Subgroups — C₄₇×#18564 #26058262

Order 12032.

1082 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3522	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3217	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3151	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1623	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1622	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1621	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1620	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1551	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#1491	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1483	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
13	C₂²×SD₁₆×C₄₇	—	2	1	no	—
14	C₄₇×#559	—	2	1	no	—
15	C₄₇×#559	C₂²	1	1	yes	characteristic, complemented
16	Q₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
17	C₄₇×#554	—	2	1	no	—
18	C₄₇×#554	C₂²	1	1	yes	characteristic, complemented
19	C₄₇×#550	C₂²	1	1	yes	characteristic, complemented
20	C₄₇×#548	—	2	1	no	—
21	C₄₇×#547	—	2	2	no	—
22	C₄₇×#547	—	2	1	no	—
23	C₄₇×#542	—	2	2	no	—
24	C₄₇×#542	—	2	1	no	—
25	C₄₇×#532	—	2	1	no	—
26	C₄₇×#532	C₂²	1	1	yes	characteristic
27	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#529	—	2	1	no	—
29	C₄₇×#529	C₂²	1	1	yes	characteristic
30	C₄₇×#520	—	2	1	no	—
31	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#496	—	2	2	no	—
33	C₄₇×#496	—	2	1	no	—
34	C₄₇×#493	C₂²	1	2	yes	—
35	C₄₇×#492	C₂²	1	2	yes	—
36	C₄₇×#490	—	2	1	no	—
37	C₄₇×#460	—	2	1	no	—
38	C₄₇×#460	—	2	1	no	—
39	C₄₇×#460	C₂²	1	2	yes	—
40	C₄₇×#460	C₂²	1	2	yes	—
41	C₄₇×#460	C₂²	1	2	yes	—
42	C₄₇×#459	C₂²	1	2	yes	—
43	C₄₇×#450	—	2	1	no	—
44	C₄₇×#447	—	2	1	no	—
45	C₄₇×#432	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
47	C₂×#102×C₄₇	—	2	1	no	—
48	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
49	C₄₇×#333	C₂²	1	2	yes	complemented
50	C₄₇×#331	C₂²	1	2	yes	complemented
51	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
52	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
53	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
54	C₄₇×D₄⋊C₈	C₂²	1	2	yes	complemented
55	C₂×#41×C₄₇	—	2	2	no	—
56	C₂×#41×C₄₇	—	2	1	no	—
57	C₂×#41×C₄₇	D₄	1	2	yes	—
58	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₄₇	—	2	1	no	—
60	C₂²×D₄×C₄₇	—	2	1	no	—
61	C₂²×D₄×C₄₇	—	2	1	no	—
62	AGL(1,ℤ/8)×C₄₇	—	4	8	no	—
63	C₂×SD₁₆×C₄₇	—	2	2	no	—
64	C₂×SD₁₆×C₄₇	—	2	2	no	—
65	C₂×SD₁₆×C₄₇	—	2	4	no	—
66	C₂×SD₁₆×C₄₇	—	2	2	no	—
67	C₂×SD₁₆×C₄₇	—	4	4	no	—
68	C₂×D₈×C₄₇	—	2	4	no	—
69	C₂×D₈×C₄₇	—	2	2	no	—
70	C₂×M₁₆×C₄₇	—	2	1	no	—
