Subgroups — C₄₇×#18788 #26058486

Order 12032.

1032 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3038	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1627	C₂	1	2	yes	maximal, complemented
6	C₄₇×#1626	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1623	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1622	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#1487	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1477	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₇	—	2	1	no	—
13	C₂²×SD₁₆×C₄₇	—	2	1	no	—
14	Q₈²×C₄₇	C₂²	1	1	yes	characteristic, complemented
15	C₄₇×#557	—	2	1	no	—
16	C₄₇×#557	C₂²	1	1	yes	characteristic, complemented
17	C₄₇×#554	—	2	2	no	—
18	C₄₇×#554	—	2	1	no	—
19	C₄₇×#554	—	2	1	no	—
20	C₄₇×#554	C₂²	1	1	yes	characteristic, complemented
21	D₄×Q₈×C₄₇	—	2	1	no	—
22	D₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
23	C₄₇×#533	—	2	2	no	—
24	C₄₇×#533	—	2	1	no	—
25	C₄₇×#533	—	2	1	no	—
26	C₄₇×#533	C₂²	1	1	yes	characteristic
27	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×#41×C₄₇	—	2	1	no	—
29	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
31	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#492	C₂²	1	2	yes	—
33	C₄₇×#486	C₂²	1	2	yes	—
34	C₄₇×#460	—	2	1	no	—
35	C₄₇×#460	—	2	1	no	—
36	C₄₇×#460	—	2	1	no	—
37	C₄₇×#460	—	2	2	no	—
38	C₄₇×#460	C₂²	1	2	yes	—
39	C₄₇×#460	C₂²	1	2	yes	—
40	C₄₇×#460	C₂²	1	2	yes	—
41	C₄₇×#459	—	2	1	no	—
42	C₄₇×#459	C₂²	1	2	yes	—
43	C₄₇×#450	—	2	2	no	—
44	C₄₇×#431	C₂²	1	1	yes	characteristic
45	C₄₇×#423	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
47	C₄₇×#418	—	2	1	no	—
48	C₂×#102×C₄₇	—	2	1	no	—
49	C₄₇×#333	C₂²	1	2	yes	complemented
50	C₄₇×#332	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	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₄₇	—	2	1	no	—
57	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×D₄×C₄₇	—	2	1	no	—
59	C₂²×D₄×C₄₇	—	2	1	no	—
60	C₂²×D₄×C₄₇	—	2	1	no	—
61	AGL(1,ℤ/8)×C₄₇	—	4	4	no	—
62	C₂×SD₁₆×C₄₇	—	2	2	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	—	4	4	no	—
67	C₂×D₈×C₄₇	—	2	2	no	—
68	C₂×M₁₆×C₄₇	—	2	1	no	—
69	C₂²×C₈×C₄₇	—	2	1	no	—
70	C₄⋊Q₈×C₄₇	—	2	2	no	—
71	C₄⋊Q₈×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	—	2	1	no	—
76	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
78	#127×C₄₇	—	2	1	no	—
79	#127×C₄₇	—	2	1	no	—
80	#127×C₄₇	—	2	1	no	—
81	#127×C₄₇	C₂³	1	2	yes	—
82	#125×C₄₇	—	2	2	no	—
83	#125×C₄₇	—	2	2	no	—
84	#125×C₄₇	—	2	2	no	—
85	#125×C₄₇	—	2	1	no	—
86	#125×C₄₇	—	2	1	no	—
87	#125×C₄₇	—	2	1	no	—
88	C₂²⋊Q₈×C₄₇	—	2	2	no	—
89	C₂²⋊Q₈×C₄₇	—	2	2	no	—
90	C₂²⋊Q₈×C₄₇	—	2	2	no	—
91	C₂²⋊Q₈×C₄₇	—	2	2	no	—
92	C₂²⋊Q₈×C₄₇	—	2	2	no	—
93	C₂²⋊Q₈×C₄₇	—	2	1	no	—
94	C₂²⋊Q₈×C₄₇	—	2	2	no	—
95	C₂²⋊Q₈×C₄₇	—	2	1	no	—
96	C₂²⋊Q₈×C₄₇	—	2	1	no	—
97	C₂²⋊Q₈×C₄₇	—	2	2	no	—
