Subgroups — C₄₇×#24411 #26064109

Order 12032.

1294 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3586	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#3044	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3028	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3011	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2884	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2053	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2009	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1909	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1905	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1868	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1794	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#585	C₂²	1	1	yes	characteristic
13	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#578	—	2	4	no	—
15	C₄₇×#577	—	2	4	no	—
16	C₄₇×#575	—	2	2	no	—
17	C₄₇×#575	C₂²	1	2	yes	—
18	C₂×#137×C₄₇	—	2	2	no	—
19	C₂×#137×C₄₇	C₂²	1	2	yes	—
20	C₂×#137×C₄₇	C₂²	1	1	yes	characteristic, complemented
21	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
22	C₄₇×#568	C₂²	1	1	yes	characteristic
23	C₄₇×#536	C₂²	1	1	yes	characteristic
24	C₄₇×#536	C₂²	1	2	yes	—
25	C₄₇×#519	C₂²	1	2	yes	complemented
26	C₄₇×#464	—	2	4	no	—
27	C₄₇×#462	—	2	4	no	—
28	C₄₇×#458	—	2	4	no	—
29	C₄₇×#452	—	2	2	no	—
30	C₄₇×#451	—	2	2	no	—
31	C₄₇×#424	C₂²	1	2	yes	complemented
32	C₄₇×#421	C₂²	1	2	yes	—
33	C₄₇×#419	C₂²	1	2	yes	complemented
34	C₄₇×#419	C₂²	1	1	yes	characteristic, complemented
35	C₄₇×#416	C₂²	1	2	yes	—
36	C₄₇×#416	C₂²	1	1	yes	characteristic, complemented
37	C₄₇×#410	—	2	4	no	—
38	C₄₇×#410	C₂²	1	1	yes	characteristic
39	C₄₇×#410	C₂²	1	2	yes	—
40	C₄₇×#410	C₂²	1	1	yes	characteristic
41	C₄₇×#410	C₂²	1	2	yes	complemented
42	C₄₇×#408	C₂²	1	1	yes	characteristic
43	C₄₇×#344	C₂²	1	2	yes	complemented
44	C₄₇×H(3,ℤ/4)	C₂²	1	2	yes	complemented
45	2₋¹⁺⁴×C₄₇	—	2	4	no	—
46	2₋¹⁺⁴×C₄₇	—	2	2	no	—
47	2₋¹⁺⁴×C₄₇	—	2	1	no	—
48	2₋¹⁺⁴×C₄₇	—	4	2	no	—
49	H(5,𝔽₂)×C₄₇	—	2	2	no	—
50	H(5,𝔽₂)×C₄₇	—	2	2	no	—
51	H(5,𝔽₂)×C₄₇	—	2	1	no	—
52	C₂×#41×C₄₇	—	2	2	no	—
53	C₂×#41×C₄₇	—	2	1	no	—
54	C₂×#41×C₄₇	—	2	2	no	—
55	C₂×#41×C₄₇	—	4	2	no	—
56	C₂×#41×C₄₇	D₄	1	2	yes	—
57	C₂×#41×C₄₇	C₂³	1	2	yes	—
58	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
59	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₄₇	—	2	1	no	—
61	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