71	C₂²×C₈×C₄₇	—	2	1	no	—
72	C₄⋊Q₈×C₄₇	—	2	2	no	—
73	C₄⋊Q₈×C₄₇	—	2	1	no	—
74	C₄⋊Q₈×C₄₇	—	2	1	no	—
75	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
76	#127×C₄₇	—	2	1	no	—
77	#127×C₄₇	—	2	1	no	—
78	#127×C₄₇	—	2	1	no	—
79	#127×C₄₇	—	2	1	no	—
80	#127×C₄₇	C₂³	1	2	yes	—
81	#127×C₄₇	C₂³	1	1	yes	characteristic
82	#126×C₄₇	—	2	2	no	—
83	#126×C₄₇	—	2	1	no	—
84	#125×C₄₇	—	2	1	no	—
85	#125×C₄₇	D₄	1	1	yes	characteristic
86	#124×C₄₇	—	2	2	no	—
87	#124×C₄₇	—	2	2	no	—
88	#124×C₄₇	—	2	1	no	—
89	#124×C₄₇	—	2	1	no	—
90	#124×C₄₇	D₄	1	1	yes	characteristic
91	#124×C₄₇	D₄	1	2	yes	—
92	#123×C₄₇	—	2	2	no	—
93	#123×C₄₇	—	2	2	no	—
94	#123×C₄₇	—	2	2	no	—
95	#123×C₄₇	—	2	2	no	—
96	#123×C₄₇	—	2	1	no	—
97	C₂²⋊Q₈×C₄₇	—	2	2	no	—
98	C₂²⋊Q₈×C₄₇	—	2	2	no	—
99	C₂²⋊Q₈×C₄₇	—	2	2	no	—
100	C₂²⋊Q₈×C₄₇	—	2	2	no	—
101	C₂²⋊Q₈×C₄₇	—	2	1	no	—
102	C₂²⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
103	#121×C₄₇	—	2	2	no	—
104	#121×C₄₇	—	2	2	no	—
105	#121×C₄₇	—	2	1	no	—
106	#121×C₄₇	—	2	1	no	—
107	#121×C₄₇	—	2	1	no	—
108	#121×C₄₇	D₄	1	1	yes	characteristic
109	C₄×Q₈×C₄₇	—	2	1	no	—
110	C₄×Q₈×C₄₇	—	2	1	no	—
111	C₄×Q₈×C₄₇	—	2	1	no	—
112	C₄×Q₈×C₄₇	—	2	1	no	—
113	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
114	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
115	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
116	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
117	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
118	C₄×D₄×C₄₇	—	2	2	no	—
119	C₄×D₄×C₄₇	—	2	2	no	—
120	C₄×D₄×C₄₇	—	2	2	no	—
121	C₄×D₄×C₄₇	—	2	1	no	—
122	C₄×D₄×C₄₇	—	2	1	no	—
123	C₄×D₄×C₄₇	—	2	1	no	—
124	C₄×D₄×C₄₇	—	2	1	no	—
125	C₄×D₄×C₄₇	—	2	1	no	—
126	C₄×D₄×C₄₇	—	2	1	no	—
127	C₄×D₄×C₄₇	—	2	1	no	—
128	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
129	#117×C₄₇	—	2	2	no	—
130	#117×C₄₇	—	2	1	no	—
131	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
132	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
133	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
134	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
135	C₂×C₄²×C₄₇	—	2	1	no	—
136	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
137	C₄⋊C₈×C₄₇	—	2	1	no	—
138	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
139	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
140	#102×C₄₇	—	2	2	no	—
141	#102×C₄₇	—	2	2	no	—
142	#102×C₄₇	—	2	2	no	—
143	#102×C₄₇	—	2	2	no	—
144	C₂²⋊C₈×C₄₇	—	2	1	no	—
145	#97×C₄₇	—	2	1	no	—
146	#97×C₄₇	—	2	1	no	—
147	C₄×C₈×C₄₇	C₂³	1	2	yes	—
148	C₂⁴×C₄₇	—	4	1	no	—
149	#41×C₄₇	—	4	4	no	—
150	#41×C₄₇	—	4	2	no	—
151	#41×C₄₇	—	4	2	no	—
