Subgroups — #26370823

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689564	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689540	C₂	1	1	yes	maximal, characteristic, complemented
3	#20253058	C₂	1	1	yes	maximal, characteristic
4	#20245503	C₂	1	2	yes	maximal, complemented
5	#20245494	C₂	1	2	yes	maximal, complemented
6	#20245486	C₂	1	2	yes	maximal, complemented
7	#20245477	C₂	1	2	yes	maximal, complemented
8	#20245177	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245093	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15688927	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#15689701	—	2	1	no	—
14	#15689564	—	2	4	no	—
15	C₂×#13093970	C₂²	1	2	yes	—
16	C₂×#13093967	C₂²	1	2	yes	—
17	#15689540	—	2	4	no	—
18	C₂×#13093964	C₂²	1	2	yes	—
19	C₂×#13093964	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
22	#15689369	—	2	2	no	—
23	#15689364	C₂²	1	2	yes	—
24	#15689359	C₂²	1	2	yes	—
25	C₂×#13093958	C₂²	1	1	yes	characteristic
26	#15689125	—	2	1	no	—
27	#15688950	C₂²	1	2	yes	—
28	#15688948	C₂²	1	2	yes	—
29	#15688927	C₂²	1	2	yes	—
30	#15688927	C₂²	1	2	yes	—
31	#15688926	C₂²	1	1	yes	characteristic
32	#15688926	C₂²	1	1	yes	characteristic
33	#15688923	C₂²	1	1	yes	characteristic
34	#15688923	C₂²	1	1	yes	characteristic
35	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
37	C₄₇×#536	—	2	1	no	—
38	C₄₇×#470	—	2	2	no	—
39	C₄₇×#467	C₂²	1	2	yes	—
40	C₄₇×#467	C₂²	1	2	yes	—
41	C₄₇×#427	C₂²	1	1	yes	characteristic
42	C₄₇×#419	—	2	1	no	—
43	C₂×#10625350	—	2	1	no	—
44	#13094046	—	2	2	no	—
45	#13094046	—	2	2	no	—
46	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
47	C₂×#10625325	—	2	1	no	—
48	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
49	#13094003	—	2	2	no	—
50	#13094000	—	4	1	no	—
51	#13093998	—	2	2	no	—
52	#13093996	—	4	2	no	—
53	#13093995	—	4	1	no	—
54	C₂×C₄⋊Dic₄₇	—	2	1	no	—
55	#13093970	—	2	4	no	—
56	#13093967	—	2	4	no	—
57	#13093964	—	2	4	no	—
58	#13093964	D₄	1	4	yes	complemented
59	C₂×C₄₇⋊C₁₆	—	2	2	no	—
60	C₂×C₄₇⋊C₁₆	—	2	4	no	—
61	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
64	#13093958	—	2	1	no	—
65	#13093958	C₂³	1	2	yes	—
66	#13093949	—	2	2	no	—
67	#13093946	—	2	2	no	—
68	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
69	C₂²×Q₈×C₄₇	—	2	1	no	—
70	C₂×Q₁₆×C₄₇	—	2	2	no	—
71	C₂×Q₁₆×C₄₇	—	2	1	no	—
72	C₂×Q₁₆×C₄₇	—	2	2	no	—
73	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
74	#131×C₄₇	—	2	2	no	—
75	#131×C₄₇	—	2	2	no	—
76	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
77	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
79	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₄₇	—	4	1	no	—
81	#124×C₄₇	—	4	1	no	—
82	#123×C₄₇	—	4	2	no	—
83	C₂²⋊Q₈×C₄₇	—	2	2	no	—
84	C₂²⋊Q₈×C₄₇	—	2	2	no	—
85	#117×C₄₇	—	2	1	no	—
86	#107×C₄₇	C₂³	1	1	yes	characteristic
87	#107×C₄₇	C₂³	1	1	yes	characteristic
88	#106×C₄₇	—	2	1	no	—
89	#103×C₄₇	—	2	2	no	—
90	#103×C₄₇	—	2	2	no	—
91	#41×C₄₇	—	2	2	no	—
92	#41×C₄₇	—	2	2	no	—
93	C₂×Q₈×C₄₇	—	2	2	no	—
94	C₂×Q₈×C₄₇	—	4	1	no	—