98	C₂²⋊Q₈×C₄₇	D₄	1	2	yes	—
99	C₄×Q₈×C₄₇	—	2	2	no	—
100	C₄×Q₈×C₄₇	—	2	2	no	—
101	C₄×Q₈×C₄₇	—	2	2	no	—
102	C₄×Q₈×C₄₇	—	2	1	no	—
103	C₄×Q₈×C₄₇	—	2	1	no	—
104	C₄×Q₈×C₄₇	—	2	1	no	—
105	C₄×Q₈×C₄₇	—	2	1	no	—
106	C₄×Q₈×C₄₇	—	2	1	no	—
107	C₄×Q₈×C₄₇	—	2	1	no	—
108	C₄×Q₈×C₄₇	—	2	1	no	—
109	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
110	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
111	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
112	C₄×Q₈×C₄₇	D₄	1	2	yes	—
113	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
114	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
115	C₄×Q₈×C₄₇	C₂³	1	1	yes	characteristic
116	C₄×D₄×C₄₇	—	2	2	no	—
117	C₄×D₄×C₄₇	—	2	2	no	—
118	C₄×D₄×C₄₇	—	2	1	no	—
119	C₄×D₄×C₄₇	—	2	1	no	—
120	C₄×D₄×C₄₇	—	2	1	no	—
121	C₄×D₄×C₄₇	—	2	1	no	—
122	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
123	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
124	#117×C₄₇	—	2	2	no	—
125	#117×C₄₇	—	2	1	no	—
126	#117×C₄₇	—	2	1	no	—
127	#117×C₄₇	D₄	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
129	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
130	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
131	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
132	C₂×C₄²×C₄₇	—	2	2	no	—
133	C₂×C₄²×C₄₇	—	2	1	no	—
134	C₂×C₄²×C₄₇	—	2	1	no	—
135	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
136	C₄⋊C₈×C₄₇	—	2	1	no	—
137	C₄⋊C₈×C₄₇	—	2	1	no	—
138	C₄⋊C₈×C₄₇	—	2	1	no	—
139	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
140	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
141	#102×C₄₇	—	2	4	no	—
142	#102×C₄₇	—	2	2	no	—
143	#102×C₄₇	—	2	2	no	—
144	#102×C₄₇	—	2	2	no	—
145	#102×C₄₇	—	2	2	no	—
146	C₂²⋊C₈×C₄₇	—	2	1	no	—
147	C₄×C₈×C₄₇	C₂³	1	2	yes	—
148	C₂⁴×C₄₇	—	4	1	no	—
149	#41×C₄₇	—	2	2	no	—
150	#41×C₄₇	SD₁₆	1	4	yes	complemented
151	C₂×Q₈×C₄₇	—	2	2	no	—
152	C₂×Q₈×C₄₇	—	2	2	no	—
153	C₂×Q₈×C₄₇	—	2	1	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	—	4	2	no	—
158	C₂×Q₈×C₄₇	—	4	2	no	—
159	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
160	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
161	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
162	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
163	C₂×D₄×C₄₇	—	2	1	no	—
164	C₂×D₄×C₄₇	—	2	2	no	—
165	C₂×D₄×C₄₇	—	2	2	no	—
166	C₂×D₄×C₄₇	—	2	2	no	—
167	C₂×D₄×C₄₇	—	4	2	no	—
168	C₂×D₄×C₄₇	—	4	2	no	—
169	C₂×D₄×C₄₇	—	4	1	no	—
170	C₂×D₄×C₄₇	—	4	1	no	—
171	C₂×D₄×C₄₇	—	4	1	no	—
172	C₂×D₄×C₄₇	—	4	2	no	—
173	C₂×D₄×C₄₇	—	4	2	no	—
174	C₂×D₄×C₄₇	—	8	2	no	—
175	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₄₇	—	2	1	no	—
177	C₂²×C₄×C₄₇	—	2	2	no	—
178	C₂²×C₄×C₄₇	—	2	1	no	—
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	1	no	—
183	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
186	SD₁₆×C₄₇	—	4	4	no	—
