Subgroups — C₄₇×#23870 #26063568

Order 12032.

1806 conjugacy classes of subgroups.

The subgroup lattice has 700 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×#550	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#452	C₂	1	2	yes	maximal, complemented
4	C₂×C₄₇×#448	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2409	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2014	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2000	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1895	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	C₂²×#41×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×SD₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#550	—	2	4	no	—
22	C₄₇×#550	—	2	4	no	—
23	C₂×#127×C₄₇	—	2	1	no	—
24	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₂×#121×C₄₇	—	2	1	no	—
27	C₂×#121×C₄₇	—	2	1	no	—
28	C₂×#121×C₄₇	C₂²	1	1	yes	characteristic
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×#452	—	2	8	no	—
34	C₄₇×#448	—	2	4	no	—
35	C₂×#103×C₄₇	C₂²	1	2	yes	—
36	C₂×#103×C₄₇	C₂²	1	2	yes	—
37	C₂×#102×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	2	no	—
51	C₂×#41×C₄₇	—	2	2	no	—
52	C₂×#41×C₄₇	—	2	4	no	—
53	C₂×#41×C₄₇	—	2	4	no	—
54	C₂×#41×C₄₇	—	4	2	no	—
55	C₂×#41×C₄₇	—	4	2	no	—
56	C₂×#41×C₄₇	D₄	1	4	yes	—
57	C₂²×Q₈×C₄₇	—	2	1	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	C₂³	1	2	yes	—
63	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₄₇	—	2	1	no	—
65	C₂²×D₄×C₄₇	—	2	1	no	—
66	C₂²×D₄×C₄₇	—	2	1	no	—
67	C₂²×D₄×C₄₇	—	2	1	no	—
68	C₂²×D₄×C₄₇	—	2	1	no	—
69	C₂²×D₄×C₄₇	—	2	1	no	—
70	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
71	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₄₇	—	2	1	no	—
74	C₂³×C₄×C₄₇	—	2	1	no	—
75	C₂³×C₄×C₄₇	—	2	1	no	—
76	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
78	#137×C₄₇	—	2	32	no	—
79	C₂×Q₁₆×C₄₇	—	2	4	no	—
80	C₂×Q₁₆×C₄₇	—	2	4	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×SD₁₆×C₄₇	—	2	4	no	—
86	C₂×SD₁₆×C₄₇	—	2	4	no	—
87	C₂×SD₁₆×C₄₇	—	4	2	no	—
88	C₂×M₁₆×C₄₇	—	2	2	no	—
89	C₂×M₁₆×C₄₇	—	2	4	no	—
90	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
91	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
92	#127×C₄₇	—	2	8	no	—
93	#127×C₄₇	—	2	2	no	—
94	#127×C₄₇	—	2	2	no	—
95	C₂²⋊Q₈×C₄₇	—	2	4	no	—
96	C₂²⋊Q₈×C₄₇	—	4	2	no	—
97	#121×C₄₇	—	2	8	no	—
98	#121×C₄₇	—	2	8	no	—
99	#121×C₄₇	—	2	2	no	—
100	#121×C₄₇	—	2	2	no	—
101	#121×C₄₇	—	2	2	no	—
102	#121×C₄₇	D₄	1	4	yes	—
103	C₄×Q₈×C₄₇	—	2	2	no	—
104	C₄×Q₈×C₄₇	—	2	2	no	—
105	C₄×D₄×C₄₇	—	2	8	no	—
106	C₄×D₄×C₄₇	—	2	2	no	—
107	C₄×D₄×C₄₇	—	2	2	no	—
108	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
109	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
