Subgroups — C₄₇×#23867 #26063565

Order 12032.

1748 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#137×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₇×#548	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#451	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₇×#451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2409	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2015	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2000	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1894	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1878	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1870	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1788	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#137×C₄₇	—	2	4	no	—
16	C₂×#137×C₄₇	—	2	8	no	—
17	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#548	—	2	4	no	—
22	C₄₇×#548	—	2	4	no	—
23	C₂×#124×C₄₇	—	2	1	no	—
24	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
26	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#121×C₄₇	—	2	1	no	—
29	C₂×#121×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×D₄×C₄₇	—	2	1	no	—
32	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
33	C₄₇×#451	—	2	8	no	—
34	C₄₇×#451	—	2	4	no	—
35	C₂×#103×C₄₇	C₂²	1	2	yes	—
36	C₂×#103×C₄₇	C₂²	1	2	yes	—
37	C₂×#103×C₄₇	C₂²	1	2	yes	—
38	C₂×#102×C₄₇	C₂²	1	2	yes	—
39	C₂×C₂²⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
41	C₄₇×#395	—	2	1	no	—
42	C₄₇×#393	—	2	1	no	—
43	C₄₇×#391	C₂²	1	2	yes	—
44	C₄₇×#386	—	2	1	no	—
45	C₄₇×#385	—	2	1	no	—
46	C₄₇×#380	C₂²	1	1	yes	characteristic
47	C₄₇×#340	C₂²	1	2	yes	—
48	C₄₇×#336	C₂²	1	2	yes	—
49	C₂×#41×C₄₇	—	2	2	no	—
50	C₂×#41×C₄₇	—	2	4	no	—
51	C₂×#41×C₄₇	—	4	2	no	—
52	C₂×#41×C₄₇	D₄	1	4	yes	—
53	C₂²×Q₈×C₄₇	—	2	2	no	—
54	C₂²×Q₈×C₄₇	—	2	2	no	—
55	C₂²×Q₈×C₄₇	—	2	1	no	—
56	C₂²×Q₈×C₄₇	—	2	1	no	—
57	C₂²×Q₈×C₄₇	—	2	4	no	—
58	C₂²×Q₈×C₄₇	—	2	4	no	—
59	C₂²×Q₈×C₄₇	—	2	2	no	—
60	C₂²×Q₈×C₄₇	—	2	2	no	—
61	C₂²×Q₈×C₄₇	—	4	2	no	—
62	C₂²×Q₈×C₄₇	—	4	2	no	—
63	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
65	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₄₇	—	2	1	no	—
67	C₂²×D₄×C₄₇	—	2	1	no	—
68	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₄₇	—	2	1	no	—
71	C₂³×C₄×C₄₇	—	2	1	no	—
72	C₂³×C₄×C₄₇	—	2	1	no	—
73	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
75	#137×C₄₇	—	2	32	no	—
76	C₂×Q₁₆×C₄₇	—	2	4	no	—
77	C₂×Q₁₆×C₄₇	—	2	4	no	—
78	C₂×Q₁₆×C₄₇	—	2	4	no	—
79	C₂×Q₁₆×C₄₇	—	2	4	no	—
80	C₂×Q₁₆×C₄₇	—	4	2	no	—
81	C₂×Q₁₆×C₄₇	D₄	1	8	yes	complemented
82	C₂×SD₁₆×C₄₇	—	2	4	no	—
83	C₂×SD₁₆×C₄₇	—	2	4	no	—
