Subgroups — C₄₇×#25358 #26065056

Order 12032.

764 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3141	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#474	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3036	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2078	C₂	1	2	yes	maximal, complemented
6	C₄₇×#2051	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1950	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1907	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1816	C₂	1	2	yes	maximal, complemented
11	C₄×#100×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	—
13	C₂×#131×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#532	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#489	—	2	2	no	—
18	C₄₇×#487	—	2	1	no	—
19	C₄₇×#486	—	2	1	no	—
20	C₄₇×#474	—	2	1	no	—
21	C₄₇×#474	—	2	2	no	—
22	C₄₇×#474	—	2	1	no	—
23	C₄₇×#463	—	2	2	no	—
24	C₄₇×#461	—	2	1	no	—
25	C₄₇×#459	—	2	1	no	—
26	C₄₇×#436	—	2	1	no	—
27	C₄₇×#435	—	2	2	no	—
28	C₈×D₄×C₄₇	—	2	1	no	—
29	C₂×#108×C₄₇	C₂²	1	2	yes	—
30	C₂×#107×C₄₇	C₂²	1	2	yes	—
31	C₄₇×#423	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#417	C₂²	1	2	yes	—
34	C₄₇×#417	C₂²	1	2	yes	—
35	C₄₇×#417	C₂²	1	2	yes	—
36	C₂×#102×C₄₇	C₂²	1	2	yes	—
37	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
39	C₄₇×#408	C₂²	1	2	yes	—
40	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
42	C₄₇×#343	C₂²	1	1	yes	characteristic
43	C₄₇×#343	C₂²	1	1	yes	characteristic
44	C₄₇×#343	C₂²	1	2	yes	—
45	C₄₇×#343	C₂²	1	2	yes	—
46	C₄₇×#341	C₂²	1	2	yes	—
47	C₂×#41×C₄₇	C₂³	1	2	yes	—
48	C₂²×Q₈×C₄₇	—	2	1	no	—
49	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₇	—	2	1	no	—
51	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
52	AGL(1,ℤ/8)×C₄₇	—	4	2	no	—
53	C₂×SD₁₆×C₄₇	—	2	2	no	—
54	C₂×D₈×C₄₇	—	2	2	no	—
55	#131×C₄₇	—	2	4	no	—
56	#131×C₄₇	—	4	2	no	—
57	C₂×M₁₆×C₄₇	—	2	2	no	—
58	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
59	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	—	2	2	no	—
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₇	—	2	1	no	—
66	C₄⋊Q₈×C₄₇	—	2	1	no	—
67	C₄⋊Q₈×C₄₇	—	2	1	no	—
68	C₄⋊Q₈×C₄₇	—	2	1	no	—
69	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
70	#127×C₄₇	D₄	1	1	yes	characteristic
71	#124×C₄₇	D₄	1	2	yes	—
72	#121×C₄₇	—	2	1	no	—
73	#121×C₄₇	—	2	1	no	—
74	C₄×D₄×C₄₇	—	2	1	no	—
75	C₄×D₄×C₄₇	—	2	1	no	—
76	C₄×D₄×C₄₇	—	2	1	no	—
77	C₄×D₄×C₄₇	—	2	1	no	—
78	C₄×D₄×C₄₇	—	2	1	no	—
79	C₄×D₄×C₄₇	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
84	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
85	#108×C₄₇	—	4	2	no	—
86	#107×C₄₇	—	2	2	no	—
87	#107×C₄₇	—	2	2	no	—
88	C₄⋊C₈×C₄₇	—	2	1	no	—
89	C₄⋊C₈×C₄₇	—	2	1	no	—
90	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
91	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
92	C₄⋊C₈×C₄₇	D₄	1	1	yes	characteristic
93	#103×C₄₇	—	2	2	no	—
94	#103×C₄₇	—	2	2	no	—
95	#103×C₄₇	—	2	2	no	—
