Subgroups — #26372991

Order 12032.

674 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	C₂×Q₃₂×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#20255123	C₂	1	2	yes	maximal, complemented
3	#20245646	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245479	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245471	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245448	C₂	1	2	yes	maximal, complemented
7	#20245328	C₂	1	2	yes	maximal, complemented
8	#20245303	C₂	1	2	yes	maximal, complemented
9	Q₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₆×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₇×#508	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₇	C₂²	1	2	yes	—
13	#15689780	—	2	2	no	—
14	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
15	C₂×#13093970	C₂²	1	2	yes	—
16	Q₃₂×D₄₇	—	2	4	no	—
17	C₂×#13093968	C₂²	1	2	yes	—
18	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
19	#15689506	C₂²	1	2	yes	—
20	#15689364	C₂²	1	2	yes	—
21	#15689356	C₂²	1	2	yes	—
22	#15689343	—	2	2	no	—
23	#15689285	—	2	2	no	—
24	#15689260	—	2	2	no	—
25	Q₁₆×Dic₄₇	C₂²	1	2	yes	—
26	C₈×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
27	#15688949	C₂²	1	1	yes	characteristic
28	Dic₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
29	#15688928	C₂²	1	2	yes	—
30	#15688927	C₂²	1	2	yes	—
31	#15688924	C₂²	1	2	yes	—
32	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	1	yes	characteristic
33	(C₂×D₄₇)⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂×#135×C₄₇	C₂²	1	2	yes	—
36	C₄₇×#508	—	2	4	no	—
37	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
38	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
39	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
40	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₄₇	—	2	2	no	—
42	C₂×Dic₁₈₈	C₂³	1	2	yes	—
43	C₂×#10625337	—	2	2	no	—
44	Q₁₆×D₄₇	—	2	4	no	—
45	Q₁₆×D₄₇	—	4	2	no	—
46	C₂×#10625335	—	2	2	no	—
47	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
48	#13094011	—	2	2	no	—
49	#13094011	—	4	2	no	—
50	#13094003	—	2	2	no	—
51	#13094003	—	4	2	no	—
52	#13093995	—	2	2	no	—
53	#13093995	—	4	2	no	—
54	Q₈×Dic₄₇	—	2	2	no	—
55	C₄×C₂²⋊D₄₇	—	4	2	no	—
56	C₄×C₂²⋊D₄₇	D₄	1	1	yes	characteristic
57	Dic₃₇₆	—	2	2	no	—
58	#13093970	—	2	4	no	—
59	#13093968	—	4	2	no	—
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₁₆×D₄₇	—	2	2	no	—
63	#13093958	C₂³	1	2	yes	—
64	Dic₄₇⋊C₈	D₄	1	1	yes	characteristic
65	#13093949	—	2	2	no	—
66	#13093948	—	2	2	no	—
67	#13093945	—	2	2	no	—
68	C₂²⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
69	(C₂×D₄₇)⋊C₈	D₄	1	1	yes	characteristic
70	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
71	C₂×#41×C₄₇	—	2	2	no	—
72	#135×C₄₇	—	2	4	no	—
73	#135×C₄₇	—	4	2	no	—
74	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₄₇	—	2	2	no	—
