Subgroups — #26369631

Order 12032.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 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
2	#20252265	C₂	1	1	yes	maximal, characteristic
3	#20245257	C₂	1	2	yes	maximal, complemented
4	#20245250	C₂	1	2	yes	maximal, complemented
5	C₄×#13093970	C₂	1	2	yes	maximal, complemented
6	C₄×#13093967	C₂	1	2	yes	maximal, complemented
7	#20245177	C₂	1	1	yes	maximal, characteristic, complemented
8	#20245093	C₂	1	1	yes	maximal, characteristic
9	C₂×#15688927	C₂	1	1	yes	maximal, characteristic, complemented
10	#20244985	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	1	yes	characteristic
13	#15689603	—	2	1	no	—
14	#15689564	C₄	1	8	yes	complemented
15	C₂×#13093970	C₂²	1	2	yes	—
16	C₂×#13093967	C₂²	1	2	yes	—
17	C₂×#13093964	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
19	#15689225	—	2	2	no	—
20	C₄×Dic₁₈₈	C₂²	1	2	yes	—
21	C₄×D₃₇₆	C₂²	1	2	yes	—
22	C₂×#13093958	C₂²	1	1	yes	characteristic
23	#15689125	—	2	1	no	—
24	#15688950	C₂²	1	2	yes	—
25	#15688948	C₂²	1	2	yes	—
26	#15688927	C₂²	1	2	yes	—
27	#15688927	C₂²	1	2	yes	—
28	#15688926	C₂²	1	1	yes	characteristic
29	#15688926	C₂²	1	1	yes	characteristic
30	#15688923	C₂²	1	1	yes	characteristic
31	#15688923	C₂²	1	1	yes	characteristic
32	#15688899	C₂²	1	2	yes	—
33	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
34	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#519	—	2	1	no	—
36	C₄₇×#441	—	2	2	no	—
37	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₄₇	C₂²	1	2	yes	—
39	C₄₇×#427	C₂²	1	1	yes	characteristic
40	C₄₇×#419	—	2	1	no	—
41	C₄₇×#405	C₂²	1	1	yes	characteristic
42	C₂×#10625350	—	2	1	no	—
43	#13094046	—	2	2	no	—
44	#13094046	C₂×C₄	1	4	yes	—
45	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
46	C₂×#10625325	—	2	1	no	—
47	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
48	C₄×Dic₉₄	—	2	2	no	—
49	C₄×D₁₈₈	—	2	2	no	—
50	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
51	#13093982	—	4	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	C₄⋊C₄×D₄₇	—	4	1	no	—
54	#13093970	C₂×C₄	1	8	yes	complemented
55	#13093967	C₂×C₄	1	8	yes	complemented
56	#13093964	C₂×C₄	1	4	yes	complemented
57	C₂×C₄₇⋊C₁₆	—	2	2	no	—
58	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
61	#13093958	—	2	1	no	—
62	#13093958	C₂³	1	2	yes	—
63	#13093949	—	2	2	no	—
64	#13093946	—	2	2	no	—
65	C₂²×Q₈×C₄₇	—	2	1	no	—
66	C₂×Q₁₆×C₄₇	—	2	1	no	—
67	C₂×Q₁₆×C₄₇	—	2	2	no	—
68	C₂×Q₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
69	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
70	C₂×M₁₆×C₄₇	D₄	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	C₄×Q₈×C₄₇	—	4	1	no	—
75	C₄×Q₈×C₄₇	—	4	1	no	—
76	#117×C₄₇	—	2	1	no	—
77	#117×C₄₇	—	4	2	no	—
78	#117×C₄₇	D₄	1	1	yes	characteristic
79	#107×C₄₇	C₂³	1	1	yes	characteristic
