Subgroups — #26316118

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251785	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20244527	C₂	1	4	yes	maximal
4	#20244191	C₂	1	1	yes	maximal, characteristic
5	#20243516	C₂	1	1	yes	maximal, characteristic
6	#20243515	C₂	1	1	yes	maximal, characteristic
7	#20243391	C₂	1	4	yes	maximal
8	C₄×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1956	C₂	1	1	yes	maximal, characteristic
10	C₂×#13094023	—	2	1	no	—
11	C₂×#13094023	—	2	1	no	—
12	C₂×#13094023	C₂²	1	2	yes	—
13	C₂×Q₈×Dic₄₇	C₂²	1	2	yes	—
14	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×#13093948	C₂²	1	4	yes	—
16	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	#15689030	C₂²	1	2	yes	—
18	#15689022	—	2	1	no	—
19	#15689022	—	2	1	no	—
20	#15688993	C₂²	1	1	yes	characteristic
21	#15688993	C₂²	1	1	yes	characteristic
22	#15688992	—	2	1	no	—
23	#15688992	—	2	1	no	—
24	#15688992	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₄₇	—	2	1	no	—
26	C₄⋊C₄×Dic₄₇	—	2	1	no	—
27	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
28	#15688886	C₂²	1	4	yes	—
29	#15688885	C₂²	1	4	yes	—
30	#15688877	C₂²	1	2	yes	—
31	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#103×C₄₇	C₂²	1	4	yes	—
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄₇×#384	C₂²	1	1	yes	characteristic
35	C₂²×Dic₉₄	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
37	#13094023	—	4	4	no	—
38	#13094023	—	4	4	no	—
39	#13094023	—	4	4	no	—
40	Q₈×Dic₄₇	—	2	8	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₄⋊Dic₄₇	—	2	1	no	—
43	C₂×C₄⋊Dic₄₇	—	2	1	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	2	2	no	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×Dic₄₇⋊C₄	—	2	2	no	—
49	C₂×Dic₄₇⋊C₄	—	2	2	no	—
50	C₂×Dic₄₇⋊C₄	—	2	2	no	—
51	C₂×C₄×Dic₄₇	—	2	2	no	—
52	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
55	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
56	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
58	Dic₄₇⋊C₈	—	2	4	no	—
59	#13093948	—	4	4	no	—
60	C₄×C₄₇⋊C₈	—	2	2	no	—
61	C₄₇⋊#95	—	2	2	no	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₂²×Q₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₇	—	2	2	no	—
66	C₄⋊Q₈×C₄₇	Q₈	1	4	yes	complemented
67	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#103×C₄₇	—	4	4	no	—
74	C₄×C₈×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	—	2	2	no	—
76	C₂×Q₈×C₄₇	—	2	4	no	—
77	C₂×Q₈×C₄₇	—	2	4	no	—
78	C₂×Q₈×C₄₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
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	1	yes	characteristic
89	C₂²×Dic₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₉₄	—	4	4	no	—
91	C₂×Dic₉₄	—	4	4	no	—
92	C₂×Dic₉₄	—	4	4	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₄⋊C₄×C₄₇	—	2	4	no	—
97	C₄⋊C₄×C₄₇	—	2	4	no	—
98	C₄⋊C₄×C₄₇	—	2	4	no	—
99	C₄⋊C₄×C₄₇	—	4	2	no	—
100	C₄⋊C₄×C₄₇	—	4	2	no	—
101	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₄₇	#41	1	2	yes	—
103	C₄⋊Dic₄₇	—	2	4	no	—
104	C₄⋊Dic₄₇	—	2	4	no	—
105	C₄⋊Dic₄₇	—	4	2	no	—
106	C₄⋊Dic₄₇	—	4	2	no	—
107	C₄⋊Dic₄₇	—	4	2	no	—
108	C₄⋊Dic₄₇	—	4	2	no	—
109	C₄⋊Dic₄₇	—	4	4	no	—
110	Dic₄₇⋊C₄	—	4	4	no	—
111	Dic₄₇⋊C₄	—	4	4	no	—
112	Dic₄₇⋊C₄	—	4	4	no	—
113	C₄×Dic₄₇	—	2	2	no	—
114	C₄×Dic₄₇	—	2	2	no	—
115	C₄×Dic₄₇	—	2	2	no	—
116	C₄×Dic₄₇	—	4	4	no	—
117	C₄×Dic₄₇	SD₁₆	1	4	yes	—
118	C₄×Dic₄₇	Q₁₆	1	4	yes	—
119	C₄×Dic₄₇	#41	1	4	yes	—
120	C₂×C₄₇⋊C₈	—	2	2	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	2	2	no	—
123	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₄₇	—	2	8	no	—
