Subgroups — #26225070

Order 12032.

854 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689520	C₂	1	1	yes	maximal, characteristic
2	C₄×#13094023	C₂	1	2	yes	maximal
3	#20244347	C₂	1	2	yes	maximal
4	#20244132	C₂	1	1	yes	maximal, characteristic
5	#20244130	C₂	1	2	yes	maximal
6	#20244129	C₂	1	1	yes	maximal, characteristic
7	#20243546	C₂	1	1	yes	maximal, characteristic
8	#20243286	C₂	1	2	yes	maximal
9	#20243285	C₂	1	1	yes	maximal, characteristic
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1776	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094023	C₂²	1	2	yes	—
13	C₂×C₄×Dic₉₄	—	2	2	no	—
14	#15689520	C₄	1	8	yes	complemented
15	C₂×#13093958	C₂²	1	1	yes	characteristic
16	C₂×#13093957	C₂²	1	2	yes	—
17	C₂×#13093956	C₂²	1	1	yes	characteristic
18	C₂×#13093955	C₂²	1	2	yes	—
19	C₂×#13093935	C₂²	1	1	yes	characteristic
20	C₂×#13093934	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
24	#15689027	—	2	2	no	—
25	#15689022	C₂²	1	2	yes	—
26	#15689020	—	2	2	no	—
27	#15689019	C₂²	1	2	yes	—
28	#15688992	C₂²	1	2	yes	—
29	C₄×Dic₄₇⋊C₄	—	2	2	no	—
30	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
31	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
32	#15688886	C₂²	1	2	yes	—
33	#15688885	C₂²	1	2	yes	—
34	#15688885	C₂²	1	2	yes	—
35	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#106×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
39	C₄₇×#340	C₂²	1	2	yes	—
40	C₂²×Dic₉₄	—	2	2	no	—
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
43	#13094023	—	2	4	no	—
44	#13094023	C₂×C₄	1	8	yes	—
45	C₄×Dic₉₄	—	4	8	no	—
46	C₂×C₄⋊Dic₄₇	—	2	2	no	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
49	C₂×Dic₄₇⋊C₄	—	2	2	no	—
50	C₂×Dic₄₇⋊C₄	—	2	2	no	—
51	C₂×Dic₄₇⋊C₄	—	2	2	no	—
52	C₂×Dic₄₇⋊C₄	—	2	2	no	—
53	C₂×C₄×Dic₄₇	—	2	2	no	—
54	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
55	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
56	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
58	#13093958	—	2	2	no	—
59	#13093957	—	2	4	no	—
60	#13093956	C₂×C₄	1	4	yes	—
61	#13093955	C₂×C₄	1	8	yes	—
62	#13093935	Q₈	1	4	yes	complemented
63	#13093934	—	2	2	no	—
64	#13093934	—	2	2	no	—
65	C₈×Dic₄₇	—	2	2	no	—
66	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
67	C₄₇⋊#95	—	2	2	no	—
68	C₄₇⋊#95	—	2	2	no	—
69	C₄₇⋊#95	—	2	2	no	—
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₄₇	—	2	2	no	—
75	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
76	#107×C₄₇	—	2	2	no	—
77	#106×C₄₇	C₂×C₄	1	4	yes	—
78	#97×C₄₇	Q₈	1	4	yes	complemented
79	#95×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
84	C₂²×Dic₄₇	—	2	2	no	—
85	C₂²×Dic₄₇	—	2	2	no	—
86	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
89	C₂×Dic₉₄	—	2	8	no	—
90	C₂×Dic₉₄	—	4	4	no	—
91	C₂×Dic₉₄	—	4	4	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₄₇	#41	1	2	yes	—
98	C₄⋊C₄×C₄₇	—	2	2	no	—
99	C₄⋊C₄×C₄₇	—	2	2	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
102	C₄²×C₄₇	—	4	4	no	—
103	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
104	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
105	C₄⋊Dic₄₇	—	2	2	no	—
106	C₄⋊Dic₄₇	—	2	2	no	—
107	C₄⋊Dic₄₇	—	2	2	no	—
108	C₄⋊Dic₄₇	—	4	4	no	—
109	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
110	Dic₄₇⋊C₄	—	2	8	no	—
111	Dic₄₇⋊C₄	—	4	4	no	—
112	Dic₄₇⋊C₄	—	4	4	no	—
113	Dic₄₇⋊C₄	—	4	4	no	—
114	C₄×Dic₄₇	—	2	1	no	—
115	C₄×Dic₄₇	—	2	1	no	—
116	C₄×Dic₄₇	—	2	4	no	—
117	C₄×Dic₄₇	—	2	4	no	—
118	C₄×Dic₄₇	—	4	4	no	—
119	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
120	C₄×Dic₄₇	#41	1	2	yes	—
121	C₂×C₄₇⋊C₈	—	2	1	no	—
122	C₂×C₄₇⋊C₈	—	2	1	no	—