187	SD₁₆×C₄₇	—	4	4	no	—
188	SD₁₆×C₄₇	—	4	4	no	—
189	SD₁₆×C₄₇	—	8	4	no	—
190	D₈×C₄₇	—	4	4	no	—
191	M₁₆×C₄₇	—	4	2	no	—
192	C₂×C₈×C₄₇	—	2	2	no	—
193	C₂×C₈×C₄₇	—	2	2	no	—
194	C₂×C₈×C₄₇	—	2	2	no	—
195	C₂×C₈×C₄₇	—	4	2	no	—
196	C₄⋊C₄×C₄₇	—	2	2	no	—
197	C₄⋊C₄×C₄₇	—	2	2	no	—
198	C₄⋊C₄×C₄₇	—	2	2	no	—
199	C₄⋊C₄×C₄₇	—	2	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	2	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	2	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	—	2	1	no	—
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	C₄⋊C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
220	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
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	1	no	—
226	C₂²⋊C₄×C₄₇	—	2	1	no	—
227	C₄²×C₄₇	—	2	2	no	—
228	C₄²×C₄₇	—	2	2	no	—
229	C₄²×C₄₇	—	2	1	no	—
230	C₄²×C₄₇	—	2	2	no	—
231	C₄²×C₄₇	—	2	1	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	—	4	2	no	—
243	C₂³×C₄₇	—	8	2	no	—
244	C₂³×C₄₇	—	8	2	no	—
245	C₂³×C₄₇	—	8	2	no	—
246	C₂³×C₄₇	#120	1	1	yes	characteristic
247	Q₈×C₄₇	—	2	1	no	—
248	Q₈×C₄₇	—	2	1	no	—
249	Q₈×C₄₇	—	2	1	no	—
250	Q₈×C₄₇	—	2	1	no	—
251	Q₈×C₄₇	—	4	2	no	—
252	Q₈×C₄₇	—	4	2	no	—
253	Q₈×C₄₇	—	4	4	no	—
254	Q₈×C₄₇	—	4	1	no	—
255	Q₈×C₄₇	—	4	1	no	—
256	Q₈×C₄₇	—	8	2	no	—
257	Q₈×C₄₇	C₂×SD₁₆	1	2	yes	complemented
258	Q₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	complemented
259	D₄×C₄₇	—	2	4	no	—
260	D₄×C₄₇	—	2	1	no	—
261	D₄×C₄₇	—	4	4	no	—
262	D₄×C₄₇	—	4	2	no	—
263	D₄×C₄₇	—	4	2	no	—
264	D₄×C₄₇	—	4	2	no	—
265	D₄×C₄₇	—	4	2	no	—
266	D₄×C₄₇	—	8	2	no	—
267	D₄×C₄₇	—	8	2	no	—
268	D₄×C₄₇	—	8	2	no	—
269	D₄×C₄₇	C₂×SD₁₆	1	2	yes	complemented
270	C₂×C₄×C₄₇	—	2	1	no	—
271	C₂×C₄×C₄₇	—	2	1	no	—
272	C₂×C₄×C₄₇	—	2	1	no	—
273	C₂×C₄×C₄₇	—	2	1	no	—
274	C₂×C₄×C₄₇	—	2	4	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	2	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	—	4	1	no	—
283	C₂×C₄×C₄₇	—	4	2	no	—
284	C₂×C₄×C₄₇	—	4	4	no	—
285	C₂×C₄×C₄₇	—	4	1	no	—
286	C₂×C₄×C₄₇	—	4	1	no	—
287	C₂×C₄×C₄₇	—	8	1	no	—
288	C₂×C₄×C₄₇	—	8	1	no	—
289	C₂×C₄×C₄₇	—	8	2	no	—
290	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
291	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
292	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
293	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
294	C₂×C₄×C₄₇	#137	1	2	yes	—
295	C₂×C₄×C₄₇	C₂²×D₄	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	C₈×C₄₇	—	4	2	no	—
299	C₈×C₄₇	—	4	2	no	—
300	C₈×C₄₇	—	4	2	no	—
301	#18788	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
302	C₂²×C₄₇	—	2	1	no	—
303	C₂²×C₄₇	—	2	1	no	—
304	C₂²×C₄₇	—	2	2	no	—
305	C₂²×C₄₇	—	4	1	no	—
306	C₂²×C₄₇	—	4	1	no	—
307	C₂²×C₄₇	—	8	2	no	—
308	C₂²×C₄₇	—	8	2	no	—
309	C₂²×C₄₇	—	8	2	no	—
310	C₂²×C₄₇	—	16	4	no	—
311	C₂²×C₄₇	#448	1	2	yes	—