95	C₂×Q₈×C₄₇	—	4	2	no	—
96	C₂×Q₈×C₄₇	—	4	2	no	—
97	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
98	C₂×D₄×C₄₇	D₈	1	2	yes	—
99	C₂×D₄×C₄₇	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₄₇	—	4	1	no	—
101	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₂²⋊D₄₇	—	4	1	no	—
104	C₂×C₂²⋊D₄₇	—	4	2	no	—
105	C₂²×Dic₄₇	—	4	1	no	—
106	#10625350	—	4	2	no	—
107	#10625350	—	4	2	no	—
108	C₂×D₁₈₈	—	2	1	no	—
109	C₂×C₄×D₄₇	—	4	1	no	—
110	C₂×Dic₉₄	—	2	1	no	—
111	Q₁₆×C₄₇	—	2	2	no	—
112	Q₁₆×C₄₇	—	2	2	no	—
113	Q₁₆×C₄₇	—	4	2	no	—
114	M₁₆×C₄₇	—	2	2	no	—
115	M₁₆×C₄₇	—	2	4	no	—
116	C₂×C₈×C₄₇	—	2	2	no	—
117	C₂×C₈×C₄₇	—	2	2	no	—
118	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₄₇	#41	1	2	yes	—
122	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₄₇	—	2	1	no	—
124	C₄⋊C₄×C₄₇	—	2	1	no	—
125	C₄⋊C₄×C₄₇	—	4	2	no	—
126	C₄⋊C₄×C₄₇	—	4	2	no	—
127	C₂²⋊C₄×C₄₇	—	4	1	no	—
128	C₂²⋊C₄×C₄₇	—	4	2	no	—
129	C₂²⋊C₄×C₄₇	—	4	2	no	—
130	C₄²×C₄₇	—	4	1	no	—
131	C₂²⋊Dic₄₇	—	4	2	no	—
132	(C₂×D₄₇)⋊C₄	—	4	2	no	—
133	C₄⋊Dic₄₇	—	2	2	no	—
134	C₄⋊Dic₄₇	—	4	1	no	—
135	Dic₄₇⋊C₄	—	4	2	no	—
136	Dic₁₈₈	—	2	1	no	—
137	Dic₁₈₈	—	2	1	no	—
138	D₃₇₆	—	2	1	no	—
139	D₃₇₆	—	2	1	no	—
140	#10625325	—	4	2	no	—
141	C₄₇⋊C₁₆	—	2	1	no	—
142	C₄₇⋊C₁₆	—	2	1	no	—
143	C₄₇⋊C₁₆	C₂×D₄	1	4	yes	—
144	C₂³×C₄₇	C₂×D₈	1	2	yes	—
145	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
146	C₂²×D₄₇	—	4	1	no	—
147	Q₈×C₄₇	—	2	2	no	—
148	Q₈×C₄₇	—	4	2	no	—
149	Q₈×C₄₇	—	4	2	no	—
150	Q₈×C₄₇	—	8	2	no	—
151	D₄×C₄₇	—	2	4	no	—
152	D₄×C₄₇	—	2	2	no	—
153	C₂×C₄×C₄₇	—	2	2	no	—
154	C₂×C₄×C₄₇	—	2	2	no	—
155	C₂×C₄×C₄₇	—	4	1	no	—
156	C₂×C₄×C₄₇	—	4	1	no	—
157	C₂×C₄×C₄₇	—	4	2	no	—
158	C₂×C₄×C₄₇	—	8	2	no	—
159	C₂×C₄×C₄₇	#121	1	2	yes	—
160	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₄₇	#135	1	2	yes	—
163	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
164	C₂²⋊D₄₇	—	8	2	no	—
165	C₂²⋊D₄₇	—	16	2	no	—
166	C₂×Dic₄₇	—	4	1	no	—
167	C₂×Dic₄₇	—	4	2	no	—
168	C₂×Dic₄₇	—	16	1	no	—
169	D₁₈₈	—	4	1	no	—
170	D₁₈₈	—	4	1	no	—
171	C₄×D₄₇	—	8	2	no	—
172	Dic₉₄	—	4	1	no	—
173	Dic₉₄	—	4	1	no	—
174	C₈×C₄₇	—	2	2	no	—
175	C₈×C₄₇	#121	1	2	yes	—
176	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
177	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
178	#33613	—	47	1	no	maximal, Sylow
179	C₂²×C₄₇	—	2	1	no	—
180	C₂²×C₄₇	—	2	1	no	—
181	C₂²×C₄₇	—	2	2	no	—
182	C₂²×C₄₇	—	2	2	no	—
183	C₂²×C₄₇	—	2	2	no	—
184	C₂²×C₄₇	#466	1	2	yes	—
185	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
186	C₂×D₄₇	—	8	1	no	—
187	C₂×D₄₇	—	8	1	no	—
188	C₂×D₄₇	—	8	1	no	—
189	C₄×C₄₇	—	2	2	no	—
190	C₄×C₄₇	—	8	1	no	—
191	C₄×C₄₇	—	8	1	no	—
192	C₄×C₄₇	—	8	2	no	—
193	C₄×C₄₇	#466	1	2	yes	—
194	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
195	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
196	Dic₄₇	—	8	2	no	—
197	Dic₄₇	—	8	1	no	—
198	C₂×#510	—	47	1	no	—
199	C₂×#504	—	47	1	no	—
200	#3061	—	47	1	no	—
201	#3059	D₄₇	1	1	yes	characteristic, complemented
202	#2220	—	47	2	no	—
203	#2219	—	47	2	no	—
204	#2217	—	47	2	no	—
205	#2216	—	47	2	no	—