152	#41×C₄₇	—	8	4	no	—
153	C₂×Q₈×C₄₇	—	2	2	no	—
154	C₂×Q₈×C₄₇	—	2	1	no	—
155	C₂×Q₈×C₄₇	—	2	2	no	—
156	C₂×Q₈×C₄₇	—	4	1	no	—
157	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
159	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
160	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
161	C₂×D₄×C₄₇	—	2	2	no	—
162	C₂×D₄×C₄₇	—	2	2	no	—
163	C₂×D₄×C₄₇	—	2	1	no	—
164	C₂×D₄×C₄₇	—	2	2	no	—
165	C₂×D₄×C₄₇	—	2	1	no	—
166	C₂×D₄×C₄₇	—	2	1	no	—
167	C₂×D₄×C₄₇	—	2	2	no	—
168	C₂×D₄×C₄₇	—	2	2	no	—
169	C₂×D₄×C₄₇	—	2	2	no	—
170	C₂×D₄×C₄₇	—	4	2	no	—
171	C₂×D₄×C₄₇	—	4	4	no	—
172	C₂×D₄×C₄₇	—	4	2	no	—
173	C₂×D₄×C₄₇	—	4	1	no	—
174	C₂×D₄×C₄₇	—	4	1	no	—
175	C₂×D₄×C₄₇	—	4	1	no	—
176	C₂×D₄×C₄₇	—	8	2	no	—
177	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₄₇	—	2	2	no	—
180	C₂²×C₄×C₄₇	—	2	1	no	—
181	C₂²×C₄×C₄₇	—	2	1	no	—
182	C₂²×C₄×C₄₇	—	2	2	no	—
183	C₂²×C₄×C₄₇	—	2	2	no	—
184	C₂²×C₄×C₄₇	—	2	1	no	—
185	C₂²×C₄×C₄₇	—	2	2	no	—
186	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
189	SD₁₆×C₄₇	—	4	8	no	—
190	SD₁₆×C₄₇	—	4	4	no	—
191	SD₁₆×C₄₇	—	4	4	no	—
192	SD₁₆×C₄₇	—	8	2	no	—
193	D₈×C₄₇	—	4	8	no	—
194	D₈×C₄₇	—	8	2	no	—
195	M₁₆×C₄₇	—	2	4	no	—
196	C₂×C₈×C₄₇	—	2	2	no	—
197	C₂×C₈×C₄₇	—	2	2	no	—
198	C₂×C₈×C₄₇	—	2	2	no	—
199	C₂×C₈×C₄₇	—	4	2	no	—
200	C₄⋊C₄×C₄₇	—	2	2	no	—
201	C₄⋊C₄×C₄₇	—	2	2	no	—
202	C₄⋊C₄×C₄₇	—	2	2	no	—
203	C₄⋊C₄×C₄₇	—	2	2	no	—
204	C₄⋊C₄×C₄₇	—	2	1	no	—
205	C₄⋊C₄×C₄₇	—	2	1	no	—
206	C₄⋊C₄×C₄₇	—	2	1	no	—
207	C₄⋊C₄×C₄₇	—	2	1	no	—
208	C₄⋊C₄×C₄₇	—	2	1	no	—
209	C₄⋊C₄×C₄₇	—	2	1	no	—
210	C₄⋊C₄×C₄₇	—	2	1	no	—
211	C₄⋊C₄×C₄₇	—	2	1	no	—
212	C₄⋊C₄×C₄₇	—	2	1	no	—
213	C₄⋊C₄×C₄₇	—	2	1	no	—
214	C₄⋊C₄×C₄₇	—	2	1	no	—
215	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
216	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
217	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
218	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
219	C₂²⋊C₄×C₄₇	—	2	2	no	—
220	C₂²⋊C₄×C₄₇	—	2	2	no	—
221	C₂²⋊C₄×C₄₇	—	2	2	no	—
222	C₂²⋊C₄×C₄₇	—	2	2	no	—
223	C₂²⋊C₄×C₄₇	—	2	2	no	—
224	C₂²⋊C₄×C₄₇	—	2	1	no	—
225	C₂²⋊C₄×C₄₇	—	2	2	no	—
226	C₂²⋊C₄×C₄₇	—	2	1	no	—
227	C₂²⋊C₄×C₄₇	—	2	1	no	—
228	C₂²⋊C₄×C₄₇	—	2	2	no	—
229	C₂²⋊C₄×C₄₇	—	2	1	no	—
230	C₂²⋊C₄×C₄₇	—	2	1	no	—
231	C₄²×C₄₇	—	2	2	no	—
232	C₄²×C₄₇	—	2	1	no	—
233	C₄²×C₄₇	—	2	1	no	—
234	C₄²×C₄₇	—	2	1	no	—
235	C₄²×C₄₇	—	2	1	no	—
236	C₄²×C₄₇	—	2	1	no	—
237	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
239	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
240	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
241	C₂³×C₄₇	—	2	1	no	—