62	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
64	#137×C₄₇	—	2	8	no	—
65	#137×C₄₇	—	2	4	no	—
66	#137×C₄₇	—	2	2	no	—
67	#137×C₄₇	—	2	4	no	—
68	#137×C₄₇	—	2	4	no	—
69	#137×C₄₇	—	2	2	no	—
70	#137×C₄₇	—	4	2	no	—
71	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
72	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
73	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
74	#135×C₄₇	—	2	8	no	—
75	#135×C₄₇	—	2	4	no	—
76	C₂×Q₁₆×C₄₇	—	2	4	no	—
77	C₂×Q₁₆×C₄₇	—	2	1	no	—
78	C₂×Q₁₆×C₄₇	—	2	2	no	—
79	C₂×Q₁₆×C₄₇	—	2	4	no	—
80	C₂×Q₁₆×C₄₇	D₄	1	4	yes	—
81	C₂×SD₁₆×C₄₇	—	2	4	no	—
82	C₂×SD₁₆×C₄₇	—	2	1	no	—
83	C₂×SD₁₆×C₄₇	D₄	1	4	yes	—
84	#131×C₄₇	—	2	4	no	—
85	C₂×M₁₆×C₄₇	—	2	2	no	—
86	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
88	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
89	C₄⋊Q₈×C₄₇	—	2	2	no	—
90	C₄⋊Q₈×C₄₇	—	2	1	no	—
91	#124×C₄₇	—	2	2	no	—
92	#124×C₄₇	—	2	1	no	—
93	#123×C₄₇	—	2	2	no	—
94	#123×C₄₇	—	2	4	no	—
95	C₂²⋊Q₈×C₄₇	—	2	2	no	—
96	C₂²⋊Q₈×C₄₇	—	4	2	no	—
97	C₄×Q₈×C₄₇	—	2	4	no	—
98	C₄×Q₈×C₄₇	—	4	2	no	—
99	#117×C₄₇	—	2	4	no	—
100	#117×C₄₇	C₂³	1	2	yes	—
101	#117×C₄₇	C₂³	1	1	yes	characteristic
102	#117×C₄₇	C₂³	1	2	yes	—
103	C₄⋊C₈×C₄₇	—	2	4	no	—
104	#104×C₄₇	—	4	2	no	—
105	#103×C₄₇	—	2	1	no	—
106	#103×C₄₇	—	2	2	no	—
107	#103×C₄₇	—	2	2	no	—
108	#103×C₄₇	—	2	4	no	—
109	#102×C₄₇	—	2	1	no	—
110	#102×C₄₇	—	2	2	no	—
111	#99×C₄₇	—	2	2	no	—
112	#99×C₄₇	—	2	4	no	—
113	#99×C₄₇	—	2	4	no	—
114	#99×C₄₇	D₄	1	4	yes	—
115	C₂²⋊C₈×C₄₇	—	2	2	no	—
116	#41×C₄₇	—	2	2	no	—
117	#41×C₄₇	—	2	2	no	—
118	#41×C₄₇	—	2	4	no	—
119	#41×C₄₇	—	2	4	no	—
120	#41×C₄₇	—	2	2	no	—
121	#41×C₄₇	—	2	2	no	—
122	#41×C₄₇	—	2	2	no	—
123	#41×C₄₇	—	2	2	no	—
124	#41×C₄₇	—	2	2	no	—
125	#41×C₄₇	—	2	2	no	—
126	#41×C₄₇	—	4	2	no	—
127	#41×C₄₇	—	4	8	no	—
128	#41×C₄₇	—	4	4	no	—
129	#41×C₄₇	—	4	2	no	—
130	#41×C₄₇	—	4	4	no	—
131	#41×C₄₇	—	4	2	no	—
132	#41×C₄₇	—	4	2	no	—
133	#41×C₄₇	—	4	2	no	—
134	C₂×Q₈×C₄₇	—	2	2	no	—
135	C₂×Q₈×C₄₇	—	2	1	no	—
136	C₂×Q₈×C₄₇	—	2	2	no	—
137	C₂×Q₈×C₄₇	—	2	2	no	—
138	C₂×Q₈×C₄₇	—	2	4	no	—
139	C₂×Q₈×C₄₇	—	2	2	no	—
140	C₂×Q₈×C₄₇	—	2	1	no	—
141	C₂×Q₈×C₄₇	—	2	1	no	—
142	C₂×Q₈×C₄₇	—	2	2	no	—
143	C₂×Q₈×C₄₇	—	4	2	no	—
144	C₂×Q₈×C₄₇	—	4	4	no	—
145	C₂×Q₈×C₄₇	—	4	2	no	—
146	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
147	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
148	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
