Subgroups — C₄₇×#23890 #26063588

Order 12032.

1706 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#577	C₂	1	2	yes	maximal, complemented
2	C₂×C₄₇×#453	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3012	C₂	1	2	yes	maximal, complemented
4	C₄₇×#2019	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2014	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2007	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1907	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1876	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1793	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	2	yes	complemented
12	C₄₇×#577	—	2	8	no	—
13	C₄₇×#577	—	2	8	no	—
14	C₂×#137×C₄₇	—	2	2	no	—
15	C₂×#137×C₄₇	C₂²	1	2	yes	complemented
16	C₂×AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
17	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	—
18	C₂×AGL(1,ℤ/8)×C₄₇	C₂²	1	2	yes	—
19	C₂×#135×C₄₇	—	2	2	no	—
20	C₂×#135×C₄₇	C₂²	1	2	yes	—
21	C₂²×SD₁₆×C₄₇	—	2	2	no	—
22	C₂²×SD₁₆×C₄₇	C₂²	1	2	yes	complemented
23	C₂×#131×C₄₇	C₂²	1	2	yes	complemented
24	C₄₇×#537	C₂²	1	1	yes	characteristic
25	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#453	—	2	8	no	—
27	C₄₇×#452	—	2	2	no	—
28	C₄₇×#450	—	2	2	no	—
29	C₄₇×#449	—	2	2	no	—
30	C₄₇×#447	—	2	2	no	—
31	C₂×#104×C₄₇	C₂²	1	2	yes	complemented
32	C₂×#104×C₄₇	C₂²	1	2	yes	complemented
33	C₄₇×#417	C₂²	1	2	yes	complemented
34	C₂×#102×C₄₇	C₂²	1	2	yes	complemented
35	C₂×#101×C₄₇	C₂²	1	1	yes	characteristic, complemented
36	C₂×#100×C₄₇	C₂²	1	2	yes	complemented
37	C₄₇×#408	C₂²	1	2	yes	complemented
38	C₄₇×#391	C₂²	1	1	yes	characteristic, complemented
39	C₄₇×#343	C₂²	1	1	yes	characteristic, complemented
40	C₄₇×#341	C₂²	1	2	yes	complemented
41	C₄₇×#336	C₂²	1	1	yes	characteristic, complemented
42	2₋¹⁺⁴×C₄₇	—	2	4	no	—
43	2₋¹⁺⁴×C₄₇	—	4	1	no	—
44	2₋¹⁺⁴×C₄₇	D₄	1	4	yes	complemented
45	H(5,𝔽₂)×C₄₇	—	2	4	no	—
46	H(5,𝔽₂)×C₄₇	—	2	4	no	—
47	H(5,𝔽₂)×C₄₇	—	2	4	no	—
48	H(5,𝔽₂)×C₄₇	—	4	2	no	—
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	—	4	1	no	—
53	C₂×#41×C₄₇	C₂³	1	2	yes	—
54	C₂×#41×C₄₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₄₇	—	2	2	no	—
56	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₄₇	—	2	2	no	—
58	C₂²×D₄×C₄₇	—	2	2	no	—
59	C₂²×D₄×C₄₇	—	4	1	no	—
60	C₂²×D₄×C₄₇	—	4	2	no	—
61	C₂²×D₄×C₄₇	C₂³	1	2	yes	—
62	#137×C₄₇	—	2	16	no	—
63	#137×C₄₇	—	2	4	no	—
64	#137×C₄₇	—	2	4	no	—
65	AGL(1,ℤ/8)×C₄₇	—	2	16	no	—
66	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
