Subgroups — #26369812

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252365	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245221	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245215	C₂	1	2	yes	maximal, complemented
5	#20245207	C₂	1	2	yes	maximal, complemented
6	#20245207	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245194	C₂	1	2	yes	maximal, complemented
8	#20245190	C₂	1	1	yes	maximal, characteristic, complemented
9	#20244986	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689660	C₄	1	1	yes	characteristic
13	C₂×#13093963	C₂²	1	2	yes	—
14	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
16	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
17	#15689254	C₄	1	1	yes	characteristic
18	#15689200	C₄	1	1	yes	characteristic
19	#15689193	C₄	1	2	yes	—
20	C₈×D₁₈₈	C₂²	1	2	yes	—
21	C₈×D₁₈₈	C₂²	1	1	yes	characteristic
22	#15689183	C₄	1	2	yes	—
23	#15689179	C₄	1	1	yes	characteristic
24	#15689077	C₂²	1	2	yes	—
25	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
26	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
28	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
29	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
30	(C₂×D₄₇)⋊C₁₆	C₂²	1	2	yes	—
31	#15688898	C₂²	1	2	yes	—
32	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄₇×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×#10625324	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₄₇	C₂³	1	2	yes	—
45	#13094019	C₈	1	2	yes	—
46	#13094015	C₈	1	1	yes	characteristic
47	#13094013	C₈	1	1	yes	characteristic
48	#13094009	C₈	1	2	yes	—
49	#13094000	C₈	1	1	yes	characteristic
50	#13093999	C₈	1	1	yes	characteristic
51	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
52	C₄×D₁₈₈	C₂×C₄	1	1	yes	characteristic
53	#13093981	C₂×C₄	1	2	yes	—
54	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
55	#13093963	—	2	4	no	—
56	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₁₆×D₄₇	—	2	2	no	—
60	C₁₆×D₄₇	—	2	2	no	—
61	C₁₆×D₄₇	—	2	4	no	—
62	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
63	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
64	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
65	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
66	(C₂×D₄₇)⋊C₈	C₂×C₄	1	2	yes	—
67	#13093935	C₂×C₄	1	2	yes	—
68	#13093934	C₂×C₄	1	2	yes	—
69	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
75	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	2	yes	—
80	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
81	C₂×D₁₈₈	C₂×C₈	1	2	yes	—
82	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₄₇	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
91	C₄⋊C₄×C₄₇	C₂×C₈	1	2	yes	—
92	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
94	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
95	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
96	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
97	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
98	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
100	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
101	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
102	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
103	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₄₇	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
109	#10625324	—	2	4	no	—
