Subgroups — C₄₇×#23893 #26063591

Order 12032.

1610 conjugacy classes of subgroups.

The subgroup lattice has 564 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	2	yes	maximal, complemented
2	C₄₇×#3020	C₂	1	2	yes	maximal, complemented
3	C₄₇×#3011	C₂	1	2	yes	maximal, complemented
4	C₄₇×#2020	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2017	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2009	C₂	1	2	yes	maximal, complemented
7	C₄₇×#1902	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1881	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1794	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₇×#585	C₂²	1	1	yes	characteristic
11	C₂×2₋¹⁺⁴×C₄₇	C₂²	1	2	yes	—
12	C₄₇×#578	—	2	4	no	—
13	C₄₇×#578	—	2	4	no	—
14	C₄₇×#577	—	2	4	no	—
15	C₄₇×#577	—	2	4	no	—
16	C₄₇×#575	—	2	4	no	—
17	C₄₇×#575	C₂²	1	2	yes	—
18	C₄₇×#575	C₂²	1	2	yes	—
19	C₄₇×#575	C₂²	1	1	yes	characteristic, complemented
20	C₂×#137×C₄₇	—	2	4	no	—
21	C₂×#137×C₄₇	C₂²	1	2	yes	—
22	C₂×#137×C₄₇	C₂²	1	2	yes	—
23	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₄₇×#568	C₂²	1	2	yes	—
25	C₄₇×#536	C₂²	1	2	yes	complemented
26	C₄₇×#456	—	2	4	no	—
27	C₄₇×#454	—	2	4	no	—
28	C₄₇×#452	—	2	4	no	—
29	C₄₇×#451	—	2	4	no	—
30	C₄₇×#421	C₂²	1	2	yes	—
31	C₄₇×#421	C₂²	1	2	yes	complemented
32	C₄₇×#419	C₂²	1	2	yes	complemented
33	C₄₇×#416	C₂²	1	2	yes	—
34	C₂×#101×C₄₇	C₂²	1	2	yes	complemented
35	C₄₇×#410	C₂²	1	1	yes	characteristic
36	C₄₇×#408	C₂²	1	1	yes	characteristic
37	C₄₇×#408	C₂²	1	2	yes	complemented
38	C₄₇×#344	C₂²	1	2	yes	complemented
39	C₄₇×#341	C₂²	1	1	yes	characteristic, complemented
40	C₄₇×#338	C₂²	1	1	yes	characteristic, complemented
41	2₋¹⁺⁴×C₄₇	—	2	4	no	—
42	2₋¹⁺⁴×C₄₇	—	2	4	no	—
43	2₋¹⁺⁴×C₄₇	—	2	2	no	—
44	2₋¹⁺⁴×C₄₇	—	2	4	no	—
45	2₋¹⁺⁴×C₄₇	—	4	2	no	—
46	2₋¹⁺⁴×C₄₇	D₄	1	2	yes	—
47	H(5,𝔽₂)×C₄₇	—	2	2	no	—
48	H(5,𝔽₂)×C₄₇	—	4	1	no	—
49	H(5,𝔽₂)×C₄₇	D₄	1	2	yes	—
50	C₂×#41×C₄₇	—	2	4	no	—
51	C₂×#41×C₄₇	—	2	2	no	—
52	C₂×#41×C₄₇	—	2	2	no	—
53	C₂×#41×C₄₇	—	4	2	no	—
54	C₂×#41×C₄₇	—	4	1	no	—
55	C₂×#41×C₄₇	C₂³	1	2	yes	—
56	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×#41×C₄₇	C₂³	1	2	yes	—
58	C₂²×Q₈×C₄₇	—	2	2	no	—
59	C₂²×Q₈×C₄₇	—	2	2	no	—
60	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
