Subgroups — C₄₇×#22461 #26062159

Order 12032.

1316 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3584	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#2943	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#2901	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#1975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1906	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1881	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#1794	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#585	C₂²	1	2	yes	—
13	C₄₇×#578	—	2	1	no	—
14	C₄₇×#578	C₄	1	2	yes	complemented
15	C₄₇×#577	—	2	2	no	—
16	C₄₇×#577	C₄	1	4	yes	complemented
17	C₄₇×#576	—	2	1	no	—
18	C₄₇×#576	C₄	1	2	yes	complemented
19	C₄₇×#575	—	2	2	no	—
20	C₄₇×#575	C₂²	1	2	yes	—
21	C₂×#135×C₄₇	—	2	2	no	—
22	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#568	C₂²	1	2	yes	complemented
25	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
26	C₄×#41×C₄₇	C₂²	1	2	yes	complemented
27	C₄₇×#444	—	2	2	no	—
28	C₄₇×#443	—	2	2	no	—
29	C₄₇×#442	—	2	1	no	—
30	C₄₇×#441	—	2	1	no	—
31	C₄₇×#440	—	2	2	no	—
32	C₂×#108×C₄₇	C₂²	1	1	yes	characteristic, complemented
33	C₄₇×#427	C₂²	1	1	yes	characteristic, complemented
34	C₄₇×#421	—	2	4	no	—
35	C₄₇×#421	C₂²	1	2	yes	—
36	C₄₇×#421	C₂²	1	2	yes	complemented
37	C₄₇×#421	C₂²	1	2	yes	complemented
38	C₂×#104×C₄₇	—	2	4	no	—
39	C₂×#104×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×#104×C₄₇	C₂²	1	2	yes	complemented
41	C₄₇×#416	C₂²	1	1	yes	characteristic
42	C₄₇×#416	C₂²	1	2	yes	complemented
43	C₄₇×#410	C₂²	1	2	yes	complemented
44	C₄₇×#408	C₂²	1	2	yes	complemented
45	C₄₇×#405	C₂²	1	1	yes	characteristic, complemented
46	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic, complemented
47	C₄₇×#344	C₂²	1	2	yes	complemented
48	C₄₇×#341	C₂²	1	1	yes	characteristic, complemented
49	C₄₇×#339	C₂²	1	1	yes	characteristic, complemented
50	2₋¹⁺⁴×C₄₇	—	2	2	no	—
51	2₋¹⁺⁴×C₄₇	—	2	2	no	—
52	2₋¹⁺⁴×C₄₇	C₂×C₄	1	4	yes	complemented
53	H(5,𝔽₂)×C₄₇	—	2	2	no	—
54	H(5,𝔽₂)×C₄₇	—	2	2	no	—
55	H(5,𝔽₂)×C₄₇	—	4	2	no	—
56	H(5,𝔽₂)×C₄₇	C₂×C₄	1	4	yes	complemented
57	C₂×#41×C₄₇	—	2	4	no	—
58	C₂×#41×C₄₇	—	2	4	no	—
59	C₂×#41×C₄₇	—	4	2	no	—
60	C₂×#41×C₄₇	C₂³	1	2	yes	—
61	C₂×#41×C₄₇	C₂³	1	2	yes	—
62	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
63	#137×C₄₇	—	2	2	no	—
64	#137×C₄₇	—	2	4	no	—
65	#137×C₄₇	—	2	4	no	—
66	#137×C₄₇	C₂×C₄	1	4	yes	complemented
67	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
68	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
69	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
70	AGL(1,ℤ/8)×C₄₇	C₂×C₄	1	4	yes	complemented
71	#135×C₄₇	—	2	4	no	—
72	#135×C₄₇	—	2	2	no	—
73	#135×C₄₇	—	2	4	no	—
