Subgroups — C₄₇×#17508 #26057206

Order 12032.

980 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3475	C₂	1	1	yes	maximal, characteristic
2	C₈×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1494	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#1488	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1482	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#1476	C₂	1	2	yes	maximal, complemented
11	C₂×C₄₇×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#556	C₄	1	1	yes	characteristic
13	C₄₇×#555	C₄	1	1	yes	characteristic
14	C₄₇×#554	C₄	1	1	yes	characteristic
15	C₄₇×#553	—	2	1	no	—
16	C₄₇×#552	—	2	1	no	—
17	C₄₇×#552	C₄	1	2	yes	—
18	C₄₇×#551	—	2	2	no	—
19	C₄₇×#547	C₄	1	1	yes	characteristic
20	C₄₇×#546	C₄	1	2	yes	—
21	C₄₇×#533	C₂²	1	1	yes	characteristic
22	C₄₇×#529	—	2	2	no	—
23	C₄₇×#529	C₂²	1	2	yes	—
24	C₄₇×#525	—	2	1	no	—
25	C₄×#41×C₄₇	C₂²	1	2	yes	—
26	C₄×#41×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₇	—	2	1	no	—
29	C₈×Q₈×C₄₇	—	2	1	no	—
30	C₈×D₄×C₄₇	—	2	1	no	—
31	C₈×D₄×C₄₇	—	2	2	no	—
32	C₄₇×#433	—	2	2	no	—
33	C₄₇×#432	C₂²	1	2	yes	complemented
34	C₄₇×#431	C₂²	1	1	yes	characteristic
35	C₄₇×#431	C₂²	1	2	yes	complemented
36	C₄₇×#431	C₂²	1	2	yes	complemented
37	C₄₇×#423	C₂²	1	1	yes	characteristic
38	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#408	—	2	1	no	—
40	C₄₇×#405	—	2	1	no	—
41	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	complemented
43	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
45	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₄₇×Q₈⋊C₈	C₂²	1	2	yes	complemented
47	C₄₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
49	C₄₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₄₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₄₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
55	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₄₇	—	2	1	no	—
57	C₂×M₁₆×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	—	2	1	no	—
59	C₂²×C₈×C₄₇	—	2	2	no	—
60	C₂²×C₈×C₄₇	—	2	1	no	—
61	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	#126×C₄₇	—	2	2	no	—
64	#126×C₄₇	—	2	2	no	—
65	#126×C₄₇	—	2	2	no	—
66	#125×C₄₇	—	2	2	no	—
67	#125×C₄₇	—	2	1	no	—
68	#125×C₄₇	C₂×C₄	1	2	yes	—
69	#124×C₄₇	—	2	2	no	—
70	#124×C₄₇	C₂×C₄	1	2	yes	—
71	#123×C₄₇	—	2	2	no	—
72	#123×C₄₇	—	2	2	no	—
73	#123×C₄₇	—	2	2	no	—
74	#123×C₄₇	—	2	2	no	—
75	C₂²⋊Q₈×C₄₇	—	2	1	no	—
76	C₂²⋊Q₈×C₄₇	—	2	1	no	—
77	C₂²⋊Q₈×C₄₇	—	2	1	no	—
78	C₂²⋊Q₈×C₄₇	—	2	2	no	—
79	C₂²⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
80	C₂²⋊Q₈×C₄₇	C₂×C₄	1	2	yes	—
81	#121×C₄₇	—	2	1	no	—
82	#121×C₄₇	—	2	1	no	—
83	#121×C₄₇	—	2	2	no	—
84	#121×C₄₇	C₂×C₄	1	2	yes	—
85	#120×C₄₇	—	2	2	no	—
86	C₄×Q₈×C₄₇	—	2	1	no	—
87	C₄×Q₈×C₄₇	C₂×C₄	1	2	yes	—
88	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
