Subgroups — #26220970

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	#20244329	C₂	1	1	yes	maximal, characteristic
3	#20244111	C₂	1	1	yes	maximal, characteristic
4	#20244109	C₂	1	1	yes	maximal, characteristic
5	#20243953	C₂	1	2	yes	maximal
6	#20243951	C₂	1	2	yes	maximal
7	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
8	#20243236	C₂	1	2	yes	maximal
9	#20243132	C₂	1	2	yes	maximal
10	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic
11	C₄₇×#1985	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₄₇	C₄	1	2	yes	—
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
15	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×#13093934	C₂²	1	1	yes	characteristic
18	C₂×#13093934	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
23	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
24	C₄⋊C₄×Dic₄₇	C₄	1	2	yes	—
25	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊Dic₄₇	C₂²	1	2	yes	—
27	C₄²×Dic₄₇	C₂²	1	2	yes	—
28	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
29	#15688878	C₂²	1	2	yes	—
30	#15688878	C₂²	1	2	yes	—
31	#15688877	C₂²	1	2	yes	—
32	#15688877	C₂²	1	2	yes	—
33	#15688877	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#336	C₂²	1	2	yes	—
38	C₄₇×#336	C₂²	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
42	Q₈×Dic₄₇	—	2	8	no	—
43	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
56	Dic₄₇⋊C₈	—	2	4	no	—
57	Dic₄₇⋊C₈	—	2	4	no	—
58	#13093935	—	2	2	no	—
59	#13093934	—	2	2	no	—
60	#13093934	—	2	2	no	—
61	C₈×Dic₄₇	—	2	2	no	—
62	C₈×Dic₄₇	—	2	4	no	—
63	C₈×Dic₄₇	Q₈	1	4	yes	complemented
64	C₄₇⋊#95	C₂×C₄	1	2	yes	—
65	C₄₇⋊#95	C₂×C₄	1	1	yes	characteristic
66	C₄₇⋊#95	C₂×C₄	1	2	yes	—
67	C₄₇⋊#95	C₂×C₄	1	1	yes	characteristic
68	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₄₇	—	2	2	no	—
73	C₄×Q₈×C₄₇	—	2	2	no	—
74	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
76	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
77	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₄₇	—	2	2	no	—
79	#97×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	2	2	no	—
81	C₂×Q₈×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
89	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
90	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	2	1	no	—
93	C₂×C₈×C₄₇	—	2	1	no	—
94	C₂×C₈×C₄₇	—	2	1	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	2	2	no	—
98	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₄₇	#41	1	2	yes	—
100	C₄⋊C₄×C₄₇	—	2	4	no	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	2	2	no	—
103	C₄²×C₄₇	—	2	2	no	—
104	C₄²×C₄₇	#41	1	2	yes	—
105	C₄²×C₄₇	#41	1	2	yes	—
106	C₄⋊Dic₄₇	—	2	4	no	—
107	C₄⋊Dic₄₇	—	2	4	no	—
108	C₄⋊Dic₄₇	—	2	4	no	—
109	C₄×Dic₄₇	—	2	1	no	—
110	C₄×Dic₄₇	—	2	1	no	—
111	C₄×Dic₄₇	—	2	2	no	—
112	C₄×Dic₄₇	—	2	2	no	—
113	C₄×Dic₄₇	—	2	4	no	—
114	C₄×Dic₄₇	—	2	4	no	—
115	C₄×Dic₄₇	—	2	4	no	—
116	C₄×Dic₄₇	—	2	4	no	—
117	C₄×Dic₄₇	—	2	4	no	—
118	C₄×Dic₄₇	M₁₆	1	8	yes	—
119	C₄×Dic₄₇	C₂×Q₈	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	—	2	1	no	—
124	C₂×C₄₇⋊C₈	—	2	1	no	—
125	C₂×C₄₇⋊C₈	—	2	2	no	—
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
129	C₂×C₄₇⋊C₈	#41	1	2	yes	—
130	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	Q₈×C₄₇	—	2	8	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	2	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	#117	1	2	yes	—