123	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
125	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
126	C₂×C₄₇⋊C₈	#41	1	2	yes	—
127	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	—	2	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	2	4	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	4	2	no	—
134	C₂×C₄×C₄₇	—	4	2	no	—
135	C₂×C₄×C₄₇	—	4	2	no	—
136	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
137	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₄₇	—	2	4	no	—
144	C₂×Dic₄₇	—	2	1	no	—
145	C₂×Dic₄₇	—	2	1	no	—
146	C₂×Dic₄₇	—	2	2	no	—
147	C₂×Dic₄₇	—	2	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₂×Dic₄₇	—	4	2	no	—
153	C₂×Dic₄₇	C₄×D₄	1	4	yes	—
154	C₂×Dic₄₇	AGL(1,ℤ/8)	1	4	yes	—
155	C₂×Dic₄₇	#137	1	4	yes	—
156	Dic₉₄	—	4	16	no	—
157	C₈×C₄₇	—	2	2	no	—
158	C₈×C₄₇	C₄×Q₈	1	4	yes	complemented
159	C₄₇⋊C₈	—	2	2	no	—
160	C₄₇⋊C₈	C₄×Q₈	1	4	yes	complemented
161	#21727	—	47	1	no	maximal, Sylow
162	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
164	C₂²×C₄₇	#532	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
166	C₂²×C₄₇	C₂×#137	1	2	yes	—
167	C₄×C₄₇	—	2	4	no	—
168	C₄×C₄₇	—	4	4	no	—
169	C₄×C₄₇	—	8	2	no	—
170	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
171	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
172	C₄×C₄₇	#533	1	1	yes	characteristic
173	C₄×C₄₇	#533	1	1	yes	characteristic
174	Dic₄₇	—	4	1	no	—
175	Dic₄₇	—	4	1	no	—
176	Dic₄₇	—	4	4	no	—
177	Dic₄₇	—	8	2	no	—
178	Dic₄₇	#440	1	8	yes	—
179	C₂×#498	—	47	1	no	—
180	C₄×C₄⋊Q₈	—	47	2	no	—
181	#1990	—	47	2	no	—
182	#1944	—	47	1	no	—
183	#1944	—	47	1	no	—
184	#1944	—	47	2	no	—
185	#1845	—	47	1	no	—
186	#1776	—	47	1	no	—
187	#1776	—	47	2	no	—
188	#1776	D₄₇	1	1	yes	characteristic
189	C₄×#97	—	47	1	no	—
190	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
191	C₂×C₄₇	C₂×#440	1	2	yes	—
192	C₂×C₄₇	#3165	1	2	yes	—
193	C₂×C₄₇	#3184	1	1	yes	characteristic
194	C₂×C₄₇	#3185	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	—	47	2	no	—
196	C₂×C₄×Q₈	—	94	2	no	—
197	#498	—	47	8	no	—
198	C₂×#107	—	47	1	no	—
199	C₂×#107	—	47	2	no	—
200	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
201	C₂×#106	—	47	1	no	—
202	C₂×#106	—	47	2	no	—
203	C₂×#106	C₂×D₄₇	1	1	yes	characteristic
204	C₂×#97	—	47	1	no	—
205	C₂×#97	—	47	1	no	—
206	C₂×#97	—	47	1	no	—
207	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
208	C₂×C₄×C₈	—	47	1	no	—
209	C₂×C₄×C₈	—	47	1	no	—
210	#391	—	94	2	no	—
211	#389	—	47	2	no	—
212	#389	—	47	2	no	—
213	#389	—	94	2	no	—
214	#384	—	47	2	no	—
215	C₄×C₄⋊C₄	—	47	2	no	—
216	C₄×C₄⋊C₄	—	94	2	no	—
217	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
218	C₄³	—	47	1	no	—
219	#340	—	47	2	no	—
220	#340	—	47	2	no	—
221	#340	—	47	2	no	—
222	#340	C₂×D₄₇	1	2	yes	—
223	C₄₇	#21727	1	1	yes	characteristic, complemented, Sylow
224	C₂²×Q₈	—	94	2	no	—
225	C₂²×C₈	—	47	1	no	—
226	C₂²×C₈	—	47	1	no	—
227	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
229	C₄⋊Q₈	—	47	8	no	—
230	C₄⋊Q₈	—	94	4	no	—
231	C₄×Q₈	—	188	8	no	—
232	C₂×C₄⋊C₄	—	47	2	no	—
233	C₂×C₄⋊C₄	—	47	2	no	—
234	C₂×C₄⋊C₄	—	94	2	no	—
235	C₂×C₄⋊C₄	—	94	2	no	—
236	C₂×C₄⋊C₄	—	94	2	no	—
237	C₂×C₄⋊C₄	—	94	2	no	—
238	C₂×C₄⋊C₄	—	94	2	no	—
239	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
240	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	—	47	1	no	—
243	C₂×C₄²	—	47	2	no	—
244	C₂×C₄²	—	47	2	no	—
245	C₂×C₄²	—	47	1	no	—
246	C₂×C₄²	—	94	2	no	—
247	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
248	#107	—	2	2	no	—
249	#107	—	94	4	no	—
250	#107	—	94	2	no	—
251	#106	—	47	4	no	—
252	#106	—	47	8	no	—
253	#106	C₄×D₄₇	1	4	yes	—
254	#97	—	47	4	no	—
255	#97	—	94	2	no	—