61	#137×C₄₇	—	2	8	no	—
62	#137×C₄₇	—	2	8	no	—
63	#137×C₄₇	—	2	8	no	—
64	#137×C₄₇	—	2	4	no	—
65	#137×C₄₇	—	2	4	no	—
66	#137×C₄₇	—	2	2	no	—
67	#137×C₄₇	—	2	4	no	—
68	#137×C₄₇	—	2	2	no	—
69	#137×C₄₇	—	2	2	no	—
70	#137×C₄₇	—	2	2	no	—
71	#137×C₄₇	—	2	4	no	—
72	#137×C₄₇	D₄	1	4	yes	—
73	AGL(1,ℤ/8)×C₄₇	—	2	8	no	—
74	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
75	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
76	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
77	AGL(1,ℤ/8)×C₄₇	—	2	2	no	—
78	AGL(1,ℤ/8)×C₄₇	D₄	1	4	yes	—
79	#135×C₄₇	—	2	8	no	—
80	#135×C₄₇	—	2	8	no	—
81	#135×C₄₇	—	2	4	no	—
82	#135×C₄₇	—	2	4	no	—
83	#135×C₄₇	—	4	1	no	—
84	C₂×Q₁₆×C₄₇	—	2	8	no	—
85	C₂×Q₁₆×C₄₇	—	2	2	no	—
86	C₂×Q₁₆×C₄₇	—	2	4	no	—
87	C₂×Q₁₆×C₄₇	—	4	1	no	—
88	C₂×Q₁₆×C₄₇	D₄	1	4	yes	—
89	C₂×SD₁₆×C₄₇	—	2	8	no	—
90	C₂×SD₁₆×C₄₇	—	2	2	no	—
91	C₂×SD₁₆×C₄₇	D₄	1	4	yes	—
92	#131×C₄₇	—	2	4	no	—
93	#131×C₄₇	—	2	4	no	—
94	C₂×M₁₆×C₄₇	—	2	4	no	—
95	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
96	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
97	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
98	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
99	C₄⋊Q₈×C₄₇	—	2	2	no	—
100	#124×C₄₇	—	2	2	no	—
101	#123×C₄₇	—	4	2	no	—
102	C₂²⋊Q₈×C₄₇	—	2	4	no	—
103	#117×C₄₇	C₂³	1	2	yes	—
104	#104×C₄₇	—	2	4	no	—
105	#104×C₄₇	—	2	4	no	—
106	#103×C₄₇	—	2	2	no	—
107	#103×C₄₇	—	2	4	no	—
108	#102×C₄₇	—	2	2	no	—
109	#101×C₄₇	—	2	4	no	—
110	#99×C₄₇	—	4	1	no	—
111	C₂²⋊C₈×C₄₇	—	2	4	no	—
112	C₂²⋊C₈×C₄₇	—	4	1	no	—
113	#41×C₄₇	—	2	2	no	—
114	#41×C₄₇	—	2	2	no	—
115	#41×C₄₇	—	2	8	no	—
116	#41×C₄₇	—	2	4	no	—
117	#41×C₄₇	—	2	4	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	—	2	2	no	—
123	#41×C₄₇	—	2	2	no	—
124	#41×C₄₇	—	4	2	no	—
125	#41×C₄₇	—	4	2	no	—
126	#41×C₄₇	—	4	4	no	—
127	#41×C₄₇	—	4	4	no	—
128	#41×C₄₇	—	4	8	no	—
129	#41×C₄₇	—	4	4	no	—
130	#41×C₄₇	—	4	2	no	—
131	#41×C₄₇	—	4	8	no	—
132	#41×C₄₇	—	4	4	no	—
133	#41×C₄₇	C₂×D₄	1	4	yes	—
134	#41×C₄₇	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₄₇	—	2	2	no	—
136	C₂×Q₈×C₄₇	—	2	8	no	—
137	C₂×Q₈×C₄₇	—	2	4	no	—
138	C₂×Q₈×C₄₇	—	2	2	no	—
139	C₂×Q₈×C₄₇	—	2	2	no	—
140	C₂×Q₈×C₄₇	—	2	2	no	—
141	C₂×Q₈×C₄₇	—	2	2	no	—
142	C₂×Q₈×C₄₇	—	4	2	no	—
143	C₂×Q₈×C₄₇	—	4	4	no	—