74	#135×C₄₇	—	2	2	no	—
75	#135×C₄₇	—	2	1	no	—
76	#135×C₄₇	—	2	1	no	—
77	#135×C₄₇	—	2	1	no	—
78	#135×C₄₇	—	2	1	no	—
79	#135×C₄₇	C₂×C₄	1	2	yes	—
80	#135×C₄₇	C₂×C₄	1	2	yes	—
81	#135×C₄₇	D₄	1	2	yes	—
82	#135×C₄₇	D₄	1	2	yes	—
83	C₂×Q₁₆×C₄₇	—	2	2	no	—
84	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
85	C₂×Q₁₆×C₄₇	D₄	1	1	yes	characteristic
86	C₂×SD₁₆×C₄₇	—	2	4	no	—
87	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
88	C₂×SD₁₆×C₄₇	D₄	1	2	yes	—
89	C₂×D₈×C₄₇	—	2	2	no	—
90	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂×D₈×C₄₇	D₄	1	1	yes	characteristic
92	#131×C₄₇	—	2	1	no	—
93	#131×C₄₇	—	2	1	no	—
94	#131×C₄₇	—	2	4	no	—
95	#131×C₄₇	—	4	2	no	—
96	#131×C₄₇	C₂×C₄	1	2	yes	complemented
97	#131×C₄₇	C₂×C₄	1	2	yes	complemented
98	C₂×M₁₆×C₄₇	—	2	1	no	—
99	C₂×M₁₆×C₄₇	—	2	4	no	—
100	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
101	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
102	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₄₇	—	2	1	no	—
104	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
106	C₄×Q₈×C₄₇	—	2	2	no	—
107	C₄×D₄×C₄₇	—	2	2	no	—
108	C₄×D₄×C₄₇	—	4	2	no	—
109	#117×C₄₇	—	2	2	no	—
110	#117×C₄₇	C₂³	1	2	yes	—
111	#117×C₄₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₄₇	—	2	2	no	—
113	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
114	#108×C₄₇	—	2	1	no	—
115	#108×C₄₇	—	2	1	no	—
116	#107×C₄₇	—	2	1	no	—
117	#106×C₄₇	—	2	1	no	—
118	#104×C₄₇	—	2	4	no	—
119	#104×C₄₇	—	2	4	no	—
120	#104×C₄₇	—	2	4	no	—
121	#104×C₄₇	—	2	2	no	—
122	#104×C₄₇	—	2	2	no	—
123	#104×C₄₇	D₄	1	4	yes	—
124	#103×C₄₇	—	2	2	no	—
125	#103×C₄₇	—	2	1	no	—
126	#102×C₄₇	—	2	2	no	—
127	#102×C₄₇	—	2	1	no	—
128	#99×C₄₇	—	4	2	no	—
129	C₂²⋊C₈×C₄₇	—	4	2	no	—
130	#97×C₄₇	—	2	1	no	—
131	#97×C₄₇	—	2	1	no	—
132	#97×C₄₇	—	2	1	no	—
133	C₄×C₈×C₄₇	—	2	1	no	—
134	#41×C₄₇	—	2	1	no	—
135	#41×C₄₇	—	2	2	no	—
136	#41×C₄₇	—	2	2	no	—
137	#41×C₄₇	—	2	1	no	—
138	#41×C₄₇	—	2	4	no	—
139	#41×C₄₇	—	2	4	no	—
140	#41×C₄₇	—	2	4	no	—
141	#41×C₄₇	—	2	4	no	—
142	#41×C₄₇	—	2	2	no	—
143	#41×C₄₇	—	2	2	no	—
144	#41×C₄₇	—	2	2	no	—
145	#41×C₄₇	—	2	2	no	—
146	#41×C₄₇	—	4	4	no	—
147	#41×C₄₇	—	4	4	no	—
148	#41×C₄₇	—	4	4	no	—
149	#41×C₄₇	—	4	4	no	—
150	#41×C₄₇	—	4	2	no	—
151	#41×C₄₇	—	4	2	no	—
152	#41×C₄₇	C₂²×C₄	1	4	yes	—
153	#41×C₄₇	C₂²×C₄	1	2	yes	—
154	#41×C₄₇	C₂×D₄	1	4	yes	—
155	#41×C₄₇	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₄₇	—	2	4	no	—
157	C₂×Q₈×C₄₇	—	2	4	no	—
158	C₂×Q₈×C₄₇	—	4	2	no	—
159	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
160	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
161	C₂×Q₈×C₄₇	#41	1	1	yes	characteristic
162	C₂×D₄×C₄₇	—	2	2	no	—
163	C₂×D₄×C₄₇	—	2	1	no	—
164	C₂×D₄×C₄₇	—	2	4	no	—
