Subgroups — #26124183

Order 12032.

670 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251952	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688893	C₂	1	2	yes	maximal
3	#20243127	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688892	C₂	1	2	yes	maximal
5	C₄₇×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689150	—	2	4	no	—
9	#15689147	—	2	2	no	—
10	#15688893	—	2	4	no	—
11	#15688892	—	2	4	no	—
12	#15688892	—	2	1	no	—
13	#15688892	—	2	1	no	—
14	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#389	—	2	2	no	—
18	C₄₇×#385	C₄	1	4	yes	—
19	C₄₇×#384	—	2	1	no	—
20	C₄₇×#382	—	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	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	—	2	2	no	—
29	C₂×#10625329	—	2	2	no	—
30	C₂×#10625329	—	2	2	no	—
31	C₂×#10625329	D₄	1	4	yes	—
32	C₂×#10625329	Q₈	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	2	no	—
34	C₂²×C₄₇⋊C₈	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	#117×C₄₇	—	2	2	no	—
39	#117×C₄₇	—	2	1	no	—
40	#117×C₄₇	—	2	1	no	—
41	#117×C₄₇	—	2	2	no	—
42	#117×C₄₇	D₄	1	2	yes	—
43	#117×C₄₇	Q₈	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	—	2	1	no	—
55	C₂×C₄²×C₄₇	D₄	1	2	yes	—
56	#95×C₄₇	—	2	4	no	—
57	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	8	2	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	2	2	no	—
77	C₄⋊C₄×C₄₇	—	2	2	no	—
78	C₄⋊C₄×C₄₇	—	4	1	no	—
79	C₄⋊C₄×C₄₇	—	4	2	no	—
80	C₄⋊C₄×C₄₇	—	4	1	no	—
81	C₄⋊C₄×C₄₇	—	8	4	no	—
82	C₂²⋊C₄×C₄₇	—	2	2	no	—
83	C₂²⋊C₄×C₄₇	—	2	4	no	—
84	C₂²⋊C₄×C₄₇	C₄⋊C₄	1	4	yes	—
85	C₄²×C₄₇	—	2	2	no	—
86	C₄²×C₄₇	—	2	2	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	#10625329	—	2	4	no	—
90	#10625329	—	2	4	no	—
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	#10625329	—	4	2	no	—
94	#10625329	—	4	2	no	—
95	#10625329	—	4	2	no	—
96	#10625329	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	—	4	2	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂×C₄₇⋊C₈	—	4	2	no	—
103	C₂³×C₄₇	—	4	1	no	—
104	C₂³×C₄₇	—	4	1	no	—
105	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₄₇	#117	1	1	yes	characteristic
108	C₂³×C₄₇	C₄×D₄	1	2	yes	—
109	C₂³×C₄₇	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	—	2	4	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	2	2	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	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	2	no	—
119	C₂×C₄×C₄₇	—	8	4	no	—
120	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#121	1	2	yes	—
122	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#121	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
131	C₂×C₄×C₄₇	#123	1	2	yes	—
132	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#125	1	2	yes	—
134	C₂×C₄×C₄₇	#125	1	2	yes	—
135	C₂×C₄×C₄₇	#126	1	2	yes	—
136	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
138	C₄₇⋊C₈	—	4	4	no	—
139	C₄₇⋊C₈	—	4	4	no	—
140	#11206	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	4	1	no	—
146	C₂²×C₄₇	—	4	1	no	—
147	C₂²×C₄₇	—	4	2	no	—
148	C₂²×C₄₇	#380	1	1	yes	characteristic
149	C₂²×C₄₇	#386	1	1	yes	characteristic
150	C₂²×C₄₇	#387	1	2	yes	—
151	C₂²×C₄₇	#389	1	2	yes	—
152	C₂²×C₄₇	#391	1	1	yes	characteristic
153	C₄×C₄₇	—	4	4	no	—
154	C₄×C₄₇	—	4	2	no	—
155	C₄×C₄₇	—	4	2	no	—
156	C₄×C₄₇	—	8	2	no	—
157	C₄×C₄₇	#394	1	1	yes	characteristic
158	C₄×C₄₇	#395	1	1	yes	characteristic
159	C₄×C₄₇	#396	1	2	yes	—
160	C₄×C₄₇	#398	1	1	yes	characteristic
161	C₄×C₄₇	#398	1	2	yes	—
162	C₄×C₄₇	#400	1	1	yes	characteristic
163	#2905	—	47	1	no	—
164	#2317	D₄₇	1	1	yes	characteristic
165	C₂×#344	—	47	2	no	—
166	#1742	—	47	1	no	—
167	C₂×#343	—	47	2	no	—
168	C₂×C₄₇	—	2	1	no	—
169	C₂×C₄₇	—	2	2	no	—
170	C₂×C₄₇	—	2	2	no	—
171	C₂×C₄₇	—	2	1	no	—
172	C₂×C₄₇	#1441	1	1	yes	characteristic
173	C₂×C₄₇	#1914	1	1	yes	characteristic
174	C₂×C₄₇	#1915	1	1	yes	characteristic
175	C₂²×M₁₆	—	47	2	no	—
176	C₂²×M₁₆	—	47	2	no	—
