Subgroups — C₄₇×#18771 #26058469

Order 12032.

1234 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₇×#550	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₇×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#3077	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1641	C₂	1	4	yes	maximal, complemented
6	C₄₇×#1631	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₇×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₄₇	—	2	1	no	—
11	C₄₇×#550	—	2	2	no	—
12	C₄₇×#550	—	2	2	no	—
13	C₄₇×#550	—	2	2	no	—
14	C₄₇×#550	C₂²	1	4	yes	—
15	C₄₇×#533	C₂²	1	1	yes	characteristic
16	C₂×#127×C₄₇	—	2	1	no	—
17	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₄₇	—	2	2	no	—
19	C₂×#121×C₄₇	—	2	1	no	—
20	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₇	—	2	1	no	—
22	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#492	C₂²	1	2	yes	complemented
24	C₄₇×#486	C₂²	1	2	yes	complemented
25	C₄₇×#484	—	2	1	no	—
26	C₄₇×#478	—	2	2	no	—
27	C₄₇×#477	C₂²	1	2	yes	complemented
28	C₄₇×#476	C₂²	1	2	yes	complemented
29	C₄₇×#474	C₂²	1	2	yes	complemented
30	C₄₇×#462	C₂²	1	2	yes	complemented
31	C₄₇×#460	—	2	2	no	—
32	C₄₇×#451	—	2	1	no	—
33	C₄₇×#450	—	2	2	no	—
34	C₄×Q₁₆×C₄₇	—	2	2	no	—
35	C₄₇×#431	C₂²	1	1	yes	characteristic
36	C₄₇×#431	C₂²	1	1	yes	characteristic
37	C₄₇×#428	—	2	1	no	—
38	C₄₇×#423	C₂²	1	1	yes	characteristic
39	C₄₇×#418	—	2	1	no	—
40	C₄₇×#331	C₂²	1	4	yes	complemented
41	C₄₇×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₄₇×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂×#41×C₄₇	—	2	4	no	—
44	C₂×#41×C₄₇	—	2	4	no	—
45	C₂×#41×C₄₇	—	4	8	no	—
46	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂²×D₄×C₄₇	—	2	2	no	—
49	C₂²×D₄×C₄₇	—	2	1	no	—
50	C₂²×D₄×C₄₇	—	2	1	no	—
51	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂²×D₄×C₄₇	D₄	1	2	yes	—
53	C₂³×C₄×C₄₇	—	2	2	no	—
54	C₂³×C₄×C₄₇	—	2	1	no	—
55	C₂×Q₁₆×C₄₇	—	2	2	no	—
56	C₂×SD₁₆×C₄₇	—	2	4	no	—
57	C₂×M₁₆×C₄₇	—	2	1	no	—
58	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₄₇	C₂³	1	1	yes	characteristic
60	#127×C₄₇	—	2	4	no	—
61	#127×C₄₇	—	2	4	no	—
62	#127×C₄₇	—	2	1	no	—
63	#127×C₄₇	—	2	1	no	—
64	#127×C₄₇	D₄	1	2	yes	—
65	#127×C₄₇	C₂³	1	2	yes	—
66	#125×C₄₇	—	2	1	no	—
67	C₂²⋊Q₈×C₄₇	—	2	1	no	—
68	C₂²⋊Q₈×C₄₇	—	2	1	no	—
69	#121×C₄₇	—	2	4	no	—
70	#121×C₄₇	—	2	8	no	—
71	#121×C₄₇	—	2	8	no	—
72	#121×C₄₇	—	2	4	no	—
73	C₄×Q₈×C₄₇	—	2	1	no	—
74	C₄×Q₈×C₄₇	—	2	1	no	—
75	C₄×Q₈×C₄₇	—	2	1	no	—
