Subgroups — C₄₇×#24413 #26064111

Order 12032.

1580 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₇×#3484	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#3056	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₇×#455	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#100×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2435	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#2037	C₂	1	2	yes	maximal, complemented
7	C₄₇×#2004	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1885	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#1871	C₂	1	2	yes	maximal, complemented
10	C₄₇×#1856	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1792	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#137×C₄₇	C₂²	1	2	yes	complemented
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	complemented
16	C₄₇×#546	—	2	2	no	—
17	C₄₇×#546	—	2	2	no	—
18	D₄²×C₄₇	—	2	2	no	—
19	D₄²×C₄₇	—	2	2	no	—
20	C₄₇×#535	C₂²	1	2	yes	—
21	C₄₇×#534	—	2	2	no	—
22	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#124×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₄₇×#525	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×#120×C₄₇	—	2	2	no	—
27	C₄₇×#518	C₂²	1	1	yes	characteristic
28	C₄₇×#470	—	2	2	no	—
29	C₄₇×#468	—	2	2	no	—
30	C₄₇×#467	—	2	2	no	—
31	C₄₇×#465	—	2	2	no	—
32	C₄₇×#455	—	2	4	no	—
33	C₄₇×#428	C₂²	1	2	yes	complemented
34	C₂×#104×C₄₇	C₂²	1	2	yes	complemented
35	C₂×#104×C₄₇	C₂²	1	2	yes	complemented
36	C₄₇×#419	C₂²	1	2	yes	complemented
37	C₄₇×#417	C₂²	1	2	yes	complemented
38	C₂×#100×C₄₇	—	2	4	no	—
39	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
40	C₂×#100×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#100×C₄₇	C₂²	1	2	yes	complemented
42	C₄₇×#394	—	2	2	no	—
43	C₄₇×#390	C₂²	1	1	yes	characteristic
44	C₄₇×#390	C₂²	1	1	yes	characteristic, complemented
45	C₄₇×#379	—	2	2	no	—
46	C₄×C₂²⋊C₄×C₄₇	C₂²	1	1	yes	characteristic, complemented
47	C₄₇×#339	C₂²	1	2	yes	complemented
48	C₄₇×#339	C₂²	1	2	yes	complemented
49	C₂⁵×C₄₇	D₄	1	2	yes	—
50	H(5,𝔽₂)×C₄₇	—	2	2	no	—
51	H(5,𝔽₂)×C₄₇	—	2	2	no	—
52	H(5,𝔽₂)×C₄₇	—	4	2	no	—
53	C₂×#41×C₄₇	—	2	2	no	—
54	C₂×#41×C₄₇	C₂³	1	2	yes	—
55	C₂²×Q₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×D₄×C₄₇	—	2	2	no	—
57	C₂²×D₄×C₄₇	—	2	2	no	—
58	C₂²×D₄×C₄₇	—	2	2	no	—
59	C₂²×D₄×C₄₇	—	2	2	no	—
60	C₂²×D₄×C₄₇	—	2	2	no	—
61	C₂²×D₄×C₄₇	—	2	2	no	—
62	C₂²×D₄×C₄₇	—	2	2	no	—
63	C₂²×D₄×C₄₇	—	2	2	no	—
64	C₂²×D₄×C₄₇	—	4	2	no	—
65	C₂²×D₄×C₄₇	D₄	1	2	yes	—
66	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₄₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
69	#137×C₄₇	—	2	4	no	—
70	#137×C₄₇	—	4	2	no	—
71	C₂×Q₁₆×C₄₇	—	2	2	no	—