110	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
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	#102×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	—	2	1	no	—
132	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
133	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
134	#41×C₄₇	—	2	8	no	—
135	#41×C₄₇	—	4	4	no	—
136	#41×C₄₇	—	4	8	no	—
137	#41×C₄₇	—	4	8	no	—
138	#41×C₄₇	—	4	4	no	—
139	#41×C₄₇	—	4	4	no	—
140	C₂×Q₈×C₄₇	—	2	1	no	—
141	C₂×Q₈×C₄₇	—	2	2	no	—
142	C₂×Q₈×C₄₇	—	2	2	no	—
143	C₂×Q₈×C₄₇	—	2	8	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	8	no	—
150	C₂×Q₈×C₄₇	—	4	4	no	—
151	C₂×Q₈×C₄₇	—	4	2	no	—
152	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
153	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
154	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₄₇	—	2	4	no	—
156	C₂×D₄×C₄₇	—	2	4	no	—
157	C₂×D₄×C₄₇	—	2	2	no	—
158	C₂×D₄×C₄₇	—	2	4	no	—
159	C₂×D₄×C₄₇	—	2	4	no	—
160	C₂×D₄×C₄₇	—	2	4	no	—
161	C₂×D₄×C₄₇	—	2	4	no	—
162	C₂×D₄×C₄₇	—	2	2	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	1	no	—
167	C₂×D₄×C₄₇	—	2	2	no	—
168	C₂×D₄×C₄₇	—	2	2	no	—
169	C₂×D₄×C₄₇	—	4	1	no	—
170	C₂×D₄×C₄₇	—	4	4	no	—
171	C₂×D₄×C₄₇	—	4	4	no	—
172	C₂×D₄×C₄₇	—	4	2	no	—
173	C₂×D₄×C₄₇	—	4	2	no	—
174	C₂×D₄×C₄₇	—	8	2	no	—
175	C₂×D₄×C₄₇	—	8	2	no	—
176	C₂×D₄×C₄₇	—	8	2	no	—
177	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
178	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₄₇	—	2	1	no	—
180	C₂²×C₄×C₄₇	—	2	2	no	—
181	C₂²×C₄×C₄₇	—	2	2	no	—
182	C₂²×C₄×C₄₇	—	2	4	no	—
183	C₂²×C₄×C₄₇	—	2	1	no	—
184	C₂²×C₄×C₄₇	—	2	2	no	—
185	C₂²×C₄×C₄₇	—	2	4	no	—
186	C₂²×C₄×C₄₇	—	2	1	no	—
187	C₂²×C₄×C₄₇	—	2	1	no	—
188	C₂²×C₄×C₄₇	—	2	2	no	—
189	C₂²×C₄×C₄₇	—	2	2	no	—
190	C₂²×C₄×C₄₇	—	2	2	no	—
191	C₂²×C₄×C₄₇	—	4	4	no	—
192	C₂²×C₄×C₄₇	—	4	4	no	—
193	C₂²×C₄×C₄₇	—	4	4	no	—
194	C₂²×C₄×C₄₇	—	4	2	no	—
195	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
199	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
200	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
202	Q₁₆×C₄₇	—	2	16	no	—
203	SD₁₆×C₄₇	—	2	16	no	—
204	SD₁₆×C₄₇	—	4	8	no	—
205	M₁₆×C₄₇	—	2	8	no	—
206	C₂×C₈×C₄₇	—	2	2	no	—
207	C₂×C₈×C₄₇	—	2	2	no	—
208	C₂×C₈×C₄₇	—	2	2	no	—
209	C₂×C₈×C₄₇	—	2	2	no	—
210	C₂×C₈×C₄₇	—	2	2	no	—
211	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
212	C₄⋊C₄×C₄₇	—	2	4	no	—
213	C₄⋊C₄×C₄₇	—	2	2	no	—
214	C₄⋊C₄×C₄₇	—	2	1	no	—
215	C₄⋊C₄×C₄₇	—	2	2	no	—
216	C₄⋊C₄×C₄₇	—	2	2	no	—
217	C₄⋊C₄×C₄₇	—	4	4	no	—
218	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
219	C₂²⋊C₄×C₄₇	—	2	4	no	—
220	C₂²⋊C₄×C₄₇	—	2	2	no	—
221	C₂²⋊C₄×C₄₇	—	2	4	no	—
222	C₂²⋊C₄×C₄₇	—	2	2	no	—
223	C₂²⋊C₄×C₄₇	—	4	4	no	—
224	C₂²⋊C₄×C₄₇	—	4	2	no	—
225	C₄²×C₄₇	—	2	4	no	—