96	#102×C₄₇	—	2	2	no	—
97	#102×C₄₇	—	2	2	no	—
98	#102×C₄₇	—	2	2	no	—
99	#102×C₄₇	—	2	2	no	—
100	#102×C₄₇	—	2	2	no	—
101	#100×C₄₇	—	2	2	no	—
102	#100×C₄₇	—	2	2	no	—
103	C₂²⋊C₈×C₄₇	—	2	2	no	—
104	C₂²⋊C₈×C₄₇	—	2	2	no	—
105	#97×C₄₇	—	2	1	no	—
106	#97×C₄₇	—	2	1	no	—
107	C₄×C₈×C₄₇	—	2	1	no	—
108	C₄×C₈×C₄₇	—	2	1	no	—
109	C₂⁴×C₄₇	—	2	1	no	—
110	#41×C₄₇	—	2	2	no	—
111	#41×C₄₇	—	2	2	no	—
112	#41×C₄₇	—	2	2	no	—
113	#41×C₄₇	—	2	2	no	—
114	C₂×Q₈×C₄₇	—	4	1	no	—
115	C₂×Q₈×C₄₇	—	4	1	no	—
116	C₂×Q₈×C₄₇	—	8	2	no	—
117	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₄₇	—	2	2	no	—
119	C₂×D₄×C₄₇	—	2	1	no	—
120	C₂×D₄×C₄₇	—	2	1	no	—
121	C₂×D₄×C₄₇	—	2	1	no	—
122	C₂×D₄×C₄₇	—	2	1	no	—
123	C₂×D₄×C₄₇	—	2	1	no	—
124	C₂×D₄×C₄₇	—	2	1	no	—
125	C₂×D₄×C₄₇	—	2	1	no	—
126	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₄₇	#41	1	2	yes	—
128	C₂²×C₄×C₄₇	—	2	2	no	—
129	C₂²×C₄×C₄₇	—	2	1	no	—
130	C₂²×C₄×C₄₇	—	2	2	no	—
131	C₂²×C₄×C₄₇	—	4	1	no	—
132	C₂²×C₄×C₄₇	—	4	1	no	—
133	C₂²×C₄×C₄₇	—	8	2	no	—
134	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
135	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
136	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
137	SD₁₆×C₄₇	—	4	4	no	—
138	D₈×C₄₇	—	4	4	no	—
139	M₁₆×C₄₇	—	2	1	no	—
140	M₁₆×C₄₇	—	2	2	no	—
141	M₁₆×C₄₇	—	2	1	no	—
142	M₁₆×C₄₇	—	2	1	no	—
143	M₁₆×C₄₇	—	2	1	no	—
144	M₁₆×C₄₇	—	2	1	no	—
145	M₁₆×C₄₇	—	2	2	no	—
146	M₁₆×C₄₇	—	2	1	no	—
147	M₁₆×C₄₇	—	2	1	no	—
148	M₁₆×C₄₇	—	4	4	no	—
149	M₁₆×C₄₇	#41	1	2	yes	—
150	C₂×C₈×C₄₇	—	2	1	no	—
151	C₂×C₈×C₄₇	—	2	2	no	—
152	C₂×C₈×C₄₇	—	2	2	no	—
153	C₂×C₈×C₄₇	—	2	1	no	—
154	C₂×C₈×C₄₇	—	2	1	no	—
155	C₂×C₈×C₄₇	—	2	2	no	—
156	C₂×C₈×C₄₇	—	2	1	no	—
157	C₂×C₈×C₄₇	—	4	4	no	—
158	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₄₇	—	2	1	no	—
161	C₄⋊C₄×C₄₇	—	2	1	no	—
162	C₄⋊C₄×C₄₇	—	2	1	no	—
163	C₄⋊C₄×C₄₇	—	2	2	no	—
164	C₄⋊C₄×C₄₇	—	2	1	no	—
165	C₄⋊C₄×C₄₇	—	2	2	no	—
166	C₄⋊C₄×C₄₇	—	4	1	no	—
167	C₄⋊C₄×C₄₇	—	4	1	no	—
168	C₂²⋊C₄×C₄₇	—	2	2	no	—
169	C₂²⋊C₄×C₄₇	—	2	2	no	—
170	C₂²⋊C₄×C₄₇	—	4	1	no	—
171	C₂²⋊C₄×C₄₇	—	4	1	no	—
172	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
176	C₂³×C₄₇	—	2	2	no	—
177	C₂³×C₄₇	—	2	2	no	—
178	C₂³×C₄₇	—	4	1	no	—
179	C₂³×C₄₇	—	4	1	no	—
180	C₂³×C₄₇	—	8	1	no	—
181	C₂³×C₄₇	—	8	1	no	—
182	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
183	Q₈×C₄₇	—	2	2	no	—
184	Q₈×C₄₇	—	2	2	no	—
185	Q₈×C₄₇	—	8	2	no	—
186	Q₈×C₄₇	—	8	2	no	—
187	D₄×C₄₇	—	2	2	no	—
188	D₄×C₄₇	—	2	2	no	—
189	D₄×C₄₇	—	2	2	no	—
190	D₄×C₄₇	—	2	2	no	—
191	D₄×C₄₇	—	4	2	no	—
192	D₄×C₄₇	—	4	2	no	—
193	D₄×C₄₇	—	4	1	no	—
194	D₄×C₄₇	—	4	1	no	—
195	D₄×C₄₇	—	4	2	no	—
196	D₄×C₄₇	—	8	1	no	—
197	C₂×C₄×C₄₇	—	2	2	no	—
198	C₂×C₄×C₄₇	—	2	2	no	—
199	C₂×C₄×C₄₇	—	4	1	no	—
200	C₂×C₄×C₄₇	—	4	2	no	—
201	C₂×C₄×C₄₇	—	4	2	no	—
202	C₂×C₄×C₄₇	—	4	1	no	—
203	C₂×C₄×C₄₇	—	4	2	no	—
204	C₂×C₄×C₄₇	—	8	2	no	—
205	C₂×C₄×C₄₇	—	8	2	no	—
206	C₂×C₄×C₄₇	—	8	1	no	—
207	C₂×C₄×C₄₇	—	8	2	no	—
