Subgroups — #26109112

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20243365	C₂	1	2	yes	maximal
3	C₂×#15688873	C₂	1	2	yes	maximal, complemented
4	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₄₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
14	C₄×C₈×D₄₇	—	2	4	no	—
15	#15688877	—	2	2	no	—
16	#15688873	—	2	4	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₄₇	—	2	2	no	—
26	C₂×C₈×D₄₇	—	2	2	no	—
27	C₂×C₈×D₄₇	—	2	2	no	—
28	C₂×C₈×D₄₇	—	2	2	no	—
29	C₂×C₈×D₄₇	—	2	2	no	—
30	C₂×C₈×D₄₇	C₈	1	4	yes	complemented
31	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
34	C₄²×D₄₇	—	2	2	no	—
35	C₄²×D₄₇	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
38	(C₂×D₄₇)⋊C₈	—	4	4	no	—
39	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
40	C₈×Dic₄₇	—	2	2	no	—
41	C₈×Dic₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₄₇	—	2	2	no	—
47	C₄⋊C₈×C₄₇	—	2	2	no	—
48	C₄×C₈×C₄₇	—	2	2	no	—
49	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×D₄₇	—	2	2	no	—
57	C₂×C₄×D₄₇	—	2	1	no	—
58	C₂×C₄×D₄₇	—	2	1	no	—
59	C₂×C₄×D₄₇	—	2	2	no	—
60	C₂×C₄×D₄₇	—	2	2	no	—
61	C₂×C₄×D₄₇	—	2	1	no	—
62	C₂×C₄×D₄₇	—	2	1	no	—
63	C₂×C₄×D₄₇	—	2	1	no	—
64	C₂×C₄×D₄₇	—	2	1	no	—
65	C₂×C₄×D₄₇	—	2	2	no	—
66	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₄₇	M₁₆	1	2	yes	—
68	C₂×C₄×D₄₇	SD₁₆	1	4	yes	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	4	2	no	—
74	C₂×C₈×C₄₇	—	4	2	no	—
75	C₂×C₈×C₄₇	—	4	2	no	—
76	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
78	C₄²×C₄₇	C₂×D₄	1	2	yes	—
79	C₄²×C₄₇	#41	1	2	yes	—
80	C₄×Dic₄₇	—	2	1	no	—
81	C₄×Dic₄₇	—	2	1	no	—
82	C₄×Dic₄₇	—	2	2	no	—
83	C₄×Dic₄₇	—	2	2	no	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	C₂×C₄₇⋊C₈	—	2	1	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
90	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
91	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
92	C₂×C₄₇⋊C₈	M₁₆	1	2	yes	—
93	C₈×D₄₇	—	2	8	no	—
94	C₈×D₄₇	—	2	8	no	—
95	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂²×D₄₇	—	2	1	no	—
97	C₂²×D₄₇	—	2	1	no	—
98	C₂²×D₄₇	—	2	1	no	—
99	C₂²×D₄₇	—	2	1	no	—
100	C₂²×D₄₇	C₄⋊C₈	1	2	yes	—
101	C₂²×D₄₇	#106	1	2	yes	—
102	C₂²×D₄₇	#108	1	2	yes	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
116	C₂×Dic₄₇	—	2	1	no	—
117	C₂×Dic₄₇	—	2	1	no	—
118	C₂×Dic₄₇	—	2	1	no	—
119	C₂×Dic₄₇	—	2	1	no	—
120	C₂×Dic₄₇	—	2	1	no	—
121	C₂×Dic₄₇	—	2	1	no	—
122	C₂×Dic₄₇	C₄⋊C₈	1	2	yes	—
123	C₂×Dic₄₇	#106	1	2	yes	—
124	C₂×Dic₄₇	#108	1	2	yes	—
125	C₄×D₄₇	—	2	8	no	—
126	C₄×D₄₇	—	2	2	no	—
127	C₄×D₄₇	—	2	2	no	—
128	C₄×D₄₇	—	2	2	no	—
129	C₄×D₄₇	—	2	2	no	—
130	C₄×D₄₇	—	2	4	no	—
131	C₄×D₄₇	—	2	4	no	—
132	C₈×C₄₇	—	2	2	no	—
133	C₈×C₄₇	—	2	2	no	—
134	C₈×C₄₇	—	8	2	no	—
135	C₄₇⋊C₈	—	2	2	no	—
136	C₄₇⋊C₈	—	8	2	no	—
137	C₄₇⋊C₈	C₂²⋊C₈	1	4	yes	complemented
138	#8595	—	47	1	no	maximal, Sylow
139	C₂²×C₄₇	#380	1	1	yes	characteristic
140	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂²×C₄₇	#423	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