226	C₄²×C₄₇	—	2	1	no	—
227	C₄²×C₄₇	—	2	1	no	—
228	C₂³×C₄₇	—	2	2	no	—
229	C₂³×C₄₇	—	2	1	no	—
230	C₂³×C₄₇	—	2	1	no	—
231	C₂³×C₄₇	—	2	2	no	—
232	C₂³×C₄₇	—	2	1	no	—
233	C₂³×C₄₇	—	2	2	no	—
234	C₂³×C₄₇	—	2	2	no	—
235	C₂³×C₄₇	—	2	2	no	—
236	C₂³×C₄₇	—	2	1	no	—
237	C₂³×C₄₇	—	2	1	no	—
238	C₂³×C₄₇	—	2	2	no	—
239	C₂³×C₄₇	—	4	2	no	—
240	C₂³×C₄₇	—	4	2	no	—
241	C₂³×C₄₇	—	4	1	no	—
242	C₂³×C₄₇	—	4	1	no	—
243	C₂³×C₄₇	—	4	1	no	—
244	C₂³×C₄₇	—	4	1	no	—
245	C₂³×C₄₇	—	4	2	no	—
246	C₂³×C₄₇	—	4	2	no	—
247	C₂³×C₄₇	#120	1	2	yes	—
248	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
249	Q₈×C₄₇	—	2	2	no	—
250	Q₈×C₄₇	—	2	4	no	—
251	Q₈×C₄₇	—	2	4	no	—
252	Q₈×C₄₇	—	2	4	no	—
253	Q₈×C₄₇	—	4	8	no	—
254	Q₈×C₄₇	—	4	2	no	—
255	Q₈×C₄₇	—	4	2	no	—
256	Q₈×C₄₇	#120	1	4	yes	—
257	D₄×C₄₇	—	2	2	no	—
258	D₄×C₄₇	—	2	2	no	—
259	D₄×C₄₇	—	2	4	no	—
260	D₄×C₄₇	—	4	4	no	—
261	D₄×C₄₇	—	4	8	no	—
262	D₄×C₄₇	—	4	8	no	—
263	D₄×C₄₇	—	4	8	no	—
264	D₄×C₄₇	—	4	1	no	—
265	D₄×C₄₇	—	4	1	no	—
266	D₄×C₄₇	—	4	2	no	—
267	D₄×C₄₇	—	8	4	no	—
268	D₄×C₄₇	—	8	4	no	—
269	D₄×C₄₇	—	8	4	no	—
270	C₂×C₄×C₄₇	—	2	1	no	—
271	C₂×C₄×C₄₇	—	2	2	no	—
272	C₂×C₄×C₄₇	—	2	2	no	—
273	C₂×C₄×C₄₇	—	2	4	no	—
274	C₂×C₄×C₄₇	—	2	1	no	—
275	C₂×C₄×C₄₇	—	2	2	no	—
276	C₂×C₄×C₄₇	—	2	2	no	—
277	C₂×C₄×C₄₇	—	2	4	no	—
278	C₂×C₄×C₄₇	—	2	1	no	—
279	C₂×C₄×C₄₇	—	2	2	no	—
280	C₂×C₄×C₄₇	—	2	1	no	—
281	C₂×C₄×C₄₇	—	2	4	no	—
282	C₂×C₄×C₄₇	—	2	2	no	—
283	C₂×C₄×C₄₇	—	2	2	no	—
284	C₂×C₄×C₄₇	—	4	8	no	—
285	C₂×C₄×C₄₇	—	4	8	no	—
286	C₂×C₄×C₄₇	—	4	8	no	—
287	C₂×C₄×C₄₇	—	4	2	no	—
288	C₂×C₄×C₄₇	—	4	2	no	—
289	C₂×C₄×C₄₇	—	4	2	no	—
290	C₂×C₄×C₄₇	—	4	1	no	—
291	C₂×C₄×C₄₇	—	4	1	no	—
292	C₂×C₄×C₄₇	—	4	2	no	—
293	C₂×C₄×C₄₇	—	4	2	no	—
294	C₂×C₄×C₄₇	—	4	2	no	—
295	C₂×C₄×C₄₇	—	4	2	no	—
296	C₂×C₄×C₄₇	#120	1	2	yes	—
297	C₂×C₄×C₄₇	#137	1	2	yes	complemented
298	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
302	C₈×C₄₇	—	2	4	no	—
303	C₈×C₄₇	—	4	2	no	—
304	#23870	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂²×C₄₇	—	2	2	no	—
306	C₂²×C₄₇	—	2	2	no	—
307	C₂²×C₄₇	—	2	2	no	—
308	C₂²×C₄₇	—	4	2	no	—
309	C₂²×C₄₇	—	4	2	no	—
310	C₂²×C₄₇	—	4	2	no	—
311	C₂²×C₄₇	—	4	2	no	—
312	C₂²×C₄₇	—	4	1	no	—
313	C₂²×C₄₇	—	4	1	no	—
314	C₂²×C₄₇	—	4	2	no	—
315	C₂²×C₄₇	—	4	2	no	—
316	C₂²×C₄₇	—	4	1	no	—
317	C₂²×C₄₇	—	4	2	no	—
318	C₂²×C₄₇	—	4	2	no	—
319	C₂²×C₄₇	—	4	2	no	—
320	C₂²×C₄₇	—	4	2	no	—
321	C₂²×C₄₇	—	8	2	no	—
322	C₂²×C₄₇	—	8	2	no	—
323	C₂²×C₄₇	—	8	2	no	—
324	C₂²×C₄₇	—	8	2	no	—
325	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
326	C₂²×C₄₇	D₄²	1	2	yes	—
327	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
328	C₂²×C₄₇	#575	1	1	yes	characteristic
329	C₂²×C₄₇	#577	1	2	yes	—
330	C₄×C₄₇	—	2	2	no	—
331	C₄×C₄₇	—	2	4	no	—
332	C₄×C₄₇	—	2	2	no	—
333	C₄×C₄₇	—	4	1	no	—
