Subgroups — C₄₇×#25439 #26065137

Order 12032.

960 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3057	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3042	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#3041	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#2046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#2045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#2003	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#2002	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1977	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#1946	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄₇×#1887	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄₇×#1885	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₇×#1765	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#536	C₂²	1	1	yes	characteristic
24	C₄₇×#535	C₂²	1	1	yes	characteristic
25	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic
27	C₄₇×#518	C₂²	1	1	yes	characteristic
28	C₄₇×#518	C₂²	1	1	yes	characteristic
29	C₄₇×#496	—	2	4	no	—
30	C₄₇×#472	—	2	4	no	—
31	C₄₇×#468	—	2	2	no	—
32	C₄₇×#465	—	2	2	no	—
33	C₄₇×#463	—	2	2	no	—
34	C₄₇×#463	—	2	2	no	—
35	C₄₇×#461	—	2	2	no	—
36	C₄₇×#459	—	2	2	no	—
37	C₄₇×#442	—	2	2	no	—
38	C₄×D₈×C₄₇	—	2	2	no	—
39	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic
40	C₄₇×#427	C₂²	1	1	yes	characteristic
41	C₄₇×#424	C₂²	1	1	yes	characteristic
42	C₄₇×#424	C₂²	1	1	yes	characteristic
43	C₂×#104×C₄₇	C₂²	1	1	yes	characteristic
44	C₄₇×#419	C₂²	1	1	yes	characteristic
45	C₄₇×#419	C₂²	1	1	yes	characteristic
46	C₄₇×#418	C₂²	1	1	yes	characteristic
47	C₄₇×#418	C₂²	1	1	yes	characteristic
48	C₄₇×#418	C₂²	1	1	yes	characteristic
49	C₄₇×#418	C₂²	1	1	yes	characteristic
50	C₄₇×#416	C₂²	1	1	yes	characteristic
51	C₂×#101×C₄₇	C₂²	1	1	yes	characteristic
52	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
53	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
54	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
55	C₄₇×#405	C₂²	1	1	yes	characteristic
56	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
57	C₄₇×#341	C₂²	1	1	yes	characteristic
58	C₄₇×#339	C₂²	1	1	yes	characteristic
59	C₄₇×#339	C₂²	1	1	yes	characteristic
60	C₄₇×#339	C₂²	1	1	yes	characteristic
61	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₄₇	—	4	1	no	—
64	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
67	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
68	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
69	AGL(1,ℤ/8)×C₄₇	—	4	1	no	—