76	C₄×Q₈×C₄₇	—	2	1	no	—
77	C₄×Q₈×C₄₇	D₄	1	2	yes	—
78	C₄×Q₈×C₄₇	C₂³	1	2	yes	—
79	C₄×D₄×C₄₇	—	2	4	no	—
80	C₄×D₄×C₄₇	—	2	4	no	—
81	C₄×D₄×C₄₇	—	2	1	no	—
82	C₄×D₄×C₄₇	—	2	1	no	—
83	C₄×D₄×C₄₇	D₄	1	2	yes	—
84	C₄×D₄×C₄₇	C₂³	1	2	yes	—
85	#117×C₄₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
91	C₂×C₄²×C₄₇	—	2	1	no	—
92	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
93	#107×C₄₇	—	2	2	no	—
94	#107×C₄₇	—	2	2	no	—
95	#106×C₄₇	—	2	2	no	—
96	C₄⋊C₈×C₄₇	—	2	1	no	—
97	C₄⋊C₈×C₄₇	—	2	1	no	—
98	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
99	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
100	#103×C₄₇	—	2	2	no	—
101	#103×C₄₇	—	2	2	no	—
102	#102×C₄₇	—	2	4	no	—
103	#102×C₄₇	—	2	2	no	—
104	#102×C₄₇	—	2	2	no	—
105	C₂²⋊C₈×C₄₇	—	2	1	no	—
106	C₂²⋊C₈×C₄₇	—	2	1	no	—
107	C₄×C₈×C₄₇	—	2	1	no	—
108	C₄×C₈×C₄₇	C₂³	1	2	yes	—
109	C₂⁴×C₄₇	—	2	1	no	—
110	C₂⁴×C₄₇	—	2	2	no	—
111	#41×C₄₇	—	4	16	no	—
112	#41×C₄₇	—	4	8	no	—
113	#41×C₄₇	—	4	8	no	—
114	C₂×Q₈×C₄₇	—	2	4	no	—
115	C₂×Q₈×C₄₇	—	2	1	no	—
116	C₂×Q₈×C₄₇	—	2	1	no	—
117	C₂×Q₈×C₄₇	—	2	1	no	—
118	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₄₇	—	2	8	no	—
120	C₂×D₄×C₄₇	—	2	8	no	—
121	C₂×D₄×C₄₇	—	2	1	no	—
122	C₂×D₄×C₄₇	—	2	2	no	—
123	C₂×D₄×C₄₇	—	2	2	no	—
124	C₂×D₄×C₄₇	—	2	2	no	—
125	C₂×D₄×C₄₇	—	2	2	no	—
126	C₂×D₄×C₄₇	—	2	2	no	—
127	C₂×D₄×C₄₇	—	2	2	no	—
128	C₂×D₄×C₄₇	—	2	2	no	—
129	C₂×D₄×C₄₇	—	4	8	no	—
130	C₂×D₄×C₄₇	—	4	4	no	—
131	C₂×D₄×C₄₇	—	4	2	no	—
132	C₂×D₄×C₄₇	—	4	2	no	—
133	C₂×D₄×C₄₇	—	8	2	no	—
134	C₂×D₄×C₄₇	—	8	4	no	—
135	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₄₇	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₄₇	—	2	1	no	—
138	C₂²×C₄×C₄₇	—	2	8	no	—
139	C₂²×C₄×C₄₇	—	2	1	no	—
140	C₂²×C₄×C₄₇	—	2	1	no	—
141	C₂²×C₄×C₄₇	—	2	2	no	—
142	C₂²×C₄×C₄₇	—	2	2	no	—
143	C₂²×C₄×C₄₇	—	4	8	no	—
144	C₂²×C₄×C₄₇	—	4	4	no	—
145	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
148	Q₁₆×C₄₇	—	4	4	no	—
149	SD₁₆×C₄₇	—	8	4	no	—
150	M₁₆×C₄₇	—	4	2	no	—
151	C₂×C₈×C₄₇	—	2	2	no	—
152	C₂×C₈×C₄₇	—	2	2	no	—
153	C₂×C₈×C₄₇	—	2	2	no	—
154	C₄⋊C₄×C₄₇	—	2	2	no	—
155	C₄⋊C₄×C₄₇	—	2	1	no	—
156	C₄⋊C₄×C₄₇	—	2	1	no	—
157	C₄⋊C₄×C₄₇	—	2	2	no	—
158	C₄⋊C₄×C₄₇	—	2	1	no	—
159	C₄⋊C₄×C₄₇	—	2	1	no	—
160	C₄⋊C₄×C₄₇	—	2	1	no	—
161	C₄⋊C₄×C₄₇	—	2	1	no	—
162	C₄⋊C₄×C₄₇	—	4	1	no	—
163	C₄⋊C₄×C₄₇	—	4	1	no	—
164	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
166	C₂²⋊C₄×C₄₇	—	2	8	no	—
167	C₂²⋊C₄×C₄₇	—	2	2	no	—
168	C₂²⋊C₄×C₄₇	—	2	2	no	—