67	AGL(1,ℤ/8)×C₄₇	—	2	4	no	—
68	AGL(1,ℤ/8)×C₄₇	—	4	2	no	—
69	AGL(1,ℤ/8)×C₄₇	D₄	1	8	yes	complemented
70	#135×C₄₇	—	2	16	no	—
71	#135×C₄₇	—	2	4	no	—
72	#135×C₄₇	—	2	4	no	—
73	C₂×Q₁₆×C₄₇	—	2	2	no	—
74	C₂×Q₁₆×C₄₇	D₄	1	2	yes	complemented
75	C₂×SD₁₆×C₄₇	—	2	16	no	—
76	C₂×SD₁₆×C₄₇	—	2	2	no	—
77	C₂×SD₁₆×C₄₇	—	2	4	no	—
78	C₂×SD₁₆×C₄₇	—	2	2	no	—
79	C₂×SD₁₆×C₄₇	—	2	4	no	—
80	C₂×SD₁₆×C₄₇	—	2	2	no	—
81	C₂×SD₁₆×C₄₇	—	2	2	no	—
82	C₂×SD₁₆×C₄₇	—	2	2	no	—
83	C₂×SD₁₆×C₄₇	D₄	1	2	yes	complemented
84	C₂×SD₁₆×C₄₇	D₄	1	2	yes	complemented
85	C₂×D₈×C₄₇	—	2	2	no	—
86	C₂×D₈×C₄₇	—	2	2	no	—
87	C₂×D₈×C₄₇	D₄	1	2	yes	complemented
88	#131×C₄₇	—	2	4	no	—
89	#131×C₄₇	—	2	4	no	—
90	C₂×M₁₆×C₄₇	—	2	2	no	—
91	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
92	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
93	C₂²×C₈×C₄₇	—	2	2	no	—
94	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
95	#127×C₄₇	—	2	4	no	—
96	C₂²⋊Q₈×C₄₇	—	2	2	no	—
97	#121×C₄₇	—	2	2	no	—
98	#121×C₄₇	—	4	1	no	—
99	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
101	#104×C₄₇	—	2	4	no	—
102	#104×C₄₇	—	2	4	no	—
103	#103×C₄₇	—	2	2	no	—
104	#102×C₄₇	—	2	2	no	—
105	#102×C₄₇	—	2	2	no	—
106	#102×C₄₇	—	2	2	no	—
107	#101×C₄₇	—	4	1	no	—
108	#100×C₄₇	—	2	4	no	—
109	C₂²⋊C₈×C₄₇	—	2	2	no	—
110	C₂²⋊C₈×C₄₇	—	2	2	no	—
111	#95×C₄₇	—	4	1	no	—
112	C₂⁴×C₄₇	—	2	2	no	—
113	C₂⁴×C₄₇	—	4	2	no	—
114	#41×C₄₇	—	2	2	no	—
115	#41×C₄₇	—	2	2	no	—
116	#41×C₄₇	—	2	8	no	—
117	#41×C₄₇	—	2	8	no	—
118	#41×C₄₇	—	2	2	no	—
119	#41×C₄₇	—	2	2	no	—
120	#41×C₄₇	—	2	2	no	—
121	#41×C₄₇	—	2	2	no	—
122	#41×C₄₇	—	4	4	no	—
123	#41×C₄₇	—	4	4	no	—
124	#41×C₄₇	—	4	4	no	—
125	#41×C₄₇	—	4	4	no	—
126	#41×C₄₇	—	4	4	no	—
127	#41×C₄₇	—	4	4	no	—
128	#41×C₄₇	C₂×D₄	1	4	yes	—
129	#41×C₄₇	C₂×D₄	1	4	yes	—
130	C₂×Q₈×C₄₇	—	2	1	no	—
131	C₂×Q₈×C₄₇	—	2	8	no	—
132	C₂×Q₈×C₄₇	—	2	2	no	—
133	C₂×Q₈×C₄₇	—	2	2	no	—
134	C₂×Q₈×C₄₇	—	4	1	no	—
135	C₂×Q₈×C₄₇	—	4	4	no	—
136	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
139	C₂×D₄×C₄₇	—	2	2	no	—
140	C₂×D₄×C₄₇	—	2	8	no	—
141	C₂×D₄×C₄₇	—	2	2	no	—
142	C₂×D₄×C₄₇	—	2	2	no	—
143	C₂×D₄×C₄₇	—	2	2	no	—
144	C₂×D₄×C₄₇	—	2	2	no	—
145	C₂×D₄×C₄₇	—	2	2	no	—
146	C₂×D₄×C₄₇	—	2	2	no	—
147	C₂×D₄×C₄₇	—	4	4	no	—
148	C₂×D₄×C₄₇	—	4	2	no	—
149	C₂×D₄×C₄₇	—	4	2	no	—
150	C₂×D₄×C₄₇	—	4	4	no	—
151	C₂×D₄×C₄₇	—	4	8	no	—
152	C₂×D₄×C₄₇	—	4	8	no	—
153	C₂×D₄×C₄₇	—	4	1	no	—
154	C₂×D₄×C₄₇	—	4	4	no	—
155	C₂×D₄×C₄₇	—	4	4	no	—
156	C₂×D₄×C₄₇	—	4	1	no	—
157	C₂×D₄×C₄₇	—	8	2	no	—
158	C₂×D₄×C₄₇	—	8	2	no	—
159	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