140	C₂×C₄×C₄₇	#117	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
142	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	—	2	2	no	—
154	C₂×Dic₄₇	—	2	2	no	—
155	C₂×Dic₄₇	—	2	2	no	—
156	C₂×Dic₄₇	C₄×Q₈	1	4	yes	—
157	C₂×Dic₄₇	C₂×M₁₆	1	4	yes	—
158	C₂×Dic₄₇	C₂×M₁₆	1	4	yes	—
159	C₂×Dic₄₇	#131	1	4	yes	—
160	C₈×C₄₇	—	2	2	no	—
161	C₈×C₄₇	—	2	2	no	—
162	C₈×C₄₇	—	4	2	no	—
163	C₄₇⋊C₈	—	2	2	no	—
164	C₄₇⋊C₈	—	2	2	no	—
165	C₄₇⋊C₈	—	4	2	no	—
166	#21340	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
168	C₂²×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
170	C₂²×C₄₇	C₂²×M₁₆	1	2	yes	—
171	C₂²×C₄₇	C₂×#131	1	2	yes	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	2	4	no	—
174	C₄×C₄₇	—	2	2	no	—
175	C₄×C₄₇	—	4	2	no	—
176	C₄×C₄₇	#529	1	1	yes	characteristic
177	C₄×C₄₇	#529	1	1	yes	characteristic
178	C₄×C₄₇	#533	1	1	yes	characteristic
179	C₄×C₄₇	#533	1	1	yes	characteristic
180	Dic₄₇	—	2	4	no	—
181	Dic₄₇	—	4	2	no	—
182	Dic₄₇	—	4	2	no	—
183	Dic₄₇	#446	1	8	yes	—
184	C₄²×Q₈	—	47	1	no	—
185	#1985	—	47	1	no	—
186	#1985	D₄₇	1	1	yes	characteristic
187	#1940	—	47	1	no	—
188	#1940	—	47	1	no	—
189	#1917	—	47	2	no	—
190	#1917	—	47	2	no	—
191	C₄×C₄⋊C₈	—	47	1	no	—
192	C₄×C₄⋊C₈	—	47	2	no	—
193	#1744	—	47	2	no	—
194	C₄×#97	—	47	1	no	—
195	C₂×C₄₇	#2295	1	1	yes	characteristic
196	C₂×C₄₇	#2920	1	1	yes	characteristic
197	C₂×C₄₇	#2922	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#446	1	2	yes	—
199	C₂×C₄₇	#2965	1	2	yes	—
200	C₂×C₄×Q₈	—	47	2	no	—
201	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
202	C₂×C₄⋊C₈	—	47	2	no	—
203	C₂×C₄⋊C₈	—	47	2	no	—
204	C₂×C₄⋊C₈	—	47	1	no	—
205	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
206	C₂×#97	—	47	1	no	—
207	C₂×#97	—	47	1	no	—
208	C₂×#97	—	47	1	no	—
209	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
210	C₂×C₄×C₈	—	47	1	no	—
211	C₂×C₄×C₈	—	47	1	no	—
212	C₂×C₄×C₈	—	47	2	no	—
213	C₄×C₄⋊C₄	—	47	2	no	—
214	C₄×C₄⋊C₄	—	47	2	no	—
215	C₄×C₄⋊C₄	—	47	2	no	—
216	C₄×C₄⋊C₄	—	47	2	no	—
217	C₄×C₄⋊C₄	—	47	1	no	—
218	C₄³	—	47	1	no	—
219	C₄³	—	47	2	no	—
220	#336	—	47	2	no	—
221	#336	—	47	2	no	—
222	#336	—	47	2	no	—
223	#336	—	47	2	no	—
224	#336	—	47	2	no	—
225	#336	C₂×D₄₇	1	2	yes	—
226	#336	C₂×D₄₇	1	2	yes	—
227	C₄₇	#21340	1	1	yes	characteristic, complemented, Sylow
228	C₂²×Q₈	C₄×D₄₇	1	1	yes	characteristic
229	C₂²×C₈	—	47	1	no	—
230	C₂²×C₈	—	47	1	no	—
231	C₂²×C₈	—	47	2	no	—
232	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
233	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
235	C₄×Q₈	—	2	2	no	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	94	8	no	—
238	C₂×C₄⋊C₄	—	47	2	no	—
239	C₂×C₄⋊C₄	—	47	2	no	—
240	C₂×C₄⋊C₄	—	47	2	no	—
241	C₂×C₄⋊C₄	C₄×D₄₇	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
243	C₂×C₄²	—	47	2	no	—
244	C₂×C₄²	—	47	1	no	—
245	C₂×C₄²	—	47	2	no	—
246	C₂×C₄²	—	47	2	no	—
247	C₂×C₄²	—	47	2	no	—
248	C₂×C₄²	—	47	2	no	—
249	C₂×C₄²	—	47	2	no	—
250	C₂×C₄²	—	47	1	no	—
251	C₂×C₄²	—	47	2	no	—
252	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
253	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	94	4	no	—
256	C₄⋊C₈	—	94	4	no	—
257	C₄⋊C₈	—	94	2	no	—
258	#97	—	2	2	no	—
259	#97	—	94	2	no	—
260	#97	—	94	2	no	—