169	C₂²⋊C₄×C₄₇	—	4	1	no	—
170	C₄²×C₄₇	—	2	2	no	—
171	C₄²×C₄₇	—	2	2	no	—
172	C₄²×C₄₇	—	4	1	no	—
173	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
177	C₂³×C₄₇	—	2	2	no	—
178	C₂³×C₄₇	—	2	4	no	—
179	C₂³×C₄₇	—	4	4	no	—
180	C₂³×C₄₇	—	4	2	no	—
181	C₂³×C₄₇	—	4	4	no	—
182	C₂³×C₄₇	—	4	2	no	—
183	C₂³×C₄₇	—	4	2	no	—
184	C₂³×C₄₇	—	4	4	no	—
185	C₂³×C₄₇	#120	1	1	yes	characteristic
186	Q₈×C₄₇	—	2	4	no	—
187	Q₈×C₄₇	—	2	2	no	—
188	Q₈×C₄₇	—	4	2	no	—
189	Q₈×C₄₇	—	4	2	no	—
190	Q₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
191	D₄×C₄₇	—	2	2	no	—
192	D₄×C₄₇	—	2	2	no	—
193	D₄×C₄₇	—	4	4	no	—
194	D₄×C₄₇	—	4	16	no	—
195	D₄×C₄₇	—	4	4	no	—
196	D₄×C₄₇	—	4	2	no	—
197	D₄×C₄₇	—	4	4	no	—
198	D₄×C₄₇	—	4	2	no	—
199	D₄×C₄₇	—	4	2	no	—
200	D₄×C₄₇	—	4	4	no	—
201	D₄×C₄₇	—	8	2	no	—
202	D₄×C₄₇	—	8	8	no	—
203	C₂×C₄×C₄₇	—	2	4	no	—
204	C₂×C₄×C₄₇	—	2	2	no	—
205	C₂×C₄×C₄₇	—	2	2	no	—
206	C₂×C₄×C₄₇	—	2	2	no	—
207	C₂×C₄×C₄₇	—	2	2	no	—
208	C₂×C₄×C₄₇	—	4	1	no	—
209	C₂×C₄×C₄₇	—	4	16	no	—
210	C₂×C₄×C₄₇	—	4	4	no	—
211	C₂×C₄×C₄₇	—	4	2	no	—
212	C₂×C₄×C₄₇	—	4	2	no	—
213	C₂×C₄×C₄₇	—	4	4	no	—
214	C₂×C₄×C₄₇	—	4	1	no	—
215	C₂×C₄×C₄₇	—	4	1	no	—
216	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
217	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
218	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
219	C₂×C₄×C₄₇	#135	1	2	yes	—
220	C₂×C₄×C₄₇	#137	1	2	yes	—
221	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
224	C₈×C₄₇	—	4	2	no	—
225	C₈×C₄₇	—	4	2	no	—
226	C₈×C₄₇	—	4	2	no	—
227	#18771	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂²×C₄₇	—	2	1	no	—
229	C₂²×C₄₇	—	2	1	no	—
230	C₂²×C₄₇	—	4	2	no	—
231	C₂²×C₄₇	—	4	4	no	—
232	C₂²×C₄₇	—	4	4	no	—
233	C₂²×C₄₇	—	4	2	no	—
234	C₂²×C₄₇	—	4	4	no	—
235	C₂²×C₄₇	—	4	2	no	—
236	C₂²×C₄₇	—	8	4	no	—
237	C₂²×C₄₇	—	8	8	no	—
238	C₂²×C₄₇	#452	1	2	yes	—
239	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
240	C₄×C₄₇	—	2	4	no	—
241	C₄×C₄₇	—	2	1	no	—
242	C₄×C₄₇	—	2	1	no	—
243	C₄×C₄₇	—	2	1	no	—
244	C₄×C₄₇	—	4	2	no	—
245	C₄×C₄₇	—	4	2	no	—
246	C₄×C₄₇	—	8	1	no	—
247	C₄×C₄₇	—	8	1	no	—
248	C₄×C₄₇	#452	1	2	yes	—
249	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
250	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
251	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
252	C₂×#550	C₂×C₄₇	1	1	yes	characteristic
253	#3208	C₂×C₄₇	1	1	yes	characteristic, complemented
254	#3104	C₂×C₄₇	1	1	yes	characteristic, complemented