72	C₂×SD₁₆×C₄₇	—	2	2	no	—
73	C₂×M₁₆×C₄₇	—	2	2	no	—
74	C₂×M₁₆×C₄₇	—	2	4	no	—
75	C₂×M₁₆×C₄₇	—	4	2	no	—
76	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
77	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂²×C₈×C₄₇	—	2	2	no	—
79	#127×C₄₇	—	2	2	no	—
80	#124×C₄₇	—	2	2	no	—
81	#124×C₄₇	—	2	2	no	—
82	#124×C₄₇	—	2	2	no	—
83	#124×C₄₇	—	4	2	no	—
84	#123×C₄₇	—	2	4	no	—
85	#123×C₄₇	—	2	2	no	—
86	C₂²⋊Q₈×C₄₇	—	2	2	no	—
87	C₂²⋊Q₈×C₄₇	—	2	2	no	—
88	C₂²⋊Q₈×C₄₇	D₄	1	2	yes	—
89	C₂²⋊Q₈×C₄₇	D₄	1	2	yes	—
90	#121×C₄₇	—	2	4	no	—
91	#121×C₄₇	—	2	4	no	—
92	#121×C₄₇	—	2	2	no	—
93	#121×C₄₇	—	2	2	no	—
94	#121×C₄₇	D₄	1	2	yes	—
95	#121×C₄₇	D₄	1	2	yes	—
96	#120×C₄₇	—	2	4	no	—
97	#120×C₄₇	—	2	4	no	—
98	#120×C₄₇	—	2	4	no	—
99	#120×C₄₇	—	2	2	no	—
100	#120×C₄₇	—	2	4	no	—
101	C₄×D₄×C₄₇	—	2	2	no	—
102	C₄×D₄×C₄₇	—	2	2	no	—
103	C₄×D₄×C₄₇	—	2	2	no	—
104	C₄×D₄×C₄₇	—	2	2	no	—
105	#117×C₄₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
107	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
108	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
109	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
117	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
119	#107×C₄₇	—	2	2	no	—
120	#106×C₄₇	—	2	2	no	—
121	#104×C₄₇	—	2	4	no	—
122	#104×C₄₇	—	4	2	no	—
123	#103×C₄₇	—	2	2	no	—
124	#103×C₄₇	—	2	2	no	—
125	#103×C₄₇	—	2	2	no	—
126	#102×C₄₇	—	2	2	no	—
127	#100×C₄₇	—	2	2	no	—
128	#100×C₄₇	—	2	4	no	—
129	#100×C₄₇	D₄	1	4	yes	complemented
130	#95×C₄₇	—	2	1	no	—
131	C₂⁴×C₄₇	—	2	2	no	—
132	C₂⁴×C₄₇	—	2	2	no	—
133	C₂⁴×C₄₇	—	2	2	no	—
134	C₂⁴×C₄₇	—	2	4	no	—
135	C₂⁴×C₄₇	—	4	2	no	—
136	C₂⁴×C₄₇	—	4	4	no	—
137	C₂⁴×C₄₇	—	8	2	no	—
138	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
139	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
140	C₂⁴×C₄₇	C₂×D₄	1	2	yes	—
141	#41×C₄₇	—	2	2	no	—
142	#41×C₄₇	—	2	2	no	—
143	#41×C₄₇	—	2	2	no	—
144	#41×C₄₇	—	2	2	no	—
145	#41×C₄₇	—	4	4	no	—
146	#41×C₄₇	—	4	4	no	—
147	C₂×Q₈×C₄₇	—	2	2	no	—
148	C₂×Q₈×C₄₇	—	2	1	no	—
149	C₂×Q₈×C₄₇	—	2	1	no	—
150	C₂×Q₈×C₄₇	—	2	2	no	—
151	C₂×Q₈×C₄₇	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₄₇	—	2	4	no	—
153	C₂×D₄×C₄₇	—	2	2	no	—
154	C₂×D₄×C₄₇	—	2	2	no	—
155	C₂×D₄×C₄₇	—	2	4	no	—
156	C₂×D₄×C₄₇	—	2	2	no	—
157	C₂×D₄×C₄₇	—	2	4	no	—
158	C₂×D₄×C₄₇	—	2	4	no	—
159	C₂×D₄×C₄₇	—	2	4	no	—
160	C₂×D₄×C₄₇	—	2	2	no	—
161	C₂×D₄×C₄₇	—	2	2	no	—
162	C₂×D₄×C₄₇	—	2	1	no	—
163	C₂×D₄×C₄₇	—	2	1	no	—
164	C₂×D₄×C₄₇	—	2	2	no	—
165	C₂×D₄×C₄₇	—	2	2	no	—
166	C₂×D₄×C₄₇	—	4	4	no	—
167	C₂×D₄×C₄₇	—	4	4	no	—
168	C₂×D₄×C₄₇	—	4	4	no	—
169	C₂×D₄×C₄₇	—	4	4	no	—
170	C₂×D₄×C₄₇	—	4	2	no	—
171	C₂×D₄×C₄₇	—	4	4	no	—
172	C₂×D₄×C₄₇	—	4	4	no	—
173	C₂×D₄×C₄₇	—	4	4	no	—