334	C₄×C₄₇	—	4	4	no	—
335	C₄×C₄₇	—	4	1	no	—
336	C₄×C₄₇	—	4	2	no	—
337	C₄×C₄₇	—	8	2	no	—
338	C₄×C₄₇	C₂×#120	1	1	yes	characteristic
339	C₄×C₄₇	#534	1	1	yes	characteristic
340	C₄×C₄₇	D₄²	1	2	yes	—
341	C₂²×#137	C₂×C₄₇	1	1	yes	characteristic, complemented
342	C₂×#550	C₂×C₄₇	1	1	yes	characteristic
343	C₂×#452	C₂×C₄₇	1	2	yes	complemented
344	C₂×#448	C₂×C₄₇	1	1	yes	characteristic, complemented
345	#2409	C₂×C₄₇	1	1	yes	characteristic, complemented
346	#2014	C₂×C₄₇	1	2	yes	complemented
347	#2000	C₂×C₄₇	1	2	yes	complemented
348	#1895	C₂×C₄₇	1	1	yes	characteristic, complemented
349	#1878	C₂×C₄₇	1	2	yes	complemented
350	#1870	C₂×C₄₇	1	1	yes	characteristic, complemented
351	#1788	C₂×C₄₇	1	1	yes	characteristic, complemented
352	C₂×C₄₇	—	4	2	no	—
353	C₂×C₄₇	—	8	1	no	—
354	C₂×C₄₇	—	8	2	no	—
355	C₂×C₄₇	—	8	2	no	—
356	C₂×C₄₇	#2403	1	1	yes	characteristic
357	C₂×C₄₇	#3007	1	1	yes	characteristic
358	C₂×C₄₇	#3009	1	1	yes	characteristic
359	C₂×C₄₇	#3271	1	2	yes	—
360	C₂×C₄₇	#3278	1	2	yes	—
361	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
362	C₂²×#41	C₂²×C₄₇	1	1	yes	characteristic
363	C₂³×Q₈	C₂²×C₄₇	1	1	yes	characteristic
364	C₂×#137	—	2	4	no	—
365	C₂×#137	—	2	8	no	—
366	C₂²×Q₁₆	C₂²×C₄₇	1	2	yes	—
367	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	—
368	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	—
369	C₂²×M₁₆	C₂²×C₄₇	1	1	yes	characteristic
370	#550	—	2	4	no	—
371	#550	—	2	4	no	—
372	C₂×#127	—	2	1	no	—
373	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
374	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	2	yes	—
375	C₂×#121	—	2	1	no	—
376	C₂×#121	—	2	1	no	—
377	C₂×#121	C₂²×C₄₇	1	1	yes	characteristic
378	C₂×#121	C₂²×C₄₇	1	1	yes	characteristic
379	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
380	C₂×C₄×D₄	—	2	1	no	—
381	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
382	#452	—	2	8	no	—
383	#448	—	2	4	no	—
384	C₂×#103	C₂²×C₄₇	1	2	yes	—
385	C₂×#103	C₂²×C₄₇	1	2	yes	—
386	C₂×#102	C₂²×C₄₇	1	2	yes	—
387	C₂×#102	C₂²×C₄₇	1	2	yes	—
388	C₂×C₂²⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
389	C₂×C₂²⋊C₈	C₂²×C₄₇	1	2	yes	—
390	#395	—	2	1	no	—
391	#393	—	2	1	no	—
392	#391	C₂²×C₄₇	1	2	yes	—
393	#386	—	2	1	no	—
394	#385	—	2	1	no	—
395	#380	C₂²×C₄₇	1	1	yes	characteristic
396	#340	C₂²×C₄₇	1	2	yes	—
397	#336	C₂²×C₄₇	1	2	yes	—
398	C₄₇	#23870	1	1	yes	characteristic, direct product factor, Sylow
399	C₂×#41	—	2	2	no	—
400	C₂×#41	—	2	2	no	—
401	C₂×#41	—	2	2	no	—
402	C₂×#41	—	2	4	no	—
403	C₂×#41	—	2	4	no	—
404	C₂×#41	—	4	2	no	—
405	C₂×#41	—	4	2	no	—
406	C₂×#41	D₄×C₄₇	1	4	yes	—
407	C₂²×Q₈	—	2	1	no	—
408	C₂²×Q₈	—	2	4	no	—
409	C₂²×Q₈	—	2	2	no	—
410	C₂²×Q₈	—	2	2	no	—
411	C₂²×Q₈	—	4	2	no	—