76	C₂×D₈×C₄₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	Q₃₂×C₄₇	D₄	1	4	yes	complemented
79	SD₃₂×C₄₇	—	2	4	no	—
80	D₁₆×C₄₇	—	2	2	no	—
81	C₂×C₁₆×C₄₇	—	2	2	no	—
82	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
84	#41×C₄₇	—	4	4	no	—
85	#41×C₄₇	—	8	2	no	—
86	C₂×Q₈×C₄₇	—	2	2	no	—
87	C₂×D₄×C₄₇	—	2	2	no	—
88	C₂×D₄×C₄₇	—	4	2	no	—
89	C₂²×C₄×C₄₇	—	4	2	no	—
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₄₇	—	4	2	no	—
92	C₂×C₂²⋊D₄₇	D₈	1	1	yes	characteristic
93	Q₈×D₄₇	—	4	4	no	—
94	Q₈×D₄₇	—	8	2	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₉₄	—	2	2	no	—
98	C₂×Dic₉₄	—	4	2	no	—
99	Q₁₆×C₄₇	—	2	2	no	—
100	Q₁₆×C₄₇	C₂×D₄	1	4	yes	—
101	SD₁₆×C₄₇	—	4	4	no	—
102	D₈×C₄₇	—	2	2	no	—
103	D₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	1	no	—
105	C₂×C₈×C₄₇	—	2	1	no	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₂²⋊Dic₄₇	—	4	2	no	—
109	C₂²⋊Dic₄₇	—	4	2	no	—
110	C₂²⋊Dic₄₇	—	4	2	no	—
111	C₂²⋊Dic₄₇	—	4	2	no	—
112	C₂²⋊Dic₄₇	D₈	1	1	yes	characteristic
113	#10625337	—	4	4	no	—
114	#10625335	—	8	2	no	—
115	(C₂×D₄₇)⋊C₄	D₈	1	1	yes	characteristic
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	4	2	no	—
118	Dic₄₇⋊C₄	—	4	2	no	—
119	Dic₄₇⋊C₄	D₈	1	1	yes	characteristic
120	C₄×Dic₄₇	—	4	2	no	—
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
124	Dic₁₈₈	—	2	2	no	—
125	Dic₁₈₈	—	2	2	no	—
126	C₈×D₄₇	—	2	1	no	—
127	C₈×D₄₇	—	2	1	no	—
128	C₁₆×C₄₇	—	2	1	no	—
129	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
130	C₄₇⋊C₁₆	—	2	1	no	—
131	C₄₇⋊C₁₆	—	2	1	no	—
132	C₂³×C₄₇	—	4	2	no	—
133	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₄₇	—	2	4	no	—
136	Q₈×C₄₇	—	4	2	no	—
137	D₄×C₄₇	—	4	2	no	—
138	D₄×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	8	4	no	—
140	C₂×C₄×C₄₇	—	2	1	no	—
141	C₂×C₄×C₄₇	—	2	1	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	8	4	no	—
144	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₄₇	—	2	2	no	—
147	C₂²⋊D₄₇	—	8	4	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
151	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
152	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
153	C₄×D₄₇	—	2	1	no	—
154	C₄×D₄₇	—	2	1	no	—
155	C₄×D₄₇	—	8	4	no	—
156	Dic₉₄	—	4	2	no	—
157	Dic₉₄	—	4	2	no	—
158	Dic₉₄	—	8	4	no	—
159	C₈×C₄₇	—	2	1	no	—
160	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₄₇⋊C₈	—	2	1	no	—
163	C₄₇⋊C₈	—	2	1	no	—
164	#33851	—	47	1	no	maximal, Sylow
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	—	2	1	no	—
167	C₂²×C₄₇	—	2	1	no	—
168	C₂²×C₄₇	—	8	2	no	—
169	C₂²×C₄₇	—	8	2	no	—
170	C₂²×C₄₇	—	8	2	no	—
171	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₄₇	—	2	1	no	—
173	C₂×D₄₇	—	2	1	no	—
174	C₂×D₄₇	#508	1	2	yes	complemented
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	4	4	no	—
177	C₄×C₄₇	D₄²	1	1	yes	characteristic
178	C₄×C₄₇	#576	1	1	yes	characteristic
179	Dic₄₇	—	2	1	no	—