174	C₂×D₄×C₄₇	—	4	4	no	—
175	C₂×D₄×C₄₇	—	8	4	no	—
176	C₂×D₄×C₄₇	—	8	4	no	—
177	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
178	C₂×D₄×C₄₇	C₂×D₄	1	2	yes	—
179	C₂×D₄×C₄₇	#41	1	2	yes	—
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	—	2	2	no	—
186	C₂²×C₄×C₄₇	—	2	2	no	—
187	C₂²×C₄×C₄₇	—	2	1	no	—
188	C₂²×C₄×C₄₇	—	2	1	no	—
189	C₂²×C₄×C₄₇	—	2	1	no	—
190	C₂²×C₄×C₄₇	—	2	1	no	—
191	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
193	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
195	Q₁₆×C₄₇	—	4	4	no	—
196	SD₁₆×C₄₇	—	4	4	no	—
197	M₁₆×C₄₇	—	2	2	no	—
198	M₁₆×C₄₇	—	2	2	no	—
199	M₁₆×C₄₇	—	2	2	no	—
200	M₁₆×C₄₇	—	4	4	no	—
201	M₁₆×C₄₇	C₂×D₄	1	4	yes	complemented
202	C₂×C₈×C₄₇	—	2	2	no	—
203	C₂×C₈×C₄₇	—	2	2	no	—
204	C₂×C₈×C₄₇	—	4	4	no	—
205	C₄⋊C₄×C₄₇	—	2	2	no	—
206	C₄⋊C₄×C₄₇	—	2	2	no	—
207	C₄⋊C₄×C₄₇	—	2	2	no	—
208	C₄⋊C₄×C₄₇	—	4	2	no	—
209	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₄₇	C₂×D₄	1	2	yes	—
211	C₂²⋊C₄×C₄₇	—	2	4	no	—
212	C₂²⋊C₄×C₄₇	—	2	4	no	—
213	C₂²⋊C₄×C₄₇	—	2	4	no	—
214	C₂²⋊C₄×C₄₇	—	2	2	no	—
215	C₂²⋊C₄×C₄₇	—	2	2	no	—
216	C₂²⋊C₄×C₄₇	—	2	2	no	—
217	C₂²⋊C₄×C₄₇	—	2	2	no	—
218	C₂²⋊C₄×C₄₇	—	2	2	no	—
219	C₂²⋊C₄×C₄₇	—	4	2	no	—
220	C₂²⋊C₄×C₄₇	—	4	4	no	—
221	C₂²⋊C₄×C₄₇	—	4	1	no	—
222	C₂²⋊C₄×C₄₇	—	4	1	no	—
223	C₂²⋊C₄×C₄₇	—	4	4	no	—
224	C₂²⋊C₄×C₄₇	—	8	2	no	—
225	C₂²⋊C₄×C₄₇	—	8	2	no	—
226	C₄²×C₄₇	—	2	2	no	—
227	C₄²×C₄₇	—	2	1	no	—
228	C₄²×C₄₇	—	2	1	no	—
229	C₄²×C₄₇	—	2	2	no	—
230	C₂³×C₄₇	—	2	2	no	—
231	C₂³×C₄₇	—	2	1	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	2	no	—
237	C₂³×C₄₇	—	2	2	no	—
238	C₂³×C₄₇	—	2	2	no	—
239	C₂³×C₄₇	—	4	1	no	—
240	C₂³×C₄₇	—	4	4	no	—
241	C₂³×C₄₇	—	4	4	no	—
242	C₂³×C₄₇	—	4	2	no	—
243	C₂³×C₄₇	—	4	2	no	—
244	C₂³×C₄₇	—	4	2	no	—
245	C₂³×C₄₇	—	4	2	no	—
246	C₂³×C₄₇	—	4	4	no	—
247	C₂³×C₄₇	—	4	4	no	—
248	C₂³×C₄₇	—	4	4	no	—
249	C₂³×C₄₇	—	8	2	no	—
250	C₂³×C₄₇	—	8	2	no	—
251	C₂³×C₄₇	—	8	2	no	—
252	C₂³×C₄₇	—	8	2	no	—
253	C₂³×C₄₇	—	8	8	no	—
254	C₂³×C₄₇	—	8	8	no	—
255	C₂³×C₄₇	#120	1	1	yes	characteristic
256	C₂³×C₄₇	#120	1	1	yes	characteristic
257	C₂³×C₄₇	#121	1	4	yes	—
258	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₄₇	—	2	2	no	—
260	Q₈×C₄₇	—	2	2	no	—
261	Q₈×C₄₇	—	4	2	no	—
262	Q₈×C₄₇	—	4	2	no	—
263	D₄×C₄₇	—	2	4	no	—
264	D₄×C₄₇	—	2	2	no	—
265	D₄×C₄₇	—	2	4	no	—
266	D₄×C₄₇	—	2	2	no	—
267	D₄×C₄₇	—	4	8	no	—
268	D₄×C₄₇	—	4	8	no	—
269	D₄×C₄₇	—	4	4	no	—
270	D₄×C₄₇	—	4	2	no	—
271	D₄×C₄₇	—	4	2	no	—
272	D₄×C₄₇	—	4	4	no	—
273	D₄×C₄₇	—	4	2	no	—
274	D₄×C₄₇	—	4	2	no	—
275	D₄×C₄₇	—	8	4	no	—
276	D₄×C₄₇	—	8	4	no	—
277	C₂×C₄×C₄₇	—	2	1	no	—
278	C₂×C₄×C₄₇	—	2	1	no	—
279	C₂×C₄×C₄₇	—	2	2	no	—
280	C₂×C₄×C₄₇	—	2	2	no	—
281	C₂×C₄×C₄₇	—	2	2	no	—