242	C₂³×C₄₇	—	2	2	no	—
243	C₂³×C₄₇	—	4	2	no	—
244	C₂³×C₄₇	—	8	2	no	—
245	C₂³×C₄₇	—	8	2	no	—
246	C₂³×C₄₇	—	8	2	no	—
247	C₂³×C₄₇	#120	1	1	yes	characteristic
248	Q₈×C₄₇	—	2	1	no	—
249	Q₈×C₄₇	—	2	1	no	—
250	Q₈×C₄₇	—	2	1	no	—
251	Q₈×C₄₇	—	2	2	no	—
252	Q₈×C₄₇	—	4	2	no	—
253	Q₈×C₄₇	—	4	2	no	—
254	Q₈×C₄₇	—	8	2	no	—
255	Q₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	complemented
256	D₄×C₄₇	—	2	2	no	—
257	D₄×C₄₇	—	4	2	no	—
258	D₄×C₄₇	—	4	4	no	—
259	D₄×C₄₇	—	4	4	no	—
260	D₄×C₄₇	—	4	4	no	—
261	D₄×C₄₇	—	4	2	no	—
262	D₄×C₄₇	—	4	1	no	—
263	D₄×C₄₇	—	4	2	no	—
264	D₄×C₄₇	—	4	1	no	—
265	D₄×C₄₇	—	4	2	no	—
266	D₄×C₄₇	—	4	2	no	—
267	D₄×C₄₇	—	4	2	no	—
268	D₄×C₄₇	—	4	2	no	—
269	D₄×C₄₇	—	8	2	no	—
270	D₄×C₄₇	—	8	2	no	—
271	D₄×C₄₇	—	8	2	no	—
272	D₄×C₄₇	—	8	2	no	—
273	C₂×C₄×C₄₇	—	2	1	no	—
274	C₂×C₄×C₄₇	—	2	2	no	—
275	C₂×C₄×C₄₇	—	2	1	no	—
276	C₂×C₄×C₄₇	—	2	1	no	—
277	C₂×C₄×C₄₇	—	2	2	no	—
278	C₂×C₄×C₄₇	—	2	1	no	—
279	C₂×C₄×C₄₇	—	2	1	no	—
280	C₂×C₄×C₄₇	—	2	1	no	—
281	C₂×C₄×C₄₇	—	2	1	no	—
282	C₂×C₄×C₄₇	—	2	2	no	—
283	C₂×C₄×C₄₇	—	2	1	no	—
284	C₂×C₄×C₄₇	—	2	1	no	—
285	C₂×C₄×C₄₇	—	4	2	no	—
286	C₂×C₄×C₄₇	—	4	4	no	—
287	C₂×C₄×C₄₇	—	4	2	no	—
288	C₂×C₄×C₄₇	—	4	1	no	—
289	C₂×C₄×C₄₇	—	4	1	no	—
290	C₂×C₄×C₄₇	—	8	2	no	—
291	C₂×C₄×C₄₇	—	8	2	no	—
292	C₂×C₄×C₄₇	—	8	2	no	—
293	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
294	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
295	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
296	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
299	C₈×C₄₇	—	2	4	no	—
300	C₈×C₄₇	—	4	2	no	—
301	C₈×C₄₇	—	4	2	no	—
302	#18564	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
303	C₂²×C₄₇	—	2	1	no	—
304	C₂²×C₄₇	—	2	1	no	—
305	C₂²×C₄₇	—	4	2	no	—
306	C₂²×C₄₇	—	4	1	no	—
307	C₂²×C₄₇	—	4	1	no	—
308	C₂²×C₄₇	—	4	2	no	—
309	C₂²×C₄₇	—	4	2	no	—
310	C₂²×C₄₇	—	4	1	no	—
311	C₂²×C₄₇	—	8	2	no	—
312	C₂²×C₄₇	—	8	2	no	—
313	C₂²×C₄₇	—	8	2	no	—
314	C₂²×C₄₇	—	16	4	no	—
315	C₂²×C₄₇	#453	1	2	yes	—
316	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
317	C₄×C₄₇	—	2	1	no	—
318	C₄×C₄₇	—	2	2	no	—
319	C₄×C₄₇	—	2	1	no	—
320	C₄×C₄₇	—	2	1	no	—
321	C₄×C₄₇	—	2	1	no	—
322	C₄×C₄₇	—	4	1	no	—
323	C₄×C₄₇	—	4	1	no	—
324	C₄×C₄₇	—	4	1	no	—
325	C₄×C₄₇	—	4	1	no	—
326	C₄×C₄₇	—	4	1	no	—
327	C₄×C₄₇	—	4	2	no	—
328	C₄×C₄₇	—	4	1	no	—
329	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₄×C₄₇	#575	1	1	yes	characteristic
331	C₄×C₄₇	#577	1	2	yes	—
332	#3522	C₂×C₄₇	1	1	yes	characteristic, complemented
333	#3217	C₂×C₄₇	1	1	yes	characteristic, complemented
334	#3151	C₂×C₄₇	1	1	yes	characteristic, complemented