165	C₂×D₄×C₄₇	—	2	4	no	—
166	C₂×D₄×C₄₇	—	2	2	no	—
167	C₂×D₄×C₄₇	—	4	4	no	—
168	C₂×D₄×C₄₇	—	4	2	no	—
169	C₂×D₄×C₄₇	—	4	4	no	—
170	C₂×D₄×C₄₇	—	4	4	no	—
171	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
172	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
173	C₂×D₄×C₄₇	#41	1	1	yes	characteristic
174	C₂²×C₄×C₄₇	—	2	2	no	—
175	C₂²×C₄×C₄₇	—	2	1	no	—
176	C₂²×C₄×C₄₇	—	2	1	no	—
177	C₂²×C₄×C₄₇	—	2	2	no	—
178	C₂²×C₄×C₄₇	—	4	4	no	—
179	C₂²×C₄×C₄₇	—	4	2	no	—
180	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₄₇	—	2	4	no	—
182	Q₁₆×C₄₇	—	2	1	no	—
183	Q₁₆×C₄₇	—	2	1	no	—
184	Q₁₆×C₄₇	C₂²⋊C₄	1	2	yes	—
185	Q₁₆×C₄₇	C₂²⋊C₄	1	2	yes	—
186	SD₁₆×C₄₇	—	2	4	no	—
187	SD₁₆×C₄₇	—	2	4	no	—
188	SD₁₆×C₄₇	—	2	2	no	—
189	SD₁₆×C₄₇	—	2	2	no	—
190	SD₁₆×C₄₇	C₂²⋊C₄	1	4	yes	—
191	SD₁₆×C₄₇	C₂²⋊C₄	1	4	yes	—
192	D₈×C₄₇	—	2	4	no	—
193	D₈×C₄₇	—	2	1	no	—
194	D₈×C₄₇	—	2	1	no	—
195	D₈×C₄₇	C₂²⋊C₄	1	2	yes	—
196	D₈×C₄₇	C₂²⋊C₄	1	2	yes	—
197	M₁₆×C₄₇	—	2	1	no	—
198	M₁₆×C₄₇	—	2	1	no	—
199	M₁₆×C₄₇	—	2	2	no	—
200	M₁₆×C₄₇	—	2	2	no	—
201	M₁₆×C₄₇	—	2	2	no	—
202	M₁₆×C₄₇	—	2	2	no	—
203	M₁₆×C₄₇	—	4	4	no	—
204	M₁₆×C₄₇	C₂²×C₄	1	2	yes	—
205	M₁₆×C₄₇	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₄₇	—	2	1	no	—
207	C₂×C₈×C₄₇	—	2	1	no	—
208	C₂×C₈×C₄₇	—	2	2	no	—
209	C₂×C₈×C₄₇	—	2	2	no	—
210	C₂×C₈×C₄₇	—	2	1	no	—
211	C₂×C₈×C₄₇	—	2	1	no	—
212	C₂×C₈×C₄₇	—	2	1	no	—
213	C₂×C₈×C₄₇	—	2	2	no	—
214	C₂×C₈×C₄₇	—	4	4	no	—
215	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
216	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
217	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
218	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
219	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
220	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
221	C₄⋊C₄×C₄₇	—	2	2	no	—
222	C₄⋊C₄×C₄₇	—	2	1	no	—
223	C₄⋊C₄×C₄₇	—	4	2	no	—
224	C₂²⋊C₄×C₄₇	—	2	2	no	—
225	C₂²⋊C₄×C₄₇	—	2	1	no	—
226	C₂²⋊C₄×C₄₇	—	4	2	no	—
227	C₄²×C₄₇	—	2	1	no	—
228	C₄²×C₄₇	—	2	2	no	—
229	C₄²×C₄₇	—	2	1	no	—
230	C₄²×C₄₇	—	4	2	no	—
231	C₄²×C₄₇	C₂×D₄	1	2	yes	—
232	C₂³×C₄₇	—	2	2	no	—
233	C₂³×C₄₇	—	2	1	no	—
234	C₂³×C₄₇	—	2	2	no	—
235	C₂³×C₄₇	—	4	4	no	—
236	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
237	Q₈×C₄₇	—	2	1	no	—
238	Q₈×C₄₇	—	2	2	no	—
239	Q₈×C₄₇	—	2	2	no	—
240	Q₈×C₄₇	—	2	2	no	—
241	Q₈×C₄₇	—	4	4	no	—
242	Q₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
243	Q₈×C₄₇	C₄×D₄	1	2	yes	—
244	D₄×C₄₇	—	2	2	no	—
245	D₄×C₄₇	—	2	4	no	—
246	D₄×C₄₇	—	2	2	no	—
247	D₄×C₄₇	—	2	4	no	—
248	D₄×C₄₇	—	2	2	no	—
249	D₄×C₄₇	—	2	2	no	—
250	D₄×C₄₇	—	2	1	no	—
251	D₄×C₄₇	—	2	2	no	—
252	D₄×C₄₇	—	4	4	no	—
253	D₄×C₄₇	—	4	4	no	—
254	D₄×C₄₇	—	4	4	no	—
255	D₄×C₄₇	—	4	2	no	—
256	D₄×C₄₇	—	4	2	no	—