161	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₄₇	—	2	2	no	—
163	C₂²×C₄×C₄₇	—	2	1	no	—
164	C₂²×C₄×C₄₇	—	2	1	no	—
165	C₂²×C₄×C₄₇	—	2	2	no	—
166	C₂²×C₄×C₄₇	—	2	1	no	—
167	C₂²×C₄×C₄₇	—	4	2	no	—
168	C₂²×C₄×C₄₇	—	4	2	no	—
169	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
170	Q₁₆×C₄₇	—	2	8	no	—
171	Q₁₆×C₄₇	—	2	4	no	—
172	SD₁₆×C₄₇	—	2	8	no	—
173	SD₁₆×C₄₇	—	2	8	no	—
174	SD₁₆×C₄₇	—	2	8	no	—
175	SD₁₆×C₄₇	—	2	4	no	—
176	SD₁₆×C₄₇	—	2	8	no	—
177	SD₁₆×C₄₇	—	2	4	no	—
178	SD₁₆×C₄₇	—	4	4	no	—
179	D₈×C₄₇	—	2	8	no	—
180	D₈×C₄₇	—	2	4	no	—
181	D₈×C₄₇	—	4	4	no	—
182	M₁₆×C₄₇	—	2	8	no	—
183	M₁₆×C₄₇	—	2	2	no	—
184	M₁₆×C₄₇	—	2	2	no	—
185	M₁₆×C₄₇	—	2	2	no	—
186	M₁₆×C₄₇	C₂×D₄	1	4	yes	—
187	C₂×C₈×C₄₇	—	2	8	no	—
188	C₂×C₈×C₄₇	—	2	2	no	—
189	C₂×C₈×C₄₇	—	2	2	no	—
190	C₂×C₈×C₄₇	—	2	2	no	—
191	C₂×C₈×C₄₇	—	2	2	no	—
192	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
193	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₄₇	—	2	2	no	—
195	C₄⋊C₄×C₄₇	—	4	1	no	—
196	C₂²⋊C₄×C₄₇	—	4	2	no	—
197	C₄²×C₄₇	—	2	2	no	—
198	C₂³×C₄₇	—	2	2	no	—
199	C₂³×C₄₇	—	2	2	no	—
200	C₂³×C₄₇	—	2	4	no	—
201	C₂³×C₄₇	—	4	8	no	—
202	C₂³×C₄₇	—	4	2	no	—
203	C₂³×C₄₇	—	4	2	no	—
204	C₂³×C₄₇	—	8	2	no	—
205	C₂³×C₄₇	—	8	2	no	—
206	C₂³×C₄₇	—	16	2	no	—
207	C₂³×C₄₇	—	16	2	no	—
208	C₂³×C₄₇	H(5,𝔽₂)	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	2	no	—
213	Q₈×C₄₇	—	4	4	no	—
214	Q₈×C₄₇	—	4	2	no	—
215	Q₈×C₄₇	#120	1	4	yes	—
216	D₄×C₄₇	—	2	4	no	—
217	D₄×C₄₇	—	2	4	no	—
218	D₄×C₄₇	—	2	4	no	—
219	D₄×C₄₇	—	2	2	no	—
220	D₄×C₄₇	—	2	2	no	—
221	D₄×C₄₇	—	2	2	no	—
222	D₄×C₄₇	—	2	4	no	—
223	D₄×C₄₇	—	4	8	no	—
224	D₄×C₄₇	—	4	4	no	—
225	D₄×C₄₇	—	4	8	no	—
226	D₄×C₄₇	—	4	2	no	—
227	D₄×C₄₇	—	4	2	no	—
228	D₄×C₄₇	—	4	2	no	—
229	D₄×C₄₇	—	4	2	no	—
230	D₄×C₄₇	—	8	8	no	—
231	D₄×C₄₇	—	16	1	no	—
232	D₄×C₄₇	#120	1	4	yes	—
233	C₂×C₄×C₄₇	—	2	2	no	—
234	C₂×C₄×C₄₇	—	2	4	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	2	no	—
239	C₂×C₄×C₄₇	—	4	8	no	—
240	C₂×C₄×C₄₇	—	4	8	no	—
241	C₂×C₄×C₄₇	—	4	1	no	—
242	C₂×C₄×C₄₇	—	4	1	no	—
243	C₂×C₄×C₄₇	—	4	1	no	—
244	C₂×C₄×C₄₇	—	4	1	no	—
245	C₂×C₄×C₄₇	—	4	2	no	—
246	C₂×C₄×C₄₇	—	4	2	no	—
247	C₂×C₄×C₄₇	—	4	1	no	—
248	C₂×C₄×C₄₇	—	8	1	no	—
249	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
253	C₈×C₄₇	—	2	2	no	—
254	C₈×C₄₇	—	2	4	no	—
255	C₈×C₄₇	—	2	2	no	—
256	C₈×C₄₇	—	4	2	no	—
257	#23890	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₄₇	—	2	1	no	—
259	C₂²×C₄₇	—	2	1	no	—
260	C₂²×C₄₇	—	2	4	no	—
261	C₂²×C₄₇	—	4	2	no	—