143	C₂²×C₄₇	#427	1	1	yes	characteristic
144	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×#108	1	1	yes	characteristic
146	C₂×D₄₇	—	2	2	no	—
147	C₂×D₄₇	—	2	2	no	—
148	C₂×D₄₇	—	2	2	no	—
149	C₂×D₄₇	—	2	2	no	—
150	C₂×D₄₇	—	2	2	no	—
151	C₂×D₄₇	—	2	2	no	—
152	C₂×D₄₇	#334	1	4	yes	complemented
153	C₄×C₄₇	—	2	2	no	—
154	C₄×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₄×C₄₇	#407	1	1	yes	characteristic
156	C₄×C₄₇	#431	1	1	yes	characteristic
157	C₄×C₄₇	#432	1	1	yes	characteristic
158	C₄×C₄₇	#465	1	2	yes	—
159	C₄×C₄₇	#465	1	2	yes	—
160	Dic₄₇	—	2	2	no	—
161	Dic₄₇	—	2	2	no	—
162	Dic₄₇	—	2	2	no	—
163	C₂²×C₄×C₈	—	47	1	no	—
164	#1797	—	47	2	no	—
165	C₂×#334	—	47	2	no	—
166	#1277	—	47	1	no	—
167	#1277	D₄₇	1	1	yes	characteristic, complemented
168	C₂×C₄₇	C₂×#334	1	1	yes	characteristic
169	C₂×C₄₇	#1564	1	1	yes	characteristic
170	C₂×C₄₇	#1566	1	1	yes	characteristic
171	C₂×C₄₇	#1568	1	1	yes	characteristic
172	C₂×C₄₇	#1797	1	1	yes	characteristic
173	C₂×C₄₇	#1807	1	1	yes	characteristic
174	C₂×C₄₇	#1810	1	1	yes	characteristic
175	D₄₇	—	2	4	no	—
176	C₂³×C₈	—	47	1	no	—
177	C₂³×C₈	—	47	1	no	—
178	C₂²×C₄²	—	47	1	no	—
179	C₂×C₄⋊C₈	—	47	2	no	—
180	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
181	C₂×C₂²⋊C₈	—	94	2	no	—
182	C₂×C₄×C₈	—	47	1	no	—
183	C₂×C₄×C₈	—	47	1	no	—
184	C₂×C₄×C₈	—	47	1	no	—
185	C₂×C₄×C₈	—	94	4	no	—
186	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
187	#336	—	94	2	no	—
188	#334	—	94	4	no	—
189	C₄₇	#8595	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	—	47	1	no	—
191	C₂³×C₄	—	47	1	no	—
192	C₂³×C₄	—	47	1	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	47	1	no	—
195	C₂²×C₈	—	47	1	no	—
196	C₂²×C₈	—	47	4	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂²×C₈	—	94	2	no	—
200	C₂²×C₈	—	94	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₂²×C₈	—	94	2	no	—
203	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
204	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
205	C₂×C₄²	—	47	1	no	—
206	C₂×C₄²	—	47	1	no	—
207	C₂×C₄²	—	47	1	no	—
208	C₂×C₄²	—	94	2	no	—
209	C₂×C₄²	—	94	2	no	—
210	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	94	2	no	—
214	C₄⋊C₈	—	94	2	no	—
215	C₂²⋊C₈	—	188	4	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	—	47	4	no	—
218	C₄×C₈	—	94	2	no	—
219	C₄×C₈	—	94	2	no	—
220	C₂⁴	—	47	1	no	—
221	C₂²×C₄	—	47	1	no	—
222	C₂²×C₄	—	47	1	no	—
223	C₂²×C₄	—	47	1	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	2	no	—
226	C₂²×C₄	—	47	4	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	—	94	2	no	—
229	C₂²×C₄	—	94	1	no	—
230	C₂²×C₄	—	94	1	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	2	no	—
233	C₂²×C₄	—	94	1	no	—
234	C₂²×C₄	—	94	1	no	—
235	C₂²×C₄	—	94	1	no	—
236	C₂²×C₄	—	94	1	no	—
237	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
239	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	47	2	no	—
248	C₂×C₈	—	47	2	no	—
249	C₂×C₈	—	47	2	no	—
250	C₂×C₈	—	47	2	no	—
251	C₂×C₈	—	94	8	no	—