261	#97	—	94	2	no	—
262	C₄×C₈	—	47	4	no	—
263	C₄×C₈	—	94	2	no	—
264	C₄×C₈	—	94	4	no	—
265	#95	—	47	1	no	—
266	#95	—	47	2	no	—
267	#95	—	47	2	no	—
268	#95	—	47	1	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	4	no	—
271	C₂²×C₄	—	47	2	no	—
272	C₂²×C₄	—	47	2	no	—
273	C₂²×C₄	—	47	2	no	—
274	C₂²×C₄	—	47	2	no	—
275	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
276	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
279	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	47	2	no	—
288	C₂×C₈	—	47	2	no	—
289	C₂×C₈	—	94	1	no	—
290	C₂×C₈	—	94	1	no	—
291	C₂×C₈	—	94	1	no	—
292	C₂×C₈	—	94	1	no	—
293	C₂×C₈	—	94	2	no	—
294	C₂×C₈	—	94	2	no	—
295	C₂×C₈	—	94	2	no	—
296	C₂×C₈	Q₈×D₄₇	1	2	yes	—
297	C₂×C₈	#10625354	1	2	yes	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	94	4	no	—
301	C₄⋊C₄	—	94	4	no	—
302	C₄⋊C₄	—	94	4	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	47	8	no	—
306	C₄²	—	47	2	no	—
307	C₄²	—	47	2	no	—
308	C₄²	—	94	4	no	—
309	C₄²	—	94	4	no	—
310	C₄²	—	94	4	no	—
311	C₄²	—	94	4	no	—
312	C₄²	—	94	4	no	—
313	C₄²	—	94	1	no	—
314	C₄²	—	94	2	no	—
315	C₄²	—	94	1	no	—
316	C₄²	—	94	2	no	—
317	C₄²	#10625352	1	2	yes	—
318	C₄²	#10625352	1	2	yes	—
319	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
320	Q₈	—	2	8	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	47	4	no	—
329	C₂×C₄	—	47	4	no	—
330	C₂×C₄	—	47	4	no	—
331	C₂×C₄	—	47	4	no	—
332	C₂×C₄	—	94	2	no	—
333	C₂×C₄	—	94	2	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	94	2	no	—
336	C₂×C₄	—	94	2	no	—
337	C₂×C₄	—	94	2	no	—
338	C₂×C₄	—	94	2	no	—
339	C₂×C₄	—	94	2	no	—
340	C₂×C₄	#13093980	1	2	yes	—
341	C₂×C₄	#13093980	1	2	yes	—
342	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
343	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
345	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
346	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
347	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
348	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	94	2	no	—
353	C₈	—	94	2	no	—
354	C₈	—	188	2	no	—
355	C₂²	C₂×#13093980	1	1	yes	characteristic
356	C₂²	C₄×Q₈×D₄₇	1	1	yes	characteristic
357	C₂²	C₄×#10625354	1	1	yes	characteristic
358	C₂²	C₂×M₁₆×D₄₇	1	2	yes	—
359	C₂²	#131×D₄₇	1	2	yes	—
360	C₄	—	2	2	no	—
361	C₄	—	2	4	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	47	8	no	—
365	C₄	—	94	4	no	—
366	C₄	—	188	2	no	—
367	C₄	—	188	2	no	—
368	C₄	#15689652	1	1	yes	characteristic
369	C₄	#15689659	1	1	yes	characteristic
370	C₄	#15689675	1	1	yes	characteristic
371	C₄	#15689678	1	1	yes	characteristic
372	C₂	#20245863	1	1	yes	characteristic
373	C₂	#20252017	1	1	yes	characteristic
374	C₂	#20252028	1	1	yes	characteristic
375	C₂	#446×D₄₇	1	2	yes	—
376	C₂	#20252353	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1985	C₂
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1985	D₄₇
hypocenter	C₄₇	#21340
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1985

Complements

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

N	H ≅ G/N	type
63. C₈×Dic₄₇	Q₈	semidirect
227. C₄₇	#21340	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×Dic₄₇	2
#20244329	2
#20244111	2
#20244109	2
#20243953	2
#20243951	2
C₄⋊C₈×Dic₄₇	2
#20243236	2
#20243132	2
#97×Dic₄₇	2
C₄₇×#1985	2
#21340	47