70	AGL(1,ℤ/8)×C₄₇	—	4	1	no	—
71	C₂×SD₁₆×C₄₇	—	2	1	no	—
72	C₂×SD₁₆×C₄₇	—	2	1	no	—
73	C₂×SD₁₆×C₄₇	—	2	2	no	—
74	C₂×SD₁₆×C₄₇	—	2	2	no	—
75	C₂×SD₁₆×C₄₇	—	2	1	no	—
76	C₂×SD₁₆×C₄₇	—	2	1	no	—
77	C₂×SD₁₆×C₄₇	—	2	2	no	—
78	C₂×SD₁₆×C₄₇	—	2	2	no	—
79	C₂×SD₁₆×C₄₇	—	4	1	no	—
80	C₂×D₈×C₄₇	—	2	1	no	—
81	C₂×D₈×C₄₇	—	2	1	no	—
82	C₂×D₈×C₄₇	—	2	2	no	—
83	C₂×D₈×C₄₇	—	2	1	no	—
84	C₂×D₈×C₄₇	—	2	1	no	—
85	C₂×D₈×C₄₇	—	2	2	no	—
86	C₂×D₈×C₄₇	—	2	2	no	—
87	C₂×D₈×C₄₇	—	4	1	no	—
88	C₂×D₈×C₄₇	D₄	1	4	yes	complemented
89	#131×C₄₇	—	2	2	no	—
90	#131×C₄₇	—	2	2	no	—
91	C₂×M₁₆×C₄₇	—	2	1	no	—
92	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₄₇	—	2	1	no	—
97	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
99	C₄⋊Q₈×C₄₇	—	2	1	no	—
100	#127×C₄₇	—	2	1	no	—
101	#127×C₄₇	—	2	2	no	—
102	#127×C₄₇	—	2	2	no	—
103	#124×C₄₇	—	2	1	no	—
104	#124×C₄₇	—	2	1	no	—
105	#124×C₄₇	—	2	2	no	—
106	#124×C₄₇	—	2	2	no	—
107	#123×C₄₇	—	4	1	no	—
108	C₂²⋊Q₈×C₄₇	—	2	2	no	—
109	#121×C₄₇	—	2	2	no	—
110	#120×C₄₇	—	4	1	no	—
111	C₄×D₄×C₄₇	—	2	2	no	—
112	C₄×D₄×C₄₇	—	2	2	no	—
113	C₄×D₄×C₄₇	—	2	2	no	—
114	C₄×D₄×C₄₇	—	2	2	no	—
115	#117×C₄₇	C₂³	1	1	yes	characteristic
116	#117×C₄₇	C₂³	1	1	yes	characteristic
117	#117×C₄₇	C₂³	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
119	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
120	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
121	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
122	#108×C₄₇	—	2	2	no	—
123	#107×C₄₇	D₄	1	2	yes	complemented
124	#106×C₄₇	D₄	1	2	yes	complemented
125	C₄⋊C₈×C₄₇	—	2	2	no	—
126	C₄⋊C₈×C₄₇	—	2	2	no	—
127	#104×C₄₇	—	2	2	no	—
128	#103×C₄₇	—	2	2	no	—
129	#103×C₄₇	—	2	2	no	—
130	#103×C₄₇	D₄	1	2	yes	complemented
131	#102×C₄₇	—	2	2	no	—
132	#102×C₄₇	—	2	2	no	—
133	#102×C₄₇	—	2	2	no	—
134	#102×C₄₇	—	2	2	no	—
135	#102×C₄₇	D₄	1	2	yes	complemented
136	#101×C₄₇	—	4	1	no	—
137	#100×C₄₇	—	2	2	no	—
138	#100×C₄₇	—	2	2	no	—
139	#100×C₄₇	—	4	1	no	—
140	#97×C₄₇	—	2	2	no	—
141	#97×C₄₇	—	2	1	no	—
142	C₄×C₈×C₄₇	—	2	1	no	—
143	C₂⁴×C₄₇	—	2	1	no	—
144	C₂⁴×C₄₇	—	2	1	no	—
145	C₂⁴×C₄₇	—	2	1	no	—
146	#41×C₄₇	—	2	1	no	—
147	#41×C₄₇	—	2	1	no	—
148	#41×C₄₇	—	2	1	no	—
149	#41×C₄₇	—	2	1	no	—
150	C₂×Q₈×C₄₇	—	2	1	no	—
151	C₂×Q₈×C₄₇	—	2	1	no	—
152	C₂×Q₈×C₄₇	—	2	1	no	—
153	C₂×Q₈×C₄₇	—	2	1	no	—
154	C₂×Q₈×C₄₇	—	2	1	no	—
155	C₂×Q₈×C₄₇	—	2	1	no	—
156	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
157	C₂×D₄×C₄₇	—	2	1	no	—
158	C₂×D₄×C₄₇	—	2	1	no	—
159	C₂×D₄×C₄₇	—	2	1	no	—
160	C₂×D₄×C₄₇	—	2	1	no	—
161	C₂×D₄×C₄₇	—	2	1	no	—
162	C₂×D₄×C₄₇	—	2	1	no	—
163	C₂×D₄×C₄₇	—	2	1	no	—