262	C₂²×C₄₇	—	4	2	no	—
263	C₂²×C₄₇	—	4	2	no	—
264	C₂²×C₄₇	—	4	4	no	—
265	C₂²×C₄₇	—	4	2	no	—
266	C₂²×C₄₇	—	4	2	no	—
267	C₂²×C₄₇	—	8	4	no	—
268	C₂²×C₄₇	—	8	2	no	—
269	C₂²×C₄₇	—	8	2	no	—
270	C₂²×C₄₇	—	16	2	no	—
271	C₂²×C₄₇	—	16	8	no	—
272	C₂²×C₄₇	#534	1	1	yes	characteristic
273	C₂²×C₄₇	D₄²	1	1	yes	characteristic
274	C₂²×C₄₇	D₄²	1	1	yes	characteristic
275	C₄×C₄₇	—	2	4	no	—
276	C₄×C₄₇	—	4	2	no	—
277	C₄×C₄₇	—	4	2	no	—
278	C₄×C₄₇	—	4	2	no	—
279	C₄×C₄₇	—	8	1	no	—
280	C₄×C₄₇	C₂×#120	1	2	yes	—
281	C₄×C₄₇	D₄²	1	2	yes	—
282	C₂×#577	C₂×C₄₇	1	2	yes	complemented
283	C₂×#453	C₂×C₄₇	1	2	yes	complemented
284	#3012	C₂×C₄₇	1	2	yes	complemented
285	#2019	C₂×C₄₇	1	2	yes	complemented
286	#2014	C₂×C₄₇	1	1	yes	characteristic, complemented
287	#2007	C₂×C₄₇	1	1	yes	characteristic, complemented
288	#1907	C₂×C₄₇	1	2	yes	complemented
289	#1876	C₂×C₄₇	1	2	yes	complemented
290	#1793	C₂×C₄₇	1	1	yes	characteristic, complemented
291	C₂×C₄₇	—	2	1	no	—
292	C₂×C₄₇	—	2	1	no	—
293	C₂×C₄₇	—	4	4	no	—
294	C₂×C₄₇	—	8	2	no	—
295	C₂×C₄₇	—	8	4	no	—
296	C₂×C₄₇	#2403	1	1	yes	characteristic
297	C₂×C₄₇	#3288	1	1	yes	characteristic
298	C₂×C₄₇	#3291	1	1	yes	characteristic
299	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	1	yes	characteristic
300	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	2	yes	complemented
301	#577	—	2	8	no	—
302	#577	—	2	8	no	—
303	C₂×#137	—	2	2	no	—
304	C₂×#137	C₂²×C₄₇	1	2	yes	complemented
305	C₂×AGL(1,ℤ/8)	—	2	2	no	—
306	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	—
307	C₂×AGL(1,ℤ/8)	C₂²×C₄₇	1	2	yes	—
308	C₂×#135	—	2	2	no	—
309	C₂×#135	C₂²×C₄₇	1	2	yes	—
310	C₂²×SD₁₆	—	2	2	no	—
311	C₂²×SD₁₆	C₂²×C₄₇	1	2	yes	complemented
312	C₂×#131	C₂²×C₄₇	1	2	yes	complemented
313	#537	C₂²×C₄₇	1	1	yes	characteristic
314	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
315	#453	—	2	8	no	—
316	#452	—	2	2	no	—
317	#450	—	2	2	no	—
318	#449	—	2	2	no	—
319	#447	—	2	2	no	—
320	C₂×#104	C₂²×C₄₇	1	2	yes	complemented
321	C₂×#104	C₂²×C₄₇	1	2	yes	complemented
322	#417	C₂²×C₄₇	1	2	yes	complemented
323	C₂×#102	C₂²×C₄₇	1	2	yes	complemented
324	C₂×#101	C₂²×C₄₇	1	1	yes	characteristic, complemented
325	C₂×#100	C₂²×C₄₇	1	2	yes	complemented
326	#408	C₂²×C₄₇	1	2	yes	complemented
327	#391	C₂²×C₄₇	1	1	yes	characteristic, complemented
328	#343	C₂²×C₄₇	1	1	yes	characteristic, complemented
329	#341	C₂²×C₄₇	1	2	yes	complemented
330	#336	C₂²×C₄₇	1	1	yes	characteristic, complemented
331	C₄₇	#23890	1	1	yes	characteristic, direct product factor, Sylow