282	C₂×C₄×C₄₇	—	2	2	no	—
283	C₂×C₄×C₄₇	—	2	2	no	—
284	C₂×C₄×C₄₇	—	2	2	no	—
285	C₂×C₄×C₄₇	—	2	2	no	—
286	C₂×C₄×C₄₇	—	4	4	no	—
287	C₂×C₄×C₄₇	—	4	2	no	—
288	C₂×C₄×C₄₇	—	4	2	no	—
289	C₂×C₄×C₄₇	—	4	1	no	—
290	C₂×C₄×C₄₇	—	4	1	no	—
291	C₂×C₄×C₄₇	—	4	1	no	—
292	C₂×C₄×C₄₇	—	4	1	no	—
293	C₂×C₄×C₄₇	—	4	2	no	—
294	C₂×C₄×C₄₇	—	4	1	no	—
295	C₂×C₄×C₄₇	—	4	1	no	—
296	C₂×C₄×C₄₇	—	4	1	no	—
297	C₂×C₄×C₄₇	—	4	1	no	—
298	C₂×C₄×C₄₇	—	4	1	no	—
299	C₂×C₄×C₄₇	—	4	1	no	—
300	C₂×C₄×C₄₇	—	8	2	no	—
301	C₂×C₄×C₄₇	—	8	2	no	—
302	C₂×C₄×C₄₇	#120	1	2	yes	—
303	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
306	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
307	C₈×C₄₇	—	2	4	no	—
308	C₈×C₄₇	—	4	2	no	—
309	#24413	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
310	C₂²×C₄₇	—	2	1	no	—
311	C₂²×C₄₇	—	2	1	no	—
312	C₂²×C₄₇	—	2	1	no	—
313	C₂²×C₄₇	—	2	1	no	—
314	C₂²×C₄₇	—	2	1	no	—
315	C₂²×C₄₇	—	2	4	no	—
316	C₂²×C₄₇	—	2	1	no	—
317	C₂²×C₄₇	—	2	1	no	—
318	C₂²×C₄₇	—	4	2	no	—
319	C₂²×C₄₇	—	4	2	no	—
320	C₂²×C₄₇	—	4	4	no	—
321	C₂²×C₄₇	—	4	4	no	—
322	C₂²×C₄₇	—	4	2	no	—
323	C₂²×C₄₇	—	4	2	no	—
324	C₂²×C₄₇	—	4	2	no	—
325	C₂²×C₄₇	—	4	2	no	—
326	C₂²×C₄₇	—	4	2	no	—
327	C₂²×C₄₇	—	4	2	no	—
328	C₂²×C₄₇	—	4	2	no	—
329	C₂²×C₄₇	—	4	2	no	—
330	C₂²×C₄₇	—	8	2	no	—
331	C₂²×C₄₇	—	8	8	no	—
332	C₂²×C₄₇	—	8	2	no	—
333	C₂²×C₄₇	—	8	8	no	—
334	C₂²×C₄₇	—	16	4	no	—
335	C₂²×C₄₇	#455	1	2	yes	complemented
336	C₂²×C₄₇	C₂×#120	1	1	yes	characteristic
337	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
338	C₂²×C₄₇	#534	1	1	yes	characteristic
339	C₄×C₄₇	—	2	2	no	—
340	C₄×C₄₇	—	4	2	no	—
341	C₄×C₄₇	—	4	2	no	—
342	C₄×C₄₇	—	4	2	no	—
343	C₄×C₄₇	—	8	1	no	—
344	C₄×C₄₇	—	8	1	no	—
345	C₄×C₄₇	—	8	1	no	—
346	C₄×C₄₇	—	8	1	no	—
347	C₄×C₄₇	D₄²	1	2	yes	—
348	C₄×C₄₇	#546	1	2	yes	—
349	#3484	C₂×C₄₇	1	1	yes	characteristic
350	#3056	C₂×C₄₇	1	2	yes	complemented
351	C₂×#455	C₂×C₄₇	1	1	yes	characteristic, complemented
352	C₂²×#100	C₂×C₄₇	1	1	yes	characteristic, complemented
353	#2435	C₂×C₄₇	1	1	yes	characteristic, complemented
354	#2037	C₂×C₄₇	1	2	yes	complemented
355	#2004	C₂×C₄₇	1	2	yes	complemented
356	#1885	C₂×C₄₇	1	1	yes	characteristic, complemented
357	#1871	C₂×C₄₇	1	2	yes	complemented
358	#1856	C₂×C₄₇	1	1	yes	characteristic, complemented
359	#1792	C₂×C₄₇	1	1	yes	characteristic, complemented
360	C₂×C₄₇	—	2	1	no	—
361	C₂×C₄₇	—	2	1	no	—
362	C₂×C₄₇	—	2	2	no	—
363	C₂×C₄₇	—	4	1	no	—
364	C₂×C₄₇	—	4	4	no	—
365	C₂×C₄₇	—	8	2	no	—
366	C₂×C₄₇	—	8	2	no	—
367	C₂×C₄₇	#2410	1	1	yes	characteristic
368	C₂×C₄₇	C₂×#455	1	1	yes	characteristic
369	C₂×C₄₇	#3019	1	1	yes	characteristic
370	C₂×H(5,𝔽₂)	C₂²×C₄₇	1	1	yes	characteristic