255	#3077	C₂×C₄₇	1	1	yes	characteristic, complemented
256	#1641	C₂×C₄₇	1	4	yes	complemented
257	#1631	C₂×C₄₇	1	4	yes	complemented
258	#1545	C₂×C₄₇	1	1	yes	characteristic, complemented
259	#1487	C₂×C₄₇	1	1	yes	characteristic, complemented
260	#1478	C₂×C₄₇	1	1	yes	characteristic, complemented
261	C₂×C₄₇	—	2	2	no	—
262	C₂×C₄₇	—	8	2	no	—
263	C₂×C₄₇	—	8	4	no	—
264	C₂×C₄₇	C₂×#452	1	1	yes	characteristic
265	C₂×C₄₇	#3003	1	1	yes	characteristic
266	C₂×C₄₇	#3019	1	1	yes	characteristic
267	C₂²×#41	—	2	1	no	—
268	#550	—	2	2	no	—
269	#550	—	2	2	no	—
270	#550	—	2	2	no	—
271	#550	C₂²×C₄₇	1	4	yes	—
272	#533	C₂²×C₄₇	1	1	yes	characteristic
273	C₂×#127	—	2	1	no	—
274	C₂×#127	C₂²×C₄₇	1	1	yes	characteristic
275	C₂×#121	—	2	2	no	—
276	C₂×#121	—	2	1	no	—
277	C₂×C₄×Q₈	C₂²×C₄₇	1	1	yes	characteristic
278	C₂×C₄×D₄	—	2	1	no	—
279	C₂×C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
280	#492	C₂²×C₄₇	1	2	yes	complemented
281	#486	C₂²×C₄₇	1	2	yes	complemented
282	#484	—	2	1	no	—
283	#478	—	2	2	no	—
284	#477	C₂²×C₄₇	1	2	yes	complemented
285	#476	C₂²×C₄₇	1	2	yes	complemented
286	#474	C₂²×C₄₇	1	2	yes	complemented
287	#462	C₂²×C₄₇	1	2	yes	complemented
288	#460	—	2	2	no	—
289	#451	—	2	1	no	—
290	#450	—	2	2	no	—
291	C₄×Q₁₆	—	2	2	no	—
292	#431	C₂²×C₄₇	1	1	yes	characteristic
293	#431	C₂²×C₄₇	1	1	yes	characteristic
294	#428	—	2	1	no	—
295	#423	C₂²×C₄₇	1	1	yes	characteristic
296	#418	—	2	1	no	—
297	#331	C₂²×C₄₇	1	4	yes	complemented
298	Q₈⋊C₈	C₂²×C₄₇	1	4	yes	complemented
299	D₄⋊C₈	C₂²×C₄₇	1	4	yes	complemented
300	C₄₇	#18771	1	1	yes	characteristic, direct product factor, Sylow
301	C₂×#41	—	2	4	no	—
302	C₂×#41	—	2	4	no	—
303	C₂×#41	—	4	8	no	—
304	C₂²×Q₈	D₄×C₄₇	1	1	yes	characteristic
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	—	2	1	no	—
308	C₂²×D₄	—	2	1	no	—
309	C₂²×D₄	D₄×C₄₇	1	1	yes	characteristic
310	C₂²×D₄	D₄×C₄₇	1	2	yes	—
311	C₂³×C₄	—	2	2	no	—
312	C₂³×C₄	—	2	1	no	—
313	C₂×Q₁₆	—	2	2	no	—
314	C₂×SD₁₆	—	2	4	no	—
315	C₂×M₁₆	—	2	1	no	—
316	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
317	C₄⋊Q₈	C₂³×C₄₇	1	1	yes	characteristic
318	#127	—	2	4	no	—
319	#127	—	2	4	no	—
320	#127	—	2	1	no	—
321	#127	—	2	1	no	—
322	#127	D₄×C₄₇	1	2	yes	—
323	#127	C₂³×C₄₇	1	2	yes	—
324	#125	—	2	1	no	—
325	C₂²⋊Q₈	—	2	1	no	—
326	C₂²⋊Q₈	—	2	1	no	—
327	#121	—	2	4	no	—
328	#121	—	2	8	no	—
329	#121	—	2	8	no	—
330	#121	—	2	4	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×Q₈	—	2	1	no	—
333	C₄×Q₈	—	2	1	no	—