208	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
209	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
210	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
211	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
212	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
218	C₈×C₄₇	—	2	2	no	—
219	C₈×C₄₇	—	2	2	no	—
220	C₈×C₄₇	—	2	2	no	—
221	C₈×C₄₇	—	4	1	no	—
222	C₈×C₄₇	—	4	2	no	—
223	C₈×C₄₇	—	4	1	no	—
224	C₈×C₄₇	—	4	1	no	—
225	#25358	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂²×C₄₇	—	2	1	no	—
227	C₂²×C₄₇	—	2	1	no	—
228	C₂²×C₄₇	—	4	2	no	—
229	C₂²×C₄₇	—	4	2	no	—
230	C₂²×C₄₇	—	4	2	no	—
231	C₂²×C₄₇	—	4	2	no	—
232	C₂²×C₄₇	—	8	2	no	—
233	C₂²×C₄₇	—	8	2	no	—
234	C₂²×C₄₇	—	8	1	no	—
235	C₂²×C₄₇	—	8	1	no	—
236	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
237	C₂²×C₄₇	#529	1	1	yes	characteristic
238	C₂²×C₄₇	#542	1	1	yes	characteristic
239	C₄×C₄₇	—	2	1	no	—
240	C₄×C₄₇	—	2	1	no	—
241	C₄×C₄₇	—	2	1	no	—
242	C₄×C₄₇	—	4	2	no	—
243	C₄×C₄₇	—	8	1	no	—
244	C₄×C₄₇	—	8	2	no	—
245	C₄×C₄₇	#472	1	2	yes	—
246	C₄×C₄₇	#546	1	1	yes	characteristic
247	C₄×C₄₇	#546	1	1	yes	characteristic
248	C₄×C₄₇	#547	1	1	yes	characteristic
249	C₄×C₄₇	#548	1	1	yes	characteristic
250	#3141	C₂×C₄₇	1	1	yes	characteristic, complemented
251	C₂×#474	C₂×C₄₇	1	1	yes	characteristic, complemented
252	#3036	C₂×C₄₇	1	1	yes	characteristic
253	#2935	C₂×C₄₇	1	1	yes	characteristic, complemented
254	#2078	C₂×C₄₇	1	2	yes	complemented
255	#2051	C₂×C₄₇	1	2	yes	complemented
256	#1997	C₂×C₄₇	1	1	yes	characteristic, complemented
257	#1950	C₂×C₄₇	1	1	yes	characteristic, complemented
258	#1907	C₂×C₄₇	1	2	yes	complemented
259	#1816	C₂×C₄₇	1	2	yes	complemented
260	C₄×#100	C₂×C₄₇	1	1	yes	characteristic, complemented
261	C₂×C₄₇	—	2	1	no	—
262	C₂×C₄₇	—	2	1	no	—
263	C₂×C₄₇	—	8	2	no	—
264	C₂×C₄₇	—	8	2	no	—
265	C₂×C₄₇	#2427	1	1	yes	characteristic
266	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
267	C₂×C₄₇	#3069	1	1	yes	characteristic
268	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	—
269	C₂×#131	C₂²×C₄₇	1	2	yes	—
270	#532	C₂²×C₄₇	1	1	yes	characteristic
271	C₂×C₄⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
272	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
273	#489	—	2	2	no	—
274	#487	—	2	1	no	—
275	#486	—	2	1	no	—
276	#474	—	2	1	no	—
277	#474	—	2	2	no	—
278	#474	—	2	1	no	—
279	#463	—	2	2	no	—
280	#461	—	2	1	no	—
281	#459	—	2	1	no	—
282	#436	—	2	1	no	—
283	#435	—	2	2	no	—
284	C₈×D₄	—	2	1	no	—
285	C₂×#108	C₂²×C₄₇	1	2	yes	—
286	C₂×#107	C₂²×C₄₇	1	2	yes	—
287	#423	C₂²×C₄₇	1	1	yes	characteristic
288	C₂×C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
289	#417	C₂²×C₄₇	1	2	yes	—
290	#417	C₂²×C₄₇	1	2	yes	—
291	#417	C₂²×C₄₇	1	2	yes	—
292	C₂×#102	C₂²×C₄₇	1	2	yes	—
293	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
294	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
295	#408	C₂²×C₄₇	1	2	yes	—