144	C₂×Q₈×C₄₇	—	4	4	no	—
145	C₂×Q₈×C₄₇	—	4	1	no	—
146	C₂×Q₈×C₄₇	—	4	4	no	—
147	C₂×Q₈×C₄₇	C₂×D₄	1	4	yes	—
148	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
151	C₂×D₄×C₄₇	—	2	2	no	—
152	C₂×D₄×C₄₇	—	2	4	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	1	no	—
157	C₂×D₄×C₄₇	—	4	4	no	—
158	C₂×D₄×C₄₇	—	4	2	no	—
159	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₄₇	—	2	2	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	—	4	2	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	Q₁₆×C₄₇	—	2	4	no	—
173	Q₁₆×C₄₇	—	2	8	no	—
174	Q₁₆×C₄₇	—	4	4	no	—
175	Q₁₆×C₄₇	—	4	1	no	—
176	SD₁₆×C₄₇	—	2	8	no	—
177	SD₁₆×C₄₇	—	2	4	no	—
178	SD₁₆×C₄₇	—	2	4	no	—
179	SD₁₆×C₄₇	—	2	4	no	—
180	SD₁₆×C₄₇	—	2	4	no	—
181	SD₁₆×C₄₇	—	2	8	no	—
182	SD₁₆×C₄₇	—	4	2	no	—
183	D₈×C₄₇	—	2	4	no	—
184	D₈×C₄₇	—	2	4	no	—
185	D₈×C₄₇	—	4	1	no	—
186	M₁₆×C₄₇	—	2	8	no	—
187	M₁₆×C₄₇	—	2	2	no	—
188	M₁₆×C₄₇	—	2	2	no	—
189	M₁₆×C₄₇	—	2	2	no	—
190	M₁₆×C₄₇	—	2	2	no	—
191	M₁₆×C₄₇	C₂×D₄	1	4	yes	—
192	C₂×C₈×C₄₇	—	2	8	no	—
193	C₂×C₈×C₄₇	—	2	2	no	—
194	C₂×C₈×C₄₇	—	2	2	no	—
195	C₂×C₈×C₄₇	—	2	1	no	—
196	C₂×C₈×C₄₇	—	2	1	no	—
197	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
198	C₄⋊C₄×C₄₇	—	2	2	no	—
199	C₄⋊C₄×C₄₇	—	4	4	no	—
200	C₂²⋊C₄×C₄₇	—	2	2	no	—
201	C₂²⋊C₄×C₄₇	—	4	4	no	—
202	C₄²×C₄₇	—	2	2	no	—
203	C₂³×C₄₇	—	2	2	no	—
204	C₂³×C₄₇	—	2	2	no	—
205	C₂³×C₄₇	—	2	1	no	—
206	C₂³×C₄₇	—	4	2	no	—
207	C₂³×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
208	Q₈×C₄₇	—	2	2	no	—
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	—	2	4	no	—
214	Q₈×C₄₇	—	4	2	no	—
215	Q₈×C₄₇	—	4	8	no	—
216	Q₈×C₄₇	—	4	2	no	—
217	Q₈×C₄₇	—	4	2	no	—
218	Q₈×C₄₇	#120	1	4	yes	—
219	D₄×C₄₇	—	2	4	no	—
220	D₄×C₄₇	—	2	4	no	—
221	D₄×C₄₇	—	2	2	no	—
222	D₄×C₄₇	—	2	2	no	—
223	D₄×C₄₇	—	2	2	no	—
224	D₄×C₄₇	—	2	2	no	—
225	D₄×C₄₇	—	4	4	no	—
226	D₄×C₄₇	—	4	2	no	—
227	D₄×C₄₇	—	4	8	no	—
228	D₄×C₄₇	—	4	2	no	—
229	D₄×C₄₇	—	4	2	no	—
230	D₄×C₄₇	—	4	2	no	—
231	D₄×C₄₇	#120	1	4	yes	—
232	C₂×C₄×C₄₇	—	2	2	no	—
233	C₂×C₄×C₄₇	—	2	4	no	—
234	C₂×C₄×C₄₇	—	2	4	no	—
235	C₂×C₄×C₄₇	—	2	1	no	—
236	C₂×C₄×C₄₇	—	2	4	no	—
237	C₂×C₄×C₄₇	—	2	2	no	—
238	C₂×C₄×C₄₇	—	4	2	no	—
239	C₂×C₄×C₄₇	—	4	4	no	—
240	C₂×C₄×C₄₇	—	4	8	no	—
241	C₂×C₄×C₄₇	—	4	8	no	—
242	C₂×C₄×C₄₇	—	4	2	no	—