84	C₂×SD₁₆×C₄₇	—	2	4	no	—
85	C₂×M₁₆×C₄₇	—	2	2	no	—
86	C₂×M₁₆×C₄₇	—	2	4	no	—
87	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
88	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
89	#124×C₄₇	—	2	8	no	—
90	#124×C₄₇	—	2	2	no	—
91	#124×C₄₇	—	2	2	no	—
92	C₂²⋊Q₈×C₄₇	—	2	8	no	—
93	C₂²⋊Q₈×C₄₇	—	2	2	no	—
94	C₂²⋊Q₈×C₄₇	—	2	4	no	—
95	C₂²⋊Q₈×C₄₇	—	4	2	no	—
96	C₂²⋊Q₈×C₄₇	D₄	1	4	yes	—
97	#121×C₄₇	—	2	8	no	—
98	#121×C₄₇	—	2	2	no	—
99	#121×C₄₇	—	2	2	no	—
100	C₄×Q₈×C₄₇	—	2	2	no	—
101	C₄×Q₈×C₄₇	—	2	2	no	—
102	C₄×D₄×C₄₇	—	2	8	no	—
103	C₄×D₄×C₄₇	—	2	2	no	—
104	C₄×D₄×C₄₇	—	2	2	no	—
105	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
106	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
111	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₄₇	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₄₇	—	2	1	no	—
120	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
121	#103×C₄₇	—	2	4	no	—
122	#103×C₄₇	—	2	4	no	—
123	#103×C₄₇	—	2	4	no	—
124	#102×C₄₇	—	4	2	no	—
125	C₂²⋊C₈×C₄₇	—	2	2	no	—
126	C₂²⋊C₈×C₄₇	—	2	4	no	—
127	#95×C₄₇	—	2	1	no	—
128	#95×C₄₇	—	2	2	no	—
129	#95×C₄₇	—	2	2	no	—
130	C₂⁴×C₄₇	—	2	1	no	—
131	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
132	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
133	#41×C₄₇	—	2	8	no	—
134	#41×C₄₇	—	4	4	no	—
135	#41×C₄₇	—	4	8	no	—
136	C₂×Q₈×C₄₇	—	2	4	no	—
137	C₂×Q₈×C₄₇	—	2	1	no	—
138	C₂×Q₈×C₄₇	—	2	1	no	—
139	C₂×Q₈×C₄₇	—	2	2	no	—
140	C₂×Q₈×C₄₇	—	2	2	no	—
141	C₂×Q₈×C₄₇	—	2	8	no	—
142	C₂×Q₈×C₄₇	—	2	2	no	—
143	C₂×Q₈×C₄₇	—	2	2	no	—
144	C₂×Q₈×C₄₇	—	2	2	no	—
145	C₂×Q₈×C₄₇	—	2	2	no	—
146	C₂×Q₈×C₄₇	—	2	2	no	—
147	C₂×Q₈×C₄₇	—	2	2	no	—
148	C₂×Q₈×C₄₇	—	4	4	no	—
149	C₂×Q₈×C₄₇	—	4	4	no	—
150	C₂×Q₈×C₄₇	—	4	8	no	—
151	C₂×Q₈×C₄₇	—	4	4	no	—
152	C₂×Q₈×C₄₇	—	4	8	no	—
153	C₂×Q₈×C₄₇	—	4	4	no	—
154	C₂×Q₈×C₄₇	—	4	4	no	—
155	C₂×Q₈×C₄₇	—	4	2	no	—
156	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
157	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
158	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
159	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₄₇	—	2	4	no	—
161	C₂×D₄×C₄₇	—	2	2	no	—
162	C₂×D₄×C₄₇	—	2	4	no	—
163	C₂×D₄×C₄₇	—	2	2	no	—
164	C₂×D₄×C₄₇	—	2	2	no	—
165	C₂×D₄×C₄₇	—	2	1	no	—
166	C₂×D₄×C₄₇	—	2	2	no	—
167	C₂×D₄×C₄₇	—	4	1	no	—
168	C₂×D₄×C₄₇	—	4	4	no	—
169	C₂×D₄×C₄₇	—	4	2	no	—
170	C₂×D₄×C₄₇	—	4	2	no	—
171	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₄₇	—	2	1	no	—
173	C₂²×C₄×C₄₇	—	2	2	no	—
174	C₂²×C₄×C₄₇	—	2	2	no	—
175	C₂²×C₄×C₄₇	—	2	4	no	—
176	C₂²×C₄×C₄₇	—	2	1	no	—
177	C₂²×C₄×C₄₇	—	2	1	no	—
178	C₂²×C₄×C₄₇	—	2	2	no	—