257	D₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
258	D₄×C₄₇	C₄×D₄	1	2	yes	—
259	C₂×C₄×C₄₇	—	2	2	no	—
260	C₂×C₄×C₄₇	—	2	1	no	—
261	C₂×C₄×C₄₇	—	2	2	no	—
262	C₂×C₄×C₄₇	—	2	4	no	—
263	C₂×C₄×C₄₇	—	2	4	no	—
264	C₂×C₄×C₄₇	—	2	2	no	—
265	C₂×C₄×C₄₇	—	2	2	no	—
266	C₂×C₄×C₄₇	—	2	2	no	—
267	C₂×C₄×C₄₇	—	4	4	no	—
268	C₂×C₄×C₄₇	—	4	4	no	—
269	C₂×C₄×C₄₇	—	4	4	no	—
270	C₂×C₄×C₄₇	—	4	2	no	—
271	C₂×C₄×C₄₇	—	4	2	no	—
272	C₂×C₄×C₄₇	—	4	2	no	—
273	C₂×C₄×C₄₇	—	4	1	no	—
274	C₂×C₄×C₄₇	—	8	1	no	—
275	C₂×C₄×C₄₇	—	8	4	no	—
276	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
277	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
281	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
282	C₈×C₄₇	—	2	2	no	—
283	C₈×C₄₇	—	2	2	no	—
284	C₈×C₄₇	—	4	2	no	—
285	C₈×C₄₇	—	4	1	no	—
286	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
287	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
288	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
289	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
290	#22461	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂²×C₄₇	—	2	2	no	—
292	C₂²×C₄₇	—	2	4	no	—
293	C₂²×C₄₇	—	4	1	no	—
294	C₂²×C₄₇	—	4	2	no	—
295	C₂²×C₄₇	—	8	2	no	—
296	C₂²×C₄₇	—	8	1	no	—
297	C₂²×C₄₇	—	8	2	no	—
298	C₂²×C₄₇	—	16	4	no	—
299	C₂²×C₄₇	#518	1	1	yes	characteristic
300	C₂²×C₄₇	D₄²	1	1	yes	characteristic
301	C₂²×C₄₇	#546	1	1	yes	characteristic
302	C₄×C₄₇	—	2	2	no	—
303	C₄×C₄₇	—	2	4	no	—
304	C₄×C₄₇	—	4	2	no	—
305	C₄×C₄₇	—	4	2	no	—
306	C₄×C₄₇	—	8	1	no	—
307	C₄×C₄₇	—	8	2	no	—
308	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
309	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
310	C₄×C₄₇	D₄²	1	1	yes	characteristic
311	C₄×C₄₇	#546	1	1	yes	characteristic
312	#3584	C₂×C₄₇	1	1	yes	characteristic
313	#2957	C₂×C₄₇	1	1	yes	characteristic, complemented
314	#2943	C₂×C₄₇	1	1	yes	characteristic, complemented
315	#2901	C₂×C₄₇	1	2	yes	complemented
316	#1976	C₂×C₄₇	1	1	yes	characteristic, complemented
317	#1975	C₂×C₄₇	1	1	yes	characteristic, complemented
318	#1906	C₂×C₄₇	1	2	yes	complemented
319	#1881	C₂×C₄₇	1	2	yes	complemented
320	#1814	C₂×C₄₇	1	1	yes	characteristic, complemented
321	#1794	C₂×C₄₇	1	2	yes	complemented
322	#1765	C₂×C₄₇	1	1	yes	characteristic, complemented
323	C₂×C₄₇	—	2	1	no	—
324	C₂×C₄₇	—	2	1	no	—
325	C₂×C₄₇	—	2	1	no	—
326	C₂×C₄₇	—	4	4	no	—
327	C₂×C₄₇	—	4	2	no	—
328	C₂×C₄₇	—	8	2	no	—
329	C₂×C₄₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
330	#585	C₂²×C₄₇	1	2	yes	—
331	#578	—	2	1	no	—
332	#578	C₄×C₄₇	1	2	yes	complemented
333	#577	—	2	2	no	—
334	#577	C₄×C₄₇	1	4	yes	complemented
335	#576	—	2	1	no	—
336	#576	C₄×C₄₇	1	2	yes	complemented
337	#575	—	2	2	no	—
338	#575	C₂²×C₄₇	1	2	yes	—
