Subgroups — #26247207

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20244864	C₂	1	2	yes	maximal
3	#20244860	C₂	1	2	yes	maximal
4	#20244718	C₂	1	2	yes	maximal
5	#20244716	C₂	1	2	yes	maximal
6	#20243286	C₂	1	4	yes	maximal
7	#20243137	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₉₄	—	2	1	no	—
11	C₂×#13093957	C₂²	1	4	yes	—
12	C₂×#13093955	C₂²	1	4	yes	—
13	C₂×#13093935	C₂²	1	2	yes	—
14	#15689029	—	2	1	no	—
15	#15689021	—	2	1	no	—
16	#15689021	C₂²	1	2	yes	—
17	#15688993	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊Dic₄₇	—	2	1	no	—
19	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
20	#15688885	C₂²	1	4	yes	—
21	#15688885	C₂²	1	4	yes	—
22	#15688878	C₂²	1	2	yes	—
23	#15688878	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₄₇	—	2	1	no	—
26	C₄₇×#391	—	2	1	no	—
27	C₄₇×#389	—	2	1	no	—
28	C₄₇×#389	C₂²	1	2	yes	—
29	C₄₇×#384	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
32	C₄³×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×Dic₉₄	—	2	1	no	—
34	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
35	#13094024	—	2	4	no	—
36	C₄×Dic₉₄	—	4	4	no	—
37	C₂×C₄⋊Dic₄₇	—	2	1	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇⋊C₄	—	2	2	no	—
43	C₂×C₄×Dic₄₇	—	2	2	no	—
44	#13093957	—	2	8	no	—
45	#13093955	—	2	8	no	—
46	#13093935	Q₈	1	8	yes	complemented
47	C₄₇⋊#95	—	2	2	no	—
48	C₄₇⋊#95	—	2	2	no	—
49	C₂²×Q₈×C₄₇	—	2	1	no	—
50	C₄⋊Q₈×C₄₇	—	2	4	no	—
51	C₄×Q₈×C₄₇	—	4	4	no	—
52	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₄₇	—	2	2	no	—
59	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	#95×C₄₇	—	2	2	no	—
67	#95×C₄₇	—	2	2	no	—
68	C₂×Q₈×C₄₇	—	2	4	no	—
69	C₂×Q₈×C₄₇	—	4	2	no	—
70	C₂×Q₈×C₄₇	—	4	2	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	2	2	no	—
81	C₂²×Dic₄₇	—	2	2	no	—
82	C₂×Dic₉₄	—	2	4	no	—
83	C₂×Dic₉₄	—	4	2	no	—
84	C₂×Dic₉₄	—	4	2	no	—
85	C₄⋊C₄×C₄₇	—	2	4	no	—
86	C₄⋊C₄×C₄₇	—	2	2	no	—
87	C₄⋊C₄×C₄₇	—	2	4	no	—
88	C₄⋊C₄×C₄₇	—	4	2	no	—
89	C₄⋊C₄×C₄₇	—	4	2	no	—
90	C₄⋊C₄×C₄₇	—	4	4	no	—
91	C₄⋊C₄×C₄₇	#41	1	4	yes	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	2	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	—	4	4	no	—
99	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
100	C₄⋊Dic₄₇	—	2	4	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	2	4	no	—
103	C₄⋊Dic₄₇	—	4	2	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	#41	1	4	yes	—
106	Dic₄₇⋊C₄	—	4	4	no	—
107	C₄×Dic₄₇	—	4	4	no	—
108	C₂×C₄₇⋊C₈	—	2	4	no	—
109	C₂×C₄₇⋊C₈	—	2	4	no	—
110	C₂×C₄₇⋊C₈	C₂×Q₈	1	8	yes	—
111	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₄₇	—	4	8	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	4	no	—
126	C₂×C₄×C₄₇	—	4	2	no	—
127	C₂×C₄×C₄₇	—	4	2	no	—
128	C₂×C₄×C₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	4	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
135	C₂×C₄×C₄₇	#137	1	2	yes	—
136	C₂×C₄×C₄₇	#137	1	2	yes	—
137	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
138	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×Dic₄₇	—	2	4	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	Dic₉₄	—	4	8	no	—
147	C₄₇⋊C₈	—	2	8	no	—
148	#23839	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	#525	1	1	yes	characteristic
150	C₂²×C₄₇	#525	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
155	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	—	2	2	no	—
160	C₄×C₄₇	—	2	2	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	4	4	no	—
163	C₄×C₄₇	—	8	2	no	—
164	C₄×C₄₇	#533	1	4	yes	—
165	Dic₄₇	—	4	4	no	—
166	Dic₄₇	—	8	2	no	—
167	C₄×C₄⋊Q₈	—	47	1	no	—
168	C₄×C₄⋊Q₈	D₄₇	1	1	yes	characteristic
169	#2084	—	47	2	no	—
170	#2083	—	47	2	no	—
171	#2058	—	47	2	no	—
172	#2057	—	47	2	no	—
173	#1776	—	47	4	no	—
174	#1745	—	47	1	no	—
175	C₂×C₄₇	#2399	1	1	yes	characteristic
176	C₂×C₄₇	#3109	1	1	yes	characteristic