179	C₂²×C₄×C₄₇	—	2	4	no	—
180	C₂²×C₄×C₄₇	—	2	1	no	—
181	C₂²×C₄×C₄₇	—	2	1	no	—
182	C₂²×C₄×C₄₇	—	2	2	no	—
183	C₂²×C₄×C₄₇	—	2	2	no	—
184	C₂²×C₄×C₄₇	—	2	2	no	—
185	C₂²×C₄×C₄₇	—	4	4	no	—
186	C₂²×C₄×C₄₇	—	4	4	no	—
187	C₂²×C₄×C₄₇	—	4	4	no	—
188	C₂²×C₄×C₄₇	—	4	2	no	—
189	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
196	Q₁₆×C₄₇	—	2	16	no	—
197	Q₁₆×C₄₇	—	4	8	no	—
198	SD₁₆×C₄₇	—	2	16	no	—
199	M₁₆×C₄₇	—	2	8	no	—
200	C₂×C₈×C₄₇	—	2	2	no	—
201	C₂×C₈×C₄₇	—	2	2	no	—
202	C₂×C₈×C₄₇	—	2	2	no	—
203	C₂×C₈×C₄₇	—	2	2	no	—
204	C₂×C₈×C₄₇	—	2	2	no	—
205	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
206	C₄⋊C₄×C₄₇	—	2	4	no	—
207	C₄⋊C₄×C₄₇	—	2	4	no	—
208	C₄⋊C₄×C₄₇	—	2	2	no	—
209	C₄⋊C₄×C₄₇	—	2	1	no	—
210	C₄⋊C₄×C₄₇	—	2	2	no	—
211	C₄⋊C₄×C₄₇	—	2	2	no	—
212	C₄⋊C₄×C₄₇	—	4	4	no	—
213	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
214	C₂²⋊C₄×C₄₇	—	2	4	no	—
215	C₂²⋊C₄×C₄₇	—	2	4	no	—
216	C₂²⋊C₄×C₄₇	—	2	4	no	—
217	C₂²⋊C₄×C₄₇	—	2	4	no	—
218	C₂²⋊C₄×C₄₇	—	2	2	no	—
219	C₂²⋊C₄×C₄₇	—	2	2	no	—
220	C₂²⋊C₄×C₄₇	—	4	4	no	—
221	C₂²⋊C₄×C₄₇	—	4	2	no	—
222	C₄²×C₄₇	—	2	4	no	—
223	C₄²×C₄₇	—	2	1	no	—
224	C₄²×C₄₇	—	2	1	no	—
225	C₂³×C₄₇	—	2	1	no	—
226	C₂³×C₄₇	—	2	2	no	—
227	C₂³×C₄₇	—	2	2	no	—
228	C₂³×C₄₇	—	2	1	no	—
229	C₂³×C₄₇	—	2	1	no	—
230	C₂³×C₄₇	—	2	1	no	—
231	C₂³×C₄₇	—	2	2	no	—
232	C₂³×C₄₇	—	2	2	no	—
233	C₂³×C₄₇	—	2	2	no	—
234	C₂³×C₄₇	—	4	2	no	—
235	C₂³×C₄₇	—	4	1	no	—
236	C₂³×C₄₇	—	4	1	no	—
237	C₂³×C₄₇	—	4	2	no	—
238	C₂³×C₄₇	#120	1	2	yes	—
239	C₂³×C₄₇	#137	1	2	yes	complemented
240	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₄₇	—	2	2	no	—
242	Q₈×C₄₇	—	2	4	no	—
243	Q₈×C₄₇	—	2	4	no	—
244	Q₈×C₄₇	—	2	4	no	—
245	Q₈×C₄₇	—	2	4	no	—
246	Q₈×C₄₇	—	4	8	no	—
247	Q₈×C₄₇	—	4	8	no	—
248	Q₈×C₄₇	—	4	1	no	—
249	Q₈×C₄₇	—	4	1	no	—
250	Q₈×C₄₇	—	4	2	no	—
251	Q₈×C₄₇	—	4	2	no	—
252	Q₈×C₄₇	#120	1	4	yes	—
253	D₄×C₄₇	—	2	2	no	—
254	D₄×C₄₇	—	2	2	no	—
255	D₄×C₄₇	—	4	8	no	—
256	D₄×C₄₇	—	4	4	no	—
257	D₄×C₄₇	—	4	8	no	—
258	D₄×C₄₇	—	4	2	no	—
259	C₂×C₄×C₄₇	—	2	2	no	—
260	C₂×C₄×C₄₇	—	2	2	no	—
261	C₂×C₄×C₄₇	—	2	4	no	—
262	C₂×C₄×C₄₇	—	2	2	no	—
263	C₂×C₄×C₄₇	—	2	2	no	—
264	C₂×C₄×C₄₇	—	2	4	no	—
265	C₂×C₄×C₄₇	—	2	1	no	—
266	C₂×C₄×C₄₇	—	2	1	no	—
267	C₂×C₄×C₄₇	—	2	2	no	—
268	C₂×C₄×C₄₇	—	2	2	no	—
269	C₂×C₄×C₄₇	—	2	1	no	—
270	C₂×C₄×C₄₇	—	2	4	no	—
271	C₂×C₄×C₄₇	—	2	2	no	—
272	C₂×C₄×C₄₇	—	2	1	no	—
273	C₂×C₄×C₄₇	—	2	1	no	—
274	C₂×C₄×C₄₇	—	2	2	no	—
275	C₂×C₄×C₄₇	—	4	8	no	—
276	C₂×C₄×C₄₇	—	4	8	no	—
277	C₂×C₄×C₄₇	—	4	8	no	—
278	C₂×C₄×C₄₇	—	4	2	no	—
279	C₂×C₄×C₄₇	—	4	2	no	—
280	C₂×C₄×C₄₇	—	4	2	no	—
281	C₂×C₄×C₄₇	—	4	1	no	—
282	C₂×C₄×C₄₇	—	4	1	no	—
283	C₂×C₄×C₄₇	—	4	1	no	—