150	C₂×Q₈×C₄₇	#41	1	1	yes	characteristic
151	C₂×D₄×C₄₇	—	2	2	no	—
152	C₂×D₄×C₄₇	—	2	2	no	—
153	C₂×D₄×C₄₇	—	2	2	no	—
154	C₂×D₄×C₄₇	—	2	1	no	—
155	C₂×D₄×C₄₇	—	4	2	no	—
156	C₂×D₄×C₄₇	—	4	4	no	—
157	C₂×D₄×C₄₇	—	4	2	no	—
158	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
159	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₄₇	#41	1	1	yes	characteristic
162	C₂²×C₄×C₄₇	—	2	1	no	—
163	C₂²×C₄×C₄₇	—	2	2	no	—
164	C₂²×C₄×C₄₇	—	2	2	no	—
165	C₂²×C₄×C₄₇	—	2	1	no	—
166	C₂²×C₄×C₄₇	—	2	2	no	—
167	C₂²×C₄×C₄₇	—	2	1	no	—
168	C₂²×C₄×C₄₇	—	4	2	no	—
169	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₄₇	—	2	4	no	—
172	Q₁₆×C₄₇	—	2	8	no	—
173	Q₁₆×C₄₇	—	4	2	no	—
174	Q₁₆×C₄₇	—	4	4	no	—
175	SD₁₆×C₄₇	—	2	4	no	—
176	SD₁₆×C₄₇	—	2	4	no	—
177	SD₁₆×C₄₇	—	2	8	no	—
178	SD₁₆×C₄₇	—	4	2	no	—
179	D₈×C₄₇	—	2	4	no	—
180	M₁₆×C₄₇	—	2	4	no	—
181	M₁₆×C₄₇	—	2	2	no	—
182	M₁₆×C₄₇	—	2	2	no	—
183	M₁₆×C₄₇	—	2	2	no	—
184	C₂×C₈×C₄₇	—	2	4	no	—
185	C₂×C₈×C₄₇	—	2	1	no	—
186	C₂×C₈×C₄₇	—	2	2	no	—
187	C₂×C₈×C₄₇	—	2	1	no	—
188	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
189	C₄⋊C₄×C₄₇	—	2	2	no	—
190	C₄⋊C₄×C₄₇	—	2	2	no	—
191	C₄⋊C₄×C₄₇	—	2	1	no	—
192	C₄⋊C₄×C₄₇	—	4	4	no	—
193	C₄⋊C₄×C₄₇	—	4	4	no	—
194	C₄⋊C₄×C₄₇	—	4	2	no	—
195	C₂²⋊C₄×C₄₇	—	2	2	no	—
196	C₂²⋊C₄×C₄₇	—	2	1	no	—
197	C₂²⋊C₄×C₄₇	—	4	4	no	—
198	C₂²⋊C₄×C₄₇	—	4	2	no	—
199	C₂²⋊C₄×C₄₇	C₂×D₄	1	4	yes	—
200	C₄²×C₄₇	—	2	2	no	—
201	C₄²×C₄₇	—	2	1	no	—
202	C₄²×C₄₇	—	2	2	no	—
203	C₄²×C₄₇	—	4	4	no	—
204	C₂³×C₄₇	—	2	1	no	—
205	C₂³×C₄₇	—	2	2	no	—
206	C₂³×C₄₇	—	4	2	no	—
207	C₂³×C₄₇	#121	1	2	yes	—
208	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₄₇	—	2	2	no	—
210	Q₈×C₄₇	—	2	4	no	—
211	Q₈×C₄₇	—	2	2	no	—
212	Q₈×C₄₇	—	2	1	no	—
213	Q₈×C₄₇	—	2	2	no	—
214	Q₈×C₄₇	—	2	1	no	—
215	Q₈×C₄₇	—	2	2	no	—
216	Q₈×C₄₇	—	4	4	no	—
217	Q₈×C₄₇	—	4	2	no	—
218	Q₈×C₄₇	—	4	1	no	—
219	Q₈×C₄₇	—	4	1	no	—
220	Q₈×C₄₇	—	4	2	no	—
221	Q₈×C₄₇	#120	1	4	yes	—
222	D₄×C₄₇	—	2	4	no	—
223	D₄×C₄₇	—	2	1	no	—
224	D₄×C₄₇	—	2	2	no	—
225	D₄×C₄₇	—	2	1	no	—
226	D₄×C₄₇	—	2	2	no	—
227	D₄×C₄₇	—	2	2	no	—
228	D₄×C₄₇	—	4	4	no	—
229	D₄×C₄₇	—	4	2	no	—
230	D₄×C₄₇	—	4	4	no	—
231	D₄×C₄₇	—	4	2	no	—
232	D₄×C₄₇	—	4	2	no	—
233	D₄×C₄₇	—	4	2	no	—
234	C₂×C₄×C₄₇	—	2	2	no	—
235	C₂×C₄×C₄₇	—	2	4	no	—
236	C₂×C₄×C₄₇	—	2	2	no	—
237	C₂×C₄×C₄₇	—	2	2	no	—
238	C₂×C₄×C₄₇	—	2	1	no	—
239	C₂×C₄×C₄₇	—	2	4	no	—
240	C₂×C₄×C₄₇	—	2	2	no	—
241	C₂×C₄×C₄₇	—	4	4	no	—
242	C₂×C₄×C₄₇	—	4	4	no	—
243	C₂×C₄×C₄₇	—	4	4	no	—