243	C₂×C₄×C₄₇	—	4	2	no	—
244	C₂×C₄×C₄₇	—	4	2	no	—
245	C₂×C₄×C₄₇	—	4	1	no	—
246	C₂×C₄×C₄₇	—	4	1	no	—
247	C₂×C₄×C₄₇	—	4	2	no	—
248	C₂×C₄×C₄₇	—	4	2	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	4	no	—
254	C₈×C₄₇	—	2	4	no	—
255	C₈×C₄₇	—	4	1	no	—
256	C₈×C₄₇	—	4	1	no	—
257	#23893	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
258	C₂²×C₄₇	—	2	4	no	—
259	C₂²×C₄₇	—	4	1	no	—
260	C₂²×C₄₇	—	4	4	no	—
261	C₂²×C₄₇	—	4	2	no	—
262	C₂²×C₄₇	—	4	1	no	—
263	C₂²×C₄₇	—	8	2	no	—
264	C₂²×C₄₇	—	8	1	no	—
265	C₂²×C₄₇	—	16	2	no	—
266	C₂²×C₄₇	#534	1	1	yes	characteristic
267	C₂²×C₄₇	D₄²	1	2	yes	—
268	C₄×C₄₇	—	2	4	no	—
269	C₄×C₄₇	—	4	1	no	—
270	C₄×C₄₇	—	4	4	no	—
271	C₄×C₄₇	—	4	2	no	—
272	C₄×C₄₇	—	8	2	no	—
273	C₄×C₄₇	C₂×#120	1	2	yes	—
274	C₄×C₄₇	D₄²	1	1	yes	characteristic
275	C₄×C₄₇	D₄²	1	1	yes	characteristic
276	#3586	C₂×C₄₇	1	2	yes	complemented
277	#3020	C₂×C₄₇	1	2	yes	complemented
278	#3011	C₂×C₄₇	1	2	yes	complemented
279	#2020	C₂×C₄₇	1	1	yes	characteristic, complemented
280	#2017	C₂×C₄₇	1	1	yes	characteristic, complemented
281	#2009	C₂×C₄₇	1	2	yes	complemented
282	#1902	C₂×C₄₇	1	2	yes	complemented
283	#1881	C₂×C₄₇	1	2	yes	complemented
284	#1794	C₂×C₄₇	1	1	yes	characteristic, complemented
285	C₂×C₄₇	—	2	2	no	—
286	C₂×C₄₇	—	2	1	no	—
287	C₂×C₄₇	—	4	4	no	—
288	C₂×C₄₇	—	8	2	no	—
289	C₂×C₄₇	—	8	1	no	—
290	C₂×C₄₇	#2403	1	1	yes	characteristic
291	#585	C₂²×C₄₇	1	1	yes	characteristic
292	C₂×2₋¹⁺⁴	C₂²×C₄₇	1	2	yes	—
293	#578	—	2	4	no	—
294	#578	—	2	4	no	—
295	#577	—	2	4	no	—
296	#577	—	2	4	no	—
297	#575	—	2	4	no	—
298	#575	C₂²×C₄₇	1	2	yes	—
299	#575	C₂²×C₄₇	1	2	yes	—
300	#575	C₂²×C₄₇	1	1	yes	characteristic, complemented
301	C₂×#137	—	2	4	no	—
302	C₂×#137	C₂²×C₄₇	1	2	yes	—
303	C₂×#137	C₂²×C₄₇	1	2	yes	—
304	C₂²×Q₁₆	C₂²×C₄₇	1	1	yes	characteristic, complemented
305	#568	C₂²×C₄₇	1	2	yes	—
306	#536	C₂²×C₄₇	1	2	yes	complemented
307	#456	—	2	4	no	—
308	#454	—	2	4	no	—
309	#452	—	2	4	no	—
310	#451	—	2	4	no	—
311	#421	C₂²×C₄₇	1	2	yes	—
312	#421	C₂²×C₄₇	1	2	yes	complemented
313	#419	C₂²×C₄₇	1	2	yes	complemented
314	#416	C₂²×C₄₇	1	2	yes	—
315	C₂×#101	C₂²×C₄₇	1	2	yes	complemented
316	#410	C₂²×C₄₇	1	1	yes	characteristic
317	#408	C₂²×C₄₇	1	1	yes	characteristic
318	#408	C₂²×C₄₇	1	2	yes	complemented