206	#2159	—	47	1	no	—
207	#2138	—	47	1	no	—
208	C₂×#358	—	47	1	no	—
209	C₂×C₄₇	—	2	2	no	—
210	C₂×C₄₇	—	4	2	no	—
211	C₂×C₄₇	C₂×#466	1	1	yes	characteristic
212	C₂×C₄₇	#3416	1	1	yes	characteristic
213	C₂×C₄₇	#3417	1	1	yes	characteristic
214	D₄₇	—	16	1	no	—
215	C₂×#135	—	47	1	no	—
216	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
217	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
218	#537	—	94	1	no	—
219	#536	—	2	1	no	—
220	#510	—	94	4	no	—
221	C₂×Q₃₂	—	47	2	no	—
222	C₂×SD₃₂	—	47	2	no	—
223	#504	—	94	4	no	—
224	C₂×M₃₂	—	47	1	no	—
225	C₂×M₃₂	—	47	2	no	—
226	C₂²×C₁₆	—	47	1	no	—
227	C₂²×C₁₆	—	47	2	no	—
228	#470	—	2	2	no	—
229	#468	—	94	2	no	—
230	#467	—	47	2	no	—
231	#467	C₂×D₄₇	1	2	yes	—
232	#467	C₂×D₄₇	1	2	yes	—
233	#466	—	47	2	no	—
234	#427	C₂×D₄₇	1	1	yes	characteristic
235	C₂×#107	—	47	1	no	—
236	#419	—	2	1	no	—
237	#417	—	94	1	no	—
238	#367	—	47	2	no	—
239	#366	—	47	2	no	—
240	#358	—	47	2	no	—
241	#358	—	47	2	no	—
242	#358	—	47	1	no	—
243	#358	—	47	1	no	—
244	#357	—	47	1	no	—
245	#357	—	47	1	no	—
246	C₄₇	#33613	1	1	yes	characteristic, complemented, Sylow
247	C₂×#41	—	94	1	no	—
248	C₂×#41	C₂²⋊D₄₇	1	1	yes	characteristic
249	C₂²×Q₈	—	2	1	no	—
250	#135	—	94	2	no	—
251	#135	—	94	2	no	—
252	C₂×Q₁₆	—	2	2	no	—
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×Q₁₆	—	2	1	no	—
255	C₂×Q₁₆	—	47	1	no	—
256	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
257	C₂×SD₁₆	—	94	1	no	—
258	C₂×D₈	—	47	1	no	—
259	#131	—	2	2	no	—
260	#131	—	2	2	no	—
261	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
262	C₂×M₁₆	C₂²⋊D₄₇	1	2	yes	—
263	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
264	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
265	C₄⋊Q₈	—	4	1	no	—
266	#124	—	4	1	no	—
267	#123	—	4	2	no	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₂²⋊Q₈	—	2	2	no	—
270	C₂²⋊Q₈	—	94	2	no	—
271	C₂²⋊Q₈	—	188	1	no	—
272	#121	—	94	2	no	—
273	#121	—	188	1	no	—
274	#121	—	188	2	no	—
275	#117	—	2	1	no	—
276	C₂×C₄⋊C₄	—	94	1	no	—
277	Q₃₂	—	94	4	no	—
278	SD₃₂	—	94	4	no	—
279	M₃₂	—	47	4	no	—
280	M₃₂	—	94	4	no	—
281	C₂×C₁₆	—	47	2	no	—
282	C₂×C₁₆	—	47	1	no	—
283	C₂×C₁₆	—	47	1	no	—
284	C₂×C₁₆	—	94	4	no	—
285	C₂×C₁₆	—	94	2	no	—
286	#107	—	47	2	no	—
287	#107	—	94	1	no	—
288	#107	C₂²×D₄₇	1	1	yes	characteristic
289	#107	C₂²×D₄₇	1	1	yes	characteristic
290	#106	—	2	1	no	—
291	#103	—	2	2	no	—
292	#103	—	2	2	no	—
293	#103	—	94	2	no	—
294	#102	—	94	2	no	—
295	#41	—	2	2	no	—
296	#41	—	2	2	no	—
297	#41	—	188	2	no	—
298	#41	—	188	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	4	1	no	—
301	C₂×Q₈	—	4	2	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	94	1	no	—
304	C₂×Q₈	#10625334	1	1	yes	characteristic
305	C₂×D₄	—	94	1	no	—
306	C₂×D₄	—	188	1	no	—
307	C₂×D₄	—	188	2	no	—
308	C₂×D₄	#10625334	1	2	yes	—
309	C₂×D₄	#10625334	1	1	yes	characteristic
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	188	1	no	—
312	C₂²×C₄	—	188	1	no	—
313	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