125	Q₈×C₄₇	—	4	4	no	—
126	C₂×C₄×C₄₇	—	2	4	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	—	4	2	no	—
132	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
135	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₄₇	#137	1	2	yes	—
138	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
142	C₂×Dic₄₇	—	2	1	no	—
143	C₂×Dic₄₇	—	2	1	no	—
144	C₂×Dic₄₇	—	2	1	no	—
145	C₂×Dic₄₇	—	2	1	no	—
146	C₂×Dic₄₇	—	2	1	no	—
147	C₂×Dic₄₇	—	2	1	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	—	4	2	no	—
154	C₂×Dic₄₇	#124	1	2	yes	—
155	C₂×Dic₄₇	#124	1	2	yes	—
156	C₂×Dic₄₇	C₂×SD₁₆	1	2	yes	—
157	C₂×Dic₄₇	C₂×SD₁₆	1	2	yes	—
158	C₂×Dic₄₇	C₂×Q₁₆	1	2	yes	—
159	C₂×Dic₄₇	C₂×Q₁₆	1	2	yes	—
160	Dic₉₄	—	8	8	no	—
161	C₈×C₄₇	—	4	2	no	—
162	C₄₇⋊C₈	—	4	2	no	—
163	#29141	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
165	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
166	C₂²×C₄₇	#535	1	1	yes	characteristic
167	C₂²×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
168	C₂²×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
169	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
171	C₄×C₄₇	—	4	4	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	#475	1	4	yes	—
174	C₄×C₄₇	#477	1	4	yes	—
175	C₄×C₄₇	#554	1	2	yes	—
176	C₄×C₄₇	#555	1	2	yes	—
177	Dic₄₇	—	2	2	no	—
178	Dic₄₇	—	2	2	no	—
179	Dic₄₇	—	2	2	no	—
180	Dic₄₇	—	8	2	no	—
181	Dic₄₇	—	8	2	no	—
182	Dic₄₇	#487	1	4	yes	—
183	#2868	—	47	1	no	—
184	C₄×C₄⋊Q₈	—	47	1	no	—
185	#2024	—	47	4	no	—
186	#1956	—	47	1	no	—
187	#1956	D₄₇	1	1	yes	characteristic
188	#1834	—	47	1	no	—
189	#1834	—	47	1	no	—
190	#1803	—	47	4	no	—
191	C₄×C₄⋊C₈	—	47	1	no	—
192	C₂×C₄₇	#2563	1	1	yes	characteristic
193	C₂×C₄₇	C₂×#475	1	1	yes	characteristic
194	C₂×C₄₇	C₂×#477	1	1	yes	characteristic
195	C₂×C₄₇	C₂×#487	1	1	yes	characteristic
196	C₂×C₄₇	#3141	1	1	yes	characteristic
197	C₂×C₄₇	#3208	1	1	yes	characteristic
198	C₂×C₄₇	#3209	1	1	yes	characteristic
199	C₂×C₄⋊Q₈	—	47	2	no	—
200	C₂×C₄⋊Q₈	—	94	1	no	—
201	C₂×C₄⋊Q₈	—	94	1	no	—
202	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
203	C₂×C₄×Q₈	—	47	2	no	—
204	C₂×C₄⋊C₈	—	47	2	no	—
205	C₂×#103	—	47	4	no	—
206	C₂×#103	C₂×D₄₇	1	4	yes	—
207	C₂×C₄×C₈	—	47	1	no	—
208	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
209	#391	—	47	2	no	—
210	#389	—	94	1	no	—
211	#389	—	94	1	no	—
212	#384	—	47	2	no	—
213	#384	—	47	1	no	—
214	#384	—	47	1	no	—
215	#384	—	94	1	no	—
216	#384	—	94	1	no	—
217	#384	C₂×D₄₇	1	1	yes	characteristic
218	C₄×C₄⋊C₄	—	94	1	no	—
219	C₄×C₄⋊C₄	—	94	1	no	—
220	C₄³	—	47	1	no	—
221	#340	—	47	4	no	—
222	#340	—	47	4	no	—
223	#336	—	47	2	no	—
224	C₄₇	#29141	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	94	2	no	—
226	C₂²×Q₈	C₂²×D₄₇	1	2	yes	—
227	C₂²×C₈	—	47	2	no	—
228	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
229	C₄⋊Q₈	—	2	2	no	—
230	C₄⋊Q₈	—	188	4	no	—
231	C₄⋊Q₈	—	188	4	no	—
232	C₄⋊Q₈	—	188	4	no	—
233	C₄⋊Q₈	Dic₉₄	1	4	yes	complemented
234	C₄×Q₈	—	94	8	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	47	2	no	—
240	C₂×C₄⋊C₄	—	47	2	no	—
241	C₂×C₄⋊C₄	—	94	2	no	—
242	C₂×C₄⋊C₄	—	94	2	no	—
243	C₂×C₄⋊C₄	—	94	2	no	—
244	C₂×C₄⋊C₄	—	94	2	no	—
245	C₂×C₄⋊C₄	—	94	1	no	—
246	C₂×C₄⋊C₄	—	94	1	no	—
247	C₂×C₄⋊C₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
250	C₂×C₄²	—	47	1	no	—
251	C₂×C₄²	—	47	1	no	—
252	C₂×C₄²	—	47	1	no	—
253	C₂×C₄²	—	47	1	no	—
254	C₂×C₄²	—	47	1	no	—
255	C₂×C₄²	—	47	1	no	—
256	C₂×C₄²	—	94	2	no	—
257	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