371	C₂³×D₄	C₂²×C₄₇	1	1	yes	characteristic
372	C₂×#137	C₂²×C₄₇	1	2	yes	complemented
373	C₂²×M₁₆	C₂²×C₄₇	1	2	yes	complemented
374	#546	—	2	2	no	—
375	#546	—	2	2	no	—
376	D₄²	—	2	2	no	—
377	D₄²	—	2	2	no	—
378	#535	C₂²×C₄₇	1	2	yes	—
379	#534	—	2	2	no	—
380	C₂×#124	C₂²×C₄₇	1	1	yes	characteristic
381	C₂×#124	C₂²×C₄₇	1	1	yes	characteristic, complemented
382	#525	C₂²×C₄₇	1	2	yes	—
383	C₂×C₂²⋊Q₈	C₂²×C₄₇	1	1	yes	characteristic, complemented
384	C₂×#120	—	2	2	no	—
385	#518	C₂²×C₄₇	1	1	yes	characteristic
386	#470	—	2	2	no	—
387	#468	—	2	2	no	—
388	#467	—	2	2	no	—
389	#465	—	2	2	no	—
390	#455	—	2	4	no	—
391	#428	C₂²×C₄₇	1	2	yes	complemented
392	C₂×#104	C₂²×C₄₇	1	2	yes	complemented
393	C₂×#104	C₂²×C₄₇	1	2	yes	complemented
394	#419	C₂²×C₄₇	1	2	yes	complemented
395	#417	C₂²×C₄₇	1	2	yes	complemented
396	C₂×#100	—	2	4	no	—
397	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
398	C₂×#100	C₂²×C₄₇	1	1	yes	characteristic
399	C₂×#100	C₂²×C₄₇	1	2	yes	complemented
400	#394	—	2	2	no	—
401	#390	C₂²×C₄₇	1	1	yes	characteristic
402	#390	C₂²×C₄₇	1	1	yes	characteristic, complemented
403	#379	—	2	2	no	—
404	C₄×C₂²⋊C₄	C₂²×C₄₇	1	1	yes	characteristic, complemented
405	#339	C₂²×C₄₇	1	2	yes	complemented
406	#339	C₂²×C₄₇	1	2	yes	complemented
407	C₄₇	#24413	1	1	yes	characteristic, direct product factor, Sylow
408	C₂⁵	D₄×C₄₇	1	2	yes	—
409	H(5,𝔽₂)	—	2	2	no	—
410	H(5,𝔽₂)	—	2	2	no	—
411	H(5,𝔽₂)	—	4	2	no	—
412	C₂×#41	—	2	2	no	—
413	C₂×#41	C₂³×C₄₇	1	2	yes	—
414	C₂²×Q₈	C₂³×C₄₇	1	1	yes	characteristic
415	C₂²×D₄	—	2	2	no	—
416	C₂²×D₄	—	2	2	no	—
417	C₂²×D₄	—	2	2	no	—
418	C₂²×D₄	—	2	2	no	—
419	C₂²×D₄	—	2	2	no	—
420	C₂²×D₄	—	2	2	no	—
421	C₂²×D₄	—	2	2	no	—
422	C₂²×D₄	—	2	2	no	—
423	C₂²×D₄	—	4	2	no	—
424	C₂²×D₄	D₄×C₄₇	1	2	yes	—
425	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
426	C₂²×D₄	C₂³×C₄₇	1	1	yes	characteristic
427	C₂³×C₄	C₂³×C₄₇	1	1	yes	characteristic
428	#137	—	2	4	no	—
429	#137	—	4	2	no	—
430	C₂×Q₁₆	—	2	2	no	—
431	C₂×SD₁₆	—	2	2	no	—
432	C₂×M₁₆	—	2	2	no	—
433	C₂×M₁₆	—	2	4	no	—
434	C₂×M₁₆	—	4	2	no	—
435	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
436	C₂×M₁₆	C₂³×C₄₇	1	2	yes	—
437	C₂²×C₈	—	2	2	no	—
438	#127	—	2	2	no	—
439	#124	—	2	2	no	—
440	#124	—	2	2	no	—
441	#124	—	2	2	no	—
442	#124	—	4	2	no	—
443	#123	—	2	4	no	—
444	#123	—	2	2	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	D₄×C₄₇	1	2	yes	—
448	C₂²⋊Q₈	D₄×C₄₇	1	2	yes	—
449	#121	—	2	4	no	—
450	#121	—	2	4	no	—
451	#121	—	2	2	no	—
452	#121	—	2	2	no	—
453	#121	D₄×C₄₇	1	2	yes	—
454	#121	D₄×C₄₇	1	2	yes	—
455	#120	—	2	4	no	—
456	#120	—	2	4	no	—
457	#120	—	2	4	no	—
458	#120	—	2	2	no	—
459	#120	—	2	4	no	—
460	C₄×D₄	—	2	2	no	—
461	C₄×D₄	—	2	2	no	—
