Subgroups — #26124200

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688992	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688892	C₂	1	1	yes	maximal, characteristic
3	#20243109	C₂	1	2	yes	maximal
4	C₂×#15688882	C₂	1	2	yes	maximal
5	C₄₇×#2316	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₄₇⋊C₄	—	2	2	no	—
9	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688992	—	2	2	no	—
11	#15688992	C₄	1	4	yes	complemented
12	#15688892	—	2	2	no	—
13	#15688885	—	2	4	no	—
14	#15688882	—	2	4	no	—
15	C₂×#117×C₄₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#389	—	2	4	no	—
18	C₄₇×#380	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	1	no	—
20	C₂³×Dic₄₇	—	2	1	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	Q₈	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×Dic₄₇⋊C₄	—	2	8	no	—
33	C₂×Dic₄₇⋊C₄	—	4	8	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄×Dic₄₇	—	2	1	no	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
40	C₂³×C₄×C₄₇	D₄	1	2	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	#117×C₄₇	—	2	2	no	—
43	#117×C₄₇	—	2	2	no	—
44	#117×C₄₇	—	2	2	no	—
45	#117×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	#95×C₄₇	—	2	4	no	—
55	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	—	2	4	no	—
59	C₂²×C₄×C₄₇	—	4	4	no	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	#41	1	2	yes	—
70	C₂²×C₄×C₄₇	#41	1	2	yes	—
71	C₂²×Dic₄₇	—	2	4	no	—
72	C₂²×Dic₄₇	—	2	4	no	—
73	C₂²×Dic₄₇	—	2	2	no	—
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	—	4	1	no	—
79	C₂²×Dic₄₇	—	4	1	no	—
80	C₄⋊C₄×C₄₇	—	2	4	no	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	2	4	no	—
84	C₄⋊C₄×C₄₇	—	8	4	no	—
85	C₂²⋊C₄×C₄₇	—	4	2	no	—
86	C₂²⋊C₄×C₄₇	—	4	2	no	—
87	C₄²×C₄₇	—	4	2	no	—
88	C₄²×C₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	2	4	no	—
90	C₄⋊Dic₄₇	—	4	2	no	—
91	Dic₄₇⋊C₄	—	4	16	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	C₄×Dic₄₇	—	2	2	no	—
94	C₄×Dic₄₇	—	2	2	no	—
95	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	complemented
96	#10625329	—	2	4	no	—
97	#10625329	—	2	4	no	—
98	#10625329	—	4	2	no	—
99	#10625329	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂×C₄₇⋊C₈	—	4	2	no	—
103	C₂³×C₄₇	—	2	2	no	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₇	#120	1	1	yes	characteristic
107	C₂³×C₄₇	#121	1	1	yes	characteristic
108	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₄₇	#123	1	1	yes	characteristic
110	C₂³×C₄₇	#124	1	1	yes	characteristic
111	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	—	2	4	no	—
113	C₂×C₄×C₄₇	—	4	4	no	—
114	C₂×C₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	—	4	4	no	—
117	C₂×C₄×C₄₇	—	4	4	no	—
118	C₂×C₄×C₄₇	—	8	2	no	—
119	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₄₇	#121	1	2	yes	—
124	C₂×C₄×C₄₇	#121	1	2	yes	—
125	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×C₄×C₄₇	#123	1	2	yes	—
131	C₂×C₄×C₄₇	#125	1	2	yes	—
132	C₂×C₄×C₄₇	#125	1	2	yes	—
133	C₂×C₄×C₄₇	#126	1	2	yes	—
134	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	2	8	no	—
137	C₂×Dic₄₇	—	2	4	no	—
138	C₂×Dic₄₇	—	2	8	no	—
