Subgroups — #26123227

Order 12032.

850 conjugacy classes of subgroups.

The subgroup lattice has 370 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	C₂²×#13093943	C₂	1	2	yes	maximal
3	C₂×#15688893	C₂	1	2	yes	maximal
4	#20243127	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#2310	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	C₂×#13093943	—	2	4	no	—
11	C₂×#13093943	—	2	4	no	—
12	C₂×#13093943	—	2	4	no	—
13	#15688893	—	2	4	no	—
14	#15688892	—	2	2	no	—
15	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#391	—	2	2	no	—
19	C₄₇×#391	—	2	2	no	—
20	C₄₇×#381	C₄	1	4	yes	—
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	D₄	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	#13093943	—	2	16	no	—
37	C₂²×Q₈×C₄₇	—	2	2	no	—
38	C₂²×Q₈×C₄₇	—	2	2	no	—
39	C₂²×Q₈×C₄₇	—	2	1	no	—
40	C₂²×Q₈×C₄₇	—	2	1	no	—
41	C₂²×Q₈×C₄₇	—	2	1	no	—
42	C₂²×Q₈×C₄₇	—	2	1	no	—
43	C₂²×Q₈×C₄₇	—	2	2	no	—
44	C₂²×Q₈×C₄₇	—	2	2	no	—
45	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
46	C₂²×Q₈×C₄₇	Q₈	1	2	yes	—
47	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
48	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	#117×C₄₇	—	2	2	no	—
50	#117×C₄₇	—	2	2	no	—
51	#117×C₄₇	—	2	2	no	—
52	#117×C₄₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₇	—	2	1	no	—
58	C₂×C₄²×C₄₇	D₄	1	2	yes	—
59	#95×C₄₇	—	2	8	no	—
60	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×Q₈×C₄₇	—	2	4	no	—
62	C₂×Q₈×C₄₇	—	2	2	no	—
63	C₂×Q₈×C₄₇	—	2	2	no	—
64	C₂×Q₈×C₄₇	—	2	2	no	—
65	C₂×Q₈×C₄₇	—	2	2	no	—
66	C₂×Q₈×C₄₇	—	2	4	no	—
67	C₂×Q₈×C₄₇	—	2	4	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	4	1	no	—
70	C₂×Q₈×C₄₇	—	4	2	no	—
71	C₂×Q₈×C₄₇	—	4	2	no	—
72	C₂×Q₈×C₄₇	—	4	2	no	—
73	C₂×Q₈×C₄₇	—	4	1	no	—
74	C₂×Q₈×C₄₇	—	4	2	no	—
75	C₂×Q₈×C₄₇	—	4	2	no	—
76	C₂×Q₈×C₄₇	—	4	1	no	—
77	C₂×Q₈×C₄₇	—	4	2	no	—
78	C₂×Q₈×C₄₇	—	4	1	no	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	—	2	4	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
86	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
93	C₂²×C₄×C₄₇	#41	1	2	yes	—
94	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₄₇	#41	1	2	yes	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	—	4	1	no	—
98	C₄⋊C₄×C₄₇	—	4	1	no	—
99	C₂²⋊C₄×C₄₇	—	2	4	no	—
100	C₂²⋊C₄×C₄₇	—	2	4	no	—
101	C₄²×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	—	4	2	no	—
104	#10625329	—	2	4	no	—
105	#10625329	—	2	4	no	—
106	#10625329	—	2	4	no	—
107	#10625329	—	2	4	no	—
108	#10625329	—	4	2	no	—
109	#10625329	—	4	2	no	—
110	#10625329	—	4	2	no	—
111	#10625329	—	4	2	no	—
112	C₂×C₄₇⋊C₈	—	2	4	no	—
113	C₂×C₄₇⋊C₈	—	2	4	no	—
114	C₂×C₄₇⋊C₈	—	4	2	no	—
115	C₂×C₄₇⋊C₈	—	4	2	no	—
116	C₂×C₄₇⋊C₈	—	4	2	no	—
117	C₂×C₄₇⋊C₈	—	4	2	no	—
118	C₂³×C₄₇	—	4	2	no	—
119	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₄₇	#117	1	1	yes	characteristic
122	C₂³×C₄₇	C₄×D₄	1	2	yes	—
123	C₂³×C₄₇	C₄×D₄	1	2	yes	—
124	Q₈×C₄₇	—	4	2	no	—
125	Q₈×C₄₇	—	4	2	no	—
126	Q₈×C₄₇	—	4	4	no	—
127	Q₈×C₄₇	—	4	4	no	—
128	Q₈×C₄₇	—	4	2	no	—
129	Q₈×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	2	8	no	—
132	C₂×C₄×C₄₇	—	2	4	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	4	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	—	4	4	no	—