462	C₄×D₄	—	2	2	no	—
463	C₄×D₄	—	2	2	no	—
464	#117	C₂³×C₄₇	1	2	yes	—
465	C₂×C₄⋊C₄	—	2	1	no	—
466	C₂×C₄⋊C₄	C₂³×C₄₇	1	1	yes	characteristic
467	C₂×C₂²⋊C₄	—	2	2	no	—
468	C₂×C₂²⋊C₄	—	2	2	no	—
469	C₂×C₂²⋊C₄	—	2	2	no	—
470	C₂×C₂²⋊C₄	—	2	1	no	—
471	C₂×C₂²⋊C₄	—	2	2	no	—
472	C₂×C₂²⋊C₄	—	2	2	no	—
473	C₂×C₂²⋊C₄	—	2	1	no	—
474	C₂×C₂²⋊C₄	—	2	2	no	—
475	C₂×C₂²⋊C₄	—	2	2	no	—
476	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
477	C₂×C₄²	C₂³×C₄₇	1	1	yes	characteristic
478	#107	—	2	2	no	—
479	#106	—	2	2	no	—
480	#104	—	2	4	no	—
481	#104	—	4	2	no	—
482	#103	—	2	2	no	—
483	#103	—	2	2	no	—
484	#103	—	2	2	no	—
485	#102	—	2	2	no	—
486	#100	—	2	2	no	—
487	#100	—	2	4	no	—
488	#100	D₄×C₄₇	1	4	yes	complemented
489	#95	—	2	1	no	—
490	C₂⁴	—	2	2	no	—
491	C₂⁴	—	2	2	no	—
492	C₂⁴	—	2	2	no	—
493	C₂⁴	—	2	4	no	—
494	C₂⁴	—	4	2	no	—
495	C₂⁴	—	4	4	no	—
496	C₂⁴	—	8	2	no	—
497	C₂⁴	C₂×D₄×C₄₇	1	1	yes	characteristic
498	C₂⁴	C₂×D₄×C₄₇	1	2	yes	—
499	C₂⁴	C₂×D₄×C₄₇	1	2	yes	—
500	#41	—	2	2	no	—
501	#41	—	2	2	no	—
502	#41	—	2	2	no	—
503	#41	—	2	2	no	—
504	#41	—	4	4	no	—
505	#41	—	4	4	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	2	no	—
510	C₂×Q₈	C₂×D₄×C₄₇	1	2	yes	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	2	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	—	2	2	no	—
516	C₂×D₄	—	2	4	no	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	2	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	2	no	—
522	C₂×D₄	—	2	1	no	—
523	C₂×D₄	—	2	2	no	—
524	C₂×D₄	—	2	2	no	—
525	C₂×D₄	—	4	4	no	—
526	C₂×D₄	—	4	4	no	—
527	C₂×D₄	—	4	4	no	—
528	C₂×D₄	—	4	4	no	—
529	C₂×D₄	—	4	2	no	—
530	C₂×D₄	—	4	4	no	—
531	C₂×D₄	—	4	4	no	—
532	C₂×D₄	—	4	4	no	—
533	C₂×D₄	—	4	4	no	—
534	C₂×D₄	—	8	4	no	—
535	C₂×D₄	—	8	4	no	—
536	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
537	C₂×D₄	C₂×D₄×C₄₇	1	2	yes	—
538	C₂×D₄	#41×C₄₇	1	2	yes	—
539	C₂²×C₄	—	2	1	no	—
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	2	no	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	2	no	—
546	C₂²×C₄	—	2	1	no	—
547	C₂²×C₄	—	2	1	no	—
548	C₂²×C₄	—	2	1	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
551	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
552	C₂²×C₄	C₂×D₄×C₄₇	1	2	yes	—
553	C₂²×C₄	C₂⁴×C₄₇	1	1	yes	characteristic
554	Q₁₆	—	4	4	no	—
555	SD₁₆	—	4	4	no	—
556	M₁₆	—	2	2	no	—
557	M₁₆	—	2	2	no	—
558	M₁₆	—	2	2	no	—
559	M₁₆	—	4	4	no	—
560	M₁₆	C₂×D₄×C₄₇	1	4	yes	complemented
561	C₂×C₈	—	2	2	no	—
562	C₂×C₈	—	2	2	no	—
563	C₂×C₈	—	4	4	no	—
564	C₄⋊C₄	—	2	2	no	—
565	C₄⋊C₄	—	2	2	no	—
566	C₄⋊C₄	—	2	2	no	—
567	C₄⋊C₄	—	4	2	no	—
568	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
569	C₄⋊C₄	C₂×D₄×C₄₇	1	2	yes	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	4	no	—
