Subgroups — #26111931

Order 12032.

644 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	C₂²×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#20251740	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688868	C₂	1	2	yes	maximal, complemented
4	#20241272	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	—	2	1	no	—
7	C₂×#13094024	C₂²	1	1	yes	characteristic
8	C₂×#13094003	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	#15689029	—	2	1	no	—
13	#15689021	—	2	1	no	—
14	#15689021	C₄	1	2	yes	—
15	#15688993	C₂²	1	1	yes	characteristic
16	#15688983	—	2	1	no	—
17	#15688983	C₄	1	2	yes	—
18	#15688868	—	2	4	no	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
22	C₄₇×#336	—	2	1	no	—
23	C₂²×Dic₉₄	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	2	4	no	—
25	C₂²×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₄₇⋊C₈	—	4	2	no	—
27	C₂²×C₄₇⋊C₈	—	4	2	no	—
28	#13094024	—	2	2	no	—
29	#13094024	—	2	2	no	—
30	#13094003	—	4	2	no	—
31	#13094003	—	4	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄⋊Dic₄₇	—	2	1	no	—
34	C₂×C₄⋊Dic₄₇	—	2	1	no	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×Dic₄₇⋊C₄	—	2	2	no	—
38	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
39	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
41	C₄⋊(C₄₇⋊C₈)	D₄	1	4	yes	—
42	C₄₇⋊#95	—	2	2	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	—	2	2	no	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	2	no	—
50	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₇	—	2	2	no	—
55	C₄⋊C₈×C₄₇	—	2	2	no	—
56	C₂²⋊C₈×C₄₇	—	4	2	no	—
57	C₂⁴×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	1	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	2	1	no	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	4	1	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	D₈	1	2	yes	—
71	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	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×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂×Dic₉₄	—	4	1	no	—
80	C₂×Dic₉₄	—	4	1	no	—
81	C₂×Dic₉₄	—	4	1	no	—
82	C₂×Dic₉₄	—	4	1	no	—
83	C₂×Dic₉₄	—	4	1	no	—
84	C₂×Dic₉₄	—	4	1	no	—
85	C₂×C₈×C₄₇	—	4	2	no	—
86	C₂×C₈×C₄₇	—	4	2	no	—
87	C₂×C₈×C₄₇	—	4	2	no	—
88	C₄²×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	1	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	1	no	—
92	C₄²×C₄₇	C₂×D₄	1	2	yes	—
93	C₄²×C₄₇	#41	1	2	yes	—
94	C₂²⋊Dic₄₇	—	4	4	no	—
95	C₄⋊Dic₄₇	—	2	4	no	—
96	C₄⋊Dic₄₇	—	4	2	no	—
97	C₄⋊Dic₄₇	—	4	1	no	—
98	C₄⋊Dic₄₇	—	4	1	no	—
99	C₄⋊Dic₄₇	—	4	2	no	—
100	C₄⋊Dic₄₇	—	4	2	no	—
101	Dic₄₇⋊C₄	—	4	4	no	—
102	C₂×C₄₇⋊C₈	—	2	4	no	—
103	C₂×C₄₇⋊C₈	—	4	8	no	—
104	C₂×C₄₇⋊C₈	—	4	2	no	—
105	C₂×C₄₇⋊C₈	—	4	2	no	—
106	C₂³×C₄₇	—	2	1	no	—
107	C₂³×C₄₇	—	2	1	no	—
108	C₂³×C₄₇	—	2	1	no	—
109	C₂³×C₄₇	—	2	1	no	—
110	C₂³×C₄₇	#101	1	2	yes	—
111	C₂³×C₄₇	#102	1	2	yes	—
112	C₂³×C₄₇	#103	1	2	yes	—
113	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	4	no	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	—	2	1	no	—
128	C₂×C₄×C₄₇	—	2	1	no	—
129	C₂×C₄×C₄₇	—	4	4	no	—
130	C₂×C₄×C₄₇	#102	1	2	yes	—
131	C₂×C₄×C₄₇	#103	1	2	yes	—
132	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#121	1	2	yes	—