137	C₂×C₄×C₄₇	—	4	4	no	—
138	C₂×C₄×C₄₇	—	4	2	no	—
139	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#120	1	2	yes	—
141	C₂×C₄×C₄₇	#121	1	2	yes	—
142	C₂×C₄×C₄₇	#121	1	2	yes	—
143	C₂×C₄×C₄₇	#121	1	2	yes	—
144	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
148	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
151	C₂×C₄×C₄₇	#123	1	2	yes	—
152	C₂×C₄×C₄₇	#123	1	2	yes	—
153	C₂×C₄×C₄₇	#124	1	2	yes	—
154	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
155	C₂×C₄×C₄₇	#126	1	2	yes	—
156	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
157	C₄₇⋊C₈	—	4	4	no	—
158	C₄₇⋊C₈	—	4	4	no	—
159	#11048	—	47	1	no	maximal, Sylow
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	2	4	no	—
162	C₂²×C₄₇	—	4	1	no	—
163	C₂²×C₄₇	—	4	1	no	—
164	C₂²×C₄₇	—	4	2	no	—
165	C₂²×C₄₇	#380	1	1	yes	characteristic
166	C₂²×C₄₇	#385	1	2	yes	—
167	C₂²×C₄₇	#386	1	1	yes	characteristic
168	C₂²×C₄₇	#388	1	2	yes	—
169	C₂²×C₄₇	#390	1	1	yes	characteristic
170	C₄×C₄₇	—	4	4	no	—
171	C₄×C₄₇	—	4	4	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	#392	1	1	yes	characteristic
174	C₄×C₄₇	#393	1	1	yes	characteristic
175	C₄×C₄₇	#394	1	2	yes	—
176	C₄×C₄₇	#396	1	2	yes	—
177	C₄×C₄₇	#397	1	1	yes	characteristic
178	C₄×C₄₇	#399	1	1	yes	characteristic
179	#2905	—	47	1	no	—
180	C₂²×#101	—	47	2	no	—
181	#2310	D₄₇	1	1	yes	characteristic
182	C₂×#344	—	47	2	no	—
183	#1742	—	47	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	—	2	1	no	—
186	C₂×C₄₇	—	2	2	no	—
187	C₂×C₄₇	—	2	2	no	—
188	C₂×C₄₇	#1439	1	1	yes	characteristic
189	C₂×C₄₇	#1915	1	2	yes	—
190	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
191	C₂²×M₁₆	—	47	2	no	—
192	C₂²×M₁₆	—	47	2	no	—
193	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
194	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
195	#424	—	94	4	no	—
196	#423	—	94	2	no	—
197	C₂×#101	—	94	4	no	—
198	C₂×#101	—	94	4	no	—
199	C₂×#101	—	94	4	no	—
200	#391	—	2	2	no	—
201	#391	—	2	2	no	—
202	#381	Dic₄₇	1	4	yes	—
203	#344	—	94	4	no	—
204	#343	—	94	2	no	—
205	C₄₇	#11048	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₈	—	2	1	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×Q₈	—	2	2	no	—
209	C₂²×Q₈	—	2	2	no	—
210	C₂²×Q₈	—	2	2	no	—
211	C₂²×Q₈	—	2	2	no	—
212	C₂²×Q₈	—	2	1	no	—
213	C₂²×Q₈	—	2	1	no	—
214	C₂²×Q₈	Dic₉₄	1	2	yes	—
215	C₂²×Q₈	D₁₈₈	1	2	yes	—
216	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
217	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×M₁₆	—	47	4	no	—
219	C₂×M₁₆	—	47	4	no	—
220	C₂×M₁₆	—	94	2	no	—
221	C₂×M₁₆	—	94	2	no	—
222	C₂×M₁₆	—	94	2	no	—
223	C₂×M₁₆	—	94	2	no	—
224	C₂×M₁₆	—	94	2	no	—
225	C₂×M₁₆	—	94	2	no	—
226	C₂×M₁₆	—	94	2	no	—
227	C₂×M₁₆	—	94	2	no	—
228	C₂×M₁₆	—	94	2	no	—
229	C₂×M₁₆	—	94	2	no	—
230	C₂²×C₈	—	94	2	no	—
231	C₂²×C₈	—	94	2	no	—
232	#117	—	2	2	no	—
233	#117	—	2	2	no	—
234	#117	—	2	2	no	—
235	#117	—	2	2	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
238	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
239	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
242	C₄⋊C₈	—	94	8	no	—
243	#101	—	94	16	no	—
244	#95	—	2	8	no	—