177	C₂×C₄₇	#3110	1	1	yes	characteristic
178	C₂×C₄₇	#3112	1	1	yes	characteristic
179	C₂×C₄₇	#3113	1	1	yes	characteristic
180	C₂×C₄₇	#3165	1	1	yes	characteristic
181	C₂×C₄₇	#3165	1	1	yes	characteristic
182	C₂×C₄⋊Q₈	—	47	1	no	—
183	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
184	C₂×C₄×Q₈	—	2	1	no	—
185	C₂×C₄×Q₈	—	94	1	no	—
186	C₂×#107	—	47	4	no	—
187	C₂×#106	—	47	4	no	—
188	C₂×#97	—	47	2	no	—
189	#391	—	2	1	no	—
190	#391	—	94	1	no	—
191	#389	—	2	1	no	—
192	#389	—	47	2	no	—
193	#389	—	94	1	no	—
194	#389	C₂×D₄₇	1	2	yes	—
195	#384	—	47	1	no	—
196	#384	C₂×D₄₇	1	1	yes	characteristic
197	C₄×C₄⋊C₄	—	2	1	no	—
198	C₄×C₄⋊C₄	—	47	2	no	—
199	C₄×C₄⋊C₄	—	94	1	no	—
200	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
201	C₄³	C₂×D₄₇	1	1	yes	characteristic
202	#340	—	47	4	no	—
203	#340	—	47	4	no	—
204	#336	—	47	2	no	—
205	#336	—	47	2	no	—
206	C₄₇	#23839	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×Q₈	—	94	1	no	—
209	C₂²×C₈	—	47	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	94	4	no	—
212	C₄×Q₈	—	4	4	no	—
213	C₄×Q₈	—	188	4	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	47	2	no	—
219	C₂×C₄⋊C₄	—	47	2	no	—
220	C₂×C₄⋊C₄	—	94	1	no	—
221	C₂×C₄⋊C₄	—	94	2	no	—
222	C₂×C₄⋊C₄	—	94	1	no	—
223	C₂×C₄⋊C₄	—	94	1	no	—
224	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	94	2	no	—
228	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
235	#107	—	94	8	no	—
236	#106	—	94	8	no	—
237	#97	—	47	8	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	94	2	no	—
241	#95	—	94	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	94	4	no	—
246	C₂×Q₈	—	188	2	no	—
247	C₂×Q₈	—	188	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	—	94	2	no	—
252	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
259	C₂×C₈	—	47	8	no	—
260	C₂×C₈	—	94	4	no	—
261	C₂×C₈	—	94	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	47	4	no	—
269	C₄⋊C₄	—	94	4	no	—
270	C₄⋊C₄	—	94	2	no	—
271	C₄⋊C₄	—	94	4	no	—
272	C₄⋊C₄	—	188	2	no	—
273	C₄⋊C₄	—	188	4	no	—
274	C₄⋊C₄	—	188	2	no	—
275	C₄⋊C₄	#10625350	1	4	yes	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	4	no	—
283	C₄²	—	188	4	no	—
284	C₄²	Q₈×D₄₇	1	4	yes	—
285	C₂³	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
286	Q₈	—	4	8	no	—
287	Q₈	—	188	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	94	4	no	—
307	C₂×C₄	—	188	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	#13094022	1	2	yes	—
313	C₂×C₄	#13094022	1	2	yes	—
314	C₂×C₄	#13094049	1	2	yes	—
315	C₂×C₄	#13094050	1	2	yes	—
316	C₂×C₄	#13094054	1	2	yes	—
317	C₂×C₄	#13094055	1	2	yes	—
318	C₂×C₄	C₂×Q₈×D₄₇	1	2	yes	—
319	C₂×C₄	#41×D₄₇	1	2	yes	—
320	C₂×C₄	C₂×#10625350	1	2	yes	—
321	C₈	—	94	8	no	—
322	C₂²	#15689630	1	1	yes	characteristic
323	C₂²	#15689634	1	1	yes	characteristic
324	C₂²	C₂×#13094022	1	1	yes	characteristic
325	C₂²	C₂×#13094049	1	1	yes	characteristic
326	C₂²	C₂×#13094050	1	1	yes	characteristic
327	C₂²	C₂×#13094054	1	1	yes	characteristic
328	C₂²	C₂×#13094055	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	4	no	—
336	C₄	—	8	2	no	—
337	C₄	—	376	2	no	—
338	C₄	—	188	4	no	—
339	C₄	#15689677	1	4	yes	—
340	C₂	#20246679	1	1	yes	characteristic
341	C₂	#20253475	1	1	yes	characteristic
342	C₂	#20253487	1	1	yes	characteristic
343	C₂	#20253510	1	1	yes	characteristic
344	C₂	#20253514	1	1	yes	characteristic
345	C₂	#20253930	1	1	yes	characteristic
346	C₂	#20253935	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×C₄⋊Q₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#23839
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
46. #13093935	Q₈	semidirect
206. C₄₇	#23839	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094024	2
#20244864	2
#20244860	2
#20244718	2
#20244716	2
#20243286	2
#20243137	2
C₄×C₄⋊Q₈×C₄₇	2
#23839	47