296	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
297	C₄×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
298	#343	C₂²×C₄₇	1	2	yes	—
299	#343	C₂²×C₄₇	1	1	yes	characteristic
300	#343	C₂²×C₄₇	1	2	yes	—
301	#343	C₂²×C₄₇	1	1	yes	characteristic
302	#341	C₂²×C₄₇	1	2	yes	—
303	C₄₇	#25358	1	1	yes	characteristic, direct product factor, Sylow
304	C₂×#41	C₂³×C₄₇	1	2	yes	—
305	C₂²×Q₈	—	2	1	no	—
306	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
307	C₂³×C₄	—	2	1	no	—
308	AGL(1,ℤ/8)	—	2	4	no	—
309	AGL(1,ℤ/8)	—	4	2	no	—
310	C₂×SD₁₆	—	2	2	no	—
311	C₂×D₈	—	2	2	no	—
312	#131	—	2	4	no	—
313	#131	—	4	2	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
316	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
317	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
318	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
319	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
320	C₂²×C₈	—	2	2	no	—
321	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
322	C₄⋊Q₈	—	2	1	no	—
323	C₄⋊Q₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	1	no	—
325	C₄⋊Q₈	—	2	1	no	—
326	C₄⋊Q₈	D₄×C₄₇	1	1	yes	characteristic
327	#127	D₄×C₄₇	1	1	yes	characteristic
328	#124	D₄×C₄₇	1	2	yes	—
329	#121	—	2	1	no	—
330	#121	—	2	1	no	—
331	C₄×D₄	—	2	1	no	—
332	C₄×D₄	—	2	1	no	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
340	C₂×C₂²⋊C₄	—	2	1	no	—
341	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
342	#108	—	4	2	no	—
343	#107	—	2	2	no	—
344	#107	—	2	2	no	—
345	C₄⋊C₈	—	2	1	no	—
346	C₄⋊C₈	—	2	1	no	—
347	C₄⋊C₈	D₄×C₄₇	1	1	yes	characteristic
348	C₄⋊C₈	D₄×C₄₇	1	2	yes	—
349	C₄⋊C₈	D₄×C₄₇	1	1	yes	characteristic
350	#103	—	2	2	no	—
351	#103	—	2	2	no	—
352	#103	—	2	2	no	—
353	#102	—	2	2	no	—
354	#102	—	2	2	no	—
355	#102	—	2	2	no	—
356	#102	—	2	2	no	—
357	#102	—	2	2	no	—
358	#100	—	2	2	no	—
359	#100	—	2	2	no	—
360	C₂²⋊C₈	—	2	2	no	—
361	C₂²⋊C₈	—	2	2	no	—
362	#97	—	2	1	no	—
363	#97	—	2	1	no	—
364	C₄×C₈	—	2	1	no	—
365	C₄×C₈	—	2	1	no	—
366	C₂⁴	—	2	1	no	—
367	#41	—	2	2	no	—
368	#41	—	2	2	no	—
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	C₂×Q₈	—	4	1	no	—
372	C₂×Q₈	—	4	1	no	—
373	C₂×Q₈	—	8	2	no	—
374	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
384	C₂×D₄	#41×C₄₇	1	2	yes	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	4	1	no	—
389	C₂²×C₄	—	4	1	no	—
390	C₂²×C₄	—	8	2	no	—
391	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
392	C₂²×C₄	#41×C₄₇	1	1	yes	characteristic
393	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
394	SD₁₆	—	4	4	no	—
395	D₈	—	4	4	no	—
396	M₁₆	—	2	1	no	—
397	M₁₆	—	2	1	no	—
398	M₁₆	—	2	1	no	—
399	M₁₆	—	2	1	no	—
400	M₁₆	—	2	2	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	1	no	—
404	M₁₆	—	2	1	no	—
405	M₁₆	—	4	4	no	—
406	M₁₆	#41×C₄₇	1	2	yes	—
407	C₂×C₈	—	2	1	no	—
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	4	4	no	—
415	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