572	C₂²⋊C₄	—	2	4	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	2	no	—
578	C₂²⋊C₄	—	4	2	no	—
579	C₂²⋊C₄	—	4	4	no	—
580	C₂²⋊C₄	—	4	1	no	—
581	C₂²⋊C₄	—	4	1	no	—
582	C₂²⋊C₄	—	4	4	no	—
583	C₂²⋊C₄	—	8	2	no	—
584	C₂²⋊C₄	—	8	2	no	—
585	C₄²	—	2	2	no	—
586	C₄²	—	2	1	no	—
587	C₄²	—	2	1	no	—
588	C₄²	—	2	2	no	—
589	C₂³	—	2	2	no	—
590	C₂³	—	2	2	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	2	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	2	2	no	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	4	1	no	—
599	C₂³	—	4	4	no	—
600	C₂³	—	4	4	no	—
601	C₂³	—	4	2	no	—
602	C₂³	—	4	2	no	—
603	C₂³	—	4	2	no	—
604	C₂³	—	4	2	no	—
605	C₂³	—	4	4	no	—
606	C₂³	—	4	4	no	—
607	C₂³	—	4	4	no	—
608	C₂³	—	8	2	no	—
609	C₂³	—	8	2	no	—
610	C₂³	—	8	2	no	—
611	C₂³	—	8	2	no	—
612	C₂³	—	8	8	no	—
613	C₂³	—	8	8	no	—
614	C₂³	#120×C₄₇	1	1	yes	characteristic
615	C₂³	#120×C₄₇	1	1	yes	characteristic
616	C₂³	#121×C₄₇	1	4	yes	—
617	C₂³	C₂²×D₄×C₄₇	1	1	yes	characteristic
618	Q₈	—	2	2	no	—
619	Q₈	—	2	2	no	—
620	Q₈	—	4	2	no	—
621	Q₈	—	4	2	no	—
622	D₄	—	2	4	no	—
623	D₄	—	2	2	no	—
624	D₄	—	2	4	no	—
625	D₄	—	2	2	no	—
626	D₄	—	4	8	no	—
627	D₄	—	4	8	no	—
628	D₄	—	4	4	no	—
629	D₄	—	4	4	no	—
630	D₄	—	4	2	no	—
631	D₄	—	4	2	no	—
632	D₄	—	4	2	no	—
633	D₄	—	4	2	no	—
634	D₄	—	8	4	no	—
635	D₄	—	8	4	no	—
636	C₂×C₄	—	2	1	no	—
637	C₂×C₄	—	2	1	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	4	4	no	—
646	C₂×C₄	—	4	2	no	—
647	C₂×C₄	—	4	2	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	4	1	no	—
652	C₂×C₄	—	4	2	no	—
653	C₂×C₄	—	4	1	no	—
654	C₂×C₄	—	4	1	no	—
655	C₂×C₄	—	4	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	1	no	—
658	C₂×C₄	—	4	1	no	—
659	C₂×C₄	—	8	2	no	—
660	C₂×C₄	—	8	2	no	—
661	C₂×C₄	#120×C₄₇	1	2	yes	—
662	C₂×C₄	C₂²×D₄×C₄₇	1	1	yes	characteristic
663	C₂×C₄	C₂²×D₄×C₄₇	1	2	yes	—
664	C₂×C₄	C₂×#41×C₄₇	1	1	yes	characteristic
665	C₂×C₄	H(5,𝔽₂)×C₄₇	1	2	yes	—
666	C₈	—	2	4	no	—
667	C₈	—	4	2	no	—
668	C₂²	—	2	1	no	—
669	C₂²	—	2	1	no	—
670	C₂²	—	2	1	no	—
671	C₂²	—	2	1	no	—
672	C₂²	—	2	1	no	—
673	C₂²	—	2	4	no	—
674	C₂²	—	2	1	no	—
675	C₂²	—	2	1	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	4	2	no	—
678	C₂²	—	4	4	no	—
679	C₂²	—	4	4	no	—
680	C₂²	—	4	2	no	—
681	C₂²	—	4	2	no	—
682	C₂²	—	4	2	no	—
683	C₂²	—	4	2	no	—
684	C₂²	—	4	2	no	—
685	C₂²	—	4	2	no	—
686	C₂²	—	4	2	no	—
687	C₂²	—	4	2	no	—
688	C₂²	—	8	2	no	—
689	C₂²	—	8	8	no	—
690	C₂²	—	8	2	no	—
691	C₂²	—	8	8	no	—
692	C₂²	—	16	4	no	—
693	C₂²	C₄₇×#455	1	2	yes	complemented
694	C₂²	C₂×#120×C₄₇	1	1	yes	characteristic
695	C₂²	C₂×#121×C₄₇	1	1	yes	characteristic
696	C₂²	C₄₇×#534	1	1	yes	characteristic