91	C₄×D₄×C₄₇	—	2	2	no	—
92	C₄×D₄×C₄₇	—	2	2	no	—
93	C₄×D₄×C₄₇	—	2	2	no	—
94	C₄×D₄×C₄₇	—	2	1	no	—
95	C₄×D₄×C₄₇	—	2	1	no	—
96	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
97	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
98	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
99	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
100	C₄×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
101	C₄×D₄×C₄₇	D₄	1	2	yes	—
102	C₄×D₄×C₄₇	D₄	1	1	yes	characteristic
103	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
104	C₄×D₄×C₄₇	C₂³	1	1	yes	characteristic
105	#117×C₄₇	—	2	1	no	—
106	#117×C₄₇	—	2	1	no	—
107	#117×C₄₇	—	2	1	no	—
108	#117×C₄₇	—	2	2	no	—
109	#117×C₄₇	—	2	2	no	—
110	#117×C₄₇	—	2	2	no	—
111	#117×C₄₇	C₂×C₄	1	2	yes	—
112	#117×C₄₇	D₄	1	1	yes	characteristic
113	#117×C₄₇	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
115	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
116	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
117	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
119	C₂×C₄²×C₄₇	—	2	2	no	—
120	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
122	C₄⋊C₈×C₄₇	—	2	1	no	—
123	C₄⋊C₈×C₄₇	—	2	1	no	—
124	C₄⋊C₈×C₄₇	—	2	1	no	—
125	C₄⋊C₈×C₄₇	—	2	2	no	—
126	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
127	C₄⋊C₈×C₄₇	C₂³	1	1	yes	characteristic
128	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
129	C₂²⋊C₈×C₄₇	—	2	2	no	—
130	C₂²⋊C₈×C₄₇	—	2	2	no	—
131	C₂²⋊C₈×C₄₇	—	2	1	no	—
132	C₂²⋊C₈×C₄₇	—	2	2	no	—
133	#97×C₄₇	—	2	2	no	—
134	C₄×C₈×C₄₇	—	2	1	no	—
135	C₄×C₈×C₄₇	—	2	2	no	—
136	C₄×C₈×C₄₇	—	2	2	no	—
137	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₄₇	C₂³	1	2	yes	—
140	C₂⁴×C₄₇	—	2	1	no	—
141	#41×C₄₇	—	2	2	no	—
142	#41×C₄₇	—	2	2	no	—
143	#41×C₄₇	—	2	4	no	—
144	#41×C₄₇	—	4	2	no	—
145	C₂×Q₈×C₄₇	—	2	1	no	—
146	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₄₇	—	2	2	no	—
149	C₂×D₄×C₄₇	—	2	2	no	—
150	C₂×D₄×C₄₇	—	2	2	no	—
151	C₂×D₄×C₄₇	—	2	1	no	—
152	C₂×D₄×C₄₇	—	4	1	no	—
153	C₂×D₄×C₄₇	C₂²⋊C₄	1	2	yes	—
154	C₂×D₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
155	C₂×D₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
156	C₂×D₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₇	—	2	1	no	—
158	C₂²×C₄×C₄₇	—	2	2	no	—
159	C₂²×C₄×C₄₇	—	2	1	no	—
160	C₂²×C₄×C₄₇	—	2	1	no	—
161	C₂²×C₄×C₄₇	—	2	1	no	—
162	C₂²×C₄×C₄₇	—	2	1	no	—
163	C₂²×C₄×C₄₇	—	2	2	no	—
164	C₂²×C₄×C₄₇	—	4	2	no	—
165	C₂²×C₄×C₄₇	—	4	2	no	—
166	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
172	M₁₆×C₄₇	—	4	2	no	—
173	C₂×C₈×C₄₇	—	2	1	no	—
174	C₂×C₈×C₄₇	—	2	2	no	—
175	C₂×C₈×C₄₇	—	2	1	no	—
176	C₂×C₈×C₄₇	—	2	1	no	—
177	C₂×C₈×C₄₇	—	2	1	no	—
178	C₂×C₈×C₄₇	—	2	2	no	—
179	C₂×C₈×C₄₇	—	4	2	no	—
180	C₂×C₈×C₄₇	—	4	4	no	—
181	C₂×C₈×C₄₇	—	4	2	no	—
182	C₄⋊C₄×C₄₇	—	2	1	no	—