412	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
413	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
414	C₂²×D₄	—	2	1	no	—
415	C₂²×D₄	—	2	1	no	—
416	C₂²×D₄	—	2	1	no	—
417	C₂²×D₄	—	2	1	no	—
418	C₂²×D₄	—	2	1	no	—
419	C₂²×D₄	—	2	1	no	—
420	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
421	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
422	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
423	C₂³×C₄	—	2	1	no	—
424	C₂³×C₄	—	2	1	no	—
425	C₂³×C₄	—	2	1	no	—
426	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
428	#137	—	2	32	no	—
429	C₂×Q₁₆	—	2	4	no	—
430	C₂×Q₁₆	—	2	4	no	—
431	C₂×Q₁₆	D₄×C₄₇	1	8	yes	complemented
432	C₂×SD₁₆	—	2	4	no	—
433	C₂×SD₁₆	—	2	4	no	—
434	C₂×SD₁₆	—	2	4	no	—
435	C₂×SD₁₆	—	2	4	no	—
436	C₂×SD₁₆	—	2	4	no	—
437	C₂×SD₁₆	—	4	2	no	—
438	C₂×M₁₆	—	2	2	no	—
439	C₂×M₁₆	—	2	4	no	—
440	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
441	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
442	#127	—	2	8	no	—
443	#127	—	2	2	no	—
444	#127	—	2	2	no	—
445	C₂²⋊Q₈	—	2	4	no	—
446	C₂²⋊Q₈	—	4	2	no	—
447	#121	—	2	8	no	—
448	#121	—	2	8	no	—
449	#121	—	2	2	no	—
450	#121	—	2	2	no	—
451	#121	—	2	2	no	—
452	#121	D₄×C₄₇	1	4	yes	—
453	C₄×Q₈	—	2	2	no	—
454	C₄×Q₈	—	2	2	no	—
455	C₄×D₄	—	2	8	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	2	no	—
458	C₂×C₄⋊C₄	—	2	1	no	—
459	C₂×C₄⋊C₄	—	2	2	no	—
460	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
461	C₂×C₄⋊C₄	C₂³×C₄₇	1	2	yes	—
462	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
463	C₂×C₂²⋊C₄	—	2	1	no	—
464	C₂×C₂²⋊C₄	—	2	1	no	—
465	C₂×C₂²⋊C₄	—	2	2	no	—
466	C₂×C₂²⋊C₄	—	2	1	no	—
467	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
468	C₂×C₂²⋊C₄	D₄×C₄₇	1	1	yes	characteristic
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
471	#103	—	2	4	no	—
472	#103	—	2	4	no	—
473	#102	—	2	4	no	—
474	#102	—	4	2	no	—
475	C₂²⋊C₈	—	2	2	no	—
476	C₂²⋊C₈	—	2	4	no	—
477	#95	—	2	1	no	—
478	#95	—	2	2	no	—
479	#95	—	2	2	no	—
480	C₂⁴	—	2	1	no	—
481	C₂⁴	—	2	1	no	—
482	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
483	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
484	#41	—	2	8	no	—
485	#41	—	4	4	no	—
486	#41	—	4	8	no	—
487	#41	—	4	8	no	—
488	#41	—	4	4	no	—
489	#41	—	4	4	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	2	no	—
492	C₂×Q₈	—	2	2	no	—
493	C₂×Q₈	—	2	8	no	—
494	C₂×Q₈	—	2	2	no	—
495	C₂×Q₈	—	2	2	no	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	4	4	no	—
499	C₂×Q₈	—	4	8	no	—
500	C₂×Q₈	—	4	4	no	—
501	C₂×Q₈	—	4	2	no	—
502	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
503	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
504	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	4	no	—
507	C₂×D₄	—	2	2	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	2	no	—