244	C₂×C₄×C₄₇	—	4	2	no	—
245	C₂×C₄×C₄₇	—	4	2	no	—
246	C₂×C₄×C₄₇	—	4	2	no	—
247	C₂×C₄×C₄₇	—	4	2	no	—
248	C₂×C₄×C₄₇	—	4	1	no	—
249	C₂×C₄×C₄₇	—	4	1	no	—
250	C₂×C₄×C₄₇	—	4	2	no	—
251	C₂×C₄×C₄₇	—	4	2	no	—
252	C₂×C₄×C₄₇	#121	1	2	yes	—
253	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
257	C₈×C₄₇	—	2	4	no	—
258	C₈×C₄₇	—	4	1	no	—
259	C₈×C₄₇	—	4	1	no	—
260	#24411	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂²×C₄₇	—	2	2	no	—
262	C₂²×C₄₇	—	4	1	no	—
263	C₂²×C₄₇	—	4	2	no	—
264	C₂²×C₄₇	—	4	1	no	—
265	C₂²×C₄₇	—	4	2	no	—
266	C₂²×C₄₇	—	8	2	no	—
267	C₂²×C₄₇	—	8	1	no	—
268	C₂²×C₄₇	—	16	2	no	—
269	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
270	C₂²×C₄₇	D₄²	1	2	yes	—
271	C₄×C₄₇	—	2	4	no	—
272	C₄×C₄₇	—	2	2	no	—
273	C₄×C₄₇	—	4	4	no	—
274	C₄×C₄₇	—	4	2	no	—
275	C₄×C₄₇	—	4	1	no	—
276	C₄×C₄₇	—	8	2	no	—
277	C₄×C₄₇	—	8	1	no	—
278	C₄×C₄₇	—	8	1	no	—
279	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
280	C₄×C₄₇	#534	1	1	yes	characteristic
281	C₄×C₄₇	#546	1	2	yes	—
282	#3586	C₂×C₄₇	1	1	yes	characteristic
283	#3044	C₂×C₄₇	1	2	yes	complemented
284	#3028	C₂×C₄₇	1	1	yes	characteristic, complemented
285	#3011	C₂×C₄₇	1	1	yes	characteristic, complemented
286	#2884	C₂×C₄₇	1	1	yes	characteristic, complemented
287	#2053	C₂×C₄₇	1	2	yes	complemented
288	#2009	C₂×C₄₇	1	2	yes	complemented
289	#1909	C₂×C₄₇	1	1	yes	characteristic, complemented
290	#1905	C₂×C₄₇	1	2	yes	complemented
291	#1868	C₂×C₄₇	1	1	yes	characteristic, complemented
292	#1794	C₂×C₄₇	1	1	yes	characteristic, complemented
293	C₂×C₄₇	—	2	2	no	—
294	C₂×C₄₇	—	2	1	no	—
295	C₂×C₄₇	—	4	2	no	—
296	C₂×C₄₇	—	8	1	no	—
297	C₂×C₄₇	—	8	2	no	—
298	C₂×C₄₇	#2410	1	1	yes	characteristic
299	#585	C₂²×C₄₇	1	1	yes	characteristic
300	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	1	yes	characteristic
301	#578	—	2	4	no	—
302	#577	—	2	4	no	—
303	#575	—	2	2	no	—
304	#575	C₂²×C₄₇	1	2	yes	—
305	C₂×#137	—	2	2	no	—
306	C₂×#137	C₂²×C₄₇	1	2	yes	—
307	C₂×#137	C₂²×C₄₇	1	1	yes	characteristic, complemented
308	C₂²×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic, complemented
309	#568	C₂²×C₄₇	1	1	yes	characteristic
310	#536	C₂²×C₄₇	1	1	yes	characteristic
311	#536	C₂²×C₄₇	1	2	yes	—
312	#519	C₂²×C₄₇	1	2	yes	complemented
313	#464	—	2	4	no	—
314	#462	—	2	4	no	—
315	#458	—	2	4	no	—
316	#452	—	2	2	no	—
317	#451	—	2	2	no	—
318	#424	C₂²×C₄₇	1	2	yes	complemented
319	#421	C₂²×C₄₇	1	2	yes	—