339	C₂×#135	—	2	2	no	—
340	C₂×#135	C₂²×C₄₇	1	1	yes	characteristic
341	C₂×#135	C₂²×C₄₇	1	1	yes	characteristic
342	#568	C₂²×C₄₇	1	2	yes	complemented
343	C₂×#131	C₂²×C₄₇	1	1	yes	characteristic
344	C₄×#41	C₂²×C₄₇	1	2	yes	complemented
345	#444	—	2	2	no	—
346	#443	—	2	2	no	—
347	#442	—	2	1	no	—
348	#441	—	2	1	no	—
349	#440	—	2	2	no	—
350	C₂×#108	C₂²×C₄₇	1	1	yes	characteristic, complemented
351	#427	C₂²×C₄₇	1	1	yes	characteristic, complemented
352	#421	—	2	4	no	—
353	#421	C₂²×C₄₇	1	2	yes	—
354	#421	C₂²×C₄₇	1	2	yes	complemented
355	#421	C₂²×C₄₇	1	2	yes	complemented
356	C₂×#104	—	2	4	no	—
357	C₂×#104	C₂²×C₄₇	1	1	yes	characteristic
358	C₂×#104	C₂²×C₄₇	1	2	yes	complemented
359	#416	C₂²×C₄₇	1	1	yes	characteristic
360	#416	C₂²×C₄₇	1	2	yes	complemented
361	#410	C₂²×C₄₇	1	2	yes	complemented
362	#408	C₂²×C₄₇	1	2	yes	complemented
363	#405	C₂²×C₄₇	1	1	yes	characteristic, complemented
364	C₂×#97	C₂²×C₄₇	1	1	yes	characteristic, complemented
365	#344	C₂²×C₄₇	1	2	yes	complemented
366	#341	C₂²×C₄₇	1	1	yes	characteristic, complemented
367	#339	C₂²×C₄₇	1	1	yes	characteristic, complemented
368	C₄₇	#22461	1	1	yes	characteristic, direct product factor, Sylow
369	2₋¹⁺⁴	—	2	2	no	—
370	2₋¹⁺⁴	—	2	2	no	—
371	2₋¹⁺⁴	C₂×C₄×C₄₇	1	4	yes	complemented
372	H(5,𝔽₂)	—	2	2	no	—
373	H(5,𝔽₂)	—	2	2	no	—
374	H(5,𝔽₂)	—	4	2	no	—
375	H(5,𝔽₂)	C₂×C₄×C₄₇	1	4	yes	complemented
376	C₂×#41	—	2	4	no	—
377	C₂×#41	—	2	4	no	—
378	C₂×#41	—	4	2	no	—
379	C₂×#41	C₂³×C₄₇	1	2	yes	—
380	C₂×#41	C₂³×C₄₇	1	2	yes	—
381	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
382	#137	—	2	2	no	—
383	#137	—	2	4	no	—
384	#137	—	2	4	no	—
385	#137	C₂×C₄×C₄₇	1	4	yes	complemented
386	AGL(1,ℤ/8)	—	2	2	no	—
387	AGL(1,ℤ/8)	—	2	4	no	—
388	AGL(1,ℤ/8)	—	2	4	no	—
389	AGL(1,ℤ/8)	C₂×C₄×C₄₇	1	4	yes	complemented
390	#135	—	2	4	no	—
391	#135	—	2	2	no	—
392	#135	—	2	4	no	—
393	#135	—	2	2	no	—
394	#135	—	2	1	no	—
395	#135	—	2	1	no	—
396	#135	—	2	1	no	—
397	#135	—	2	1	no	—
398	#135	C₂×C₄×C₄₇	1	2	yes	—
399	#135	C₂×C₄×C₄₇	1	2	yes	—
400	#135	D₄×C₄₇	1	2	yes	—
401	#135	D₄×C₄₇	1	2	yes	—
402	C₂×Q₁₆	—	2	2	no	—
403	C₂×Q₁₆	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
404	C₂×Q₁₆	D₄×C₄₇	1	1	yes	characteristic
405	C₂×SD₁₆	—	2	4	no	—
406	C₂×SD₁₆	C₂×C₄×C₄₇	1	2	yes	complemented
407	C₂×SD₁₆	D₄×C₄₇	1	2	yes	—
408	C₂×D₈	—	2	2	no	—
409	C₂×D₈	C₂×C₄×C₄₇	1	1	yes	characteristic, complemented
410	C₂×D₈	D₄×C₄₇	1	1	yes	characteristic
411	#131	—	2	1	no	—
412	#131	—	2	1	no	—
413	#131	—	2	4	no	—
414	#131	—	4	2	no	—
415	#131	C₂×C₄×C₄₇	1	2	yes	complemented
416	#131	C₂×C₄×C₄₇	1	2	yes	complemented
417	C₂×M₁₆	—	2	1	no	—
418	C₂×M₁₆	—	2	4	no	—
419	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