514	C₂×D₄	—	2	2	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	2	no	—
518	C₂×D₄	—	2	2	no	—
519	C₂×D₄	—	4	1	no	—
520	C₂×D₄	—	4	4	no	—
521	C₂×D₄	—	4	4	no	—
522	C₂×D₄	—	4	2	no	—
523	C₂×D₄	—	4	2	no	—
524	C₂×D₄	—	8	2	no	—
525	C₂×D₄	—	8	2	no	—
526	C₂×D₄	—	8	2	no	—
527	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
528	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	1	no	—
535	C₂²×C₄	—	2	4	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	1	no	—
538	C₂²×C₄	—	2	1	no	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	4	4	no	—
542	C₂²×C₄	—	4	4	no	—
543	C₂²×C₄	—	4	4	no	—
544	C₂²×C₄	—	4	2	no	—
545	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
546	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
547	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
548	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
549	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
550	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
551	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
552	Q₁₆	—	2	16	no	—
553	SD₁₆	—	2	16	no	—
554	SD₁₆	—	4	8	no	—
555	M₁₆	—	2	8	no	—
556	C₂×C₈	—	2	2	no	—
557	C₂×C₈	—	2	2	no	—
558	C₂×C₈	—	2	2	no	—
559	C₂×C₈	—	2	2	no	—
560	C₂×C₈	—	2	2	no	—
561	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
562	C₄⋊C₄	—	2	4	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	4	4	no	—
568	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
569	C₂²⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	2	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	—	4	4	no	—
574	C₂²⋊C₄	—	4	2	no	—
575	C₄²	—	2	4	no	—
576	C₄²	—	2	1	no	—
577	C₄²	—	2	1	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	1	no	—
580	C₂³	—	2	1	no	—
581	C₂³	—	2	2	no	—
582	C₂³	—	2	1	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	1	no	—
587	C₂³	—	2	1	no	—
588	C₂³	—	2	2	no	—
589	C₂³	—	4	2	no	—
590	C₂³	—	4	2	no	—
591	C₂³	—	4	1	no	—
592	C₂³	—	4	1	no	—
593	C₂³	—	4	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	2	no	—
596	C₂³	—	4	2	no	—
597	C₂³	#120×C₄₇	1	2	yes	—
598	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
599	Q₈	—	2	2	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	2	4	no	—
602	Q₈	—	2	4	no	—
603	Q₈	—	4	8	no	—
604	Q₈	—	4	2	no	—
605	Q₈	—	4	2	no	—
606	Q₈	#120×C₄₇	1	4	yes	—
607	D₄	—	2	2	no	—
608	D₄	—	2	2	no	—
609	D₄	—	2	4	no	—
610	D₄	—	4	4	no	—
611	D₄	—	4	8	no	—
612	D₄	—	4	8	no	—
613	D₄	—	4	8	no	—
614	D₄	—	4	1	no	—
615	D₄	—	4	1	no	—
616	D₄	—	4	2	no	—
617	D₄	—	8	4	no	—
618	D₄	—	8	4	no	—
619	D₄	—	8	4	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	4	8	no	—
635	C₂×C₄	—	4	8	no	—
636	C₂×C₄	—	4	8	no	—
637	C₂×C₄	—	4	2	no	—
638	C₂×C₄	—	4	2	no	—