335	C₂×#460	C₂×C₄₇	1	1	yes	characteristic, complemented
336	#1623	C₂×C₄₇	1	2	yes	complemented
337	#1622	C₂×C₄₇	1	2	yes	complemented
338	#1621	C₂×C₄₇	1	2	yes	complemented
339	#1620	C₂×C₄₇	1	2	yes	complemented
340	#1551	C₂×C₄₇	1	1	yes	characteristic, complemented
341	#1491	C₂×C₄₇	1	1	yes	characteristic, complemented
342	#1483	C₂×C₄₇	1	1	yes	characteristic, complemented
343	C₂×C₄₇	—	2	2	no	—
344	C₂×C₄₇	—	8	1	no	—
345	C₂×C₄₇	—	8	2	no	—
346	C₂×C₄₇	—	16	2	no	—
347	C₂×C₄₇	#3006	1	1	yes	characteristic
348	C₂×C₄₇	#3010	1	1	yes	characteristic
349	C₂×C₄₇	C₂×#453	1	1	yes	characteristic
350	C₂×AGL(1,ℤ/8)	—	2	2	no	—
351	C₂²×SD₁₆	—	2	1	no	—
352	#559	—	2	1	no	—
353	#559	C₂²×C₄₇	1	1	yes	characteristic, complemented
354	Q₈²	C₂²×C₄₇	1	1	yes	characteristic, complemented
355	#554	—	2	1	no	—
356	#554	C₂²×C₄₇	1	1	yes	characteristic, complemented
357	#550	C₂²×C₄₇	1	1	yes	characteristic, complemented
358	#548	—	2	1	no	—
359	#547	—	2	2	no	—
360	#547	—	2	1	no	—
361	#542	—	2	2	no	—
362	#542	—	2	1	no	—
363	#532	—	2	1	no	—
364	#532	C₂²×C₄₇	1	1	yes	characteristic
365	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
366	#529	—	2	1	no	—
367	#529	C₂²×C₄₇	1	1	yes	characteristic
368	#520	—	2	1	no	—
369	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
370	#496	—	2	2	no	—
371	#496	—	2	1	no	—
372	#493	C₂²×C₄₇	1	2	yes	—
373	#492	C₂²×C₄₇	1	2	yes	—
374	#490	—	2	1	no	—
375	#460	—	2	1	no	—
376	#460	—	2	1	no	—
377	#460	C₂²×C₄₇	1	2	yes	—
378	#460	C₂²×C₄₇	1	2	yes	—
379	#460	C₂²×C₄₇	1	2	yes	—
380	#459	C₂²×C₄₇	1	2	yes	—
381	#450	—	2	1	no	—
382	#447	—	2	1	no	—
383	#432	C₂²×C₄₇	1	1	yes	characteristic
384	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
385	C₂×#102	—	2	1	no	—
386	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
387	#333	C₂²×C₄₇	1	2	yes	complemented
388	#331	C₂²×C₄₇	1	2	yes	complemented
389	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
390	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
391	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
392	D₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
393	C₄₇	#18564	1	1	yes	characteristic, direct product factor, Sylow
394	C₂×#41	—	2	2	no	—
395	C₂×#41	—	2	1	no	—
396	C₂×#41	D₄×C₄₇	1	2	yes	—
397	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
398	C₂²×D₄	—	2	1	no	—
399	C₂²×D₄	—	2	1	no	—
400	C₂²×D₄	—	2	1	no	—
401	AGL(1,ℤ/8)	—	4	8	no	—
402	C₂×SD₁₆	—	2	2	no	—
403	C₂×SD₁₆	—	2	2	no	—
404	C₂×SD₁₆	—	2	4	no	—
405	C₂×SD₁₆	—	2	2	no	—
406	C₂×SD₁₆	—	4	4	no	—
407	C₂×D₈	—	2	4	no	—
408	C₂×D₈	—	2	2	no	—
409	C₂×M₁₆	—	2	1	no	—
410	C₂²×C₈	—	2	1	no	—