256	#97	—	94	2	no	—
257	#97	Dic₉₄	1	4	yes	complemented
258	C₄×C₈	—	47	4	no	—
259	C₄×C₈	—	94	2	no	—
260	#95	—	2	2	no	—
261	#95	—	94	2	no	—
262	#95	—	94	2	no	—
263	#95	—	94	2	no	—
264	C₂×Q₈	—	94	8	no	—
265	C₂×Q₈	—	188	4	no	—
266	C₂×Q₈	—	188	4	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	47	2	no	—
270	C₂²×C₄	—	47	1	no	—
271	C₂²×C₄	—	47	1	no	—
272	C₂²×C₄	—	94	2	no	—
273	C₂²×C₄	—	94	2	no	—
274	C₂²×C₄	D₄×D₄₇	1	2	yes	—
275	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	47	2	no	—
279	C₂×C₈	—	47	2	no	—
280	C₂×C₈	—	47	2	no	—
281	C₂×C₈	—	47	2	no	—
282	C₂×C₈	—	94	1	no	—
283	C₂×C₈	—	94	1	no	—
284	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
285	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
287	C₂×C₈	#10625350	1	2	yes	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	47	4	no	—
292	C₄⋊C₄	—	94	8	no	—
293	C₄⋊C₄	—	94	2	no	—
294	C₄⋊C₄	—	94	2	no	—
295	C₄⋊C₄	—	94	2	no	—
296	C₄⋊C₄	—	188	4	no	—
297	C₄⋊C₄	—	188	4	no	—
298	C₄⋊C₄	—	188	4	no	—
299	C₄⋊C₄	—	188	4	no	—
300	C₄⋊C₄	C₂×C₄×D₄₇	1	4	yes	—
301	C₄²	—	4	4	no	—
302	C₄²	—	47	2	no	—
303	C₄²	—	47	2	no	—
304	C₄²	—	94	4	no	—
305	C₄²	—	94	4	no	—
306	C₄²	—	94	1	no	—
307	C₄²	—	94	1	no	—
308	C₄²	—	188	4	no	—
309	C₄²	C₂×Dic₉₄	1	2	yes	—
310	C₄²	Q₈×D₄₇	1	2	yes	—
311	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
312	Q₈	—	188	16	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	47	4	no	—
322	C₂×C₄	—	47	4	no	—
323	C₂×C₄	—	47	4	no	—
324	C₂×C₄	—	94	4	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	—	94	2	no	—
327	C₂×C₄	—	94	1	no	—
328	C₂×C₄	—	94	1	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	—	188	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	—	188	2	no	—
333	C₂×C₄	—	188	2	no	—
334	C₂×C₄	#13094023	1	4	yes	—
335	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
336	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
337	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
338	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
339	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
340	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
341	C₈	—	2	2	no	—
342	C₈	—	47	4	no	—
343	C₈	—	94	2	no	—
344	C₈	C₄×Dic₉₄	1	4	yes	complemented
345	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
346	C₂²	C₂×#13094023	1	1	yes	characteristic
347	C₂²	#15689670	1	1	yes	characteristic
348	C₂²	AGL(1,ℤ/8)×D₄₇	1	2	yes	—
349	C₂²	#137×D₄₇	1	2	yes	—
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	8	2	no	—
353	C₄	—	47	8	no	—
354	C₄	—	376	2	no	—
355	C₄	—	188	4	no	—
356	C₄	—	188	1	no	—
357	C₄	—	188	1	no	—
358	C₄	C₄×Q₈×D₄₇	1	1	yes	characteristic
359	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
360	C₄	#15689676	1	1	yes	characteristic
361	C₄	#15689677	1	1	yes	characteristic
362	C₂	C₄×#13094023	1	1	yes	characteristic
363	C₂	#440×D₄₇	1	2	yes	—
364	C₂	#20253932	1	2	yes	—
365	C₂	#20254123	1	1	yes	characteristic
366	C₂	#20254130	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1776	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1776	D₄₇
hypocenter	C₄₇	#21727
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1776

Complements

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

N	H ≅ G/N	type
14. #15689520	C₄	semidirect
62. #13093935	Q₈	semidirect
78. #97×C₄₇	Q₈	semidirect
158. C₈×C₄₇	C₄×Q₈	semidirect
160. C₄₇⋊C₈	C₄×Q₈	semidirect
223. C₄₇	#21727	semidirect
257. #97	Dic₉₄	semidirect
344. C₈	C₄×Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689520	2
C₄×#13094023	2
#20244347	2
#20244132	2
#20244130	2
#20244129	2
#20243546	2
#20243286	2
#20243285	2
#97×Dic₄₇	2
C₄₇×#1776	2
#21727	47