183	C₄⋊C₄×C₄₇	—	2	1	no	—
184	C₄⋊C₄×C₄₇	—	2	2	no	—
185	C₄⋊C₄×C₄₇	—	2	2	no	—
186	C₄⋊C₄×C₄₇	—	2	1	no	—
187	C₄⋊C₄×C₄₇	—	2	2	no	—
188	C₄⋊C₄×C₄₇	—	2	2	no	—
189	C₄⋊C₄×C₄₇	—	2	2	no	—
190	C₄⋊C₄×C₄₇	—	2	1	no	—
191	C₄⋊C₄×C₄₇	—	2	2	no	—
192	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
195	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
196	C₄⋊C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
199	C₂²⋊C₄×C₄₇	—	2	2	no	—
200	C₂²⋊C₄×C₄₇	—	2	2	no	—
201	C₂²⋊C₄×C₄₇	—	2	2	no	—
202	C₂²⋊C₄×C₄₇	—	2	2	no	—
203	C₂²⋊C₄×C₄₇	—	2	2	no	—
204	C₂²⋊C₄×C₄₇	—	2	1	no	—
205	C₂²⋊C₄×C₄₇	—	2	1	no	—
206	C₂²⋊C₄×C₄₇	—	2	2	no	—
207	C₂²⋊C₄×C₄₇	—	2	2	no	—
208	C₂²⋊C₄×C₄₇	—	2	1	no	—
209	C₂²⋊C₄×C₄₇	—	2	1	no	—
210	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
211	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
212	C₂²⋊C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
213	C₄²×C₄₇	—	2	2	no	—
214	C₄²×C₄₇	—	2	2	no	—
215	C₄²×C₄₇	—	2	2	no	—
216	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
217	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
218	C₄²×C₄₇	C₂×D₄	1	2	yes	—
219	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₄₇	C₂×D₄	1	2	yes	—
222	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
223	C₂³×C₄₇	—	2	1	no	—
224	C₂³×C₄₇	—	2	1	no	—
225	C₂³×C₄₇	—	4	2	no	—
226	C₂³×C₄₇	—	4	2	no	—
227	C₂³×C₄₇	—	8	1	no	—
228	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
229	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	Q₈×C₄₇	—	2	2	no	—
231	Q₈×C₄₇	—	2	2	no	—
232	Q₈×C₄₇	—	2	2	no	—
233	Q₈×C₄₇	—	8	1	no	—
234	D₄×C₄₇	—	2	4	no	—
235	D₄×C₄₇	—	2	1	no	—
236	D₄×C₄₇	—	2	1	no	—
237	D₄×C₄₇	—	2	1	no	—
238	D₄×C₄₇	—	2	2	no	—
239	D₄×C₄₇	—	4	2	no	—
240	D₄×C₄₇	—	4	4	no	—
241	D₄×C₄₇	—	8	2	no	—
242	D₄×C₄₇	#131	1	2	yes	—
243	C₂×C₄×C₄₇	—	2	1	no	—
244	C₂×C₄×C₄₇	—	2	2	no	—
245	C₂×C₄×C₄₇	—	2	1	no	—
246	C₂×C₄×C₄₇	—	2	2	no	—
247	C₂×C₄×C₄₇	—	2	2	no	—
248	C₂×C₄×C₄₇	—	2	2	no	—
249	C₂×C₄×C₄₇	—	2	1	no	—
250	C₂×C₄×C₄₇	—	2	2	no	—
251	C₂×C₄×C₄₇	—	2	2	no	—
252	C₂×C₄×C₄₇	—	2	1	no	—
253	C₂×C₄×C₄₇	—	4	2	no	—
254	C₂×C₄×C₄₇	—	4	1	no	—
255	C₂×C₄×C₄₇	—	4	4	no	—
256	C₂×C₄×C₄₇	—	4	2	no	—
257	C₂×C₄×C₄₇	—	4	1	no	—
258	C₂×C₄×C₄₇	—	4	1	no	—
259	C₂×C₄×C₄₇	—	4	1	no	—
260	C₂×C₄×C₄₇	—	4	1	no	—
261	C₂×C₄×C₄₇	—	4	2	no	—
262	C₂×C₄×C₄₇	—	4	2	no	—
263	C₂×C₄×C₄₇	—	8	2	no	—
264	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
268	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₄₇	#135	1	2	yes	—
271	C₂×C₄×C₄₇	#135	1	2	yes	—
272	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
273	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
275	C₈×C₄₇	—	2	2	no	—
276	C₈×C₄₇	—	4	2	no	—
277	C₈×C₄₇	—	4	1	no	—
278	C₈×C₄₇	—	4	1	no	—
279	C₈×C₄₇	—	4	1	no	—
280	C₈×C₄₇	—	4	2	no	—
281	#17508	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