420	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
421	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
422	C₂²×C₈	—	2	1	no	—
423	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
424	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
425	C₄×Q₈	—	2	2	no	—
426	C₄×D₄	—	2	2	no	—
427	C₄×D₄	—	4	2	no	—
428	#117	—	2	2	no	—
429	#117	C₂³×C₄₇	1	2	yes	—
430	#117	C₂³×C₄₇	1	1	yes	characteristic
431	C₂×C₄²	—	2	2	no	—
432	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
433	#108	—	2	1	no	—
434	#108	—	2	1	no	—
435	#107	—	2	1	no	—
436	#106	—	2	1	no	—
437	#104	—	2	4	no	—
438	#104	—	2	4	no	—
439	#104	—	2	4	no	—
440	#104	—	2	2	no	—
441	#104	—	2	2	no	—
442	#104	D₄×C₄₇	1	4	yes	—
443	#103	—	2	2	no	—
444	#103	—	2	1	no	—
445	#102	—	2	2	no	—
446	#102	—	2	1	no	—
447	#99	—	4	2	no	—
448	C₂²⋊C₈	—	4	2	no	—
449	#97	—	2	1	no	—
450	#97	—	2	1	no	—
451	#97	—	2	1	no	—
452	C₄×C₈	—	2	1	no	—
453	#41	—	2	1	no	—
454	#41	—	2	2	no	—
455	#41	—	2	2	no	—
456	#41	—	2	1	no	—
457	#41	—	2	4	no	—
458	#41	—	2	4	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	2	no	—
462	#41	—	2	2	no	—
463	#41	—	2	2	no	—
464	#41	—	2	2	no	—
465	#41	—	4	4	no	—
466	#41	—	4	4	no	—
467	#41	—	4	4	no	—
468	#41	—	4	4	no	—
469	#41	—	4	2	no	—
470	#41	—	4	2	no	—
471	#41	C₂²×C₄×C₄₇	1	4	yes	—
472	#41	C₂²×C₄×C₄₇	1	2	yes	—
473	#41	C₂×D₄×C₄₇	1	4	yes	—
474	#41	C₂×D₄×C₄₇	1	2	yes	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	4	2	no	—
478	C₂×Q₈	C₂²×C₄×C₄₇	1	2	yes	—
479	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
480	C₂×Q₈	#41×C₄₇	1	1	yes	characteristic
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	2	no	—
486	C₂×D₄	—	4	4	no	—
487	C₂×D₄	—	4	2	no	—
488	C₂×D₄	—	4	4	no	—
489	C₂×D₄	—	4	4	no	—
490	C₂×D₄	C₂²×C₄×C₄₇	1	2	yes	—
491	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
492	C₂×D₄	#41×C₄₇	1	1	yes	characteristic
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	4	4	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
500	Q₁₆	—	2	4	no	—
501	Q₁₆	—	2	1	no	—
502	Q₁₆	—	2	1	no	—
503	Q₁₆	C₂²⋊C₄×C₄₇	1	2	yes	—
504	Q₁₆	C₂²⋊C₄×C₄₇	1	2	yes	—
505	SD₁₆	—	2	4	no	—
506	SD₁₆	—	2	4	no	—
507	SD₁₆	—	2	2	no	—
508	SD₁₆	—	2	2	no	—
509	SD₁₆	C₂²⋊C₄×C₄₇	1	4	yes	—
510	SD₁₆	C₂²⋊C₄×C₄₇	1	4	yes	—
511	D₈	—	2	4	no	—
512	D₈	—	2	1	no	—
513	D₈	—	2	1	no	—
514	D₈	C₂²⋊C₄×C₄₇	1	2	yes	—
515	D₈	C₂²⋊C₄×C₄₇	1	2	yes	—
516	M₁₆	—	2	1	no	—
517	M₁₆	—	2	1	no	—
518	M₁₆	—	2	2	no	—
519	M₁₆	—	2	2	no	—
520	M₁₆	—	2	2	no	—
521	M₁₆	—	2	2	no	—
522	M₁₆	—	4	4	no	—
523	M₁₆	C₂²×C₄×C₄₇	1	2	yes	—
524	M₁₆	C₂×D₄×C₄₇	1	2	yes	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	2	no	—
528	C₂×C₈	—	2	2	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	2	no	—
533	C₂×C₈	—	4	4	no	—