411	C₄⋊Q₈	—	2	2	no	—
412	C₄⋊Q₈	—	2	1	no	—
413	C₄⋊Q₈	—	2	1	no	—
414	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
415	#127	—	2	1	no	—
416	#127	—	2	1	no	—
417	#127	—	2	1	no	—
418	#127	—	2	1	no	—
419	#127	C₂³×C₄₇	1	2	yes	—
420	#127	C₂³×C₄₇	1	1	yes	characteristic
421	#126	—	2	2	no	—
422	#126	—	2	1	no	—
423	#125	—	2	1	no	—
424	#125	D₄×C₄₇	1	1	yes	characteristic
425	#124	—	2	2	no	—
426	#124	—	2	2	no	—
427	#124	—	2	1	no	—
428	#124	—	2	1	no	—
429	#124	D₄×C₄₇	1	1	yes	characteristic
430	#124	D₄×C₄₇	1	2	yes	—
431	#123	—	2	2	no	—
432	#123	—	2	2	no	—
433	#123	—	2	2	no	—
434	#123	—	2	2	no	—
435	#123	—	2	1	no	—
436	C₂²⋊Q₈	—	2	2	no	—
437	C₂²⋊Q₈	—	2	2	no	—
438	C₂²⋊Q₈	—	2	2	no	—
439	C₂²⋊Q₈	—	2	2	no	—
440	C₂²⋊Q₈	—	2	1	no	—
441	C₂²⋊Q₈	D₄×C₄₇	1	1	yes	characteristic
442	#121	—	2	2	no	—
443	#121	—	2	2	no	—
444	#121	—	2	1	no	—
445	#121	—	2	1	no	—
446	#121	—	2	1	no	—
447	#121	D₄×C₄₇	1	1	yes	characteristic
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
453	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
454	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
455	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
456	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	2	no	—
460	C₄×D₄	—	2	1	no	—
461	C₄×D₄	—	2	1	no	—
462	C₄×D₄	—	2	1	no	—
463	C₄×D₄	—	2	1	no	—
464	C₄×D₄	—	2	1	no	—
465	C₄×D₄	—	2	1	no	—
466	C₄×D₄	—	2	1	no	—
467	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
468	#117	—	2	2	no	—
469	#117	—	2	1	no	—
470	C₂×C₄⋊C₄	—	2	2	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄⋊C₄	—	2	1	no	—
473	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
474	C₂×C₄²	—	2	1	no	—
475	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
476	C₄⋊C₈	—	2	1	no	—
477	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
478	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
479	#102	—	2	2	no	—
480	#102	—	2	2	no	—
481	#102	—	2	2	no	—
482	#102	—	2	2	no	—
483	C₂²⋊C₈	—	2	1	no	—
484	#97	—	2	1	no	—
485	#97	—	2	1	no	—
486	C₄×C₈	C₂³×C₄₇	1	2	yes	—
487	C₂⁴	—	4	1	no	—
488	#41	—	4	4	no	—
489	#41	—	4	2	no	—
490	#41	—	4	2	no	—
491	#41	—	8	4	no	—
492	C₂×Q₈	—	2	2	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	—	2	2	no	—
495	C₂×Q₈	—	4	1	no	—
496	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
497	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
498	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
499	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	2	no	—
508	C₂×D₄	—	2	2	no	—
509	C₂×D₄	—	4	2	no	—
510	C₂×D₄	—	4	4	no	—
511	C₂×D₄	—	4	2	no	—
512	C₂×D₄	—	4	1	no	—