282	C₂²×C₄₇	—	2	1	no	—
283	C₂²×C₄₇	—	2	1	no	—
284	C₂²×C₄₇	—	2	2	no	—
285	C₂²×C₄₇	—	2	2	no	—
286	C₂²×C₄₇	—	2	2	no	—
287	C₂²×C₄₇	—	2	2	no	—
288	C₂²×C₄₇	—	4	1	no	—
289	C₂²×C₄₇	—	4	1	no	—
290	C₂²×C₄₇	—	4	1	no	—
291	C₂²×C₄₇	—	4	1	no	—
292	C₂²×C₄₇	—	4	1	no	—
293	C₂²×C₄₇	—	8	4	no	—
294	C₂²×C₄₇	#416	1	2	yes	—
295	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
296	C₄×C₄₇	—	2	2	no	—
297	C₄×C₄₇	—	2	1	no	—
298	C₄×C₄₇	—	2	2	no	—
299	C₄×C₄₇	—	2	1	no	—
300	C₄×C₄₇	—	2	2	no	—
301	C₄×C₄₇	—	2	1	no	—
302	C₄×C₄₇	—	4	2	no	—
303	C₄×C₄₇	—	4	1	no	—
304	C₄×C₄₇	—	4	1	no	—
305	C₄×C₄₇	—	4	2	no	—
306	C₄×C₄₇	#416	1	2	yes	—
307	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
308	C₄×C₄₇	#568	1	1	yes	characteristic
309	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
310	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
311	#3475	C₂×C₄₇	1	1	yes	characteristic
312	C₈×#41	C₂×C₄₇	1	1	yes	characteristic, complemented
313	#2925	C₂×C₄₇	1	1	yes	characteristic, complemented
314	#1494	C₂×C₄₇	1	2	yes	complemented
315	#1493	C₂×C₄₇	1	1	yes	characteristic, complemented
316	#1488	C₂×C₄₇	1	2	yes	complemented
317	#1486	C₂×C₄₇	1	1	yes	characteristic, complemented
318	#1482	C₂×C₄₇	1	2	yes	complemented
319	#1481	C₂×C₄₇	1	1	yes	characteristic, complemented
320	#1476	C₂×C₄₇	1	2	yes	complemented
321	C₂×D₄⋊C₈	C₂×C₄₇	1	1	yes	characteristic, complemented
322	C₂×C₄₇	—	2	2	no	—
323	C₂×C₄₇	—	4	2	no	—
324	C₂×C₄₇	—	4	2	no	—
325	C₂×C₄₇	—	8	1	no	—
326	C₂×C₄₇	#2880	1	1	yes	characteristic
327	C₂×C₄₇	C₂×#416	1	1	yes	characteristic
328	C₂×C₄₇	#2901	1	1	yes	characteristic
329	#556	C₄×C₄₇	1	1	yes	characteristic
330	#555	C₄×C₄₇	1	1	yes	characteristic
331	#554	C₄×C₄₇	1	1	yes	characteristic
332	#553	—	2	1	no	—
333	#552	—	2	1	no	—
334	#552	C₄×C₄₇	1	2	yes	—
335	#551	—	2	2	no	—
336	#547	C₄×C₄₇	1	1	yes	characteristic
337	#546	C₄×C₄₇	1	2	yes	—
338	#533	C₂²×C₄₇	1	1	yes	characteristic
339	#529	—	2	2	no	—
340	#529	C₂²×C₄₇	1	2	yes	—
341	#525	—	2	1	no	—
342	C₄×#41	C₂²×C₄₇	1	2	yes	—
343	C₄×#41	C₂²×C₄₇	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
345	C₂×#117	—	2	1	no	—
346	C₈×Q₈	—	2	1	no	—
347	C₈×D₄	—	2	1	no	—
348	C₈×D₄	—	2	2	no	—
349	#433	—	2	2	no	—
350	#432	C₂²×C₄₇	1	2	yes	complemented
351	#431	C₂²×C₄₇	1	1	yes	characteristic
352	#431	C₂²×C₄₇	1	2	yes	complemented
353	#431	C₂²×C₄₇	1	2	yes	complemented
354	#423	C₂²×C₄₇	1	1	yes	characteristic
355	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
356	#408	—	2	1	no	—
357	#405	—	2	1	no	—
358	C₂×C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
359	C₂×C₄×C₈	C₂²×C₄₇	1	2	yes	complemented
360	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
361	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
362	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