164	C₂×D₄×C₄₇	—	2	1	no	—
165	C₂×D₄×C₄₇	—	2	1	no	—
166	C₂×D₄×C₄₇	—	2	1	no	—
167	C₂×D₄×C₄₇	—	2	1	no	—
168	C₂×D₄×C₄₇	—	2	1	no	—
169	C₂×D₄×C₄₇	—	2	1	no	—
170	C₂×D₄×C₄₇	—	2	1	no	—
171	C₂×D₄×C₄₇	—	2	1	no	—
172	C₂×D₄×C₄₇	—	2	1	no	—
173	C₂×D₄×C₄₇	—	2	1	no	—
174	C₂×D₄×C₄₇	—	4	4	no	—
175	C₂×D₄×C₄₇	—	4	2	no	—
176	C₂×D₄×C₄₇	—	8	4	no	—
177	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
179	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
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	—	4	2	no	—
184	C₂²×C₄×C₄₇	—	4	2	no	—
185	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
186	SD₁₆×C₄₇	—	4	4	no	—
187	SD₁₆×C₄₇	—	4	2	no	—
188	SD₁₆×C₄₇	—	4	2	no	—
189	SD₁₆×C₄₇	—	4	2	no	—
190	SD₁₆×C₄₇	—	4	2	no	—
191	D₈×C₄₇	—	2	4	no	—
192	D₈×C₄₇	—	2	4	no	—
193	D₈×C₄₇	—	4	4	no	—
194	D₈×C₄₇	—	4	2	no	—
195	D₈×C₄₇	—	4	2	no	—
196	M₁₆×C₄₇	—	2	1	no	—
197	M₁₆×C₄₇	—	2	1	no	—
198	M₁₆×C₄₇	—	2	1	no	—
199	M₁₆×C₄₇	—	2	1	no	—
200	M₁₆×C₄₇	—	2	1	no	—
201	M₁₆×C₄₇	—	2	1	no	—
202	M₁₆×C₄₇	—	2	1	no	—
203	M₁₆×C₄₇	—	2	1	no	—
204	M₁₆×C₄₇	—	4	2	no	—
205	C₂×C₈×C₄₇	—	2	1	no	—
206	C₂×C₈×C₄₇	—	2	1	no	—
207	C₂×C₈×C₄₇	—	2	1	no	—
208	C₂×C₈×C₄₇	—	2	1	no	—
209	C₂×C₈×C₄₇	—	2	1	no	—
210	C₂×C₈×C₄₇	—	2	1	no	—
211	C₂×C₈×C₄₇	—	4	2	no	—
212	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
213	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
214	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
215	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
216	C₂×C₈×C₄₇	#41	1	2	yes	—
217	C₄⋊C₄×C₄₇	—	4	2	no	—
218	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
219	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
220	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
221	C₂²⋊C₄×C₄₇	—	2	1	no	—
222	C₂²⋊C₄×C₄₇	—	2	1	no	—
223	C₂²⋊C₄×C₄₇	—	2	1	no	—
224	C₂²⋊C₄×C₄₇	—	4	2	no	—
225	C₂²⋊C₄×C₄₇	—	4	2	no	—
226	C₂²⋊C₄×C₄₇	—	4	2	no	—
227	C₂²⋊C₄×C₄₇	—	4	4	no	—
228	C₂²⋊C₄×C₄₇	—	4	2	no	—
229	C₄²×C₄₇	—	2	1	no	—
230	C₄²×C₄₇	—	2	1	no	—
231	C₄²×C₄₇	—	2	1	no	—
232	C₄²×C₄₇	—	2	1	no	—
233	C₄²×C₄₇	#41	1	2	yes	—
234	C₂³×C₄₇	—	2	1	no	—
235	C₂³×C₄₇	—	2	2	no	—
236	C₂³×C₄₇	—	2	2	no	—
237	C₂³×C₄₇	—	4	1	no	—
238	C₂³×C₄₇	—	4	1	no	—
239	C₂³×C₄₇	—	4	1	no	—
240	C₂³×C₄₇	—	4	1	no	—
241	C₂³×C₄₇	—	4	1	no	—
242	C₂³×C₄₇	—	4	1	no	—
243	C₂³×C₄₇	—	4	1	no	—
244	C₂³×C₄₇	—	8	2	no	—
245	C₂³×C₄₇	—	8	2	no	—
246	C₂³×C₄₇	—	8	2	no	—
247	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
248	Q₈×C₄₇	—	2	1	no	—
249	Q₈×C₄₇	—	2	1	no	—
250	Q₈×C₄₇	—	4	1	no	—
251	Q₈×C₄₇	—	4	1	no	—
252	Q₈×C₄₇	—	4	1	no	—
253	Q₈×C₄₇	—	4	1	no	—
254	D₄×C₄₇	—	2	2	no	—