320	#419	C₂²×C₄₇	1	1	yes	characteristic, complemented
321	#419	C₂²×C₄₇	1	2	yes	complemented
322	#416	C₂²×C₄₇	1	2	yes	—
323	#416	C₂²×C₄₇	1	1	yes	characteristic, complemented
324	#410	—	2	4	no	—
325	#410	C₂²×C₄₇	1	1	yes	characteristic
326	#410	C₂²×C₄₇	1	2	yes	—
327	#410	C₂²×C₄₇	1	1	yes	characteristic
328	#410	C₂²×C₄₇	1	2	yes	complemented
329	#408	C₂²×C₄₇	1	1	yes	characteristic
330	#344	C₂²×C₄₇	1	2	yes	complemented
331	H(3,ℤ/4)	C₂²×C₄₇	1	2	yes	complemented
332	C₄₇	#24411	1	1	yes	characteristic, direct product factor, Sylow
333	2₋¹⁺⁴	—	2	4	no	—
334	2₋¹⁺⁴	—	2	2	no	—
335	2₋¹⁺⁴	—	2	1	no	—
336	2₋¹⁺⁴	—	4	2	no	—
337	H(5,𝔽₂)	—	2	2	no	—
338	H(5,𝔽₂)	—	2	2	no	—
339	H(5,𝔽₂)	—	2	1	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	4	2	no	—
344	C₂×#41	D₄×C₄₇	1	2	yes	—
345	C₂×#41	C₂³×C₄₇	1	2	yes	—
346	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
347	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
348	C₂²×Q₈	—	2	1	no	—
349	C₂²×Q₈	D₄×C₄₇	1	2	yes	—
350	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
351	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
352	#137	—	2	8	no	—
353	#137	—	2	4	no	—
354	#137	—	2	2	no	—
355	#137	—	2	4	no	—
356	#137	—	2	4	no	—
357	#137	—	2	2	no	—
358	#137	—	4	2	no	—
359	AGL(1,ℤ/8)	—	2	4	no	—
360	AGL(1,ℤ/8)	—	2	2	no	—
361	AGL(1,ℤ/8)	—	2	2	no	—
362	#135	—	2	8	no	—
363	#135	—	2	4	no	—
364	C₂×Q₁₆	—	2	4	no	—
365	C₂×Q₁₆	—	2	2	no	—
366	C₂×Q₁₆	—	2	4	no	—
367	C₂×Q₁₆	—	2	1	no	—
368	C₂×Q₁₆	D₄×C₄₇	1	4	yes	—
369	C₂×SD₁₆	—	2	4	no	—
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×SD₁₆	D₄×C₄₇	1	4	yes	—
372	#131	—	2	4	no	—
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
375	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
376	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
377	C₄⋊Q₈	—	2	2	no	—
378	C₄⋊Q₈	—	2	1	no	—
379	#124	—	2	2	no	—
380	#124	—	2	1	no	—
381	#123	—	2	2	no	—
382	#123	—	2	4	no	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	4	2	no	—
385	C₄×Q₈	—	2	4	no	—
386	C₄×Q₈	—	4	2	no	—
387	#117	—	2	4	no	—
388	#117	C₂³×C₄₇	1	2	yes	—
389	#117	C₂³×C₄₇	1	1	yes	characteristic
390	#117	C₂³×C₄₇	1	2	yes	—
391	C₄⋊C₈	—	2	4	no	—
392	#104	—	4	2	no	—
393	#103	—	2	2	no	—
394	#103	—	2	1	no	—
395	#103	—	2	2	no	—
396	#103	—	2	4	no	—
397	#102	—	2	2	no	—
398	#102	—	2	1	no	—
399	#99	—	2	2	no	—
400	#99	—	2	4	no	—
401	#99	—	2	4	no	—
402	#99	D₄×C₄₇	1	4	yes	—
403	C₂²⋊C₈	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	4	no	—