513	C₂×D₄	—	4	1	no	—
514	C₂×D₄	—	4	1	no	—
515	C₂×D₄	—	8	2	no	—
516	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
517	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
526	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
527	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
528	SD₁₆	—	4	8	no	—
529	SD₁₆	—	4	4	no	—
530	SD₁₆	—	4	4	no	—
531	SD₁₆	—	8	2	no	—
532	D₈	—	4	8	no	—
533	D₈	—	8	2	no	—
534	M₁₆	—	2	4	no	—
535	C₂×C₈	—	2	2	no	—
536	C₂×C₈	—	2	2	no	—
537	C₂×C₈	—	2	2	no	—
538	C₂×C₈	—	4	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
555	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
556	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
557	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
558	C₂²⋊C₄	—	2	2	no	—
559	C₂²⋊C₄	—	2	2	no	—
560	C₂²⋊C₄	—	2	2	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	2	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	2	no	—
565	C₂²⋊C₄	—	2	1	no	—
566	C₂²⋊C₄	—	2	1	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	1	no	—
569	C₂²⋊C₄	—	2	1	no	—
570	C₄²	—	2	2	no	—
571	C₄²	—	2	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	2	1	no	—
574	C₄²	—	2	1	no	—
575	C₄²	—	2	1	no	—
576	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
577	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
578	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
579	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
580	C₂³	—	2	1	no	—
581	C₂³	—	2	2	no	—
582	C₂³	—	4	2	no	—
583	C₂³	—	8	2	no	—
584	C₂³	—	8	2	no	—
585	C₂³	—	8	2	no	—
586	C₂³	#120×C₄₇	1	1	yes	characteristic
587	Q₈	—	2	1	no	—
588	Q₈	—	2	1	no	—
589	Q₈	—	2	1	no	—
590	Q₈	—	2	2	no	—
591	Q₈	—	4	2	no	—
592	Q₈	—	4	2	no	—
593	Q₈	—	8	2	no	—
594	Q₈	AGL(1,ℤ/8)×C₄₇	1	2	yes	complemented
595	D₄	—	2	2	no	—
596	D₄	—	4	2	no	—
597	D₄	—	4	4	no	—
598	D₄	—	4	4	no	—
599	D₄	—	4	4	no	—
600	D₄	—	4	2	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	2	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	2	no	—
605	D₄	—	4	2	no	—
606	D₄	—	4	2	no	—
607	D₄	—	4	2	no	—
608	D₄	—	8	2	no	—
609	D₄	—	8	2	no	—
610	D₄	—	8	2	no	—
611	D₄	—	8	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	4	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	8	2	no	—
630	C₂×C₄	—	8	2	no	—
631	C₂×C₄	—	8	2	no	—
632	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
633	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
634	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
635	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
636	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