255	D₄×C₄₇	—	2	2	no	—
256	D₄×C₄₇	—	2	1	no	—
257	D₄×C₄₇	—	2	1	no	—
258	D₄×C₄₇	—	2	2	no	—
259	D₄×C₄₇	—	2	2	no	—
260	D₄×C₄₇	—	4	1	no	—
261	D₄×C₄₇	—	4	1	no	—
262	D₄×C₄₇	—	4	1	no	—
263	D₄×C₄₇	—	4	1	no	—
264	D₄×C₄₇	—	4	1	no	—
265	D₄×C₄₇	—	4	1	no	—
266	D₄×C₄₇	—	4	1	no	—
267	D₄×C₄₇	—	4	1	no	—
268	D₄×C₄₇	—	4	1	no	—
269	D₄×C₄₇	—	4	1	no	—
270	D₄×C₄₇	—	8	8	no	—
271	D₄×C₄₇	—	16	2	no	—
272	C₂×C₄×C₄₇	—	2	2	no	—
273	C₂×C₄×C₄₇	—	2	2	no	—
274	C₂×C₄×C₄₇	—	2	2	no	—
275	C₂×C₄×C₄₇	—	2	1	no	—
276	C₂×C₄×C₄₇	—	2	2	no	—
277	C₂×C₄×C₄₇	—	4	2	no	—
278	C₂×C₄×C₄₇	—	4	1	no	—
279	C₂×C₄×C₄₇	—	4	1	no	—
280	C₂×C₄×C₄₇	—	4	1	no	—
281	C₂×C₄×C₄₇	—	4	1	no	—
282	C₂×C₄×C₄₇	—	4	1	no	—
283	C₂×C₄×C₄₇	—	16	2	no	—
284	C₂×C₄×C₄₇	—	16	2	no	—
285	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
288	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
290	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
291	C₈×C₄₇	—	2	2	no	—
292	C₈×C₄₇	—	4	1	no	—
293	C₈×C₄₇	—	4	1	no	—
294	C₈×C₄₇	—	4	1	no	—
295	C₈×C₄₇	—	4	1	no	—
296	C₈×C₄₇	#121	1	2	yes	—
297	C₈×C₄₇	#121	1	2	yes	—
298	#25439	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
299	C₂²×C₄₇	—	2	2	no	—
300	C₂²×C₄₇	—	4	1	no	—
301	C₂²×C₄₇	—	4	2	no	—
302	C₂²×C₄₇	—	4	2	no	—
303	C₂²×C₄₇	—	4	2	no	—
304	C₂²×C₄₇	—	8	2	no	—
305	C₂²×C₄₇	—	8	2	no	—
306	C₂²×C₄₇	—	8	2	no	—
307	C₂²×C₄₇	—	8	2	no	—
308	C₂²×C₄₇	—	8	2	no	—
309	C₂²×C₄₇	—	8	1	no	—
310	C₂²×C₄₇	—	8	2	no	—
311	C₂²×C₄₇	—	16	1	no	—
312	C₂²×C₄₇	—	16	1	no	—
313	C₂²×C₄₇	—	16	1	no	—
314	C₂²×C₄₇	#540	1	1	yes	characteristic
315	C₂²×C₄₇	D₄²	1	1	yes	characteristic
316	C₂²×C₄₇	#546	1	1	yes	characteristic
317	C₄×C₄₇	—	4	2	no	—
318	C₄×C₄₇	—	4	1	no	—
319	C₄×C₄₇	—	4	2	no	—
320	C₄×C₄₇	—	4	2	no	—
321	C₄×C₄₇	—	4	2	no	—
322	C₄×C₄₇	—	8	1	no	—
323	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
324	C₄×C₄₇	#532	1	1	yes	characteristic
325	C₄×C₄₇	D₄²	1	1	yes	characteristic
326	C₄×C₄₇	#550	1	1	yes	characteristic
327	C₂×#496	C₂×C₄₇	1	1	yes	characteristic, complemented
328	C₂×#472	C₂×C₄₇	1	1	yes	characteristic, complemented
329	#3057	C₂×C₄₇	1	1	yes	characteristic, complemented
330	#3042	C₂×C₄₇	1	1	yes	characteristic, complemented
331	#3041	C₂×C₄₇	1	1	yes	characteristic, complemented
332	#2948	C₂×C₄₇	1	1	yes	characteristic, complemented
333	#2046	C₂×C₄₇	1	1	yes	characteristic, complemented
334	#2045	C₂×C₄₇	1	1	yes	characteristic, complemented
335	#2003	C₂×C₄₇	1	1	yes	characteristic, complemented
336	#2002	C₂×C₄₇	1	1	yes	characteristic, complemented
337	#1977	C₂×C₄₇	1	1	yes	characteristic, complemented