639	C₂×C₄	—	4	2	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	2	no	—
643	C₂×C₄	—	4	2	no	—
644	C₂×C₄	—	4	2	no	—
645	C₂×C₄	—	4	2	no	—
646	C₂×C₄	#120×C₄₇	1	2	yes	—
647	C₂×C₄	#137×C₄₇	1	2	yes	complemented
648	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
649	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
650	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
651	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
652	C₈	—	2	4	no	—
653	C₈	—	4	2	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	2	2	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	2	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	1	no	—
662	C₂²	—	4	1	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	1	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	4	2	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	8	2	no	—
671	C₂²	—	8	2	no	—
672	C₂²	—	8	2	no	—
673	C₂²	—	8	2	no	—
674	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
675	C₂²	D₄²×C₄₇	1	2	yes	—
676	C₂²	C₂×#137×C₄₇	1	1	yes	characteristic
677	C₂²	C₄₇×#575	1	1	yes	characteristic
678	C₂²	C₄₇×#577	1	2	yes	—
679	C₄	—	2	2	no	—
680	C₄	—	2	4	no	—
681	C₄	—	2	2	no	—
682	C₄	—	4	1	no	—
683	C₄	—	4	4	no	—
684	C₄	—	4	1	no	—
685	C₄	—	4	2	no	—
686	C₄	—	8	2	no	—
687	C₄	C₂×#120×C₄₇	1	1	yes	characteristic
688	C₄	C₄₇×#534	1	1	yes	characteristic
689	C₄	D₄²×C₄₇	1	2	yes	—
690	C₂	—	4	2	no	—
691	C₂	—	8	1	no	—
692	C₂	—	8	2	no	—
693	C₂	—	8	2	no	—
694	C₂	C₄₇×#2403	1	1	yes	characteristic
695	C₂	C₄₇×#3007	1	1	yes	characteristic
696	C₂	C₄₇×#3009	1	1	yes	characteristic
697	C₂	C₄₇×#3271	1	2	yes	—
698	C₂	C₄₇×#3278	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₂×C₄₇×#550	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×#452	C₂	semidirect
4. C₂×C₄₇×#448	C₂	semidirect
5. C₄₇×#2409	C₂	semidirect
6. C₄₇×#2014	C₂	semidirect
7. C₄₇×#2000	C₂	semidirect
8. C₄₇×#1895	C₂	semidirect
9. C₄₇×#1878	C₂	semidirect
10. C₄₇×#1870	C₂	semidirect
11. C₄₇×#1788	C₂	semidirect
81. C₂×Q₁₆×C₄₇	D₄	semidirect
297. C₂×C₄×C₄₇	#137	semidirect
304. #23870	C₄₇	direct (N × H)
341. C₂²×#137	C₂×C₄₇	semidirect
343. C₂×#452	C₂×C₄₇	semidirect
344. C₂×#448	C₂×C₄₇	semidirect
345. #2409	C₂×C₄₇	semidirect
346. #2014	C₂×C₄₇	semidirect
347. #2000	C₂×C₄₇	semidirect
348. #1895	C₂×C₄₇	semidirect
349. #1878	C₂×C₄₇	semidirect
350. #1870	C₂×C₄₇	semidirect
351. #1788	C₂×C₄₇	semidirect
398. C₄₇	#23870	direct (N × H)
431. C₂×Q₁₆	D₄×C₄₇	semidirect
647. C₂×C₄	#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×#550	2
C₂×C₄₇×#452	2
C₂×C₄₇×#448	2
C₄₇×#2409	2
C₄₇×#2014	2
C₄₇×#2000	2
C₄₇×#1895	2
C₄₇×#1878	2
C₄₇×#1870	2
C₄₇×#1788	2
#23870	47

Subgroups — C47×#23870 #26063568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23870 #26063568