180	Dic₄₇	—	2	1	no	—
181	Dic₄₇	—	2	1	no	—
182	Dic₄₇	—	8	2	no	—
183	Dic₄₇	—	8	2	no	—
184	C₂×#508	D₄₇	1	1	yes	characteristic, complemented
185	C₂²×Q₃₂	—	47	1	no	—
186	#3285	—	47	2	no	—
187	#2249	—	47	1	no	—
188	#2216	—	47	1	no	—
189	#2215	—	47	1	no	—
190	#2211	—	47	2	no	—
191	#2189	—	47	2	no	—
192	#2185	—	47	2	no	—
193	C₄×Q₃₂	—	47	1	no	—
194	D₄×C₁₆	—	47	1	no	—
195	C₂×C₄₇	—	4	1	no	—
196	C₂×C₄₇	—	16	2	no	—
197	C₂×C₄₇	D₄×D₈	1	1	yes	characteristic
198	C₂×C₄₇	C₂×#508	1	1	yes	characteristic
199	C₂×C₄₇	#3417	1	1	yes	characteristic
200	D₄₇	—	2	2	no	—
201	C₂×#135	C₂×D₄₇	1	2	yes	—
202	C₂²×Q₁₆	—	47	2	no	—
203	#548	—	94	2	no	—
204	#508	—	2	4	no	—
205	C₂×Q₃₂	—	47	2	no	—
206	C₂×Q₃₂	—	47	1	no	—
207	C₂×Q₃₂	—	94	4	no	—
208	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
209	C₂×SD₃₂	—	47	2	no	—
210	C₂×SD₃₂	C₂×D₄₇	1	2	yes	—
211	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
212	C₂²×C₁₆	—	47	1	no	—
213	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
214	#495	—	47	2	no	—
215	#467	—	47	2	no	—
216	#466	—	47	2	no	—
217	#464	—	94	2	no	—
218	#452	—	94	2	no	—
219	#448	—	94	2	no	—
220	C₄×Q₁₆	—	47	2	no	—
221	C₈×D₄	—	47	1	no	—
222	#366	—	47	1	no	—
223	C₄⋊C₁₆	—	47	1	no	—
224	#358	—	47	2	no	—
225	#358	—	47	2	no	—
226	#357	—	47	2	no	—
227	C₂²⋊C₁₆	—	47	1	no	—
228	C₂²⋊C₁₆	—	47	1	no	—
229	C₄×C₁₆	—	47	1	no	—
230	C₄₇	#33851	1	1	yes	characteristic, complemented, Sylow
231	C₂×#41	—	2	2	no	—
232	C₂²×Q₈	—	94	2	no	—
233	#135	—	2	4	no	—
234	#135	—	4	2	no	—
235	C₂×Q₁₆	—	47	2	no	—
236	C₂×Q₁₆	—	94	2	no	—
237	C₂×Q₁₆	—	94	4	no	—
238	C₂×Q₁₆	—	188	2	no	—
239	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
240	C₂×SD₁₆	—	2	2	no	—
241	C₂×SD₁₆	—	94	2	no	—
242	C₂×D₈	C₂²×D₄₇	1	2	yes	—
243	C₂²×C₈	—	47	1	no	—
244	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	#124	—	94	2	no	—
246	#124	—	188	2	no	—
247	C₂²⋊Q₈	—	94	2	no	—
248	C₂²⋊Q₈	—	188	2	no	—
249	#121	—	94	2	no	—
250	#121	—	188	2	no	—
251	C₄×Q₈	—	94	2	no	—
252	C₄×D₄	—	47	1	no	—
253	C₄×D₄	—	188	2	no	—
254	Q₃₂	—	94	4	no	—
255	Q₃₂	—	94	2	no	—
256	Q₃₂	C₂²⋊D₄₇	1	4	yes	complemented
257	SD₃₂	—	2	4	no	—
258	SD₃₂	—	188	2	no	—
259	D₁₆	—	2	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	47	1	no	—
262	C₂×C₁₆	—	47	1	no	—
263	C₂×C₁₆	—	94	2	no	—
264	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
266	#107	—	47	2	no	—
267	C₄⋊C₈	—	47	1	no	—
268	#103	—	94	2	no	—
269	#103	—	94	2	no	—
270	#102	—	94	2	no	—
271	C₂²⋊C₈	—	47	1	no	—
272	C₂²⋊C₈	—	47	1	no	—
273	C₄×C₈	—	47	1	no	—
274	#41	—	4	4	no	—
275	#41	—	8	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	94	2	no	—
278	C₂×Q₈	—	376	2	no	—
279	C₂×Q₈	—	188	2	no	—
280	C₂×Q₈	—	188	4	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	47	1	no	—
284	C₂×D₄	—	188	2	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	—	47	1	no	—
287	C₂²×C₄	—	188	2	no	—