338	#1946	C₂×C₄₇	1	1	yes	characteristic, complemented
339	#1887	C₂×C₄₇	1	1	yes	characteristic, complemented
340	#1885	C₂×C₄₇	1	1	yes	characteristic, complemented
341	#1765	C₂×C₄₇	1	1	yes	characteristic, complemented
342	C₂×C₄₇	—	2	1	no	—
343	C₂×C₄₇	—	2	1	no	—
344	C₂×C₄₇	—	8	1	no	—
345	C₂×C₄₇	—	8	2	no	—
346	C₂×C₄₇	—	8	2	no	—
347	C₂×C₄₇	—	16	1	no	—
348	C₂×C₄₇	#2428	1	1	yes	characteristic
349	C₂×C₄₇	#3288	1	2	yes	—
350	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	1	yes	characteristic
351	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	1	yes	characteristic
352	C₂²×SD₁₆	C₂²×C₄₇	1	1	yes	characteristic
353	C₂²×SD₁₆	C₂²×C₄₇	1	1	yes	characteristic
354	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
355	C₂²×D₈	C₂²×C₄₇	1	1	yes	characteristic
356	C₂×#131	C₂²×C₄₇	1	1	yes	characteristic
357	#536	C₂²×C₄₇	1	1	yes	characteristic
358	#535	C₂²×C₄₇	1	1	yes	characteristic
359	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
360	C₂×#124	C₂²×C₄₇	1	1	yes	characteristic
361	#518	C₂²×C₄₇	1	1	yes	characteristic
362	#518	C₂²×C₄₇	1	1	yes	characteristic
363	#496	—	2	4	no	—
364	#472	—	2	4	no	—
365	#468	—	2	2	no	—
366	#465	—	2	2	no	—
367	#463	—	2	2	no	—
368	#463	—	2	2	no	—
369	#461	—	2	2	no	—
370	#459	—	2	2	no	—
371	#442	—	2	2	no	—
372	C₄×D₈	—	2	2	no	—
373	C₂×#108	C₂²×C₄₇	1	1	yes	characteristic
374	#427	C₂²×C₄₇	1	1	yes	characteristic
375	#424	C₂²×C₄₇	1	1	yes	characteristic
376	#424	C₂²×C₄₇	1	1	yes	characteristic
377	C₂×#104	C₂²×C₄₇	1	1	yes	characteristic
378	#419	C₂²×C₄₇	1	1	yes	characteristic
379	#419	C₂²×C₄₇	1	1	yes	characteristic
380	#418	C₂²×C₄₇	1	1	yes	characteristic
381	#418	C₂²×C₄₇	1	1	yes	characteristic
382	#418	C₂²×C₄₇	1	1	yes	characteristic
383	#418	C₂²×C₄₇	1	1	yes	characteristic
384	#416	C₂²×C₄₇	1	1	yes	characteristic
385	C₂×#101	C₂²×C₄₇	1	1	yes	characteristic
386	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
387	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
388	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
389	#405	C₂²×C₄₇	1	1	yes	characteristic
390	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic
391	#341	C₂²×C₄₇	1	1	yes	characteristic
392	#339	C₂²×C₄₇	1	1	yes	characteristic
393	#339	C₂²×C₄₇	1	1	yes	characteristic
394	#339	C₂²×C₄₇	1	1	yes	characteristic
395	C₄₇	#25439	1	1	yes	characteristic, direct product factor, Sylow
396	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
398	C₂²×D₄	—	4	1	no	—
399	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
400	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