258	C₄⋊C₈	—	94	4	no	—
259	#103	—	4	4	no	—
260	#103	—	188	4	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	94	2	no	—
263	#95	—	94	2	no	—
264	#95	—	94	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	4	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	188	4	no	—
269	C₂×Q₈	—	188	4	no	—
270	C₂×Q₈	—	188	4	no	—
271	C₂×Q₈	C₂×Dic₉₄	1	4	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	47	1	no	—
275	C₂²×C₄	—	47	1	no	—
276	C₂²×C₄	—	47	1	no	—
277	C₂²×C₄	—	47	1	no	—
278	C₂²×C₄	—	94	2	no	—
279	C₂²×C₄	—	94	2	no	—
280	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
282	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	94	2	no	—
288	C₂×C₈	—	94	2	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	94	4	no	—
295	C₄⋊C₄	—	94	4	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	—	188	2	no	—
301	C₄⋊C₄	—	188	2	no	—
302	C₄⋊C₄	—	188	2	no	—
303	C₄⋊C₄	—	188	2	no	—
304	C₄²	—	47	4	no	—
305	C₄²	—	47	4	no	—
306	C₄²	—	47	4	no	—
307	C₄²	—	94	2	no	—
308	C₄²	—	94	2	no	—
309	C₄²	—	94	2	no	—
310	C₄²	—	188	4	no	—
311	C₄²	C₂×Dic₉₄	1	2	yes	—
312	C₄²	#10625352	1	2	yes	—
313	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
314	Q₈	—	2	8	no	—
315	Q₈	—	4	4	no	—
316	Q₈	—	376	8	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	47	2	no	—
324	C₂×C₄	—	47	2	no	—
325	C₂×C₄	—	47	2	no	—
326	C₂×C₄	—	47	2	no	—
327	C₂×C₄	—	47	2	no	—
328	C₂×C₄	—	47	2	no	—
329	C₂×C₄	—	94	1	no	—
330	C₂×C₄	—	94	1	no	—
331	C₂×C₄	—	94	1	no	—
332	C₂×C₄	—	94	1	no	—
333	C₂×C₄	—	94	1	no	—
334	C₂×C₄	—	94	1	no	—
335	C₂×C₄	—	188	2	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	—	188	2	no	—
340	C₂×C₄	—	188	2	no	—
341	C₂×C₄	#13094001	1	2	yes	—
342	C₂×C₄	#13094001	1	2	yes	—
343	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
344	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
345	C₂×C₄	#13094051	1	2	yes	—
346	C₂×C₄	#13094056	1	2	yes	—
347	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
348	C₂×C₄	#41×D₄₇	1	2	yes	—
349	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
350	C₂×C₄	#13094069	1	2	yes	—
351	C₈	—	4	2	no	—
352	C₈	—	188	2	no	—
353	C₂²	C₂×#13094001	1	1	yes	characteristic
354	C₂²	#124×D₄₇	1	1	yes	characteristic
355	C₂²	#15689686	1	1	yes	characteristic
356	C₂²	C₂×SD₁₆×D₄₇	1	1	yes	characteristic
357	C₂²	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
358	C₂²	C₂×#13094051	1	1	yes	characteristic
359	C₂²	C₂×#13094056	1	1	yes	characteristic
360	C₄	—	4	4	no	—
361	C₄	—	8	2	no	—
362	C₄	—	47	4	no	—
363	C₄	—	94	2	no	—
364	C₄	—	94	2	no	—
365	C₄	—	94	2	no	—
366	C₄	—	376	2	no	—
367	C₄	—	376	2	no	—
368	C₄	#15689406	1	4	yes	—
369	C₄	#15689416	1	4	yes	—
370	C₄	#15689832	1	2	yes	—
371	C₄	#15689837	1	2	yes	—
372	C₂	#20248559	1	1	yes	characteristic
373	C₂	C₂×#15689406	1	1	yes	characteristic
374	C₂	C₂×#15689416	1	1	yes	characteristic
375	C₂	#487×D₄₇	1	1	yes	characteristic
376	C₂	#20253783	1	1	yes	characteristic
377	C₂	#20254325	1	1	yes	characteristic
378	C₂	#20254332	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1956	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1956	D₄₇
hypocenter	C₄₇	#29141
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

Complements

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

N	H ≅ G/N	type
66. C₄⋊Q₈×C₄₇	Q₈	semidirect
224. C₄₇	#29141	semidirect
233. C₄⋊Q₈	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20251785	2
C₄⋊Q₈×Dic₄₇	2
#20244527	2
#20244191	2
#20243516	2
#20243515	2
#20243391	2
C₄×Dic₄₇⋊C₈	2
C₄₇×#1956	2
#29141	47