312	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
313	C₄×C₄₇	—	2	2	no	—
314	C₄×C₄₇	—	2	1	no	—
315	C₄×C₄₇	—	2	2	no	—
316	C₄×C₄₇	—	2	1	no	—
317	C₄×C₄₇	—	2	1	no	—
318	C₄×C₄₇	—	4	1	no	—
319	C₄×C₄₇	—	4	2	no	—
320	C₄×C₄₇	—	4	1	no	—
321	C₄×C₄₇	—	4	1	no	—
322	C₄×C₄₇	—	4	2	no	—
323	C₄×C₄₇	—	4	1	no	—
324	C₄×C₄₇	—	4	1	no	—
325	C₄×C₄₇	#448	1	2	yes	—
326	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
327	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
328	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
329	C₄×C₄₇	#575	1	1	yes	characteristic
330	Q₈×#41	C₂×C₄₇	1	1	yes	characteristic, complemented
331	#3208	C₂×C₄₇	1	1	yes	characteristic, complemented
332	#3038	C₂×C₄₇	1	1	yes	characteristic
333	C₂×#460	C₂×C₄₇	1	1	yes	characteristic, complemented
334	#1627	C₂×C₄₇	1	2	yes	complemented
335	#1626	C₂×C₄₇	1	2	yes	complemented
336	#1623	C₂×C₄₇	1	2	yes	complemented
337	#1622	C₂×C₄₇	1	2	yes	complemented
338	#1550	C₂×C₄₇	1	1	yes	characteristic, complemented
339	#1487	C₂×C₄₇	1	1	yes	characteristic, complemented
340	#1477	C₂×C₄₇	1	1	yes	characteristic, complemented
341	C₂×C₄₇	—	2	2	no	—
342	C₂×C₄₇	—	4	2	no	—
343	C₂×C₄₇	—	16	2	no	—
344	C₂×C₄₇	C₂×#448	1	1	yes	characteristic
345	C₂×C₄₇	#3006	1	1	yes	characteristic
346	C₂×C₄₇	#3021	1	1	yes	characteristic
347	C₂×AGL(1,ℤ/8)	—	2	1	no	—
348	C₂²×SD₁₆	—	2	1	no	—
349	Q₈²	C₂²×C₄₇	1	1	yes	characteristic, complemented
350	#557	—	2	1	no	—
351	#557	C₂²×C₄₇	1	1	yes	characteristic, complemented
352	#554	—	2	2	no	—
353	#554	—	2	1	no	—
354	#554	—	2	1	no	—
355	#554	C₂²×C₄₇	1	1	yes	characteristic, complemented
356	D₄×Q₈	—	2	1	no	—
357	D₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic, complemented
358	#533	—	2	2	no	—
359	#533	—	2	1	no	—
360	#533	—	2	1	no	—
361	#533	C₂²×C₄₇	1	1	yes	characteristic
362	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
363	C₄×#41	—	2	1	no	—
364	C₄×#41	C₂²×C₄₇	1	1	yes	characteristic
365	C₂×C₄×Q₈	—	2	1	no	—
366	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
367	#492	C₂²×C₄₇	1	2	yes	—
368	#486	C₂²×C₄₇	1	2	yes	—
369	#460	—	2	2	no	—
370	#460	—	2	1	no	—
371	#460	—	2	1	no	—
372	#460	—	2	1	no	—
373	#460	C₂²×C₄₇	1	2	yes	—
374	#460	C₂²×C₄₇	1	2	yes	—
375	#460	C₂²×C₄₇	1	2	yes	—
376	#459	—	2	1	no	—
377	#459	C₂²×C₄₇	1	2	yes	—
378	#450	—	2	2	no	—
379	#431	C₂²×C₄₇	1	1	yes	characteristic
380	#423	C₂²×C₄₇	1	1	yes	characteristic
381	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
382	#418	—	2	1	no	—
383	C₂×#102	—	2	1	no	—
384	#333	C₂²×C₄₇	1	2	yes	complemented
385	#332	C₂²×C₄₇	1	2	yes	complemented
386	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
387	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
388	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
389	D₄⋊C₈	C₂²×C₄₇	1	2	yes	complemented