135	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#135	1	2	yes	—
141	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	Dic₉₄	—	8	1	no	—
150	Dic₉₄	—	8	1	no	—
151	Dic₉₄	—	8	1	no	—
152	Dic₉₄	—	8	1	no	—
153	C₈×C₄₇	—	8	2	no	—
154	C₄₇⋊C₈	—	4	4	no	—
155	#9131	—	47	1	no	maximal, Sylow
156	C₂²×C₄₇	—	2	2	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	2	2	no	—
160	C₂²×C₄₇	—	2	2	no	—
161	C₂²×C₄₇	—	2	2	no	—
162	C₂²×C₄₇	#332	1	4	yes	complemented
163	C₂²×C₄₇	#386	1	1	yes	characteristic
164	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
165	C₂²×C₄₇	#413	1	1	yes	characteristic
166	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
167	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
168	C₂²×C₄₇	#416	1	1	yes	characteristic
169	C₂²×C₄₇	#417	1	1	yes	characteristic
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	4	1	no	—
174	C₄×C₄₇	#447	1	1	yes	characteristic
175	C₄×C₄₇	#448	1	1	yes	characteristic
176	C₄×C₄₇	#465	1	2	yes	—
177	C₄×C₄₇	#467	1	2	yes	—
178	C₄×C₄₇	#480	1	1	yes	characteristic
179	C₄×C₄₇	#481	1	1	yes	characteristic
180	Dic₄₇	—	8	2	no	—
181	Dic₄₇	—	8	2	no	—
182	C₂²×C₄⋊C₈	—	47	1	no	—
183	#2857	—	47	1	no	—
184	#1797	D₄₇	1	1	yes	characteristic
185	C₂×#332	—	47	2	no	—
186	#1296	—	47	2	no	—
187	C₂×C₄₇	—	2	4	no	—
188	C₂×C₄₇	C₂×#332	1	1	yes	characteristic
189	C₂×C₄₇	#1540	1	1	yes	characteristic
190	C₂×C₄₇	#1550	1	1	yes	characteristic
191	C₂×C₄₇	#1552	1	1	yes	characteristic
192	C₂×C₄₇	#1874	1	1	yes	characteristic
193	C₂×C₄₇	#1892	1	1	yes	characteristic
194	C₂×C₄₇	#1901	1	1	yes	characteristic
195	C₂³×C₈	—	94	1	no	—
196	C₂×C₄⋊Q₈	—	47	1	no	—
197	C₂×C₂²⋊Q₈	—	94	1	no	—
198	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
199	C₂×C₄⋊C₈	—	47	2	no	—
200	C₂×C₄⋊C₈	—	94	4	no	—
201	C₂×C₄⋊C₈	—	94	1	no	—
202	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
203	C₂×C₂²⋊C₈	—	2	1	no	—
204	#391	—	94	1	no	—
205	#389	—	47	2	no	—
206	#389	—	94	1	no	—
207	#384	—	47	1	no	—
208	#380	—	47	2	no	—
209	#380	—	94	1	no	—
210	#336	—	2	1	no	—
211	#332	—	94	4	no	—
212	C₄₇	#9131	1	1	yes	characteristic, complemented, Sylow
213	C₂²×Q₈	—	94	1	no	—
214	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
215	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
216	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²×C₈	—	2	2	no	—
218	C₂²×C₈	—	94	4	no	—
219	C₂²×C₈	—	94	2	no	—
220	C₂²×C₈	—	188	2	no	—
221	C₂²×C₈	—	188	2	no	—
222	C₄⋊Q₈	—	94	2	no	—
223	C₄⋊Q₈	—	94	2	no	—
224	C₂²⋊Q₈	—	188	2	no	—
225	C₂²⋊Q₈	—	188	2	no	—
226	C₂×C₄⋊C₄	—	47	2	no	—
227	C₂×C₄⋊C₄	—	47	2	no	—
228	C₂×C₄⋊C₄	—	94	2	no	—
229	C₂×C₄⋊C₄	—	94	1	no	—
230	C₂×C₄⋊C₄	—	94	1	no	—
231	C₂×C₄⋊C₄	—	94	1	no	—
232	C₂×C₂²⋊C₄	—	94	2	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
236	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	47	4	no	—
242	C₄⋊C₈	—	94	4	no	—
243	C₄⋊C₈	—	94	2	no	—
244	C₂²⋊C₈	—	4	2	no	—
245	#95	—	94	2	no	—
246	#95	—	94	2	no	—
247	C₂⁴	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
248	C₂×Q₈	—	188	1	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×Q₈	—	188	1	no	—
251	C₂×Q₈	—	188	1	no	—
252	C₂×Q₈	—	188	1	no	—