363	Q₈⋊C₈	C₂²×C₄₇	1	2	yes	complemented
364	D₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
365	D₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
366	D₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
367	D₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
368	D₄⋊C₈	C₂²×C₄₇	1	2	yes	—
369	D₄⋊C₈	C₂²×C₄₇	1	2	yes	—
370	C₄₇	#17508	1	1	yes	characteristic, direct product factor, Sylow
371	C₂×#41	C₂×C₄×C₄₇	1	2	yes	—
372	C₂×#41	D₄×C₄₇	1	1	yes	characteristic
373	C₂²×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
374	C₂³×C₄	—	2	1	no	—
375	C₂×M₁₆	—	2	1	no	—
376	C₂²×C₈	—	2	1	no	—
377	C₂²×C₈	—	2	2	no	—
378	C₂²×C₈	—	2	1	no	—
379	C₄⋊Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
380	C₄⋊Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
381	#126	—	2	2	no	—
382	#126	—	2	2	no	—
383	#126	—	2	2	no	—
384	#125	—	2	2	no	—
385	#125	—	2	1	no	—
386	#125	C₂×C₄×C₄₇	1	2	yes	—
387	#124	—	2	2	no	—
388	#124	C₂×C₄×C₄₇	1	2	yes	—
389	#123	—	2	2	no	—
390	#123	—	2	2	no	—
391	#123	—	2	2	no	—
392	#123	—	2	2	no	—
393	C₂²⋊Q₈	—	2	1	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	C₂×C₄×C₄₇	1	2	yes	—
398	C₂²⋊Q₈	C₂×C₄×C₄₇	1	2	yes	—
399	#121	—	2	1	no	—
400	#121	—	2	1	no	—
401	#121	—	2	2	no	—
402	#121	C₂×C₄×C₄₇	1	2	yes	—
403	#120	—	2	2	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	C₂×C₄×C₄₇	1	2	yes	—
406	C₄×Q₈	C₂×C₄×C₄₇	1	1	yes	characteristic
407	C₄×Q₈	D₄×C₄₇	1	1	yes	characteristic
408	C₄×Q₈	C₂³×C₄₇	1	2	yes	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
415	C₄×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
416	C₄×D₄	C₂×C₄×C₄₇	1	2	yes	—
417	C₄×D₄	C₂×C₄×C₄₇	1	2	yes	—
418	C₄×D₄	C₂×C₄×C₄₇	1	1	yes	characteristic
419	C₄×D₄	D₄×C₄₇	1	2	yes	—
420	C₄×D₄	D₄×C₄₇	1	1	yes	characteristic
421	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
422	C₄×D₄	C₂³×C₄₇	1	1	yes	characteristic
423	#117	—	2	1	no	—
424	#117	—	2	1	no	—
425	#117	—	2	1	no	—
426	#117	—	2	2	no	—
427	#117	—	2	2	no	—
428	#117	—	2	2	no	—
429	#117	C₂×C₄×C₄₇	1	2	yes	—
430	#117	D₄×C₄₇	1	1	yes	characteristic
431	#117	D₄×C₄₇	1	2	yes	—
432	C₂×C₄⋊C₄	—	2	1	no	—
433	C₂×C₄⋊C₄	—	2	1	no	—
434	C₂×C₄⋊C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	2	no	—
437	C₂×C₄²	—	2	2	no	—
438	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₄₇	1	2	yes	—
440	C₄⋊C₈	—	2	1	no	—
441	C₄⋊C₈	—	2	1	no	—
442	C₄⋊C₈	—	2	1	no	—
443	C₄⋊C₈	—	2	2	no	—
444	C₄⋊C₈	C₂³×C₄₇	1	1	yes	characteristic
445	C₄⋊C₈	C₂³×C₄₇	1	1	yes	characteristic
446	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
447	C₂²⋊C₈	—	2	2	no	—
448	C₂²⋊C₈	—	2	2	no	—
449	C₂²⋊C₈	—	2	1	no	—
450	C₂²⋊C₈	—	2	2	no	—
451	#97	—	2	2	no	—
452	C₄×C₈	—	2	1	no	—
453	C₄×C₈	—	2	2	no	—
454	C₄×C₈	—	2	2	no	—
455	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
456	C₄×C₈	C₂³×C₄₇	1	1	yes	characteristic