390	C₄₇	#18788	1	1	yes	characteristic, direct product factor, Sylow
391	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
392	C₂²×Q₈	—	2	1	no	—
393	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
394	C₂²×D₄	—	2	1	no	—
395	C₂²×D₄	—	2	1	no	—
396	C₂²×D₄	—	2	1	no	—
397	AGL(1,ℤ/8)	—	4	4	no	—
398	C₂×SD₁₆	—	2	2	no	—
399	C₂×SD₁₆	—	2	2	no	—
400	C₂×SD₁₆	—	2	2	no	—
401	C₂×SD₁₆	—	2	4	no	—
402	C₂×SD₁₆	—	4	4	no	—
403	C₂×D₈	—	2	2	no	—
404	C₂×M₁₆	—	2	1	no	—
405	C₂²×C₈	—	2	1	no	—
406	C₄⋊Q₈	—	2	2	no	—
407	C₄⋊Q₈	—	2	1	no	—
408	C₄⋊Q₈	—	2	2	no	—
409	C₄⋊Q₈	—	2	1	no	—
410	C₄⋊Q₈	—	2	1	no	—
411	C₄⋊Q₈	—	2	1	no	—
412	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
413	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
414	#127	—	2	1	no	—
415	#127	—	2	1	no	—
416	#127	—	2	1	no	—
417	#127	C₂³×C₄₇	1	2	yes	—
418	#125	—	2	2	no	—
419	#125	—	2	2	no	—
420	#125	—	2	2	no	—
421	#125	—	2	1	no	—
422	#125	—	2	1	no	—
423	#125	—	2	1	no	—
424	C₂²⋊Q₈	—	2	2	no	—
425	C₂²⋊Q₈	—	2	2	no	—
426	C₂²⋊Q₈	—	2	2	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	1	no	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	1	no	—
432	C₂²⋊Q₈	—	2	1	no	—
433	C₂²⋊Q₈	—	2	2	no	—
434	C₂²⋊Q₈	D₄×C₄₇	1	2	yes	—
435	C₄×Q₈	—	2	2	no	—
436	C₄×Q₈	—	2	2	no	—
437	C₄×Q₈	—	2	2	no	—
438	C₄×Q₈	—	2	1	no	—
439	C₄×Q₈	—	2	1	no	—
440	C₄×Q₈	—	2	1	no	—
441	C₄×Q₈	—	2	1	no	—
442	C₄×Q₈	—	2	1	no	—
443	C₄×Q₈	—	2	1	no	—
444	C₄×Q₈	—	2	1	no	—
445	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
446	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
447	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
448	C₄×Q₈	D₄×C₄₇	1	2	yes	—
449	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
450	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
451	C₄×Q₈	C₂³×C₄₇	1	1	yes	characteristic
452	C₄×D₄	—	2	2	no	—
453	C₄×D₄	—	2	2	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	D₄×C₄₇	1	1	yes	characteristic
459	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
460	#117	—	2	2	no	—
461	#117	—	2	1	no	—
462	#117	—	2	1	no	—
463	#117	D₄×C₄₇	1	1	yes	characteristic
464	C₂×C₄⋊C₄	—	2	2	no	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	—	2	1	no	—
467	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
468	C₂×C₄²	—	2	2	no	—
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	—	2	1	no	—
471	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
472	C₄⋊C₈	—	2	1	no	—
473	C₄⋊C₈	—	2	1	no	—
474	C₄⋊C₈	—	2	1	no	—
475	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
476	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
477	#102	—	2	2	no	—
478	#102	—	2	2	no	—
479	#102	—	2	2	no	—
480	#102	—	2	2	no	—
481	#102	—	2	4	no	—
482	C₂²⋊C₈	—	2	1	no	—
483	C₄×C₈	C₂³×C₄₇	1	2	yes	—
484	C₂⁴	—	4	1	no	—
485	#41	—	2	2	no	—
486	#41	SD₁₆×C₄₇	1	4	yes	complemented
487	C₂×Q₈	—	2	2	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	—	4	1	no	—