245	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	4	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	—	4	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
270	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
271	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
273	C₂²×C₄	#10625350	1	1	yes	characteristic
274	C₂²×C₄	#10625350	1	1	yes	characteristic
275	C₂²×C₄	D₄×D₄₇	1	2	yes	—
276	C₂²×C₄	D₄×D₄₇	1	2	yes	—
277	C₂²×C₄	#10625352	1	2	yes	—
278	C₂²×C₄	#10625352	1	2	yes	—
279	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
281	M₁₆	—	94	4	no	—
282	M₁₆	—	94	4	no	—
283	M₁₆	—	94	4	no	—
284	M₁₆	—	94	4	no	—
285	M₁₆	—	188	2	no	—
286	M₁₆	—	188	2	no	—
287	M₁₆	—	188	2	no	—
288	M₁₆	—	188	2	no	—
289	C₂×C₈	—	94	4	no	—
290	C₂×C₈	—	94	4	no	—
291	C₂×C₈	—	188	2	no	—
292	C₂×C₈	—	188	2	no	—
293	C₂×C₈	—	188	2	no	—
294	C₂×C₈	—	188	2	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	2	4	no	—
299	C₂²⋊C₄	—	2	4	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
305	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
306	C₂³	#13093983	1	1	yes	characteristic
307	C₂³	D₄×Dic₄₇	1	2	yes	—
308	C₂³	D₄×Dic₄₇	1	2	yes	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	4	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#13093992	1	1	yes	characteristic
325	C₂×C₄	#13093994	1	2	yes	—
326	C₂×C₄	#13093995	1	2	yes	—
327	C₂×C₄	#13093996	1	2	yes	—
328	C₂×C₄	#13093997	1	2	yes	—
329	C₂×C₄	#13093998	1	1	yes	characteristic
330	C₂×C₄	#13093998	1	1	yes	characteristic
331	C₂×C₄	#13094001	1	2	yes	—
332	C₂×C₄	#13094001	1	2	yes	—
333	C₂×C₄	#13094003	1	1	yes	characteristic
334	C₂×C₄	#13094003	1	1	yes	characteristic
335	C₂×C₄	#13094005	1	1	yes	characteristic
336	C₂×C₄	#13094006	1	2	yes	—
337	C₂×C₄	#13094008	1	2	yes	—
338	C₂×C₄	#13094010	1	2	yes	—
339	C₂×C₄	#13094011	1	1	yes	characteristic
340	C₂×C₄	#13094017	1	2	yes	—
341	C₂×C₄	#13094020	1	1	yes	characteristic
342	C₈	—	188	4	no	—
343	C₈	—	188	4	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	4	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	2	no	—
349	C₂²	#15688983	1	1	yes	characteristic
350	C₂²	#15688998	1	2	yes	—
351	C₂²	#15689003	1	1	yes	characteristic
352	C₂²	#15689016	1	2	yes	—
353	C₂²	#15689026	1	1	yes	characteristic
354	C₄	—	4	4	no	—
355	C₄	—	4	4	no	—
356	C₄	—	8	2	no	—
357	C₄	#15689032	1	1	yes	characteristic
358	C₄	#15689035	1	1	yes	characteristic
359	C₄	#15689039	1	2	yes	—
360	C₄	#15689046	1	2	yes	—
361	C₄	#15689048	1	1	yes	characteristic
362	C₄	#15689059	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	2	2	no	—
366	C₂	—	2	2	no	—
367	C₂	#20241633	1	1	yes	characteristic
368	C₂	#20243941	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689003
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2310	C₂
socle	C₂²×C₄₇	#386
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2310	D₄₇
hypocenter	C₄₇	#11048
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2310

Complements

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

N	H ≅ G/N	type
205. C₄₇	#11048	semidirect

Maximal subgroups

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

subgroup	index
#20251952	2
C₂²×#13093943	2
C₂×#15688893	2
#20243127	2
C₄₇×#2310	2
#11048	47