252	C₂×C₈	—	94	8	no	—
253	C₂×C₈	—	94	1	no	—
254	C₂×C₈	—	94	1	no	—
255	C₂×C₈	—	188	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₂×C₈	C₈×D₄₇	1	2	yes	—
259	C₂×C₈	#10625324	1	2	yes	—
260	C₄²	—	94	1	no	—
261	C₄²	—	94	1	no	—
262	C₄²	—	94	2	no	—
263	C₄²	—	94	2	no	—
264	C₄²	#10625350	1	2	yes	—
265	C₄²	D₄×D₄₇	1	2	yes	—
266	C₂³	—	47	2	no	—
267	C₂³	—	47	2	no	—
268	C₂³	—	47	2	no	—
269	C₂³	—	94	1	no	—
270	C₂³	—	94	1	no	—
271	C₂³	—	94	1	no	—
272	C₂³	—	94	1	no	—
273	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	47	2	no	—
277	C₂×C₄	—	47	2	no	—
278	C₂×C₄	—	47	2	no	—
279	C₂×C₄	—	94	8	no	—
280	C₂×C₄	—	94	1	no	—
281	C₂×C₄	—	94	1	no	—
282	C₂×C₄	—	94	1	no	—
283	C₂×C₄	—	94	1	no	—
284	C₂×C₄	—	94	1	no	—
285	C₂×C₄	—	94	1	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	4	no	—
289	C₂×C₄	—	94	4	no	—
290	C₂×C₄	—	94	2	no	—
291	C₂×C₄	—	94	2	no	—
292	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
293	C₂×C₄	#13093981	1	1	yes	characteristic
294	C₂×C₄	#13093997	1	2	yes	—
295	C₂×C₄	#13094002	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
297	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
298	C₂×C₄	M₁₆×D₄₇	1	1	yes	characteristic
299	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
300	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
301	C₂×C₄	#13094043	1	1	yes	characteristic
302	C₂×C₄	#13094047	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	8	2	no	—
306	C₈	—	47	4	no	—
307	C₈	—	94	2	no	—
308	C₈	—	376	2	no	—
309	C₂²	—	47	4	no	—
310	C₂²	—	94	2	no	—
311	C₂²	—	94	2	no	—
312	C₂²	—	94	2	no	—
313	C₂²	—	94	2	no	—
314	C₂²	—	94	2	no	—
315	C₂²	—	94	2	no	—
316	C₂²	#15688982	1	1	yes	characteristic
317	C₂²	C₄⋊C₈×D₄₇	1	1	yes	characteristic
318	C₂²	#15689145	1	1	yes	characteristic
319	C₂²	#106×D₄₇	1	1	yes	characteristic
320	C₂²	#15689160	1	1	yes	characteristic
321	C₂²	#108×D₄₇	1	1	yes	characteristic
322	C₂²	#108×D₄₇	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	94	2	no	—
325	C₄	—	94	2	no	—
326	C₄	—	94	2	no	—
327	C₄	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
328	C₄	#15689085	1	1	yes	characteristic
329	C₄	#15689174	1	1	yes	characteristic
330	C₄	#15689178	1	1	yes	characteristic
331	C₄	#15689351	1	2	yes	—
332	C₄	#15689355	1	2	yes	—
333	C₂	—	94	4	no	—
334	C₂	#334×D₄₇	1	1	yes	characteristic
335	C₂	#20242135	1	1	yes	characteristic
336	C₂	#20242142	1	1	yes	characteristic
337	C₂	#20242150	1	1	yes	characteristic
338	C₂	#20243367	1	1	yes	characteristic
339	C₂	#20243412	1	1	yes	characteristic
340	C₂	#20243425	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#8595
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
3. C₂×#15688873	C₂	semidirect
4. #20241223	C₂	semidirect
5. C₄₇×#1277	C₂	semidirect
30. C₂×C₈×D₄₇	C₈	semidirect
137. C₄₇⋊C₈	C₂²⋊C₈	semidirect
152. C₂×D₄₇	#334	semidirect
167. #1277	D₄₇	semidirect
189. C₄₇	#8595	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₇	2
#20243365	2
C₂×#15688873	2
#20241223	2
C₄₇×#1277	2
#8595	47