319	#344	C₂²×C₄₇	1	2	yes	complemented
320	#341	C₂²×C₄₇	1	1	yes	characteristic, complemented
321	#338	C₂²×C₄₇	1	1	yes	characteristic, complemented
322	C₄₇	#23893	1	1	yes	characteristic, direct product factor, Sylow
323	2₋¹⁺⁴	—	2	4	no	—
324	2₋¹⁺⁴	—	2	4	no	—
325	2₋¹⁺⁴	—	2	2	no	—
326	2₋¹⁺⁴	—	2	4	no	—
327	2₋¹⁺⁴	—	4	2	no	—
328	2₋¹⁺⁴	D₄×C₄₇	1	2	yes	—
329	H(5,𝔽₂)	—	2	2	no	—
330	H(5,𝔽₂)	—	4	1	no	—
331	H(5,𝔽₂)	D₄×C₄₇	1	2	yes	—
332	C₂×#41	—	2	4	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	4	2	no	—
336	C₂×#41	—	4	1	no	—
337	C₂×#41	C₂³×C₄₇	1	2	yes	—
338	C₂×#41	C₂³×C₄₇	1	1	yes	characteristic
339	C₂×#41	C₂³×C₄₇	1	2	yes	—
340	C₂²×Q₈	—	2	2	no	—
341	C₂²×Q₈	—	2	2	no	—
342	C₂²×Q₈	C₂³×C₄₇	1	2	yes	—
343	#137	—	2	8	no	—
344	#137	—	2	8	no	—
345	#137	—	2	8	no	—
346	#137	—	2	4	no	—
347	#137	—	2	4	no	—
348	#137	—	2	2	no	—
349	#137	—	2	4	no	—
350	#137	—	2	2	no	—
351	#137	—	2	2	no	—
352	#137	—	2	2	no	—
353	#137	—	2	4	no	—
354	#137	D₄×C₄₇	1	4	yes	—
355	AGL(1,ℤ/8)	—	2	8	no	—
356	AGL(1,ℤ/8)	—	2	2	no	—
357	AGL(1,ℤ/8)	—	2	2	no	—
358	AGL(1,ℤ/8)	—	2	2	no	—
359	AGL(1,ℤ/8)	—	2	2	no	—
360	AGL(1,ℤ/8)	D₄×C₄₇	1	4	yes	—
361	#135	—	2	8	no	—
362	#135	—	2	8	no	—
363	#135	—	2	4	no	—
364	#135	—	2	4	no	—
365	#135	—	4	1	no	—
366	C₂×Q₁₆	—	2	8	no	—
367	C₂×Q₁₆	—	2	2	no	—
368	C₂×Q₁₆	—	2	4	no	—
369	C₂×Q₁₆	—	4	1	no	—
370	C₂×Q₁₆	D₄×C₄₇	1	4	yes	—
371	C₂×SD₁₆	—	2	8	no	—
372	C₂×SD₁₆	—	2	2	no	—
373	C₂×SD₁₆	D₄×C₄₇	1	4	yes	—
374	#131	—	2	4	no	—
375	#131	—	2	4	no	—
376	C₂×M₁₆	—	2	4	no	—
377	C₂×M₁₆	C₂³×C₄₇	1	1	yes	characteristic
378	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
379	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
380	C₂²×C₈	C₂³×C₄₇	1	1	yes	characteristic
381	C₄⋊Q₈	—	2	2	no	—
382	#124	—	2	2	no	—
383	#123	—	4	2	no	—
384	C₂²⋊Q₈	—	2	4	no	—
385	#117	C₂³×C₄₇	1	2	yes	—
386	#104	—	2	4	no	—
387	#104	—	2	4	no	—
388	#103	—	2	2	no	—
389	#103	—	2	4	no	—
390	#102	—	2	2	no	—
391	#101	—	2	4	no	—
392	#99	—	4	1	no	—
393	C₂²⋊C₈	—	2	4	no	—
394	C₂²⋊C₈	—	4	1	no	—
395	#41	—	2	2	no	—
396	#41	—	2	2	no	—
397	#41	—	2	8	no	—
398	#41	—	2	4	no	—
399	#41	—	2	4	no	—
400	#41	—	2	2	no	—
401	#41	—	2	2	no	—
402	#41	—	2	2	no	—
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	4	2	no	—
407	#41	—	4	2	no	—
408	#41	—	4	4	no	—