332	2₋¹⁺⁴	—	2	4	no	—
333	2₋¹⁺⁴	—	4	1	no	—
334	2₋¹⁺⁴	D₄×C₄₇	1	4	yes	complemented
335	H(5,𝔽₂)	—	2	4	no	—
336	H(5,𝔽₂)	—	2	4	no	—
337	H(5,𝔽₂)	—	2	4	no	—
338	H(5,𝔽₂)	—	4	2	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	4	1	no	—
343	C₂×#41	C₂³×C₄₇	1	2	yes	—
344	C₂×#41	C₂³×C₄₇	1	2	yes	—
345	C₂²×Q₈	—	2	2	no	—
346	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
347	C₂²×D₄	—	2	2	no	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	—	4	1	no	—
350	C₂²×D₄	—	4	2	no	—
351	C₂²×D₄	C₂³×C₄₇	1	2	yes	—
352	#137	—	2	16	no	—
353	#137	—	2	4	no	—
354	#137	—	2	4	no	—
355	AGL(1,ℤ/8)	—	2	16	no	—
356	AGL(1,ℤ/8)	—	2	4	no	—
357	AGL(1,ℤ/8)	—	2	4	no	—
358	AGL(1,ℤ/8)	—	4	2	no	—
359	AGL(1,ℤ/8)	D₄×C₄₇	1	8	yes	complemented
360	#135	—	2	16	no	—
361	#135	—	2	4	no	—
362	#135	—	2	4	no	—
363	C₂×Q₁₆	—	2	2	no	—
364	C₂×Q₁₆	D₄×C₄₇	1	2	yes	complemented
365	C₂×SD₁₆	—	2	16	no	—
366	C₂×SD₁₆	—	2	2	no	—
367	C₂×SD₁₆	—	2	4	no	—
368	C₂×SD₁₆	—	2	2	no	—
369	C₂×SD₁₆	—	2	4	no	—
370	C₂×SD₁₆	—	2	2	no	—
371	C₂×SD₁₆	—	2	2	no	—
372	C₂×SD₁₆	—	2	2	no	—
373	C₂×SD₁₆	D₄×C₄₇	1	2	yes	complemented
374	C₂×SD₁₆	D₄×C₄₇	1	2	yes	complemented
375	C₂×D₈	—	2	2	no	—
376	C₂×D₈	—	2	2	no	—
377	C₂×D₈	D₄×C₄₇	1	2	yes	complemented
378	#131	—	2	4	no	—
379	#131	—	2	4	no	—
380	C₂×M₁₆	—	2	2	no	—
381	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
382	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
383	C₂²×C₈	—	2	2	no	—
384	C₂²×C₈	C₂³×C₄₇	1	2	yes	—
385	#127	—	2	4	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	#121	—	2	2	no	—
388	#121	—	4	1	no	—
389	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
391	#104	—	2	4	no	—
392	#104	—	2	4	no	—
393	#103	—	2	2	no	—
394	#102	—	2	2	no	—
395	#102	—	2	2	no	—
396	#102	—	2	2	no	—
397	#101	—	4	1	no	—
398	#100	—	2	4	no	—
399	C₂²⋊C₈	—	2	2	no	—
400	C₂²⋊C₈	—	2	2	no	—
401	#95	—	4	1	no	—
402	C₂⁴	—	2	2	no	—
403	C₂⁴	—	4	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	8	no	—
407	#41	—	2	8	no	—
408	#41	—	2	2	no	—
409	#41	—	2	2	no	—
410	#41	—	2	2	no	—
411	#41	—	2	2	no	—
412	#41	—	4	4	no	—
413	#41	—	4	4	no	—
414	#41	—	4	4	no	—
415	#41	—	4	4	no	—
416	#41	—	4	4	no	—
417	#41	—	4	4	no	—
418	#41	C₂×D₄×C₄₇	1	4	yes	—
419	#41	C₂×D₄×C₄₇	1	4	yes	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	8	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	4	1	no	—
425	C₂×Q₈	—	4	4	no	—
426	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
427	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