493	C₂×Q₈	—	4	2	no	—
494	C₂×Q₈	—	4	2	no	—
495	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
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	2	yes	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	4	2	no	—
504	C₂×D₄	—	4	2	no	—
505	C₂×D₄	—	4	1	no	—
506	C₂×D₄	—	4	1	no	—
507	C₂×D₄	—	4	1	no	—
508	C₂×D₄	—	4	2	no	—
509	C₂×D₄	—	4	2	no	—
510	C₂×D₄	—	8	2	no	—
511	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	2	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
520	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
521	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
522	SD₁₆	—	4	4	no	—
523	SD₁₆	—	4	4	no	—
524	SD₁₆	—	4	4	no	—
525	SD₁₆	—	8	4	no	—
526	D₈	—	4	4	no	—
527	M₁₆	—	4	2	no	—
528	C₂×C₈	—	2	2	no	—
529	C₂×C₈	—	2	2	no	—
530	C₂×C₈	—	2	2	no	—
531	C₂×C₈	—	4	2	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	2	no	—
534	C₄⋊C₄	—	2	2	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	2	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	C₂×D₄×C₄₇	1	1	yes	characteristic
553	C₄⋊C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
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	2	yes	—
557	C₂²⋊C₄	—	2	2	no	—
558	C₂²⋊C₄	—	2	2	no	—
559	C₂²⋊C₄	—	2	2	no	—
560	C₂²⋊C₄	—	2	1	no	—
561	C₂²⋊C₄	—	2	1	no	—
562	C₂²⋊C₄	—	2	1	no	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	2	no	—
565	C₄²	—	2	1	no	—
566	C₄²	—	2	2	no	—
567	C₄²	—	2	1	no	—
568	C₄²	—	2	1	no	—
569	C₄²	—	2	1	no	—
570	C₄²	—	2	1	no	—
571	C₄²	—	2	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
574	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
575	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
576	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
577	C₂³	—	2	1	no	—
578	C₂³	—	4	2	no	—
579	C₂³	—	8	2	no	—
580	C₂³	—	8	2	no	—
581	C₂³	—	8	2	no	—
582	C₂³	#120×C₄₇	1	1	yes	characteristic
583	Q₈	—	2	1	no	—
584	Q₈	—	2	1	no	—
585	Q₈	—	2	1	no	—
586	Q₈	—	2	1	no	—
587	Q₈	—	4	2	no	—
588	Q₈	—	4	2	no	—
589	Q₈	—	4	4	no	—
590	Q₈	—	4	1	no	—
591	Q₈	—	4	1	no	—
592	Q₈	—	8	2	no	—
593	Q₈	C₂×SD₁₆×C₄₇	1	2	yes	complemented
594	Q₈	AGL(1,ℤ/8)×C₄₇	1	2	yes	complemented
595	D₄	—	2	4	no	—
596	D₄	—	2	1	no	—
597	D₄	—	4	4	no	—
598	D₄	—	4	2	no	—
599	D₄	—	4	2	no	—
600	D₄	—	4	2	no	—
601	D₄	—	4	2	no	—
602	D₄	—	8	2	no	—
603	D₄	—	8	2	no	—
604	D₄	—	8	2	no	—
605	D₄	C₂×SD₁₆×C₄₇	1	2	yes	complemented
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	4	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	8	1	no	—
624	C₂×C₄	—	8	1	no	—
625	C₂×C₄	—	8	2	no	—
626	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
627	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
628	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
629	C₂×C₄	C₂×SD₁₆×C₄₇	1	2	yes	—
630	C₂×C₄	#137×C₄₇	1	2	yes	—