457	C₄×C₈	C₂³×C₄₇	1	2	yes	—
458	C₂⁴	—	2	1	no	—
459	#41	—	2	2	no	—
460	#41	—	2	2	no	—
461	#41	—	2	4	no	—
462	#41	—	4	2	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
465	C₂×Q₈	C₂²×C₄×C₄₇	1	2	yes	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	4	1	no	—
471	C₂×D₄	C₂²⋊C₄×C₄₇	1	2	yes	—
472	C₂×D₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
473	C₂×D₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
474	C₂×D₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	4	2	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
485	C₂²×C₄	C₂²×C₄×C₄₇	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
487	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
489	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
490	M₁₆	—	4	2	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	2	2	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	4	2	no	—
498	C₂×C₈	—	4	4	no	—
499	C₂×C₈	—	4	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
511	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	1	yes	characteristic
512	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
513	C₄⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
514	C₄⋊C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×C₄×C₄₇	1	2	yes	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₂²⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
529	C₂²⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
530	C₂²⋊C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
531	C₄²	—	2	2	no	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	2	no	—
534	C₄²	C₂²×C₄×C₄₇	1	1	yes	characteristic
535	C₄²	C₂²×C₄×C₄₇	1	2	yes	—
536	C₄²	C₂×D₄×C₄₇	1	2	yes	—
537	C₄²	C₂×D₄×C₄₇	1	2	yes	—
538	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
539	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
540	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
541	C₂³	—	2	1	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	8	1	no	—
546	C₂³	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
547	C₂³	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
548	Q₈	—	2	2	no	—
549	Q₈	—	2	2	no	—
550	Q₈	—	2	2	no	—
551	Q₈	—	8	1	no	—
552	D₄	—	2	4	no	—
553	D₄	—	2	1	no	—
554	D₄	—	2	1	no	—
555	D₄	—	2	1	no	—
556	D₄	—	2	2	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	4	no	—
559	D₄	—	8	2	no	—
560	D₄	#131×C₄₇	1	2	yes	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	8	2	no	—
582	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
583	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
584	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
585	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	2	yes	—
586	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
587	C₂×C₄	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