534	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
535	C₂×C₈	C₂²×C₄×C₄₇	1	1	yes	characteristic
536	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
537	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
538	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
539	C₂×C₈	C₂×D₄×C₄₇	1	1	yes	characteristic
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	2	1	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₂²⋊C₄	—	4	2	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	2	no	—
549	C₄²	—	4	2	no	—
550	C₄²	C₂×D₄×C₄₇	1	2	yes	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	2	no	—
554	C₂³	—	4	4	no	—
555	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
556	Q₈	—	2	1	no	—
557	Q₈	—	2	2	no	—
558	Q₈	—	2	2	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	4	4	no	—
561	Q₈	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
562	Q₈	C₄×D₄×C₄₇	1	2	yes	—
563	D₄	—	2	2	no	—
564	D₄	—	2	4	no	—
565	D₄	—	2	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	2	no	—
568	D₄	—	2	2	no	—
569	D₄	—	2	1	no	—
570	D₄	—	2	2	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	4	no	—
573	D₄	—	4	4	no	—
574	D₄	—	4	2	no	—
575	D₄	—	4	2	no	—
576	D₄	C₂×C₂²⋊C₄×C₄₇	1	4	yes	—
577	D₄	C₄×D₄×C₄₇	1	2	yes	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	8	1	no	—
594	C₂×C₄	—	8	4	no	—
595	C₂×C₄	C₂³×C₄×C₄₇	1	1	yes	characteristic
596	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
599	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
600	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
601	C₈	—	2	2	no	—
602	C₈	—	2	2	no	—
603	C₈	—	4	2	no	—
604	C₈	—	4	1	no	—
605	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
606	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
607	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
608	C₈	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
609	C₂²	—	2	2	no	—
610	C₂²	—	2	4	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	8	2	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	2	no	—
616	C₂²	—	16	4	no	—
617	C₂²	C₄₇×#518	1	1	yes	characteristic
618	C₂²	D₄²×C₄₇	1	1	yes	characteristic
619	C₂²	C₄₇×#546	1	1	yes	characteristic
620	C₄	—	2	2	no	—
621	C₄	—	2	4	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	8	2	no	—
625	C₄	—	8	1	no	—
626	C₄	C₂²×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
627	C₄	C₂×C₄×D₄×C₄₇	1	1	yes	characteristic
628	C₄	D₄²×C₄₇	1	1	yes	characteristic
629	C₄	C₄₇×#546	1	1	yes	characteristic
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	2	1	no	—
633	C₂	—	4	2	no	—
634	C₂	—	4	4	no	—
635	C₂	—	8	2	no	—
636	C₂	D₄×C₂²⋊C₄×C₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₇	D₄²
commutator	C₂×C₄	C₂³×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