407	#41	—	2	4	no	—
408	#41	—	2	2	no	—
409	#41	—	2	2	no	—
410	#41	—	2	2	no	—
411	#41	—	2	2	no	—
412	#41	—	2	2	no	—
413	#41	—	2	2	no	—
414	#41	—	4	2	no	—
415	#41	—	4	8	no	—
416	#41	—	4	4	no	—
417	#41	—	4	2	no	—
418	#41	—	4	4	no	—
419	#41	—	4	2	no	—
420	#41	—	4	2	no	—
421	#41	—	4	2	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	4	2	no	—
432	C₂×Q₈	—	4	4	no	—
433	C₂×Q₈	—	4	2	no	—
434	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
435	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
436	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
437	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
438	C₂×Q₈	#41×C₄₇	1	1	yes	characteristic
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	4	2	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
447	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
448	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
449	C₂×D₄	#41×C₄₇	1	1	yes	characteristic
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
459	Q₁₆	—	2	4	no	—
460	Q₁₆	—	2	8	no	—
461	Q₁₆	—	4	2	no	—
462	Q₁₆	—	4	4	no	—
463	SD₁₆	—	2	4	no	—
464	SD₁₆	—	2	4	no	—
465	SD₁₆	—	2	8	no	—
466	SD₁₆	—	4	2	no	—
467	D₈	—	2	4	no	—
468	M₁₆	—	2	4	no	—
469	M₁₆	—	2	2	no	—
470	M₁₆	—	2	2	no	—
471	M₁₆	—	2	2	no	—
472	C₂×C₈	—	2	4	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	4	4	no	—
481	C₄⋊C₄	—	4	4	no	—
482	C₄⋊C₄	—	4	2	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	4	4	no	—
486	C₂²⋊C₄	—	4	2	no	—
487	C₂²⋊C₄	C₂×D₄×C₄₇	1	4	yes	—
488	C₄²	—	2	2	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	4	4	no	—
492	C₂³	—	2	1	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	#121×C₄₇	1	2	yes	—
496	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
497	Q₈	—	2	2	no	—
498	Q₈	—	2	4	no	—
499	Q₈	—	2	2	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	2	2	no	—
502	Q₈	—	2	1	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	4	4	no	—
505	Q₈	—	4	2	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	2	no	—
509	Q₈	#120×C₄₇	1	4	yes	—
510	D₄	—	2	4	no	—
511	D₄	—	2	1	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	1	no	—
514	D₄	—	2	2	no	—
515	D₄	—	2	2	no	—
516	D₄	—	4	4	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	4	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	#121×C₄₇	1	2	yes	—
541	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
543	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