177	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
178	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
180	#424	—	94	4	no	—
181	#423	—	94	2	no	—
182	#389	—	2	2	no	—
183	#385	Dic₄₇	1	4	yes	—
184	#384	—	2	1	no	—
185	#382	—	2	1	no	—
186	#344	—	94	4	no	—
187	#343	—	94	4	no	—
188	#343	—	94	1	no	—
189	#343	—	94	1	no	—
190	C₄₇	#11206	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
192	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
193	C₂×M₁₆	—	47	4	no	—
194	C₂×M₁₆	—	47	4	no	—
195	C₂×M₁₆	—	94	2	no	—
196	C₂×M₁₆	—	94	2	no	—
197	C₂×M₁₆	—	94	2	no	—
198	C₂×M₁₆	—	94	2	no	—
199	C₂×M₁₆	—	94	2	no	—
200	C₂×M₁₆	—	94	2	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂×M₁₆	—	94	2	no	—
203	C₂×M₁₆	—	94	2	no	—
204	C₂×M₁₆	—	94	2	no	—
205	C₂²×C₈	—	94	2	no	—
206	C₂²×C₈	—	94	2	no	—
207	#117	—	2	1	no	—
208	#117	—	2	1	no	—
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	Dic₉₄	1	2	yes	—
212	#117	D₁₈₈	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
222	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
225	C₄⋊C₈	—	94	8	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
234	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
236	C₂²×C₄	#10625350	1	1	yes	characteristic
237	C₂²×C₄	#10625350	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₇	1	2	yes	—
239	C₂²×C₄	#10625352	1	2	yes	—
240	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
241	C₂²×C₄	#10625354	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
244	M₁₆	—	94	4	no	—
245	M₁₆	—	94	4	no	—
246	M₁₆	—	94	4	no	—
247	M₁₆	—	94	4	no	—
248	M₁₆	—	188	2	no	—
249	M₁₆	—	188	2	no	—
250	M₁₆	—	188	2	no	—
251	M₁₆	—	188	2	no	—
252	C₂×C₈	—	94	4	no	—
253	C₂×C₈	—	94	4	no	—
254	C₂×C₈	—	188	2	no	—
255	C₂×C₈	—	188	2	no	—
256	C₂×C₈	—	188	2	no	—
257	C₂×C₈	—	188	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	8	4	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	C₄⋊Dic₄₇	1	4	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
277	C₂³	#13093983	1	1	yes	characteristic
278	C₂³	D₄×Dic₄₇	1	2	yes	—
279	C₂³	Q₈×Dic₄₇	1	2	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	#13093992	1	1	yes	characteristic
291	C₂×C₄	#13093996	1	2	yes	—
292	C₂×C₄	#13093998	1	1	yes	characteristic
293	C₂×C₄	#13093998	1	1	yes	characteristic
294	C₂×C₄	#13093999	1	2	yes	—
295	C₂×C₄	#13094000	1	2	yes	—
296	C₂×C₄	#13094002	1	2	yes	—
297	C₂×C₄	#13094003	1	1	yes	characteristic
298	C₂×C₄	#13094003	1	1	yes	characteristic
299	C₂×C₄	#13094004	1	2	yes	—
300	C₂×C₄	#13094005	1	1	yes	characteristic
301	C₂×C₄	#13094009	1	2	yes	—
302	C₂×C₄	#13094013	1	1	yes	characteristic
303	C₂×C₄	#13094014	1	2	yes	—
304	C₂×C₄	#13094015	1	2	yes	—
305	C₂×C₄	#13094019	1	2	yes	—
306	C₂×C₄	#13094022	1	1	yes	characteristic
307	C₂×C₄	#13094023	1	2	yes	—
308	C₈	—	188	4	no	—
309	C₈	—	188	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	#15688983	1	1	yes	characteristic
318	C₂²	#15689003	1	1	yes	characteristic
319	C₂²	#15689009	1	2	yes	—
320	C₂²	#15689022	1	2	yes	—
321	C₂²	#15689030	1	1	yes	characteristic
322	C₄	—	4	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#15689040	1	1	yes	characteristic
327	C₄	#15689043	1	1	yes	characteristic
328	C₄	#15689045	1	2	yes	—
329	C₄	#15689053	1	1	yes	characteristic
330	C₄	#15689056	1	2	yes	—
331	C₄	#15689062	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	#20241641	1	1	yes	characteristic
337	C₂	#20243936	1	1	yes	characteristic
338	C₂	#20243941	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15689003
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2317	C₂
socle	C₂²×C₄₇	#386
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2317	D₄₇
hypocenter	C₄₇	#11206
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
190. C₄₇	#11206	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
C₂×#15688893	2
#20243127	2
C₂×#15688892	2
C₄₇×#2317	2
#11206	47