401	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
402	AGL(1,ℤ/8)	—	2	2	no	—
403	AGL(1,ℤ/8)	—	2	2	no	—
404	AGL(1,ℤ/8)	—	4	1	no	—
405	AGL(1,ℤ/8)	—	4	1	no	—
406	C₂×SD₁₆	—	2	1	no	—
407	C₂×SD₁₆	—	2	1	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	2	2	no	—
410	C₂×SD₁₆	—	2	1	no	—
411	C₂×SD₁₆	—	2	1	no	—
412	C₂×SD₁₆	—	2	2	no	—
413	C₂×SD₁₆	—	2	2	no	—
414	C₂×SD₁₆	—	4	1	no	—
415	C₂×D₈	—	2	2	no	—
416	C₂×D₈	—	2	1	no	—
417	C₂×D₈	—	2	1	no	—
418	C₂×D₈	—	2	1	no	—
419	C₂×D₈	—	2	1	no	—
420	C₂×D₈	—	2	2	no	—
421	C₂×D₈	—	2	2	no	—
422	C₂×D₈	—	4	1	no	—
423	C₂×D₈	D₄×C₄₇	1	4	yes	complemented
424	#131	—	2	2	no	—
425	#131	—	2	2	no	—
426	C₂×M₁₆	—	2	1	no	—
427	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
428	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
429	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
430	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
431	C₂²×C₈	—	2	1	no	—
432	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
433	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
434	C₄⋊Q₈	—	2	1	no	—
435	#127	—	2	2	no	—
436	#127	—	2	1	no	—
437	#127	—	2	2	no	—
438	#124	—	2	1	no	—
439	#124	—	2	2	no	—
440	#124	—	2	1	no	—
441	#124	—	2	2	no	—
442	#123	—	4	1	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	#121	—	2	2	no	—
445	#120	—	4	1	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	2	no	—
448	C₄×D₄	—	2	2	no	—
449	C₄×D₄	—	2	2	no	—
450	#117	C₂³×C₄₇	1	1	yes	characteristic
451	#117	C₂³×C₄₇	1	1	yes	characteristic
452	#117	C₂³×C₄₇	1	1	yes	characteristic
453	C₂×C₂²⋊C₄	—	4	1	no	—
454	C₂×C₂²⋊C₄	—	4	1	no	—
455	C₂×C₂²⋊C₄	—	4	1	no	—
456	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
457	#108	—	2	2	no	—
458	#107	D₄×C₄₇	1	2	yes	complemented
459	#106	D₄×C₄₇	1	2	yes	complemented
460	C₄⋊C₈	—	2	2	no	—
461	C₄⋊C₈	—	2	2	no	—
462	#104	—	2	2	no	—
463	#103	—	2	2	no	—
464	#103	—	2	2	no	—
465	#103	D₄×C₄₇	1	2	yes	complemented
466	#102	—	2	2	no	—
467	#102	—	2	2	no	—
468	#102	—	2	2	no	—
469	#102	—	2	2	no	—
470	#102	D₄×C₄₇	1	2	yes	complemented
471	#101	—	4	1	no	—
472	#100	—	2	2	no	—
473	#100	—	2	2	no	—
474	#100	—	4	1	no	—
475	#97	—	2	2	no	—
476	#97	—	2	1	no	—
477	C₄×C₈	—	2	1	no	—
478	C₂⁴	—	2	1	no	—
479	C₂⁴	—	2	1	no	—
480	C₂⁴	—	2	1	no	—
481	#41	—	2	1	no	—
482	#41	—	2	1	no	—
483	#41	—	2	1	no	—
484	#41	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	4	4	no	—
510	C₂×D₄	—	4	2	no	—
511	C₂×D₄	—	8	4	no	—
512	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
513	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
514	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	2	no	—