80	#107×C₄₇	C₂³	1	1	yes	characteristic
81	#106×C₄₇	—	2	1	no	—
82	#103×C₄₇	—	2	2	no	—
83	#103×C₄₇	—	2	2	no	—
84	#97×C₄₇	D₄	1	2	yes	—
85	C₄×C₈×C₄₇	C₂³	1	2	yes	—
86	C₂×Q₈×C₄₇	—	2	4	no	—
87	C₂×Q₈×C₄₇	—	2	2	no	—
88	C₂×Q₈×C₄₇	—	4	1	no	—
89	C₂×Q₈×C₄₇	—	4	2	no	—
90	C₂²×C₄×C₄₇	—	4	1	no	—
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	—	4	1	no	—
93	C₂²×Dic₄₇	—	4	1	no	—
94	#10625350	—	2	4	no	—
95	#10625350	—	4	2	no	—
96	C₂×D₁₈₈	—	2	1	no	—
97	C₂×C₄×D₄₇	—	4	1	no	—
98	C₂×C₄×D₄₇	—	4	2	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	Q₁₆×C₄₇	—	2	4	no	—
101	Q₁₆×C₄₇	—	2	2	no	—
102	Q₁₆×C₄₇	C₂²×C₄	1	4	yes	—
103	M₁₆×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₄₇	—	2	1	no	—
110	C₄⋊C₄×C₄₇	—	2	1	no	—
111	C₄⋊C₄×C₄₇	—	4	2	no	—
112	C₄⋊C₄×C₄₇	—	4	2	no	—
113	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₄₇	D₈	1	1	yes	characteristic
115	C₂²⋊C₄×C₄₇	—	4	1	no	—
116	C₂²⋊C₄×C₄₇	D₈	1	2	yes	—
117	C₄²×C₄₇	—	4	1	no	—
118	C₄²×C₄₇	—	4	2	no	—
119	C₄²×C₄₇	—	4	2	no	—
120	C₄²×C₄₇	C₂×D₄	1	2	yes	—
121	(C₂×D₄₇)⋊C₄	—	4	2	no	—
122	C₄⋊Dic₄₇	—	2	2	no	—
123	C₄⋊Dic₄₇	—	4	1	no	—
124	Dic₄₇⋊C₄	—	4	2	no	—
125	C₄×Dic₄₇	—	4	2	no	—
126	Dic₁₈₈	—	2	1	no	—
127	Dic₁₈₈	C₂²×C₄	1	2	yes	—
128	D₃₇₆	—	2	1	no	—
129	D₃₇₆	C₂²×C₄	1	2	yes	—
130	#10625325	—	2	4	no	—
131	C₄₇⋊C₁₆	—	2	1	no	—
132	C₄₇⋊C₁₆	—	2	1	no	—
133	C₄₇⋊C₁₆	C₂²×C₄	1	4	yes	—
134	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₄₇	—	4	1	no	—
136	Q₈×C₄₇	—	2	4	no	—
137	Q₈×C₄₇	—	4	2	no	—
138	Q₈×C₄₇	—	4	4	no	—
139	C₂×C₄×C₄₇	—	4	2	no	—
140	C₂×C₄×C₄₇	—	4	4	no	—
141	C₂×C₄×C₄₇	—	4	1	no	—
142	C₂×C₄×C₄₇	—	4	1	no	—
143	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#135	1	2	yes	—
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂²⋊D₄₇	—	4	4	no	—
150	C₂×Dic₄₇	—	4	1	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₂×Dic₄₇	—	16	1	no	—
153	D₁₈₈	—	2	2	no	—
154	D₁₈₈	—	4	1	no	—
155	C₄×D₄₇	—	4	4	no	—
156	C₄×D₄₇	—	16	2	no	—
157	Dic₉₄	—	2	2	no	—
158	Dic₉₄	—	4	1	no	—
159	C₈×C₄₇	—	2	2	no	—
160	C₈×C₄₇	C₄×D₄	1	2	yes	—
161	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
163	#33455	—	47	1	no	maximal, Sylow
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	2	1	no	—
166	C₂²×C₄₇	C₄×D₈	1	2	yes	—
167	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₄₇	—	4	2	no	—
169	C₂×D₄₇	—	8	1	no	—
170	C₂×D₄₇	—	8	1	no	—
171	C₄×C₄₇	—	2	2	no	—
172	C₄×C₄₇	—	2	2	no	—
173	C₄×C₄₇	—	4	4	no	—
174	C₄×C₄₇	—	8	1	no	—
175	C₄×C₄₇	—	8	1	no	—
176	C₄×C₄₇	C₄×D₈	1	2	yes	—
177	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
179	Dic₄₇	—	4	2	no	—
180	Dic₄₇	—	8	2	no	—
181	C₂×#510	—	47	1	no	—
182	#2951	—	47	1	no	—
183	#2950	D₄₇	1	1	yes	characteristic, complemented
184	#2176	—	47	2	no	—
185	#2175	—	47	2	no	—
186	C₄×Q₃₂	—	47	2	no	—
187	C₄×SD₃₂	—	47	2	no	—
188	#2159	—	47	1	no	—
189	#2138	—	47	1	no	—
190	C₂×#358	—	47	1	no	—