428	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	8	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	4	4	no	—
438	C₂×D₄	—	4	2	no	—
439	C₂×D₄	—	4	2	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	4	8	no	—
442	C₂×D₄	—	4	8	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	—	4	1	no	—
447	C₂×D₄	—	8	2	no	—
448	C₂×D₄	—	8	2	no	—
449	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
450	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
451	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
460	Q₁₆	—	2	8	no	—
461	Q₁₆	—	2	4	no	—
462	SD₁₆	—	2	8	no	—
463	SD₁₆	—	2	8	no	—
464	SD₁₆	—	2	8	no	—
465	SD₁₆	—	2	4	no	—
466	SD₁₆	—	2	8	no	—
467	SD₁₆	—	2	4	no	—
468	SD₁₆	—	4	4	no	—
469	D₈	—	2	8	no	—
470	D₈	—	2	4	no	—
471	D₈	—	4	4	no	—
472	M₁₆	—	2	8	no	—
473	M₁₆	—	2	2	no	—
474	M₁₆	—	2	2	no	—
475	M₁₆	—	2	2	no	—
476	M₁₆	C₂×D₄×C₄₇	1	4	yes	—
477	C₂×C₈	—	2	8	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
483	C₂×C₈	C₂×D₄×C₄₇	1	2	yes	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₂²⋊C₄	—	4	2	no	—
487	C₄²	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	4	8	no	—
492	C₂³	—	4	2	no	—
493	C₂³	—	4	2	no	—
494	C₂³	—	8	2	no	—
495	C₂³	—	8	2	no	—
496	C₂³	—	16	2	no	—
497	C₂³	—	16	2	no	—
498	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
499	Q₈	—	2	2	no	—
500	Q₈	—	2	4	no	—
501	Q₈	—	2	2	no	—
502	Q₈	—	2	2	no	—
503	Q₈	—	4	4	no	—
504	Q₈	—	4	2	no	—
505	Q₈	#120×C₄₇	1	4	yes	—
506	D₄	—	2	4	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	4	no	—
509	D₄	—	2	2	no	—
510	D₄	—	2	2	no	—
511	D₄	—	2	2	no	—
512	D₄	—	2	4	no	—
513	D₄	—	4	8	no	—
514	D₄	—	4	4	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	8	8	no	—
521	D₄	—	16	1	no	—
522	D₄	#120×C₄₇	1	4	yes	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	4	8	no	—
530	C₂×C₄	—	4	8	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	1	no	—
539	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
541	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
542	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
543	C₈	—	2	2	no	—
544	C₈	—	2	4	no	—
545	C₈	—	2	2	no	—
546	C₈	—	4	2	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	2	4	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	8	4	no	—
557	C₂²	—	8	2	no	—
558	C₂²	—	8	2	no	—
559	C₂²	—	16	2	no	—
560	C₂²	—	16	8	no	—
561	C₂²	C₄₇×#534	1	1	yes	characteristic
562	C₂²	D₄²×C₄₇	1	1	yes	characteristic
563	C₂²	D₄²×C₄₇	1	1	yes	characteristic
564	C₄	—	2	4	no	—
565	C₄	—	4	2	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	2	no	—
568	C₄	—	8	1	no	—