284	C₂×C₄×C₄₇	—	4	1	no	—
285	C₂×C₄×C₄₇	—	4	2	no	—
286	C₂×C₄×C₄₇	—	4	2	no	—
287	C₂×C₄×C₄₇	—	4	2	no	—
288	C₂×C₄×C₄₇	—	4	2	no	—
289	C₂×C₄×C₄₇	#120	1	2	yes	—
290	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
294	C₈×C₄₇	—	2	4	no	—
295	C₈×C₄₇	—	4	2	no	—
296	#23867	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₂²×C₄₇	—	2	2	no	—
298	C₂²×C₄₇	—	2	2	no	—
299	C₂²×C₄₇	—	2	2	no	—
300	C₂²×C₄₇	—	2	2	no	—
301	C₂²×C₄₇	—	2	4	no	—
302	C₂²×C₄₇	—	4	1	no	—
303	C₂²×C₄₇	—	4	1	no	—
304	C₂²×C₄₇	—	4	2	no	—
305	C₂²×C₄₇	—	4	2	no	—
306	C₂²×C₄₇	—	4	2	no	—
307	C₂²×C₄₇	—	4	2	no	—
308	C₂²×C₄₇	—	4	2	no	—
309	C₂²×C₄₇	—	4	2	no	—
310	C₂²×C₄₇	—	8	2	no	—
311	C₂²×C₄₇	—	8	2	no	—
312	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
313	C₂²×C₄₇	D₄²	1	2	yes	—
314	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
315	C₂²×C₄₇	#575	1	1	yes	characteristic
316	C₂²×C₄₇	#578	1	2	yes	—
317	C₄×C₄₇	—	2	2	no	—
318	C₄×C₄₇	—	2	4	no	—
319	C₄×C₄₇	—	4	1	no	—
320	C₄×C₄₇	—	4	2	no	—
321	C₄×C₄₇	—	4	4	no	—
322	C₄×C₄₇	—	4	1	no	—
323	C₄×C₄₇	—	4	2	no	—
324	C₄×C₄₇	—	4	1	no	—
325	C₄×C₄₇	—	8	2	no	—
326	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
327	C₄×C₄₇	#534	1	1	yes	characteristic
328	C₄×C₄₇	D₄²	1	2	yes	—
329	C₂²×#137	C₂×C₄₇	1	1	yes	characteristic, complemented
330	C₂×#548	C₂×C₄₇	1	1	yes	characteristic
331	C₂×#451	C₂×C₄₇	1	2	yes	complemented
332	C₂×#451	C₂×C₄₇	1	1	yes	characteristic, complemented
333	#2409	C₂×C₄₇	1	1	yes	characteristic, complemented
334	#2015	C₂×C₄₇	1	2	yes	complemented
335	#2000	C₂×C₄₇	1	2	yes	complemented
336	#1894	C₂×C₄₇	1	1	yes	characteristic, complemented
337	#1878	C₂×C₄₇	1	2	yes	complemented
338	#1870	C₂×C₄₇	1	1	yes	characteristic, complemented
339	#1788	C₂×C₄₇	1	1	yes	characteristic, complemented
340	C₂×C₄₇	—	4	2	no	—
341	C₂×C₄₇	—	4	2	no	—
342	C₂×C₄₇	—	8	2	no	—
343	C₂×C₄₇	#2403	1	1	yes	characteristic
344	C₂×C₄₇	#3007	1	1	yes	characteristic
345	C₂×C₄₇	#3011	1	1	yes	characteristic
346	C₂×C₄₇	#3270	1	2	yes	—
347	C₂×C₄₇	#3277	1	2	yes	—
348	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
349	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
350	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
351	C₂×#137	—	2	4	no	—
352	C₂×#137	—	2	8	no	—
353	C₂²×Q₁₆	C₂²×C₄₇	1	2	yes	—
354	C₂²×Q₁₆	C₂²×C₄₇	1	2	yes	—
355	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	—
356	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
357	#548	—	2	4	no	—
358	#548	—	2	4	no	—
359	C₂×#124	—	2	1	no	—
360	C₂×#124	C₂²×C₄₇	1	1	yes	characteristic
361	C₂×C₂²⋊Q₈	—	2	1	no	—
362	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