520	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
521	SD₁₆	—	4	4	no	—
522	SD₁₆	—	4	2	no	—
523	SD₁₆	—	4	2	no	—
524	SD₁₆	—	4	2	no	—
525	SD₁₆	—	4	2	no	—
526	D₈	—	2	4	no	—
527	D₈	—	2	4	no	—
528	D₈	—	4	4	no	—
529	D₈	—	4	2	no	—
530	D₈	—	4	2	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	1	no	—
538	M₁₆	—	2	1	no	—
539	M₁₆	—	4	2	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	4	2	no	—
547	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
548	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
549	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
550	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
551	C₂×C₈	#41×C₄₇	1	2	yes	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
554	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
555	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
556	C₂²⋊C₄	—	2	1	no	—
557	C₂²⋊C₄	—	2	1	no	—
558	C₂²⋊C₄	—	2	1	no	—
559	C₂²⋊C₄	—	4	2	no	—
560	C₂²⋊C₄	—	4	2	no	—
561	C₂²⋊C₄	—	4	2	no	—
562	C₂²⋊C₄	—	4	4	no	—
563	C₂²⋊C₄	—	4	2	no	—
564	C₄²	—	2	1	no	—
565	C₄²	—	2	1	no	—
566	C₄²	—	2	1	no	—
567	C₄²	—	2	1	no	—
568	C₄²	#41×C₄₇	1	2	yes	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	2	no	—
571	C₂³	—	2	2	no	—
572	C₂³	—	4	1	no	—
573	C₂³	—	4	1	no	—
574	C₂³	—	4	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	1	no	—
578	C₂³	—	4	1	no	—
579	C₂³	—	8	2	no	—
580	C₂³	—	8	2	no	—
581	C₂³	—	8	2	no	—
582	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
583	Q₈	—	2	1	no	—
584	Q₈	—	2	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	Q₈	—	4	1	no	—
589	D₄	—	2	2	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	1	no	—
592	D₄	—	2	1	no	—
593	D₄	—	2	2	no	—
594	D₄	—	2	2	no	—
595	D₄	—	4	1	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	1	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	1	no	—
600	D₄	—	4	1	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	1	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	1	no	—
605	D₄	—	8	8	no	—
606	D₄	—	16	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	4	2	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	16	2	no	—
619	C₂×C₄	—	16	2	no	—
620	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
621	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
622	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
623	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
624	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
625	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
626	C₈	—	2	2	no	—
627	C₈	—	4	1	no	—