631	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
632	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
633	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
634	C₈	—	4	2	no	—
635	C₈	—	4	2	no	—
636	C₈	—	4	2	no	—
637	C₂²	—	2	1	no	—
638	C₂²	—	2	1	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	8	2	no	—
643	C₂²	—	8	2	no	—
644	C₂²	—	8	2	no	—
645	C₂²	—	16	4	no	—
646	C₂²	C₄₇×#448	1	2	yes	—
647	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
648	C₄	—	2	2	no	—
649	C₄	—	2	1	no	—
650	C₄	—	2	2	no	—
651	C₄	—	2	1	no	—
652	C₄	—	2	1	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	1	no	—
659	C₄	—	4	1	no	—
660	C₄	C₄₇×#448	1	2	yes	—
661	C₄	C₂²×SD₁₆×C₄₇	1	1	yes	characteristic
662	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
663	C₄	C₂×#137×C₄₇	1	1	yes	characteristic
664	C₄	C₄₇×#575	1	1	yes	characteristic
665	C₂	—	2	2	no	—
666	C₂	—	4	2	no	—
667	C₂	—	16	2	no	—
668	C₂	C₂×C₄₇×#448	1	1	yes	characteristic
669	C₂	C₄₇×#3006	1	1	yes	characteristic
670	C₂	C₄₇×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₄₇×#3038	C₂
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. Q₈×#41×C₄₇	C₂	semidirect
2. C₄₇×#3208	C₂	semidirect
4. C₂×C₄₇×#460	C₂	semidirect
5. C₄₇×#1627	C₂	semidirect
6. C₄₇×#1626	C₂	semidirect
7. C₄₇×#1623	C₂	semidirect
8. C₄₇×#1622	C₂	semidirect
9. C₄₇×#1550	C₂	semidirect
10. C₄₇×#1487	C₂	semidirect
11. C₄₇×#1477	C₂	semidirect
14. Q₈²×C₄₇	C₂²	semidirect
16. C₄₇×#557	C₂²	semidirect
20. C₄₇×#554	C₂²	semidirect
22. D₄×Q₈×C₄₇	C₂²	semidirect
49. C₄₇×#333	C₂²	semidirect
50. C₄₇×#332	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
150. #41×C₄₇	SD₁₆	semidirect
257. Q₈×C₄₇	C₂×SD₁₆	semidirect
258. Q₈×C₄₇	AGL(1,ℤ/8)	semidirect
269. D₄×C₄₇	C₂×SD₁₆	semidirect
301. #18788	C₄₇	direct (N × H)
330. Q₈×#41	C₂×C₄₇	semidirect
331. #3208	C₂×C₄₇	semidirect
333. C₂×#460	C₂×C₄₇	semidirect
334. #1627	C₂×C₄₇	semidirect
335. #1626	C₂×C₄₇	semidirect
336. #1623	C₂×C₄₇	semidirect
337. #1622	C₂×C₄₇	semidirect
338. #1550	C₂×C₄₇	semidirect
339. #1487	C₂×C₄₇	semidirect
340. #1477	C₂×C₄₇	semidirect
349. Q₈²	C₂²×C₄₇	semidirect
351. #557	C₂²×C₄₇	semidirect
355. #554	C₂²×C₄₇	semidirect
357. D₄×Q₈	C₂²×C₄₇	semidirect
384. #333	C₂²×C₄₇	semidirect
385. #332	C₂²×C₄₇	semidirect
386. Q₈⋊C₈	C₂²×C₄₇	semidirect
387. Q₈⋊C₈	C₂²×C₄₇	semidirect
388. Q₈⋊C₈	C₂²×C₄₇	semidirect
389. D₄⋊C₈	C₂²×C₄₇	semidirect
390. C₄₇	#18788	direct (N × H)
486. #41	SD₁₆×C₄₇	semidirect
593. Q₈	C₂×SD₁₆×C₄₇	semidirect
594. Q₈	AGL(1,ℤ/8)×C₄₇	semidirect
605. D₄	C₂×SD₁₆×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₄₇	2
C₄₇×#3208	2
C₄₇×#3038	2
C₂×C₄₇×#460	2
C₄₇×#1627	2
C₄₇×#1626	2
C₄₇×#1623	2
C₄₇×#1622	2
C₄₇×#1550	2
C₄₇×#1487	2
C₄₇×#1477	2
#18788	47

Subgroups — C47×#18788 #26058486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#18788 #26058486