidirect
12. C₄₇×#1776	C₂	semidirect
13. C₄₇×#1767	C₂	semidirect
21. C₄₇×#478	C₄	semidirect
69. #125×C₄₇	C₂×C₄	semidirect
104. #107×C₄₇	C₂×C₄	semidirect
106. #106×C₄₇	C₂×C₄	semidirect
108. C₄⋊C₈×C₄₇	C₂×C₄	semidirect
266. #21617	C₄₇	direct (N × H)
299. C₄×#125	C₂×C₄₇	semidirect
301. #1989	C₂×C₄₇	semidirect
302. #1949	C₂×C₄₇	semidirect
303. #1920	C₂×C₄₇	semidirect
304. #1918	C₂×C₄₇	semidirect
308. #1776	C₂×C₄₇	semidirect
309. #1776	C₂×C₄₇	semidirect
310. #1767	C₂×C₄₇	semidirect
326. #478	C₄×C₄₇	semidirect
365. C₄₇	#21617	direct (N × H)
375. #125	C₂×C₄×C₄₇	semidirect
410. #107	C₂×C₄×C₄₇	semidirect
412. #106	C₂×C₄×C₄₇	semidirect
414. C₄⋊C₈	C₂×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×#478	2
C₄×#125×C₄₇	2
C₄²×D₄×C₄₇	2
C₄₇×#1989	2
C₄₇×#1949	2
C₄₇×#1920	2
C₄₇×#1918	2
C₄₇×#1864	2
C₄₇×#1862	2
C₄₇×#1804	2
C₄₇×#1776	2
C₄₇×#1767	2
C₄₇×#1762	2
#21617	47

Subgroups — C47×#21617 #26061315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#21617 #26061315