363	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic
364	C₂×#121	—	2	1	no	—
365	C₂×#121	C₂²×C₄₇	1	1	yes	characteristic
366	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
367	C₂×C₄×D₄	—	2	1	no	—
368	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
369	#451	—	2	8	no	—
370	#451	—	2	4	no	—
371	C₂×#103	C₂²×C₄₇	1	2	yes	—
372	C₂×#103	C₂²×C₄₇	1	2	yes	—
373	C₂×#103	C₂²×C₄₇	1	2	yes	—
374	C₂×#102	C₂²×C₄₇	1	2	yes	—
375	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
376	C₂×C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
377	#395	—	2	1	no	—
378	#393	—	2	1	no	—
379	#391	C₂²×C₄₇	1	2	yes	—
380	#386	—	2	1	no	—
381	#385	—	2	1	no	—
382	#380	C₂²×C₄₇	1	1	yes	characteristic
383	#340	C₂²×C₄₇	1	2	yes	—
384	#336	C₂²×C₄₇	1	2	yes	—
385	C₄₇	#23867	1	1	yes	characteristic, direct product factor, Sylow
386	C₂×#41	—	2	2	no	—
387	C₂×#41	—	2	4	no	—
388	C₂×#41	—	4	2	no	—
389	C₂×#41	D₄×C₄₇	1	4	yes	—
390	C₂²×Q₈	—	2	2	no	—
391	C₂²×Q₈	—	2	2	no	—
392	C₂²×Q₈	—	2	1	no	—
393	C₂²×Q₈	—	2	1	no	—
394	C₂²×Q₈	—	2	4	no	—
395	C₂²×Q₈	—	2	4	no	—
396	C₂²×Q₈	—	2	2	no	—
397	C₂²×Q₈	—	2	2	no	—
398	C₂²×Q₈	—	4	2	no	—
399	C₂²×Q₈	—	4	2	no	—
400	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
402	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
403	C₂²×D₄	—	2	1	no	—
404	C₂²×D₄	—	2	1	no	—
405	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
406	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
407	C₂³×C₄	—	2	1	no	—
408	C₂³×C₄	—	2	1	no	—
409	C₂³×C₄	—	2	1	no	—
410	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
411	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
412	#137	—	2	32	no	—
413	C₂×Q₁₆	—	2	4	no	—
414	C₂×Q₁₆	—	2	4	no	—
415	C₂×Q₁₆	—	2	4	no	—
416	C₂×Q₁₆	—	2	4	no	—
417	C₂×Q₁₆	—	4	2	no	—
418	C₂×Q₁₆	D₄×C₄₇	1	8	yes	complemented
419	C₂×SD₁₆	—	2	4	no	—
420	C₂×SD₁₆	—	2	4	no	—
421	C₂×SD₁₆	—	2	4	no	—
422	C₂×M₁₆	—	2	2	no	—
423	C₂×M₁₆	—	2	4	no	—
424	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
425	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
426	#124	—	2	8	no	—
427	#124	—	2	2	no	—
428	#124	—	2	2	no	—
429	C₂²⋊Q₈	—	2	8	no	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	4	no	—
432	C₂²⋊Q₈	—	4	2	no	—
433	C₂²⋊Q₈	D₄×C₄₇	1	4	yes	—
434	#121	—	2	8	no	—
435	#121	—	2	2	no	—
436	#121	—	2	2	no	—
437	C₄×Q₈	—	2	2	no	—
438	C₄×Q₈	—	2	2	no	—
439	C₄×D₄	—	2	8	no	—
440	C₄×D₄	—	2	2	no	—
441	C₄×D₄	—	2	2	no	—
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	1	no	—
450	C₂×C₂²⋊C₄	—	2	1	no	—
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	2	no	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