409	#41	—	4	4	no	—
410	#41	—	4	8	no	—
411	#41	—	4	4	no	—
412	#41	—	4	2	no	—
413	#41	—	4	8	no	—
414	#41	—	4	4	no	—
415	#41	C₂×D₄×C₄₇	1	4	yes	—
416	#41	C₂×D₄×C₄₇	1	4	yes	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	8	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	2	2	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	4	2	no	—
425	C₂×Q₈	—	4	4	no	—
426	C₂×Q₈	—	4	4	no	—
427	C₂×Q₈	—	4	1	no	—
428	C₂×Q₈	—	4	4	no	—
429	C₂×Q₈	C₂×D₄×C₄₇	1	4	yes	—
430	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₄₇	1	1	yes	characteristic
432	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	4	2	no	—
438	C₂×D₄	—	4	1	no	—
439	C₂×D₄	—	4	4	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
442	C₂×D₄	C₂×D₄×C₄₇	1	1	yes	characteristic
443	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	4	2	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
452	Q₁₆	—	2	8	no	—
453	Q₁₆	—	2	4	no	—
454	Q₁₆	—	2	4	no	—
455	Q₁₆	—	2	8	no	—
456	Q₁₆	—	4	4	no	—
457	Q₁₆	—	4	1	no	—
458	SD₁₆	—	2	8	no	—
459	SD₁₆	—	2	4	no	—
460	SD₁₆	—	2	4	no	—
461	SD₁₆	—	2	4	no	—
462	SD₁₆	—	2	4	no	—
463	SD₁₆	—	2	8	no	—
464	SD₁₆	—	4	2	no	—
465	D₈	—	2	4	no	—
466	D₈	—	2	4	no	—
467	D₈	—	4	1	no	—
468	M₁₆	—	2	8	no	—
469	M₁₆	—	2	2	no	—
470	M₁₆	—	2	2	no	—
471	M₁₆	—	2	2	no	—
472	M₁₆	—	2	2	no	—
473	M₁₆	C₂×D₄×C₄₇	1	4	yes	—
474	C₂×C₈	—	2	8	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	C₂×D₄×C₄₇	1	4	yes	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	4	4	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	4	4	no	—
484	C₄²	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	4	2	no	—
489	C₂³	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
490	Q₈	—	2	2	no	—
491	Q₈	—	2	2	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	2	2	no	—
494	Q₈	—	2	2	no	—
495	Q₈	—	2	4	no	—
496	Q₈	—	4	2	no	—
497	Q₈	—	4	8	no	—
498	Q₈	—	4	2	no	—
499	Q₈	—	4	2	no	—
500	Q₈	#120×C₄₇	1	4	yes	—
501	D₄	—	2	4	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	2	no	—
506	D₄	—	2	2	no	—
507	D₄	—	4	4	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	8	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	2	no	—
512	D₄	—	4	2	no	—
513	D₄	#120×C₄₇	1	4	yes	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	8	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