416	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	4	1	no	—
424	C₄⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	4	1	no	—
428	C₂²⋊C₄	—	4	1	no	—
429	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
431	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	8	1	no	—
438	C₂³	—	8	1	no	—
439	C₂³	C₂×#41×C₄₇	1	1	yes	characteristic
440	Q₈	—	2	2	no	—
441	Q₈	—	2	2	no	—
442	Q₈	—	8	2	no	—
443	Q₈	—	8	2	no	—
444	D₄	—	2	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	1	no	—
452	D₄	—	4	1	no	—
453	D₄	—	8	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	2	no	—
462	C₂×C₄	—	8	2	no	—
463	C₂×C₄	—	8	1	no	—
464	C₂×C₄	—	8	2	no	—
465	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
466	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
467	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
468	C₂×C₄	#121×C₄₇	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
473	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
474	C₂×C₄	2₋¹⁺⁴×C₄₇	1	1	yes	characteristic
475	C₈	—	2	2	no	—
476	C₈	—	2	2	no	—
477	C₈	—	2	2	no	—
478	C₈	—	4	1	no	—
479	C₈	—	4	2	no	—
480	C₈	—	4	1	no	—
481	C₈	—	4	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	8	1	no	—
491	C₂²	—	8	1	no	—
492	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
493	C₂²	C₄₇×#529	1	1	yes	characteristic
494	C₂²	C₄₇×#542	1	1	yes	characteristic
495	C₄	—	2	1	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	4	2	no	—
499	C₄	—	8	1	no	—
500	C₄	—	8	2	no	—
501	C₄	C₄₇×#472	1	2	yes	—
502	C₄	C₄₇×#546	1	1	yes	characteristic
503	C₄	C₄₇×#546	1	1	yes	characteristic
504	C₄	C₄₇×#547	1	1	yes	characteristic
505	C₄	C₄₇×#548	1	1	yes	characteristic
506	C₂	—	2	1	no	—
507	C₂	—	2	1	no	—
508	C₂	—	8	2	no	—
509	C₂	—	8	2	no	—
510	C₂	C₄₇×#2427	1	1	yes	characteristic
511	C₂	C₂×C₄₇×#472	1	1	yes	characteristic
512	C₂	C₄₇×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₄₇×#3036	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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×#3141	C₂	semidirect
2. C₂×C₄₇×#474	C₂	semidirect
4. C₄₇×#2935	C₂	semidirect
5. C₄₇×#2078	C₂	semidirect
6. C₄₇×#2051	C₂	semidirect
7. C₄₇×#1997	C₂	semidirect
8. C₄₇×#1950	C₂	semidirect
9. C₄₇×#1907	C₂	semidirect
10. C₄₇×#1816	C₂	semidirect
11. C₄×#100×C₄₇	C₂	semidirect
225. #25358	C₄₇	direct (N × H)
250. #3141	C₂×C₄₇	semidirect
251. C₂×#474	C₂×C₄₇	semidirect
253. #2935	C₂×C₄₇	semidirect
254. #2078	C₂×C₄₇	semidirect
255. #2051	C₂×C₄₇	semidirect
256. #1997	C₂×C₄₇	semidirect
257. #1950	C₂×C₄₇	semidirect
258. #1907	C₂×C₄₇	semidirect
259. #1816	C₂×C₄₇	semidirect
260. C₄×#100	C₂×C₄₇	semidirect
303. C₄₇	#25358	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×#3141	2
C₂×C₄₇×#474	2
C₄₇×#3036	2
C₄₇×#2935	2
C₄₇×#2078	2
C₄₇×#2051	2
C₄₇×#1997	2
C₄₇×#1950	2
C₄₇×#1907	2
C₄₇×#1816	2
C₄×#100×C₄₇	2
#25358	47

Subgroups — C47×#25358 #26065056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#25358 #26065056