253	C₂×Q₈	—	188	1	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	94	2	no	—
264	C₂²×C₄	—	94	2	no	—
265	C₂²×C₄	#10625325	1	2	yes	—
266	C₂²×C₄	D₃₇₆	1	2	yes	—
267	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
270	C₂²×C₄	#10625335	1	2	yes	—
271	C₂²×C₄	#10625337	1	2	yes	—
272	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	94	4	no	—
279	C₂×C₈	—	188	8	no	—
280	C₂×C₈	—	188	2	no	—
281	C₂×C₈	—	188	2	no	—
282	C₄⋊C₄	—	94	4	no	—
283	C₄⋊C₄	—	188	2	no	—
284	C₄⋊C₄	—	188	4	no	—
285	C₄⋊C₄	—	188	2	no	—
286	C₄⋊C₄	—	188	1	no	—
287	C₄⋊C₄	—	188	1	no	—
288	C₄⋊C₄	—	188	2	no	—
289	C₂²⋊C₄	—	188	4	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	D₄×D₄₇	1	2	yes	—
295	C₄²	#10625352	1	2	yes	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	#13093942	1	2	yes	—
301	C₂³	#13093946	1	2	yes	—
302	C₂³	#13093947	1	2	yes	—
303	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
304	Q₈	—	376	1	no	—
305	Q₈	—	376	1	no	—
306	Q₈	—	376	1	no	—
307	Q₈	—	376	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	—	188	2	no	—
326	C₂×C₄	—	188	2	no	—
327	C₂×C₄	—	188	2	no	—
328	C₂×C₄	—	188	2	no	—
329	C₂×C₄	—	188	2	no	—
330	C₂×C₄	#13093946	1	2	yes	—
331	C₂×C₄	#13093947	1	2	yes	—
332	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
333	C₂×C₄	#13093994	1	1	yes	characteristic
334	C₂×C₄	#13093996	1	2	yes	—
335	C₂×C₄	#13094008	1	1	yes	characteristic
336	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
337	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
338	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
339	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
340	C₂×C₄	#13094044	1	2	yes	—
341	C₂×C₄	#13094052	1	1	yes	characteristic
342	C₂×C₄	#13094057	1	1	yes	characteristic
343	C₈	—	8	2	no	—
344	C₈	—	188	4	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	#15688868	1	4	yes	complemented
352	C₂²	#15689000	1	1	yes	characteristic
353	C₂²	C₂×#13093942	1	1	yes	characteristic
354	C₂²	#15689105	1	1	yes	characteristic
355	C₂²	C₂×#13093946	1	1	yes	characteristic
356	C₂²	C₂×#13093947	1	1	yes	characteristic
357	C₂²	#15689117	1	1	yes	characteristic
358	C₂²	#15689125	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	4	1	no	—
363	C₄	—	376	2	no	—
364	C₄	—	376	2	no	—
365	C₄	#15689259	1	1	yes	characteristic
366	C₄	#15689264	1	1	yes	characteristic
367	C₄	#15689350	1	2	yes	—
368	C₄	#15689362	1	2	yes	—
369	C₄	#15689437	1	1	yes	characteristic
370	C₄	#15689442	1	1	yes	characteristic
371	C₂	—	2	4	no	—
372	C₂	C₂×#15688868	1	1	yes	characteristic
373	C₂	#20242035	1	1	yes	characteristic
374	C₂	#20242079	1	1	yes	characteristic
375	C₂	#20242089	1	1	yes	characteristic
376	C₂	#20243703	1	1	yes	characteristic
377	C₂	#20243809	1	1	yes	characteristic
378	C₂	#20243854	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1797	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1797	D₄₇
hypocenter	C₄₇	#9131
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#15688868	C₂	semidirect
4. #20241272	C₂	semidirect
162. C₂²×C₄₇	#332	semidirect
212. C₄₇	#9131	semidirect
351. C₂²	#15688868	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₄₇⋊C₈)	2
#20251740	2
C₂×#15688868	2
#20241272	2
C₄₇×#1797	2
#9131	47