455	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
456	C₂×C₄²	—	2	1	no	—
457	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
458	#103	—	2	4	no	—
459	#103	—	2	4	no	—
460	#103	—	2	4	no	—
461	#102	—	4	2	no	—
462	C₂²⋊C₈	—	2	2	no	—
463	C₂²⋊C₈	—	2	4	no	—
464	#95	—	2	1	no	—
465	#95	—	2	2	no	—
466	#95	—	2	2	no	—
467	C₂⁴	—	2	1	no	—
468	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
469	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
470	#41	—	2	8	no	—
471	#41	—	4	4	no	—
472	#41	—	4	8	no	—
473	C₂×Q₈	—	2	4	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	2	no	—
477	C₂×Q₈	—	2	2	no	—
478	C₂×Q₈	—	2	8	no	—
479	C₂×Q₈	—	2	2	no	—
480	C₂×Q₈	—	2	2	no	—
481	C₂×Q₈	—	2	2	no	—
482	C₂×Q₈	—	2	2	no	—
483	C₂×Q₈	—	2	2	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	—	4	4	no	—
486	C₂×Q₈	—	4	4	no	—
487	C₂×Q₈	—	4	8	no	—
488	C₂×Q₈	—	4	4	no	—
489	C₂×Q₈	—	4	8	no	—
490	C₂×Q₈	—	4	4	no	—
491	C₂×Q₈	—	4	4	no	—
492	C₂×Q₈	—	4	2	no	—
493	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
494	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
495	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
496	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	4	1	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	—	4	2	no	—
507	C₂×D₄	—	4	2	no	—
508	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	2	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	4	4	no	—
523	C₂²×C₄	—	4	4	no	—
524	C₂²×C₄	—	4	4	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
527	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
528	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
529	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
531	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
533	Q₁₆	—	2	16	no	—
534	Q₁₆	—	4	8	no	—
535	SD₁₆	—	2	16	no	—
536	M₁₆	—	2	8	no	—
537	C₂×C₈	—	2	2	no	—
538	C₂×C₈	—	2	2	no	—
539	C₂×C₈	—	2	2	no	—
540	C₂×C₈	—	2	2	no	—
541	C₂×C₈	—	2	2	no	—
542	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	2	no	—
549	C₄⋊C₄	—	4	4	no	—
550	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	2	no	—
556	C₂²⋊C₄	—	2	2	no	—
557	C₂²⋊C₄	—	4	4	no	—
558	C₂²⋊C₄	—	4	2	no	—
559	C₄²	—	2	4	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	2	no	—
571	C₂³	—	4	2	no	—
572	C₂³	—	4	1	no	—
573	C₂³	—	4	1	no	—
574	C₂³	—	4	2	no	—
575	C₂³	#120×C₄₇	1	2	yes	—
576	C₂³	#137×C₄₇	1	2	yes	complemented
577	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
578	Q₈	—	2	2	no	—
579	Q₈	—	2	4	no	—
580	Q₈	—	2	4	no	—
581	Q₈	—	2	4	no	—
582	Q₈	—	2	4	no	—
583	Q₈	—	4	8	no	—