637	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
638	C₈	—	2	4	no	—
639	C₈	—	4	2	no	—
640	C₈	—	4	2	no	—
641	C₂²	—	2	1	no	—
642	C₂²	—	2	1	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	4	2	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	8	2	no	—
650	C₂²	—	8	2	no	—
651	C₂²	—	8	2	no	—
652	C₂²	—	16	4	no	—
653	C₂²	C₄₇×#453	1	2	yes	—
654	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
655	C₄	—	2	1	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	1	no	—
658	C₄	—	2	1	no	—
659	C₄	—	2	1	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	1	no	—
667	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
668	C₄	C₄₇×#575	1	1	yes	characteristic
669	C₄	C₄₇×#577	1	2	yes	—
670	C₂	—	2	2	no	—
671	C₂	—	8	1	no	—
672	C₂	—	8	2	no	—
673	C₂	—	16	2	no	—
674	C₂	C₄₇×#3006	1	1	yes	characteristic
675	C₂	C₄₇×#3010	1	1	yes	characteristic
676	C₂	C₂×C₄₇×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₇×#3522	C₂	semidirect
2. C₄₇×#3217	C₂	semidirect
3. C₄₇×#3151	C₂	semidirect
4. C₂×C₄₇×#460	C₂	semidirect
5. C₄₇×#1623	C₂	semidirect
6. C₄₇×#1622	C₂	semidirect
7. C₄₇×#1621	C₂	semidirect
8. C₄₇×#1620	C₂	semidirect
9. C₄₇×#1551	C₂	semidirect
10. C₄₇×#1491	C₂	semidirect
11. C₄₇×#1483	C₂	semidirect
15. C₄₇×#559	C₂²	semidirect
16. Q₈²×C₄₇	C₂²	semidirect
18. C₄₇×#554	C₂²	semidirect
19. C₄₇×#550	C₂²	semidirect
49. C₄₇×#333	C₂²	semidirect
50. C₄₇×#331	C₂²	semidirect
51. C₄₇×Q₈⋊C₈	C₂²	semidirect
52. C₄₇×Q₈⋊C₈	C₂²	semidirect
53. C₄₇×Q₈⋊C₈	C₂²	semidirect
54. C₄₇×D₄⋊C₈	C₂²	semidirect
255. Q₈×C₄₇	AGL(1,ℤ/8)	semidirect
302. #18564	C₄₇	direct (N × H)
332. #3522	C₂×C₄₇	semidirect
333. #3217	C₂×C₄₇	semidirect
334. #3151	C₂×C₄₇	semidirect
335. C₂×#460	C₂×C₄₇	semidirect
336. #1623	C₂×C₄₇	semidirect
337. #1622	C₂×C₄₇	semidirect
338. #1621	C₂×C₄₇	semidirect
339. #1620	C₂×C₄₇	semidirect
340. #1551	C₂×C₄₇	semidirect
341. #1491	C₂×C₄₇	semidirect
342. #1483	C₂×C₄₇	semidirect
353. #559	C₂²×C₄₇	semidirect
354. Q₈²	C₂²×C₄₇	semidirect
356. #554	C₂²×C₄₇	semidirect
357. #550	C₂²×C₄₇	semidirect
387. #333	C₂²×C₄₇	semidirect
388. #331	C₂²×C₄₇	semidirect
389. Q₈⋊C₈	C₂²×C₄₇	semidirect
390. Q₈⋊C₈	C₂²×C₄₇	semidirect
391. Q₈⋊C₈	C₂²×C₄₇	semidirect
392. D₄⋊C₈	C₂²×C₄₇	semidirect
393. C₄₇	#18564	direct (N × H)
594. Q₈	AGL(1,ℤ/8)×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3522	2
C₄₇×#3217	2
C₄₇×#3151	2
C₂×C₄₇×#460	2
C₄₇×#1623	2
C₄₇×#1622	2
C₄₇×#1621	2
C₄₇×#1620	2
C₄₇×#1551	2
C₄₇×#1491	2
C₄₇×#1483	2
#18564	47

Subgroups — C47×#18564 #26058262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#18564 #26058262