534	C₂×C₄	H(5,𝔽₂)×C₄₇	1	1	yes	characteristic
535	C₈	—	2	4	no	—
536	C₈	—	2	4	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	4	1	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	8	2	no	—
545	C₂²	—	8	1	no	—
546	C₂²	—	16	2	no	—
547	C₂²	C₄₇×#534	1	1	yes	characteristic
548	C₂²	D₄²×C₄₇	1	2	yes	—
549	C₄	—	2	4	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	2	no	—
553	C₄	—	8	2	no	—
554	C₄	C₂×#120×C₄₇	1	2	yes	—
555	C₄	D₄²×C₄₇	1	1	yes	characteristic
556	C₄	D₄²×C₄₇	1	1	yes	characteristic
557	C₂	—	2	2	no	—
558	C₂	—	2	1	no	—
559	C₂	—	4	4	no	—
560	C₂	—	8	2	no	—
561	C₂	—	8	1	no	—
562	C₂	C₄₇×#2403	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₇	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
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
1. C₄₇×#3586	C₂	semidirect
2. C₄₇×#3020	C₂	semidirect
3. C₄₇×#3011	C₂	semidirect
4. C₄₇×#2020	C₂	semidirect
5. C₄₇×#2017	C₂	semidirect
6. C₄₇×#2009	C₂	semidirect
7. C₄₇×#1902	C₂	semidirect
8. C₄₇×#1881	C₂	semidirect
9. C₄₇×#1794	C₂	semidirect
19. C₄₇×#575	C₂²	semidirect
23. C₂²×Q₁₆×C₄₇	C₂²	semidirect
25. C₄₇×#536	C₂²	semidirect
31. C₄₇×#421	C₂²	semidirect
32. C₄₇×#419	C₂²	semidirect
34. C₂×#101×C₄₇	C₂²	semidirect
37. C₄₇×#408	C₂²	semidirect
38. C₄₇×#344	C₂²	semidirect
39. C₄₇×#341	C₂²	semidirect
40. C₄₇×#338	C₂²	semidirect
257. #23893	C₄₇	direct (N × H)
276. #3586	C₂×C₄₇	semidirect
277. #3020	C₂×C₄₇	semidirect
278. #3011	C₂×C₄₇	semidirect
279. #2020	C₂×C₄₇	semidirect
280. #2017	C₂×C₄₇	semidirect
281. #2009	C₂×C₄₇	semidirect
282. #1902	C₂×C₄₇	semidirect
283. #1881	C₂×C₄₇	semidirect
284. #1794	C₂×C₄₇	semidirect
300. #575	C₂²×C₄₇	semidirect
304. C₂²×Q₁₆	C₂²×C₄₇	semidirect
306. #536	C₂²×C₄₇	semidirect
312. #421	C₂²×C₄₇	semidirect
313. #419	C₂²×C₄₇	semidirect
315. C₂×#101	C₂²×C₄₇	semidirect
318. #408	C₂²×C₄₇	semidirect
319. #344	C₂²×C₄₇	semidirect
320. #341	C₂²×C₄₇	semidirect
321. #338	C₂²×C₄₇	semidirect
322. C₄₇	#23893	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×#3020	2
C₄₇×#3011	2
C₄₇×#2020	2
C₄₇×#2017	2
C₄₇×#2009	2
C₄₇×#1902	2
C₄₇×#1881	2
C₄₇×#1794	2
#23893	47

Subgroups — C47×#23893 #26063591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#23893 #26063591