584	Q₈	—	4	8	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	2	no	—
588	Q₈	—	4	2	no	—
589	Q₈	#120×C₄₇	1	4	yes	—
590	D₄	—	2	2	no	—
591	D₄	—	2	2	no	—
592	D₄	—	4	8	no	—
593	D₄	—	4	4	no	—
594	D₄	—	4	8	no	—
595	D₄	—	4	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	2	no	—
623	C₂×C₄	—	4	2	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	2	no	—
626	C₂×C₄	#120×C₄₇	1	2	yes	—
627	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
629	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
630	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
631	C₈	—	2	4	no	—
632	C₈	—	4	2	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	8	2	no	—
647	C₂²	—	8	2	no	—
648	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
649	C₂²	D₄²×C₄₇	1	2	yes	—
650	C₂²	C₂×#137×C₄₇	1	1	yes	characteristic
651	C₂²	C₄₇×#575	1	1	yes	characteristic
652	C₂²	C₄₇×#578	1	2	yes	—
653	C₄	—	2	2	no	—
654	C₄	—	2	4	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	4	no	—
658	C₄	—	4	1	no	—
659	C₄	—	4	2	no	—
660	C₄	—	4	1	no	—
661	C₄	—	8	2	no	—
662	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
663	C₄	C₄₇×#534	1	1	yes	characteristic
664	C₄	D₄²×C₄₇	1	2	yes	—
665	C₂	—	4	2	no	—
666	C₂	—	4	2	no	—
667	C₂	—	8	2	no	—
668	C₂	C₄₇×#2403	1	1	yes	characteristic
669	C₂	C₄₇×#3007	1	1	yes	characteristic
670	C₂	C₄₇×#3011	1	1	yes	characteristic
671	C₂	C₄₇×#3270	1	2	yes	—
672	C₂	C₄₇×#3277	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂×C₄₇×#548	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄×D₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#137×C₄₇	C₂	semidirect
3. C₂×C₄₇×#451	C₂	semidirect
4. C₂×C₄₇×#451	C₂	semidirect
5. C₄₇×#2409	C₂	semidirect
6. C₄₇×#2015	C₂	semidirect
7. C₄₇×#2000	C₂	semidirect
8. C₄₇×#1894	C₂	semidirect
9. C₄₇×#1878	C₂	semidirect
10. C₄₇×#1870	C₂	semidirect
11. C₄₇×#1788	C₂	semidirect
81. C₂×Q₁₆×C₄₇	D₄	semidirect
239. C₂³×C₄₇	#137	semidirect
296. #23867	C₄₇	direct (N × H)
329. C₂²×#137	C₂×C₄₇	semidirect
331. C₂×#451	C₂×C₄₇	semidirect
332. C₂×#451	C₂×C₄₇	semidirect
333. #2409	C₂×C₄₇	semidirect
334. #2015	C₂×C₄₇	semidirect
335. #2000	C₂×C₄₇	semidirect
336. #1894	C₂×C₄₇	semidirect
337. #1878	C₂×C₄₇	semidirect
338. #1870	C₂×C₄₇	semidirect
339. #1788	C₂×C₄₇	semidirect
385. C₄₇	#23867	direct (N × H)
418. C₂×Q₁₆	D₄×C₄₇	semidirect
576. C₂³	#137×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#137×C₄₇	2
C₂×C₄₇×#548	2
C₂×C₄₇×#451	2
C₄₇×#2409	2
C₄₇×#2015	2
C₄₇×#2000	2
C₄₇×#1894	2
C₄₇×#1878	2
C₄₇×#1870	2
C₄₇×#1788	2
#23867	47

Subgroups — C47×#23867 #26063565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23867 #26063565