588	C₂×C₄	#135×C₄₇	1	2	yes	—
589	C₂×C₄	#135×C₄₇	1	2	yes	—
590	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
591	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
592	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
593	C₈	—	2	2	no	—
594	C₈	—	4	1	no	—
595	C₈	—	4	2	no	—
596	C₈	—	4	1	no	—
597	C₈	—	4	1	no	—
598	C₈	—	4	2	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	1	no	—
610	C₂²	—	8	4	no	—
611	C₂²	C₄₇×#416	1	2	yes	—
612	C₂²	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
613	C₄	—	2	2	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	2	no	—
618	C₄	—	2	1	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	1	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	2	no	—
623	C₄	C₄₇×#416	1	2	yes	—
624	C₄	C₂×#131×C₄₇	1	1	yes	characteristic
625	C₄	C₄₇×#568	1	1	yes	characteristic
626	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
627	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
628	C₂	—	2	2	no	—
629	C₂	—	4	2	no	—
630	C₂	—	4	2	no	—
631	C₂	—	8	1	no	—
632	C₂	C₄₇×#2880	1	1	yes	characteristic
633	C₂	C₂×C₄₇×#416	1	1	yes	characteristic
634	C₂	C₄₇×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₄²	C₂⁴×C₄₇
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
prime radical	C₄₇×#3475	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₄×#41×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×#41×C₄₇	C₂	semidirect
3. C₄₇×#2925	C₂	semidirect
4. C₄₇×#1494	C₂	semidirect
5. C₄₇×#1493	C₂	semidirect
6. C₄₇×#1488	C₂	semidirect
7. C₄₇×#1486	C₂	semidirect
8. C₄₇×#1482	C₂	semidirect
9. C₄₇×#1481	C₂	semidirect
10. C₄₇×#1476	C₂	semidirect
11. C₂×C₄₇×D₄⋊C₈	C₂	semidirect
33. C₄₇×#432	C₂²	semidirect
35. C₄₇×#431	C₂²	semidirect
36. C₄₇×#431	C₂²	semidirect
42. C₂×C₄×C₈×C₄₇	C₂²	semidirect
43. C₄₇×Q₈⋊C₈	C₂²	semidirect
44. C₄₇×Q₈⋊C₈	C₂²	semidirect
45. C₄₇×Q₈⋊C₈	C₂²	semidirect
46. C₄₇×Q₈⋊C₈	C₂²	semidirect
281. #17508	C₄₇	direct (N × H)
312. C₈×#41	C₂×C₄₇	semidirect
313. #2925	C₂×C₄₇	semidirect
314. #1494	C₂×C₄₇	semidirect
315. #1493	C₂×C₄₇	semidirect
316. #1488	C₂×C₄₇	semidirect
317. #1486	C₂×C₄₇	semidirect
318. #1482	C₂×C₄₇	semidirect
319. #1481	C₂×C₄₇	semidirect
320. #1476	C₂×C₄₇	semidirect
321. C₂×D₄⋊C₈	C₂×C₄₇	semidirect
350. #432	C₂²×C₄₇	semidirect
352. #431	C₂²×C₄₇	semidirect
353. #431	C₂²×C₄₇	semidirect
359. C₂×C₄×C₈	C₂²×C₄₇	semidirect
360. Q₈⋊C₈	C₂²×C₄₇	semidirect
361. Q₈⋊C₈	C₂²×C₄₇	semidirect
362. Q₈⋊C₈	C₂²×C₄₇	semidirect
363. Q₈⋊C₈	C₂²×C₄₇	semidirect
370. C₄₇	#17508	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×#3475	2
C₈×#41×C₄₇	2
C₄₇×#2925	2
C₄₇×#1494	2
C₄₇×#1493	2
C₄₇×#1488	2
C₄₇×#1486	2
C₄₇×#1482	2
C₄₇×#1481	2
C₄₇×#1476	2
C₂×C₄₇×D₄⋊C₈	2
#17508	47

Subgroups — C47×#17508 #26057206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#17508 #26057206