110	C₈×D₄₇	—	2	1	no	—
111	C₈×D₄₇	—	2	2	no	—
112	C₈×D₄₇	—	2	1	no	—
113	C₈×D₄₇	—	2	2	no	—
114	C₈×D₄₇	—	2	2	no	—
115	C₁₆×C₄₇	—	2	1	no	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₁₆×C₄₇	#41	1	2	yes	—
118	C₄₇⋊C₁₆	—	2	1	no	—
119	C₄₇⋊C₁₆	—	2	2	no	—
120	C₄₇⋊C₁₆	#41	1	2	yes	—
121	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
122	C₂²×D₄₇	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₄₇	—	2	2	no	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₄₇	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
133	D₁₈₈	—	2	4	no	—
134	D₁₈₈	—	2	2	no	—
135	C₄×D₄₇	—	2	2	no	—
136	C₄×D₄₇	—	2	1	no	—
137	C₄×D₄₇	—	2	2	no	—
138	C₄×D₄₇	—	2	1	no	—
139	C₄×D₄₇	—	2	2	no	—
140	C₄×D₄₇	—	2	2	no	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	—	2	2	no	—
143	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
144	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
145	C₄₇⋊C₈	—	2	1	no	—
146	C₄₇⋊C₈	—	2	2	no	—
147	C₄₇⋊C₈	#131	1	2	yes	—
148	#33480	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₄₇	—	2	1	no	—
151	C₂×D₄₇	—	2	1	no	—
152	C₂×D₄₇	—	2	2	no	—
153	C₂×D₄₇	—	2	2	no	—
154	C₂×D₄₇	—	2	2	no	—
155	C₂×D₄₇	#504	1	2	yes	—
156	C₄×C₄₇	—	2	1	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	2	2	no	—
159	C₄×C₄₇	#504	1	2	yes	—
160	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
161	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
162	Dic₄₇	—	2	1	no	—
163	Dic₄₇	—	2	1	no	—
164	Dic₄₇	—	2	1	no	—
165	Dic₄₇	—	2	2	no	—
166	#2966	—	47	1	no	—
167	#2183	D₄₇	1	1	yes	characteristic, complemented
168	Q₈×C₁₆	—	47	1	no	—
169	#2169	—	47	1	no	—
170	#2168	—	47	2	no	—
171	D₄×C₁₆	—	47	1	no	—
172	D₄×C₁₆	—	47	2	no	—
173	#2164	—	47	2	no	—
174	#2163	—	47	1	no	—
175	#2108	—	47	2	no	—
176	C₂×C₄×C₁₆	—	47	1	no	—
177	C₂×C₄₇	C₈×#41	1	1	yes	characteristic
178	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
179	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
180	D₄₇	—	2	2	no	—
181	D₄₇	—	4	2	no	—
182	#529	—	47	1	no	—
183	C₂×M₃₂	—	47	2	no	—
184	C₂²×C₁₆	—	47	1	no	—
185	C₂²×C₁₆	—	47	1	no	—
186	C₂²×C₁₆	—	47	2	no	—
187	#446	—	47	1	no	—
188	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
189	#436	—	47	1	no	—
190	#435	—	47	2	no	—
191	C₈×D₄	—	47	2	no	—
192	C₈×D₄	—	47	1	no	—
193	#433	—	47	2	no	—
194	#432	—	47	1	no	—
195	#405	—	47	2	no	—
196	C₂×C₄×C₈	—	47	1	no	—
197	C₄⋊C₁₆	—	47	1	no	—
198	C₄⋊C₁₆	—	47	2	no	—
199	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
201	C₂²⋊C₁₆	—	47	2	no	—
202	C₂²⋊C₁₆	—	47	2	no	—
203	C₂²⋊C₁₆	—	47	2	no	—
204	#346	—	47	2	no	—
205	#346	C₂×D₄₇	1	2	yes	—
206	C₄×C₁₆	—	47	1	no	—
207	C₄×C₁₆	—	47	1	no	—
208	C₄×C₁₆	—	47	1	no	—
209	C₄×C₁₆	—	47	2	no	—
210	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
211	C₄₇	#33480	1	1	yes	characteristic, complemented, Sylow
212	C₂×M₁₆	—	47	2	no	—
213	C₂²×C₈	—	47	1	no	—
214	C₂²×C₈	—	47	1	no	—
215	C₂²×C₈	—	47	2	no	—
216	#126	—	47	2	no	—
217	#125	—	47	1	no	—
218	#124	—	47	1	no	—
219	#123	—	47	2	no	—
220	C₂²⋊Q₈	—	47	1	no	—
221	#121	—	47	1	no	—
222	C₄×Q₈	C₄×D₄₇	1	1	yes	characteristic
223	C₄×D₄	—	47	2	no	—
224	C₄×D₄	—	47	1	no	—
225	#117	—	47	2	no	—
226	C₂×C₄²	—	47	1	no	—
227	M₃₂	—	94	4	no	—
228	C₂×C₁₆	—	47	1	no	—
229	C₂×C₁₆	—	47	1	no	—
230	C₂×C₁₆	—	47	2	no	—
231	C₂×C₁₆	—	94	2	no	—
232	C₂×C₁₆	—	94	4	no	—
233	C₂×C₁₆	—	94	2	no	—
234	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