628	C₈	—	4	1	no	—
629	C₈	—	4	1	no	—
630	C₈	—	4	1	no	—
631	C₈	#121×C₄₇	1	2	yes	—
632	C₈	#121×C₄₇	1	2	yes	—
633	C₂²	—	2	2	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	1	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	8	2	no	—
639	C₂²	—	8	2	no	—
640	C₂²	—	8	2	no	—
641	C₂²	—	8	2	no	—
642	C₂²	—	8	2	no	—
643	C₂²	—	8	2	no	—
644	C₂²	—	8	1	no	—
645	C₂²	—	16	1	no	—
646	C₂²	—	16	1	no	—
647	C₂²	—	16	1	no	—
648	C₂²	C₄₇×#540	1	1	yes	characteristic
649	C₂²	D₄²×C₄₇	1	1	yes	characteristic
650	C₂²	C₄₇×#546	1	1	yes	characteristic
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	1	no	—
656	C₄	—	8	1	no	—
657	C₄	C₂×#121×C₄₇	1	1	yes	characteristic
658	C₄	C₄₇×#532	1	1	yes	characteristic
659	C₄	D₄²×C₄₇	1	1	yes	characteristic
660	C₄	C₄₇×#550	1	1	yes	characteristic
661	C₂	—	2	1	no	—
662	C₂	—	2	1	no	—
663	C₂	—	8	1	no	—
664	C₂	—	8	2	no	—
665	C₂	—	8	2	no	—
666	C₂	—	16	1	no	—
667	C₂	C₄₇×#2428	1	1	yes	characteristic
668	C₂	C₄₇×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#496	C₂	semidirect
2. C₂×C₄₇×#472	C₂	semidirect
3. C₄₇×#3057	C₂	semidirect
4. C₄₇×#3042	C₂	semidirect
5. C₄₇×#3041	C₂	semidirect
6. C₄₇×#2948	C₂	semidirect
7. C₄₇×#2046	C₂	semidirect
8. C₄₇×#2045	C₂	semidirect
9. C₄₇×#2003	C₂	semidirect
10. C₄₇×#2002	C₂	semidirect
11. C₄₇×#1977	C₂	semidirect
12. C₄₇×#1946	C₂	semidirect
13. C₄₇×#1887	C₂	semidirect
14. C₄₇×#1885	C₂	semidirect
15. C₄₇×#1765	C₂	semidirect
88. C₂×D₈×C₄₇	D₄	semidirect
123. #107×C₄₇	D₄	semidirect
124. #106×C₄₇	D₄	semidirect
130. #103×C₄₇	D₄	semidirect
135. #102×C₄₇	D₄	semidirect
298. #25439	C₄₇	direct (N × H)
327. C₂×#496	C₂×C₄₇	semidirect
328. C₂×#472	C₂×C₄₇	semidirect
329. #3057	C₂×C₄₇	semidirect
330. #3042	C₂×C₄₇	semidirect
331. #3041	C₂×C₄₇	semidirect
332. #2948	C₂×C₄₇	semidirect
333. #2046	C₂×C₄₇	semidirect
334. #2045	C₂×C₄₇	semidirect
335. #2003	C₂×C₄₇	semidirect
336. #2002	C₂×C₄₇	semidirect
337. #1977	C₂×C₄₇	semidirect
338. #1946	C₂×C₄₇	semidirect
339. #1887	C₂×C₄₇	semidirect
340. #1885	C₂×C₄₇	semidirect
341. #1765	C₂×C₄₇	semidirect
395. C₄₇	#25439	direct (N × H)
423. C₂×D₈	D₄×C₄₇	semidirect
458. #107	D₄×C₄₇	semidirect
459. #106	D₄×C₄₇	semidirect
465. #103	D₄×C₄₇	semidirect
470. #102	D₄×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×#496	2
C₂×C₄₇×#472	2
C₄₇×#3057	2
C₄₇×#3042	2
C₄₇×#3041	2
C₄₇×#2948	2
C₄₇×#2046	2
C₄₇×#2045	2
C₄₇×#2003	2
C₄₇×#2002	2
C₄₇×#1977	2
C₄₇×#1946	2
C₄₇×#1887	2
C₄₇×#1885	2
C₄₇×#1765	2
#25439	47

Subgroups — C47×#25439 #26065137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#25439 #26065137