544	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
545	C₈	—	2	4	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	1	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	8	2	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	16	2	no	—
556	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
557	C₂²	D₄²×C₄₇	1	2	yes	—
558	C₄	—	2	4	no	—
559	C₄	—	2	2	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	1	no	—
563	C₄	—	8	2	no	—
564	C₄	—	8	1	no	—
565	C₄	—	8	1	no	—
566	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
567	C₄	C₄₇×#534	1	1	yes	characteristic
568	C₄	C₄₇×#546	1	2	yes	—
569	C₂	—	2	2	no	—
570	C₂	—	2	1	no	—
571	C₂	—	4	2	no	—
572	C₂	—	8	2	no	—
573	C₂	—	8	1	no	—
574	C₂	C₄₇×#2410	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	#2410
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₄₇×#3586	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#3044	C₂	semidirect
3. C₄₇×#3028	C₂	semidirect
4. C₄₇×#3011	C₂	semidirect
5. C₄₇×#2884	C₂	semidirect
6. C₄₇×#2053	C₂	semidirect
7. C₄₇×#2009	C₂	semidirect
8. C₄₇×#1909	C₂	semidirect
9. C₄₇×#1905	C₂	semidirect
10. C₄₇×#1868	C₂	semidirect
11. C₄₇×#1794	C₂	semidirect
20. C₂×#137×C₄₇	C₂²	semidirect
21. C₂²×Q₁₆×C₄₇	C₂²	semidirect
25. C₄₇×#519	C₂²	semidirect
31. C₄₇×#424	C₂²	semidirect
33. C₄₇×#419	C₂²	semidirect
34. C₄₇×#419	C₂²	semidirect
36. C₄₇×#416	C₂²	semidirect
41. C₄₇×#410	C₂²	semidirect
43. C₄₇×#344	C₂²	semidirect
44. C₄₇×H(3,ℤ/4)	C₂²	semidirect
260. #24411	C₄₇	direct (N × H)
283. #3044	C₂×C₄₇	semidirect
284. #3028	C₂×C₄₇	semidirect
285. #3011	C₂×C₄₇	semidirect
286. #2884	C₂×C₄₇	semidirect
287. #2053	C₂×C₄₇	semidirect
288. #2009	C₂×C₄₇	semidirect
289. #1909	C₂×C₄₇	semidirect
290. #1905	C₂×C₄₇	semidirect
291. #1868	C₂×C₄₇	semidirect
292. #1794	C₂×C₄₇	semidirect
307. C₂×#137	C₂²×C₄₇	semidirect
308. C₂²×Q₁₆	C₂²×C₄₇	semidirect
312. #519	C₂²×C₄₇	semidirect
318. #424	C₂²×C₄₇	semidirect
320. #419	C₂²×C₄₇	semidirect
321. #419	C₂²×C₄₇	semidirect
323. #416	C₂²×C₄₇	semidirect
328. #410	C₂²×C₄₇	semidirect
330. #344	C₂²×C₄₇	semidirect
331. H(3,ℤ/4)	C₂²×C₄₇	semidirect
332. C₄₇	#24411	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×#3586	2
C₄₇×#3044	2
C₄₇×#3028	2
C₄₇×#3011	2
C₄₇×#2884	2
C₄₇×#2053	2
C₄₇×#2009	2
C₄₇×#1909	2
C₄₇×#1905	2
C₄₇×#1868	2
C₄₇×#1794	2
#24411	47

Subgroups — C47×#24411 #26064109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#24411 #26064109