139	C₂×Dic₄₇	—	2	4	no	—
140	C₂×Dic₄₇	—	4	4	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	8	2	no	—
143	C₂×Dic₄₇	—	8	2	no	—
144	C₄₇⋊C₈	—	4	4	no	—
145	#11210	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	2	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	—	2	2	no	—
153	C₂²×C₄₇	—	2	2	no	—
154	C₂²×C₄₇	#380	1	1	yes	characteristic
155	C₂²×C₄₇	#386	1	1	yes	characteristic
156	C₂²×C₄₇	#391	1	1	yes	characteristic
157	C₂²×C₄₇	#394	1	1	yes	characteristic
158	C₂²×C₄₇	#395	1	1	yes	characteristic
159	C₂²×C₄₇	#398	1	1	yes	characteristic
160	C₂²×C₄₇	#400	1	1	yes	characteristic
161	C₂²×C₄₇	#453	1	2	yes	—
162	C₂²×C₄₇	#456	1	2	yes	—
163	C₄×C₄₇	—	4	4	no	—
164	C₄×C₄₇	—	8	4	no	—
165	C₄×C₄₇	#387	1	2	yes	—
166	C₄×C₄₇	#389	1	2	yes	—
167	C₄×C₄₇	#396	1	2	yes	—
168	C₄×C₄₇	#398	1	2	yes	—
169	Dic₄₇	—	2	8	no	—
170	Dic₄₇	—	8	2	no	—
171	#2316	D₄₇	1	1	yes	characteristic
172	C₂×#384	—	47	1	no	—
173	C₂×#343	—	47	1	no	—
174	#1736	—	47	2	no	—
175	C₂×#339	—	47	2	no	—
176	C₂×C₄₇	—	2	1	no	—
177	C₂×C₄₇	—	2	1	no	—
178	C₂×C₄₇	—	2	1	no	—
179	C₂×C₄₇	—	2	1	no	—
180	C₂×C₄₇	#1441	1	1	yes	characteristic
181	C₂×C₄₇	#1886	1	1	yes	characteristic
182	C₂×C₄₇	#1887	1	1	yes	characteristic
183	C₂×C₄₇	#2039	1	2	yes	—
184	C₂×C₄₇	#2060	1	2	yes	—
185	C₂²×M₁₆	—	47	2	no	—
186	C₂×#117	C₂×D₄₇	1	2	yes	—
187	C₂²×C₄⋊C₄	—	47	1	no	—
188	C₂²×C₄⋊C₄	—	94	2	no	—
189	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
190	C₂²×C₄²	—	47	1	no	—
191	#389	—	2	4	no	—
192	#384	—	47	4	no	—
193	#384	—	94	2	no	—
194	#380	—	2	2	no	—
195	#343	—	94	2	no	—
196	#340	—	94	4	no	—
197	#339	—	94	4	no	—
198	C₄₇	#11210	1	1	yes	characteristic, complemented, Sylow
199	C₂³×C₄	—	94	1	no	—
200	C₂³×C₄	—	94	1	no	—
201	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
202	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×M₁₆	—	47	4	no	—
204	C₂×M₁₆	—	94	2	no	—
205	C₂×M₁₆	—	94	2	no	—
206	C₂×M₁₆	—	94	2	no	—
207	C₂×M₁₆	—	94	2	no	—
208	C₂×M₁₆	—	94	2	no	—
209	C₂²×C₈	—	94	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	#117	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₄⋊C₄	—	94	2	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	94	8	no	—
221	C₂×C₄⋊C₄	—	94	1	no	—
222	C₂×C₄⋊C₄	—	188	8	no	—
223	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
224	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
225	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	—	47	2	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₂×C₄²	—	47	2	no	—
231	C₂×C₄²	—	94	1	no	—
232	C₂×C₄²	—	94	1	no	—
233	C₂×C₄²	—	94	1	no	—
234	#95	—	2	4	no	—
235	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	4	no	—
240	C₂²×C₄	—	94	4	no	—
241	C₂²×C₄	—	94	4	no	—
242	C₂²×C₄	—	94	2	no	—
243	C₂²×C₄	—	94	2	no	—
244	C₂²×C₄	—	94	2	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	—	188	4	no	—
247	C₂²×C₄	—	188	1	no	—
248	C₂²×C₄	—	188	1	no	—
249	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
250	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
252	C₂²×C₄	#10625350	1	2	yes	—
253	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
254	C₂²×C₄	#10625352	1	1	yes	characteristic