314	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
315	Q₁₆	—	2	2	no	—
316	Q₁₆	—	2	2	no	—
317	Q₁₆	—	4	2	no	—
318	Q₁₆	—	94	1	no	—
319	Q₁₆	—	94	1	no	—
320	SD₁₆	—	188	2	no	—
321	D₈	—	94	1	no	—
322	D₈	—	94	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	D₄×D₄₇	1	2	yes	—
328	C₂×C₈	#10625352	1	2	yes	—
329	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
330	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
331	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	—	94	2	no	—
337	C₄⋊C₄	—	188	2	no	—
338	C₄⋊C₄	—	188	1	no	—
339	C₂²⋊C₄	—	4	1	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	188	2	no	—
343	C₂²⋊C₄	—	188	2	no	—
344	C₄²	—	4	1	no	—
345	C₁₆	—	47	4	no	—
346	C₁₆	—	94	1	no	—
347	C₁₆	—	94	1	no	—
348	C₂³	—	188	1	no	—
349	C₂³	C₂×#10625334	1	2	yes	—
350	C₂³	C₂×#10625334	1	1	yes	characteristic
351	Q₈	—	2	2	no	—
352	Q₈	—	4	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	8	2	no	—
355	Q₈	—	188	1	no	—
356	Q₈	—	188	1	no	—
357	D₄	—	2	4	no	—
358	D₄	—	2	2	no	—
359	D₄	—	376	2	no	—
360	D₄	—	188	1	no	—
361	D₄	—	188	1	no	—
362	D₄	—	752	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	376	2	no	—
370	C₂×C₄	—	188	2	no	—
371	C₂×C₄	—	188	1	no	—
372	C₂×C₄	—	752	1	no	—
373	C₂×C₄	#13093996	1	2	yes	—
374	C₂×C₄	C₂×#10625334	1	2	yes	—
375	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
376	C₂×C₄	#13094044	1	2	yes	—
377	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	#13093996	1	2	yes	—
380	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
381	C₈	C₂×#10625352	1	1	yes	characteristic
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	376	1	no	—
388	C₂²	—	376	1	no	—
389	C₂²	—	376	1	no	—
390	C₂²	#15689357	1	2	yes	—
391	C₂²	C₂²×#10625334	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	8	1	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	1	no	—
396	C₄	—	376	2	no	—
397	C₄	—	376	1	no	—
398	C₄	#15689357	1	2	yes	—
399	C₄	C₂×#13093996	1	1	yes	characteristic
400	C₄	C₂×#13094044	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	4	2	no	—
403	C₂	—	752	1	no	—
404	C₂	C₂×#15689357	1	1	yes	characteristic
405	C₂	#20256532	1	1	yes	characteristic
406	C₂	#20256539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3059	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	#20253058	C₂
hypercenter	#3059	D₄₇
hypocenter	C₄₇	#33613
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #3059

Complements

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

N	H ≅ G/N	type
1. C₂×#15689564	C₂	semidirect
2. C₂×#15689540	C₂	semidirect
4. #20245503	C₂	semidirect
5. #20245494	C₂	semidirect
6. #20245486	C₂	semidirect
7. #20245477	C₂	semidirect
8. #20245177	C₂	semidirect
9. #20245093	C₂	semidirect
10. C₂×#15688927	C₂	semidirect
11. C₄₇×#3059	C₂	semidirect
58. #13093964	D₄	semidirect
201. #3059	D₄₇	semidirect
246. C₄₇	#33613	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689564	2
C₂×#15689540	2
#20253058	2
#20245503	2
#20245494	2
#20245486	2
#20245477	2
#20245177	2
#20245093	2
C₂×#15688927	2
C₄₇×#3059	2
#33613	47