socle	C₂×C₄₇	D₄×C₂²⋊C₄
prime radical	C₄₇×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#2957	C₂	semidirect
3. C₄₇×#2943	C₂	semidirect
4. C₄₇×#2901	C₂	semidirect
5. C₄₇×#1976	C₂	semidirect
6. C₄₇×#1975	C₂	semidirect
7. C₄₇×#1906	C₂	semidirect
8. C₄₇×#1881	C₂	semidirect
9. C₄₇×#1814	C₂	semidirect
10. C₄₇×#1794	C₂	semidirect
11. C₄₇×#1765	C₂	semidirect
14. C₄₇×#578	C₄	semidirect
16. C₄₇×#577	C₄	semidirect
18. C₄₇×#576	C₄	semidirect
24. C₄₇×#568	C₂²	semidirect
26. C₄×#41×C₄₇	C₂²	semidirect
32. C₂×#108×C₄₇	C₂²	semidirect
33. C₄₇×#427	C₂²	semidirect
36. C₄₇×#421	C₂²	semidirect
37. C₄₇×#421	C₂²	semidirect
40. C₂×#104×C₄₇	C₂²	semidirect
42. C₄₇×#416	C₂²	semidirect
43. C₄₇×#410	C₂²	semidirect
44. C₄₇×#408	C₂²	semidirect
45. C₄₇×#405	C₂²	semidirect
46. C₂×#97×C₄₇	C₂²	semidirect
47. C₄₇×#344	C₂²	semidirect
48. C₄₇×#341	C₂²	semidirect
49. C₄₇×#339	C₂²	semidirect
52. 2₋¹⁺⁴×C₄₇	C₂×C₄	semidirect
56. H(5,𝔽₂)×C₄₇	C₂×C₄	semidirect
66. #137×C₄₇	C₂×C₄	semidirect
70. AGL(1,ℤ/8)×C₄₇	C₂×C₄	semidirect
84. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
87. C₂×SD₁₆×C₄₇	C₂×C₄	semidirect
90. C₂×D₈×C₄₇	C₂×C₄	semidirect
96. #131×C₄₇	C₂×C₄	semidirect
97. #131×C₄₇	C₂×C₄	semidirect
290. #22461	C₄₇	direct (N × H)
313. #2957	C₂×C₄₇	semidirect
314. #2943	C₂×C₄₇	semidirect
315. #2901	C₂×C₄₇	semidirect
316. #1976	C₂×C₄₇	semidirect
317. #1975	C₂×C₄₇	semidirect
318. #1906	C₂×C₄₇	semidirect
319. #1881	C₂×C₄₇	semidirect
320. #1814	C₂×C₄₇	semidirect
321. #1794	C₂×C₄₇	semidirect
322. #1765	C₂×C₄₇	semidirect
332. #578	C₄×C₄₇	semidirect
334. #577	C₄×C₄₇	semidirect
336. #576	C₄×C₄₇	semidirect
342. #568	C₂²×C₄₇	semidirect
344. C₄×#41	C₂²×C₄₇	semidirect
350. C₂×#108	C₂²×C₄₇	semidirect
351. #427	C₂²×C₄₇	semidirect
354. #421	C₂²×C₄₇	semidirect
355. #421	C₂²×C₄₇	semidirect
358. C₂×#104	C₂²×C₄₇	semidirect
360. #416	C₂²×C₄₇	semidirect
361. #410	C₂²×C₄₇	semidirect
362. #408	C₂²×C₄₇	semidirect
363. #405	C₂²×C₄₇	semidirect
364. C₂×#97	C₂²×C₄₇	semidirect
365. #344	C₂²×C₄₇	semidirect
366. #341	C₂²×C₄₇	semidirect
367. #339	C₂²×C₄₇	semidirect
368. C₄₇	#22461	direct (N × H)
371. 2₋¹⁺⁴	C₂×C₄×C₄₇	semidirect
375. H(5,𝔽₂)	C₂×C₄×C₄₇	semidirect
385. #137	C₂×C₄×C₄₇	semidirect
389. AGL(1,ℤ/8)	C₂×C₄×C₄₇	semidirect
403. C₂×Q₁₆	C₂×C₄×C₄₇	semidirect
406. C₂×SD₁₆	C₂×C₄×C₄₇	semidirect
409. C₂×D₈	C₂×C₄×C₄₇	semidirect
415. #131	C₂×C₄×C₄₇	semidirect
416. #131	C₂×C₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3584	2
C₄₇×#2957	2
C₄₇×#2943	2
C₄₇×#2901	2
C₄₇×#1976	2
C₄₇×#1975	2
C₄₇×#1906	2
C₄₇×#1881	2
C₄₇×#1814	2
C₄₇×#1794	2
C₄₇×#1765	2
#22461	47

Subgroups — C47×#22461 #26062159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#22461 #26062159