Subgroups — #26116322

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243368	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241231	C₂	1	4	yes	maximal
4	C₂×C₄₇×#389	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
6	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
13	C₂²×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#389	—	2	1	no	—
16	C₄₇×#389	—	2	1	no	—
17	C₄₇×#389	—	2	1	no	—
18	C₄₇×#389	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
25	C₂²⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
26	C₄×C₄₇⋊C₈	—	4	2	no	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
40	C₂×C₄⋊C₄×C₄₇	—	4	4	no	—
41	C₂×C₄⋊C₄×C₄₇	C₈	1	4	yes	complemented
42	C₂×C₄²×C₄₇	—	4	2	no	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	#95×C₄₇	—	2	2	no	—
48	#95×C₄₇	—	2	2	no	—
49	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	4	no	—
58	C₂²×C₄×C₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	—	4	1	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	—	4	2	no	—
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
71	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₄⋊C₄×C₄₇	—	2	4	no	—
77	C₄⋊C₄×C₄₇	—	2	4	no	—
78	C₄⋊C₄×C₄₇	—	4	2	no	—
79	C₄⋊C₄×C₄₇	—	4	4	no	—
80	C₄⋊C₄×C₄₇	—	4	2	no	—
81	C₄⋊C₄×C₄₇	—	4	4	no	—
82	C₄²×C₄₇	—	2	2	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	—	4	1	no	—
85	C₄²×C₄₇	—	4	1	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂³×C₄₇	—	2	1	no	—
93	C₂³×C₄₇	—	2	1	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
98	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂³×C₄₇	#131	1	2	yes	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	4	no	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	4	2	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	4	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	8	4	no	—
126	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
128	C₂×C₄×C₄₇	#99	1	2	yes	—
129	C₂×C₄×C₄₇	#100	1	2	yes	—
130	C₂×C₄×C₄₇	#101	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
134	C₄₇⋊C₈	—	4	4	no	—
135	C₄₇⋊C₈	—	8	2	no	—
136	#9879	—	47	1	no	maximal, Sylow
137	C₂²×C₄₇	—	2	1	no	—
138	C₂²×C₄₇	—	2	1	no	—
139	C₂²×C₄₇	—	2	1	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	2	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	#391	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₄₇	#408	1	1	yes	characteristic
148	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
149	C₂²×C₄₇	#410	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
152	C₂²×C₄₇	C₈×Q₈	1	2	yes	—
153	C₂²×C₄₇	#446	1	2	yes	—
154	C₄×C₄₇	—	2	2	no	—
155	C₄×C₄₇	—	2	2	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	4	4	no	—
160	C₄×C₄₇	—	4	1	no	—
161	C₄×C₄₇	—	8	2	no	—
162	C₄×C₄₇	#324	1	4	yes	—
163	C₂×#389	D₄₇	1	1	yes	characteristic
164	#1797	—	47	1	no	—
165	C₄×C₂²⋊C₈	—	47	1	no	—
166	#1280	—	47	4	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	—	2	2	no	—
169	C₂×C₄₇	C₂×#324	1	1	yes	characteristic
170	C₂×C₄₇	#1460	1	1	yes	characteristic
171	C₂×C₄₇	#1470	1	1	yes	characteristic
172	C₂×C₄₇	#1471	1	1	yes	characteristic
173	C₂×C₄₇	#1984	1	1	yes	characteristic
174	C₂×C₄₇	#1996	1	1	yes	characteristic
175	C₂×C₄₇	#1997	1	1	yes	characteristic
176	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	Dic₄₇	1	2	yes	—
178	C₂²×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
179	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	94	1	no	—
181	C₂×C₂²⋊C₈	—	47	2	no	—
182	C₂×C₂²⋊C₈	—	47	2	no	—
183	C₂×C₄×C₈	—	94	1	no	—
184	#389	—	2	1	no	—
185	#389	—	2	1	no	—
186	#389	—	2	1	no	—
187	#389	—	2	1	no	—
188	C₂×#95	C₂×D₄₇	1	2	yes	—
189	#336	—	94	1	no	—
190	#336	—	94	1	no	—
191	C₄₇	#9879	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
195	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	—	94	2	no	—
198	C₂²×C₈	—	94	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	4	2	no	—
207	C₂×C₄⋊C₄	—	4	4	no	—
208	C₂×C₄⋊C₄	C₄₇⋊C₈	1	4	yes	complemented
209	C₂×C₄²	—	4	2	no	—
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
214	C₄⋊C₈	—	188	2	no	—
215	C₂²⋊C₈	—	47	4	no	—
216	C₂²⋊C₈	—	94	2	no	—
217	C₂²⋊C₈	—	94	4	no	—
218	C₄×C₈	—	188	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	2	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
238	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
239	C₂²×C₄	#10625329	1	2	yes	—
240	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
245	C₂²×C₄	#10625354	1	2	yes	—
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂×C₈	—	94	4	no	—
249	C₂×C₈	—	188	2	no	—
250	C₂×C₈	—	188	2	no	—
251	C₂×C₈	—	188	2	no	—
252	C₂×C₈	—	188	2	no	—
253	C₂×C₈	—	188	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	4	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
269	C₂³	Q₈×Dic₄₇	1	2	yes	—
270	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
271	C₂³	C₂×#10625329	1	1	yes	characteristic
272	C₂³	#13094031	1	2	yes	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
299	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
300	C₂×C₄	#13093939	1	2	yes	—
301	C₂×C₄	#13093941	1	2	yes	—
302	C₂×C₄	#13093943	1	2	yes	—
303	C₂×C₄	#13094000	1	2	yes	—
304	C₂×C₄	#13094013	1	1	yes	characteristic
305	C₂×C₄	#13094022	1	1	yes	characteristic
306	C₈	—	376	2	no	—
307	C₈	—	188	4	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	#15689030	1	1	yes	characteristic
317	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
318	C₂²	#15689090	1	1	yes	characteristic
319	C₂²	C₂×#13093939	1	1	yes	characteristic
320	C₂²	#15689097	1	1	yes	characteristic
321	C₂²	C₂×#13093941	1	1	yes	characteristic
322	C₂²	C₂×#13093943	1	1	yes	characteristic
323	C₂²	Q₈×C₄₇⋊C₈	1	2	yes	—
324	C₂²	#15689254	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	1	no	—
332	C₄	—	8	2	no	—
333	C₄	#15688849	1	4	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	2	2	no	—
336	C₂	C₂×#15688849	1	1	yes	characteristic
337	C₂	#20241709	1	1	yes	characteristic
338	C₂	#20241744	1	1	yes	characteristic
339	C₂	#20241748	1	1	yes	characteristic
340	C₂	#20244325	1	1	yes	characteristic
341	C₂	#20244374	1	1	yes	characteristic
342	C₂	#20244378	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#389	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#389	D₄₇
hypocenter	C₄₇	#9879
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#389

Complements

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

N	H ≅ G/N	type
41. C₂×C₄⋊C₄×C₄₇	C₈	semidirect
191. C₄₇	#9879	semidirect
208. C₂×C₄⋊C₄	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243368	2
C₄×C₂²⋊(C₄₇⋊C₈)	2
#20241231	2
C₂×C₄₇×#389	2
#9879	47