569	C₄	C₂×#120×C₄₇	1	2	yes	—
570	C₄	D₄²×C₄₇	1	2	yes	—
571	C₂	—	2	1	no	—
572	C₂	—	2	1	no	—
573	C₂	—	4	4	no	—
574	C₂	—	8	2	no	—
575	C₂	—	8	4	no	—
576	C₂	C₄₇×#2403	1	1	yes	characteristic
577	C₂	C₄₇×#3288	1	1	yes	characteristic
578	C₂	C₄₇×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂²×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₇×#577	C₂	semidirect
2. C₂×C₄₇×#453	C₂	semidirect
3. C₄₇×#3012	C₂	semidirect
4. C₄₇×#2019	C₂	semidirect
5. C₄₇×#2014	C₂	semidirect
6. C₄₇×#2007	C₂	semidirect
7. C₄₇×#1907	C₂	semidirect
8. C₄₇×#1876	C₂	semidirect
9. C₄₇×#1793	C₂	semidirect
11. C₂×H(5,𝔽₂)×C₄₇	C₂²	semidirect
15. C₂×#137×C₄₇	C₂²	semidirect
22. C₂²×SD₁₆×C₄₇	C₂²	semidirect
23. C₂×#131×C₄₇	C₂²	semidirect
31. C₂×#104×C₄₇	C₂²	semidirect
32. C₂×#104×C₄₇	C₂²	semidirect
33. C₄₇×#417	C₂²	semidirect
34. C₂×#102×C₄₇	C₂²	semidirect
35. C₂×#101×C₄₇	C₂²	semidirect
36. C₂×#100×C₄₇	C₂²	semidirect
37. C₄₇×#408	C₂²	semidirect
38. C₄₇×#391	C₂²	semidirect
39. C₄₇×#343	C₂²	semidirect
40. C₄₇×#341	C₂²	semidirect
41. C₄₇×#336	C₂²	semidirect
44. 2₋¹⁺⁴×C₄₇	D₄	semidirect
69. AGL(1,ℤ/8)×C₄₇	D₄	semidirect
74. C₂×Q₁₆×C₄₇	D₄	semidirect
83. C₂×SD₁₆×C₄₇	D₄	semidirect
84. C₂×SD₁₆×C₄₇	D₄	semidirect
87. C₂×D₈×C₄₇	D₄	semidirect
257. #23890	C₄₇	direct (N × H)
282. C₂×#577	C₂×C₄₇	semidirect
283. C₂×#453	C₂×C₄₇	semidirect
284. #3012	C₂×C₄₇	semidirect
285. #2019	C₂×C₄₇	semidirect
286. #2014	C₂×C₄₇	semidirect
287. #2007	C₂×C₄₇	semidirect
288. #1907	C₂×C₄₇	semidirect
289. #1876	C₂×C₄₇	semidirect
290. #1793	C₂×C₄₇	semidirect
300. C₂×H(5,𝔽₂)	C₂²×C₄₇	semidirect
304. C₂×#137	C₂²×C₄₇	semidirect
311. C₂²×SD₁₆	C₂²×C₄₇	semidirect
312. C₂×#131	C₂²×C₄₇	semidirect
320. C₂×#104	C₂²×C₄₇	semidirect
321. C₂×#104	C₂²×C₄₇	semidirect
322. #417	C₂²×C₄₇	semidirect
323. C₂×#102	C₂²×C₄₇	semidirect
324. C₂×#101	C₂²×C₄₇	semidirect
325. C₂×#100	C₂²×C₄₇	semidirect
326. #408	C₂²×C₄₇	semidirect
327. #391	C₂²×C₄₇	semidirect
328. #343	C₂²×C₄₇	semidirect
329. #341	C₂²×C₄₇	semidirect
330. #336	C₂²×C₄₇	semidirect
331. C₄₇	#23890	direct (N × H)
334. 2₋¹⁺⁴	D₄×C₄₇	semidirect
359. AGL(1,ℤ/8)	D₄×C₄₇	semidirect
364. C₂×Q₁₆	D₄×C₄₇	semidirect
373. C₂×SD₁₆	D₄×C₄₇	semidirect
374. C₂×SD₁₆	D₄×C₄₇	semidirect
377. C₂×D₈	D₄×C₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#577	2
C₂×C₄₇×#453	2
C₄₇×#3012	2
C₄₇×#2019	2
C₄₇×#2014	2
C₄₇×#2007	2
C₄₇×#1907	2
C₄₇×#1876	2
C₄₇×#1793	2
#23890	47

Subgroups — C47×#23890 #26063588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23890 #26063588