697	C₄	—	2	2	no	—
698	C₄	—	4	2	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	8	1	no	—
702	C₄	—	8	1	no	—
703	C₄	—	8	1	no	—
704	C₄	—	8	1	no	—
705	C₄	D₄²×C₄₇	1	2	yes	—
706	C₄	C₄₇×#546	1	2	yes	—
707	C₂	—	2	1	no	—
708	C₂	—	2	1	no	—
709	C₂	—	2	2	no	—
710	C₂	—	4	1	no	—
711	C₂	—	4	4	no	—
712	C₂	—	8	2	no	—
713	C₂	—	8	2	no	—
714	C₂	C₄₇×#2410	1	1	yes	characteristic
715	C₂	C₂×C₄₇×#455	1	1	yes	characteristic
716	C₂	C₄₇×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₇
prime radical	C₄₇×#3484	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₂³×C₄×C₄₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₇×#3056	C₂	semidirect
3. C₂×C₄₇×#455	C₂	semidirect
4. C₂²×#100×C₄₇	C₂	semidirect
5. C₄₇×#2435	C₂	semidirect
6. C₄₇×#2037	C₂	semidirect
7. C₄₇×#2004	C₂	semidirect
8. C₄₇×#1885	C₂	semidirect
9. C₄₇×#1871	C₂	semidirect
10. C₄₇×#1856	C₂	semidirect
11. C₄₇×#1792	C₂	semidirect
14. C₂×#137×C₄₇	C₂²	semidirect
15. C₂²×M₁₆×C₄₇	C₂²	semidirect
23. C₂×#124×C₄₇	C₂²	semidirect
25. C₂×C₂²⋊Q₈×C₄₇	C₂²	semidirect
33. C₄₇×#428	C₂²	semidirect
34. C₂×#104×C₄₇	C₂²	semidirect
35. C₂×#104×C₄₇	C₂²	semidirect
36. C₄₇×#419	C₂²	semidirect
37. C₄₇×#417	C₂²	semidirect
41. C₂×#100×C₄₇	C₂²	semidirect
44. C₄₇×#390	C₂²	semidirect
46. C₄×C₂²⋊C₄×C₄₇	C₂²	semidirect
47. C₄₇×#339	C₂²	semidirect
48. C₄₇×#339	C₂²	semidirect
129. #100×C₄₇	D₄	semidirect
201. M₁₆×C₄₇	C₂×D₄	semidirect
309. #24413	C₄₇	direct (N × H)
335. C₂²×C₄₇	#455	semidirect
350. #3056	C₂×C₄₇	semidirect
351. C₂×#455	C₂×C₄₇	semidirect
352. C₂²×#100	C₂×C₄₇	semidirect
353. #2435	C₂×C₄₇	semidirect
354. #2037	C₂×C₄₇	semidirect
355. #2004	C₂×C₄₇	semidirect
356. #1885	C₂×C₄₇	semidirect
357. #1871	C₂×C₄₇	semidirect
358. #1856	C₂×C₄₇	semidirect
359. #1792	C₂×C₄₇	semidirect
372. C₂×#137	C₂²×C₄₇	semidirect
373. C₂²×M₁₆	C₂²×C₄₇	semidirect
381. C₂×#124	C₂²×C₄₇	semidirect
383. C₂×C₂²⋊Q₈	C₂²×C₄₇	semidirect
391. #428	C₂²×C₄₇	semidirect
392. C₂×#104	C₂²×C₄₇	semidirect
393. C₂×#104	C₂²×C₄₇	semidirect
394. #419	C₂²×C₄₇	semidirect
395. #417	C₂²×C₄₇	semidirect
399. C₂×#100	C₂²×C₄₇	semidirect
402. #390	C₂²×C₄₇	semidirect
404. C₄×C₂²⋊C₄	C₂²×C₄₇	semidirect
405. #339	C₂²×C₄₇	semidirect
406. #339	C₂²×C₄₇	semidirect
407. C₄₇	#24413	direct (N × H)
488. #100	D₄×C₄₇	semidirect
560. M₁₆	C₂×D₄×C₄₇	semidirect
693. C₂²	C₄₇×#455	semidirect

Maximal subgroups

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

subgroup	index
C₄₇×#3484	2
C₄₇×#3056	2
C₂×C₄₇×#455	2
C₂²×#100×C₄₇	2
C₄₇×#2435	2
C₄₇×#2037	2
C₄₇×#2004	2
C₄₇×#1885	2
C₄₇×#1871	2
C₄₇×#1856	2
C₄₇×#1792	2
#24413	47

Subgroups — C47×#24413 #26064111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#24413 #26064111