191	#2108	—	47	1	no	—
192	C₂×C₄₇	—	2	2	no	—
193	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
194	C₂×C₄₇	#3416	1	1	yes	characteristic
195	C₂×C₄₇	#3417	1	1	yes	characteristic
196	D₄₇	—	8	2	no	—
197	C₂×#135	—	47	1	no	—
198	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
199	#520	—	94	1	no	—
200	#519	—	2	1	no	—
201	#510	—	47	8	no	—
202	C₂×Q₃₂	—	47	2	no	—
203	C₂×SD₃₂	—	47	2	no	—
204	C₂×M₃₂	—	47	1	no	—
205	C₂²×C₁₆	—	47	1	no	—
206	#441	—	2	2	no	—
207	#440	—	94	2	no	—
208	C₄×Q₁₆	—	47	2	no	—
209	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
210	C₄×Q₁₆	C₂×D₄₇	1	2	yes	—
211	C₄×D₈	—	47	2	no	—
212	#427	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#107	—	47	1	no	—
214	#419	—	2	1	no	—
215	#417	—	94	1	no	—
216	#405	C₂×D₄₇	1	1	yes	characteristic
217	#367	—	47	2	no	—
218	#366	—	47	2	no	—
219	#358	—	47	2	no	—
220	#358	—	47	2	no	—
221	#358	—	47	1	no	—
222	#358	—	47	1	no	—
223	#357	—	47	1	no	—
224	#357	—	47	1	no	—
225	#346	—	47	2	no	—
226	C₄×C₁₆	—	47	2	no	—
227	C₄₇	#33455	1	1	yes	characteristic, complemented, Sylow
228	C₂×#41	—	94	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	47	4	no	—
231	#135	—	94	2	no	—
232	C₂×Q₁₆	—	2	2	no	—
233	C₂×Q₁₆	—	2	1	no	—
234	C₂×Q₁₆	—	47	1	no	—
235	C₂×Q₁₆	C₄×D₄₇	1	4	yes	complemented
236	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
237	C₂×SD₁₆	—	94	1	no	—
238	C₂×D₈	—	47	1	no	—
239	C₂×M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₄×Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	4	1	no	—
244	C₄×Q₈	—	4	1	no	—
245	C₄×Q₈	—	94	2	no	—
246	C₄×D₄	—	94	2	no	—
247	C₄×D₄	—	188	2	no	—
248	#117	—	2	1	no	—
249	#117	—	4	2	no	—
250	#117	—	188	1	no	—
251	#117	C₂²⋊D₄₇	1	1	yes	characteristic
252	C₂×C₄⋊C₄	—	94	1	no	—
253	C₂×C₄⋊C₄	—	188	1	no	—
254	Q₃₂	—	47	8	no	—
255	SD₃₂	—	47	8	no	—
256	M₃₂	—	47	4	no	—
257	C₂×C₁₆	—	47	2	no	—
258	C₂×C₁₆	—	47	1	no	—
259	C₂×C₁₆	—	47	1	no	—
260	C₂×C₁₆	—	94	2	no	—
261	#107	—	47	2	no	—
262	#107	—	94	1	no	—
263	#107	C₂²×D₄₇	1	1	yes	characteristic
264	#107	C₂²×D₄₇	1	1	yes	characteristic
265	#106	—	2	1	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	94	2	no	—
269	#102	—	94	2	no	—
270	#97	C₂²⋊D₄₇	1	2	yes	—
271	C₄×C₈	C₂²×D₄₇	1	2	yes	—
272	#41	—	94	4	no	—
273	#41	—	188	2	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	94	1	no	—
279	C₂×D₄	—	94	1	no	—
280	C₂×D₄	—	188	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	188	1	no	—
283	C₂²×C₄	—	188	1	no	—
284	C₂²×C₄	—	188	2	no	—
285	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
286	Q₁₆	—	2	4	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	47	2	no	—
289	Q₁₆	—	94	1	no	—
290	Q₁₆	C₂×C₄×D₄₇	1	4	yes	—
291	SD₁₆	—	94	4	no	—
292	D₈	—	47	2	no	—
293	D₈	—	94	1	no	—
294	M₁₆	—	2	2	no	—
295	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
296	C₂×C₈	#10625350	1	2	yes	—