334	C₄×Q₈	—	2	1	no	—
335	C₄×Q₈	D₄×C₄₇	1	2	yes	—
336	C₄×Q₈	C₂³×C₄₇	1	2	yes	—
337	C₄×D₄	—	2	4	no	—
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	1	no	—
340	C₄×D₄	—	2	1	no	—
341	C₄×D₄	D₄×C₄₇	1	2	yes	—
342	C₄×D₄	C₂³×C₄₇	1	2	yes	—
343	#117	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
346	C₂×C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₂²⋊C₄	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
351	#107	—	2	2	no	—
352	#107	—	2	2	no	—
353	#106	—	2	2	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	2	1	no	—
356	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
357	C₄⋊C₈	C₂³×C₄₇	1	2	yes	—
358	#103	—	2	2	no	—
359	#103	—	2	2	no	—
360	#102	—	2	2	no	—
361	#102	—	2	2	no	—
362	#102	—	2	4	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	C₂²⋊C₈	—	2	1	no	—
365	C₄×C₈	—	2	1	no	—
366	C₄×C₈	C₂³×C₄₇	1	2	yes	—
367	C₂⁴	—	2	1	no	—
368	C₂⁴	—	2	2	no	—
369	#41	—	4	16	no	—
370	#41	—	4	8	no	—
371	#41	—	4	8	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	8	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	4	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	—	8	2	no	—
392	C₂×D₄	—	8	4	no	—
393	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
394	C₂×D₄	C₂×D₄×C₄₇	1	4	yes	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	4	8	no	—
402	C₂²×C₄	—	4	4	no	—
403	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
406	Q₁₆	—	4	4	no	—
407	SD₁₆	—	8	4	no	—
408	M₁₆	—	4	2	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	4	1	no	—
421	C₄⋊C₄	—	4	1	no	—
422	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
423	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	4	1	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	4	1	no	—
431	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₄₇	1	1	yes	characteristic
434	C₄²	C₂⁴×C₄₇	1	1	yes	characteristic
435	C₂³	—	2	2	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	4	4	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	4	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	4	no	—
443	C₂³	#120×C₄₇	1	1	yes	characteristic
444	Q₈	—	2	4	no	—
445	Q₈	—	2	2	no	—
446	Q₈	—	4	2	no	—
447	Q₈	—	4	2	no	—
448	Q₈	AGL(1,ℤ/8)×C₄₇	1	4	yes	—
449	D₄	—	2	2	no	—
450	D₄	—	2	2	no	—
451	D₄	—	4	4	no	—
452	D₄	—	4	16	no	—
453	D₄	—	4	4	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	4	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	4	no	—
458	D₄	—	4	2	no	—
459	D₄	—	8	2	no	—