288	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	94	2	no	—
291	Q₁₆	—	94	2	no	—
292	Q₁₆	—	188	4	no	—
293	Q₁₆	C₂×C₂²⋊D₄₇	1	4	yes	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	376	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	47	1	no	—
301	C₂×C₈	—	47	1	no	—
302	C₂×C₈	—	94	1	no	—
303	C₂×C₈	—	94	1	no	—
304	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
305	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
306	C₄⋊C₄	—	47	1	no	—
307	C₄⋊C₄	—	94	2	no	—
308	C₄⋊C₄	—	188	2	no	—
309	C₄⋊C₄	—	188	2	no	—
310	C₂²⋊C₄	—	47	1	no	—
311	C₂²⋊C₄	—	47	1	no	—
312	C₂²⋊C₄	—	188	2	no	—
313	C₂²⋊C₄	—	188	2	no	—
314	C₂²⋊C₄	—	188	2	no	—
315	C₂²⋊C₄	—	188	2	no	—
316	C₄²	—	47	1	no	—
317	C₄²	—	188	2	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	94	1	no	—
320	C₁₆	—	94	1	no	—
321	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	—	47	1	no	—
324	C₂³	D₈×D₄₇	1	1	yes	characteristic
325	Q₈	—	2	4	no	—
326	Q₈	—	4	2	no	—
327	Q₈	—	376	4	no	—
328	Q₈	—	188	2	no	—
329	Q₈	—	188	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	8	4	no	—
333	D₄	—	94	2	no	—
334	D₄	—	376	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	8	4	no	—
339	C₂×C₄	—	47	1	no	—
340	C₂×C₄	—	47	1	no	—
341	C₂×C₄	—	47	1	no	—
342	C₂×C₄	—	94	1	no	—
343	C₂×C₄	—	94	1	no	—
344	C₂×C₄	—	376	4	no	—
345	C₂×C₄	—	188	2	no	—
346	C₂×C₄	—	188	2	no	—
347	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	94	1	no	—
351	C₈	—	94	1	no	—
352	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
353	C₈	#13094069	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	47	2	no	—
361	C₂²	—	94	1	no	—
362	C₂²	—	94	1	no	—
363	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	4	4	no	—
366	C₄	—	94	1	no	—
367	C₄	—	94	1	no	—
368	C₄	—	94	1	no	—
369	C₄	—	376	2	no	—
370	C₄	—	376	2	no	—
371	C₄	D₄×C₂²⋊D₄₇	1	1	yes	characteristic
372	C₄	#15689924	1	1	yes	characteristic
373	C₂	—	4	1	no	—
374	C₂	—	16	2	no	—
375	C₂	—	94	2	no	—
376	C₂	D₈×C₂²⋊D₄₇	1	1	yes	characteristic
377	C₂	#508×D₄₇	1	1	yes	characteristic
378	C₂	#20256538	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#508	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#508	D₄₇
hypocenter	C₄₇	#33851
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₄₇	C₂	semidirect
2. #20255123	C₂	semidirect
3. #20245646	C₂	semidirect
4. #20245479	C₂	semidirect
5. #20245471	C₂	semidirect
6. #20245448	C₂	semidirect
7. #20245328	C₂	semidirect
8. #20245303	C₂	semidirect
9. Q₃₂×Dic₄₇	C₂	semidirect
10. C₁₆×C₂²⋊D₄₇	C₂	semidirect
11. C₂×C₄₇×#508	C₂	semidirect
78. Q₃₂×C₄₇	D₄	semidirect
174. C₂×D₄₇	#508	semidirect
184. C₂×#508	D₄₇	semidirect
230. C₄₇	#33851	semidirect
256. Q₃₂	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₄₇	2
#20255123	2
#20245646	2
#20245479	2
#20245471	2
#20245448	2
#20245328	2
#20245303	2
Q₃₂×Dic₄₇	2
C₁₆×C₂²⋊D₄₇	2
C₂×C₄₇×#508	2
#33851	47