236	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
237	C₄⋊C₈	—	47	1	no	—
238	C₄⋊C₈	—	47	2	no	—
239	C₄⋊C₈	C₄×D₄₇	1	1	yes	characteristic
240	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
241	C₂²⋊C₈	—	47	2	no	—
242	C₂²⋊C₈	—	47	2	no	—
243	C₂²⋊C₈	—	47	2	no	—
244	#97	—	47	2	no	—
245	#97	—	47	2	no	—
246	C₄×C₈	—	47	1	no	—
247	C₄×C₈	—	47	1	no	—
248	C₄×C₈	—	47	1	no	—
249	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₄₇	1	2	yes	—
251	C₂×Q₈	C₈×D₄₇	1	1	yes	characteristic
252	C₂×D₄	—	47	2	no	—
253	C₂×D₄	—	47	1	no	—
254	C₂²×C₄	—	47	1	no	—
255	C₂²×C₄	—	47	1	no	—
256	C₂²×C₄	—	47	1	no	—
257	C₂²×C₄	—	47	2	no	—
258	M₁₆	—	94	4	no	—
259	C₂×C₈	—	47	1	no	—
260	C₂×C₈	—	47	1	no	—
261	C₂×C₈	—	47	2	no	—
262	C₂×C₈	—	94	2	no	—
263	C₂×C₈	—	94	1	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	1	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
269	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
270	C₄⋊C₄	—	47	2	no	—
271	C₄⋊C₄	—	47	2	no	—
272	C₄⋊C₄	—	47	1	no	—
273	C₄⋊C₄	—	47	2	no	—
274	C₄⋊C₄	C₈×D₄₇	1	1	yes	characteristic
275	C₄⋊C₄	C₈×D₄₇	1	2	yes	—
276	C₂²⋊C₄	—	47	2	no	—
277	C₂²⋊C₄	—	47	2	no	—
278	C₂²⋊C₄	—	47	2	no	—
279	C₂²⋊C₄	—	47	2	no	—
280	C₂²⋊C₄	—	47	2	no	—
281	C₄²	—	47	1	no	—
282	C₄²	—	47	1	no	—
283	C₄²	—	47	1	no	—
284	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₄₇	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	47	2	no	—
289	C₁₆	—	94	1	no	—
290	C₁₆	—	94	2	no	—
291	C₁₆	#10625354	1	2	yes	—
292	C₂³	—	47	1	no	—
293	C₂³	—	47	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	94	4	no	—
296	D₄	—	94	2	no	—
297	C₂×C₄	—	47	1	no	—
298	C₂×C₄	—	47	1	no	—
299	C₂×C₄	—	47	1	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	94	2	no	—
302	C₂×C₄	—	94	2	no	—
303	C₂×C₄	—	94	1	no	—
304	C₂×C₄	—	94	2	no	—
305	C₂×C₄	—	94	1	no	—
306	C₂×C₄	—	94	2	no	—
307	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
309	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	47	2	no	—
315	C₈	—	94	1	no	—
316	C₈	—	94	2	no	—
317	C₈	#41×D₄₇	1	1	yes	characteristic
318	C₈	C₂×#10625354	1	1	yes	characteristic
319	C₂²	—	47	2	no	—
320	C₂²	—	94	1	no	—
321	C₂²	—	94	1	no	—
322	C₂²	—	94	2	no	—
323	C₂²	—	94	2	no	—
324	C₂²	—	94	2	no	—
325	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	94	1	no	—
330	C₄	—	94	2	no	—
331	C₄	—	94	1	no	—
332	C₄	—	94	1	no	—
333	C₄	#15689541	1	2	yes	—
334	C₄	C₄×#10625354	1	1	yes	characteristic
335	C₄	#131×D₄₇	1	1	yes	characteristic
336	C₂	—	94	2	no	—
337	C₂	—	188	2	no	—
338	C₂	C₈×#10625354	1	1	yes	characteristic
339	C₂	#504×D₄₇	1	1	yes	characteristic
340	C₂	C₂×#15689541	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₄₇×#2183	C₂
prime radical	#13093999	C₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2183	D₄₇
hypocenter	C₄₇	#33480
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄₇⋊C₁₆	C₂	semidirect
3. #20245221	C₂	semidirect
4. #20245215	C₂	semidirect
5. #20245207	C₂	semidirect
6. #20245207	C₂	semidirect
7. #20245194	C₂	semidirect
8. #20245190	C₂	semidirect
9. #20244986	C₂	semidirect
10. C₄×C₁₆×D₄₇	C₂	semidirect
11. C₄₇×#2183	C₂	semidirect
167. #2183	D₄₇	semidirect
211. C₄₇	#33480	semidirect

Maximal subgroups

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

subgroup	index
#20252365	2
Q₈×C₄₇⋊C₁₆	2
#20245221	2
#20245215	2
#20245207	2
#20245194	2
#20245190	2
#20244986	2
C₄×C₁₆×D₄₇	2
C₄₇×#2183	2
#33480	47