297	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
298	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	188	2	no	—
306	C₄⋊C₄	—	188	1	no	—
307	C₄⋊C₄	#10625334	1	1	yes	characteristic
308	C₄⋊C₄	#10625334	1	1	yes	characteristic
309	C₂²⋊C₄	—	4	1	no	—
310	C₂²⋊C₄	—	188	2	no	—
311	C₂²⋊C₄	#10625334	1	2	yes	—
312	C₄²	—	4	1	no	—
313	C₄²	—	4	2	no	—
314	C₄²	—	4	2	no	—
315	C₄²	—	188	2	no	—
316	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
317	C₁₆	—	47	4	no	—
318	C₁₆	—	94	1	no	—
319	C₁₆	—	94	1	no	—
320	C₂³	—	188	1	no	—
321	C₂³	C₂×#10625334	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	94	2	no	—
326	Q₈	—	188	1	no	—
327	D₄	—	94	2	no	—
328	D₄	—	188	4	no	—
329	D₄	—	188	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	188	2	no	—
335	C₂×C₄	—	188	1	no	—
336	C₂×C₄	—	188	4	no	—
337	C₂×C₄	—	752	2	no	—
338	C₂×C₄	—	752	1	no	—
339	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
340	C₂×C₄	C₂×#10625334	1	2	yes	—
341	C₂×C₄	C₂×#10625334	1	2	yes	—
342	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
343	C₂×C₄	#13094044	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
347	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
348	C₈	C₂×#10625350	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	376	1	no	—
352	C₂²	—	376	1	no	—
353	C₂²	—	188	2	no	—
354	C₂²	C₄×#10625334	1	2	yes	—
355	C₂²	C₂²×#10625334	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	4	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	376	2	no	—
362	C₄	—	188	2	no	—
363	C₄	C₄×#10625334	1	2	yes	—
364	C₄	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
365	C₄	C₂×#13094044	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	376	2	no	—
368	C₂	C₂×C₄×#10625334	1	1	yes	characteristic
369	C₂	#20256532	1	1	yes	characteristic
370	C₂	#20256539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2950	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×#13094046	C₂²
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2950	D₄₇
hypocenter	C₄₇	#33455
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2950

Complements

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

N	H ≅ G/N	type
3. #20245257	C₂	semidirect
4. #20245250	C₂	semidirect
5. C₄×#13093970	C₂	semidirect
6. C₄×#13093967	C₂	semidirect
7. #20245177	C₂	semidirect
9. C₂×#15688927	C₂	semidirect
10. #20244985	C₂	semidirect
11. C₄₇×#2950	C₂	semidirect
14. #15689564	C₄	semidirect
54. #13093970	C₂×C₄	semidirect
55. #13093967	C₂×C₄	semidirect
56. #13093964	C₂×C₄	semidirect
68. C₂×Q₁₆×C₄₇	C₂×C₄	semidirect
183. #2950	D₄₇	semidirect
227. C₄₇	#33455	semidirect
235. C₂×Q₁₆	C₄×D₄₇	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
#20252265	2
#20245257	2
#20245250	2
C₄×#13093970	2
C₄×#13093967	2
#20245177	2
#20245093	2
C₂×#15688927	2
#20244985	2
C₄₇×#2950	2
#33455	47