255	C₂²×C₄	#10625352	1	2	yes	—
256	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
257	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
258	C₂²×C₄	#10625354	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
260	M₁₆	—	94	4	no	—
261	M₁₆	—	94	4	no	—
262	M₁₆	—	188	2	no	—
263	M₁₆	—	188	2	no	—
264	C₂×C₈	—	94	4	no	—
265	C₂×C₈	—	188	2	no	—
266	C₂×C₈	—	188	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	8	4	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	188	16	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	47	4	no	—
280	C₄²	—	94	2	no	—
281	C₄²	—	94	2	no	—
282	C₄²	—	94	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
286	C₂³	#13093993	1	1	yes	characteristic
287	C₂³	#13093995	1	1	yes	characteristic
288	C₂³	#13094000	1	1	yes	characteristic
289	C₂³	#13094007	1	1	yes	characteristic
290	C₂³	#13094011	1	1	yes	characteristic
291	C₂³	#13094022	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	94	4	no	—
300	C₂×C₄	—	94	4	no	—
301	C₂×C₄	—	94	8	no	—
302	C₂×C₄	—	94	8	no	—
303	C₂×C₄	—	94	4	no	—
304	C₂×C₄	—	376	2	no	—
305	C₂×C₄	—	376	2	no	—
306	C₂×C₄	—	188	4	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
309	C₂×C₄	#13093982	1	1	yes	characteristic
310	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
311	C₂×C₄	#13093989	1	2	yes	—
312	C₂×C₄	#13093996	1	2	yes	—
313	C₂×C₄	#13093998	1	2	yes	—
314	C₂×C₄	#13093999	1	1	yes	characteristic
315	C₂×C₄	#13094000	1	2	yes	—
316	C₂×C₄	#13094002	1	2	yes	—
317	C₂×C₄	#13094004	1	2	yes	—
318	C₂×C₄	#13094004	1	1	yes	characteristic
319	C₂×C₄	#13094005	1	2	yes	—
320	C₂×C₄	#13094014	1	2	yes	—
321	C₂×C₄	#13094014	1	2	yes	—
322	C₂×C₄	#13094019	1	2	yes	—
323	C₂×C₄	#13094022	1	2	yes	—
324	C₈	—	188	4	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	#15688980	1	1	yes	characteristic
334	C₂²	#15689002	1	1	yes	characteristic
335	C₂²	#15689028	1	1	yes	characteristic
336	C₂²	#15689037	1	1	yes	characteristic
337	C₂²	#15689042	1	1	yes	characteristic
338	C₂²	#15689054	1	1	yes	characteristic
339	C₂²	#15689061	1	1	yes	characteristic
340	C₂²	#15689291	1	2	yes	—
341	C₂²	#15689306	1	2	yes	—
342	C₄	—	4	4	no	—
343	C₄	—	8	4	no	—
344	C₄	—	94	8	no	—
345	C₄	—	376	2	no	—
346	C₄	#15689004	1	2	yes	—
347	C₄	#15689018	1	2	yes	—
348	C₄	#15689045	1	2	yes	—
349	C₄	#15689053	1	2	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	#20241638	1	1	yes	characteristic
355	C₂	#20243781	1	1	yes	characteristic
356	C₂	#20243786	1	1	yes	characteristic
357	C₂	#20244609	1	2	yes	—
358	C₂	#20244730	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093993
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2316	C₂
socle	C₂³×C₄₇	#120
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2316	D₄₇
hypocenter	C₄₇	#11210
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #2316

Complements

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

N	H ≅ G/N	type
11. #15688992	C₄	semidirect
95. C₄×Dic₄₇	C₄⋊C₄	semidirect
198. C₄₇	#11210	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688992	2
C₂×#15688892	2
#20243109	2
C₂×#15688882	2
C₄₇×#2316	2
#11210	47