460	D₄	—	8	8	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	16	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
475	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
476	C₂×C₄	#120×C₄₇	1	1	yes	characteristic
477	C₂×C₄	#135×C₄₇	1	2	yes	—
478	C₂×C₄	#137×C₄₇	1	2	yes	—
479	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
482	C₈	—	4	2	no	—
483	C₈	—	4	2	no	—
484	C₈	—	4	2	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	4	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	8	4	no	—
494	C₂²	—	8	8	no	—
495	C₂²	C₄₇×#452	1	2	yes	—
496	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
497	C₄	—	2	4	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	8	1	no	—
504	C₄	—	8	1	no	—
505	C₄	C₄₇×#452	1	2	yes	—
506	C₄	C₂×#135×C₄₇	1	1	yes	characteristic
507	C₄	C₂×AGL(1,ℤ/8)×C₄₇	1	2	yes	—
508	C₄	C₂×#137×C₄₇	1	1	yes	characteristic
509	C₂	—	2	2	no	—
510	C₂	—	8	2	no	—
511	C₂	—	8	4	no	—
512	C₂	C₂×C₄₇×#452	1	1	yes	characteristic
513	C₂	C₄₇×#3003	1	1	yes	characteristic
514	C₂	C₄₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₇
prime radical	C₂×C₄₇×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#3208	C₂	semidirect
3. C₄₇×#3104	C₂	semidirect
4. C₄₇×#3077	C₂	semidirect
5. C₄₇×#1641	C₂	semidirect
6. C₄₇×#1631	C₂	semidirect
7. C₄₇×#1545	C₂	semidirect
8. C₄₇×#1487	C₂	semidirect
9. C₄₇×#1478	C₂	semidirect
23. C₄₇×#492	C₂²	semidirect
24. C₄₇×#486	C₂²	semidirect
27. C₄₇×#477	C₂²	semidirect
28. C₄₇×#476	C₂²	semidirect
29. C₄₇×#474	C₂²	semidirect
30. C₄₇×#462	C₂²	semidirect
40. C₄₇×#331	C₂²	semidirect
41. C₄₇×Q₈⋊C₈	C₂²	semidirect
42. C₄₇×D₄⋊C₈	C₂²	semidirect
227. #18771	C₄₇	direct (N × H)
253. #3208	C₂×C₄₇	semidirect
254. #3104	C₂×C₄₇	semidirect
255. #3077	C₂×C₄₇	semidirect
256. #1641	C₂×C₄₇	semidirect
257. #1631	C₂×C₄₇	semidirect
258. #1545	C₂×C₄₇	semidirect
259. #1487	C₂×C₄₇	semidirect
260. #1478	C₂×C₄₇	semidirect
280. #492	C₂²×C₄₇	semidirect
281. #486	C₂²×C₄₇	semidirect
284. #477	C₂²×C₄₇	semidirect
285. #476	C₂²×C₄₇	semidirect
286. #474	C₂²×C₄₇	semidirect
287. #462	C₂²×C₄₇	semidirect
297. #331	C₂²×C₄₇	semidirect
298. Q₈⋊C₈	C₂²×C₄₇	semidirect
299. D₄⋊C₈	C₂²×C₄₇	semidirect
300. C₄₇	#18771	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₇×#550	2
C₄₇×#3208	2
C₄₇×#3104	2
C₄₇×#3077	2
C₄₇×#1641	2
C₄₇×#1631	2
C₄₇×#1545	2
C₄₇×#1487	2
C₄₇×#1478	2
#18771	47

Subgroups — C47×#18771 #26058469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#18771 #26058469