Subgroups — #26186418

Order 12032.

638 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253397	C₂	1	1	yes	maximal, characteristic
2	#20252485	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252411	C₂	1	1	yes	maximal, characteristic
4	#20242244	C₂	1	2	yes	maximal, complemented
5	#20242223	C₂	1	2	yes	maximal, complemented
6	#20242199	C₂	1	2	yes	maximal, complemented
7	#20242187	C₂	1	2	yes	maximal, complemented
8	#20241830	C₂	1	1	yes	maximal, characteristic
9	#20241808	C₂	1	1	yes	maximal, characteristic, complemented
10	#20241683	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1572	C₂	1	1	yes	maximal, characteristic, complemented
12	#15689680	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
14	#15689454	C₄	1	2	yes	—
15	#15689414	C₄	1	2	yes	—
16	#15689329	C₄	1	2	yes	—
17	#15689276	C₄	1	2	yes	—
18	#15689251	—	2	2	no	—
19	#15689250	C₂²	1	2	yes	complemented
20	#15689247	C₂²	1	2	yes	complemented
21	C₄×#10625325	C₂²	1	2	yes	—
22	C₄×#10625325	C₂²	1	2	yes	—
23	C₈×D₁₈₈	—	2	2	no	—
24	#15689188	C₂²	1	2	yes	—
25	#15689188	C₂²	1	2	yes	—
26	#15689181	—	2	2	no	—
27	#15689176	C₂²	1	1	yes	characteristic
28	C₂²⋊C₈×D₄₇	—	2	2	no	—
29	C₄×#10625329	C₂²	1	1	yes	characteristic
30	#15688853	C₂²	1	2	yes	complemented
31	#15688853	C₂²	1	2	yes	complemented
32	#15688850	C₂²	1	2	yes	—
33	#15688850	C₂²	1	2	yes	—
34	#15688845	C₂²	1	1	yes	characteristic, complemented
35	#15688845	C₂²	1	1	yes	characteristic, complemented
36	#15688844	C₂²	1	1	yes	characteristic, complemented
37	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#432	C₂²	1	1	yes	characteristic
39	C₄₇×#431	C₂²	1	1	yes	characteristic
40	C₄₇×#431	C₂²	1	1	yes	characteristic
41	C₄₇×#335	C₂²	1	2	yes	complemented
42	C₄₇×#334	C₂²	1	2	yes	complemented
43	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×D₄₇	—	2	1	no	—
45	C₂×#10625325	C₂×C₄	1	2	yes	—
46	C₂×#10625325	C₂×C₄	1	2	yes	—
47	C₂×#10625329	—	2	1	no	—
48	C₂×C₈×D₄₇	—	2	2	no	—
49	C₂×C₈×D₄₇	—	2	2	no	—
50	#13094024	C₂×C₄	1	1	yes	characteristic
51	#13094024	C₂×C₄	1	1	yes	characteristic
52	#13094016	—	2	1	no	—
53	#13094003	—	2	1	no	—
54	#13094003	C₂×C₄	1	2	yes	—
55	C₄×Dic₉₄	—	2	1	no	—
56	C₄×Dic₉₄	C₂³	1	2	yes	—
57	C₄×D₁₈₈	—	2	1	no	—
58	C₄×D₁₈₈	—	2	1	no	—
59	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
60	C₄×D₁₈₈	C₂³	1	2	yes	—
61	#13093983	—	2	1	no	—
62	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
63	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
64	#13093956	C₂×C₄	1	2	yes	—
65	#13093956	C₂×C₄	1	2	yes	—
66	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	#13093949	C₂×C₄	1	2	yes	—
70	#13093949	C₂×C₄	1	2	yes	—
71	#13093946	C₂×C₄	1	2	yes	—
72	#13093946	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
74	(C₂×D₄₇)⋊C₈	—	2	2	no	—
75	(C₂×D₄₇)⋊C₈	—	2	2	no	—
76	#13093935	—	2	1	no	—
77	#13093934	—	2	2	no	—
78	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₄₇	—	2	2	no	—
83	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
85	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
86	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
87	C₂²⋊C₈×C₄₇	—	2	1	no	—
88	C₂²⋊C₈×C₄₇	—	2	1	no	—
89	C₂²⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
90	#97×C₄₇	—	2	1	no	—
91	C₄×C₈×C₄₇	—	2	1	no	—
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
94	C₂³×D₄₇	—	2	1	no	—
95	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
98	C₂²×Dic₄₇	—	2	1	no	—
99	C₂×D₁₈₈	—	2	1	no	—
100	C₂×D₁₈₈	—	2	4	no	—
101	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
102	C₂×C₄×D₄₇	—	2	2	no	—
103	C₂×C₄×D₄₇	—	2	2	no	—
104	C₂×C₄×D₄₇	—	8	2	no	—
105	C₂×Dic₉₄	—	2	1	no	—
106	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₄₇	—	2	2	no	—
108	C₂×C₈×C₄₇	—	2	2	no	—
109	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
111	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	#41	1	1	yes	characteristic
114	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
115	C₂²⋊Dic₄₇	—	2	2	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	(C₂×D₄₇)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	1	no	—
119	C₄⋊Dic₄₇	—	2	1	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	—	2	1	no	—
122	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
123	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
124	Dic₄₇⋊C₄	—	2	2	no	—
125	Dic₄₇⋊C₄	—	2	2	no	—
126	C₄×Dic₄₇	—	2	2	no	—
127	#10625329	—	4	2	no	—
128	C₂×C₄₇⋊C₈	—	2	1	no	—
129	C₂×C₄₇⋊C₈	—	2	1	no	—
130	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
131	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
132	#10625325	—	2	4	no	—
133	#10625325	—	2	4	no	—
134	C₈×D₄₇	—	8	2	no	—
135	C₈×D₄₇	—	8	2	no	—
136	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₄₇	—	2	2	no	—
138	C₂²×D₄₇	—	2	4	no	—
139	C₂²×D₄₇	—	8	1	no	—
140	C₂²×D₄₇	—	8	1	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	2	no	—
143	C₂×C₄×C₄₇	—	4	1	no	—
144	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	C₂×Dic₄₇	—	2	2	no	—
153	C₂×Dic₄₇	—	8	1	no	—
154	D₁₈₈	—	2	4	no	—
155	D₁₈₈	—	2	2	no	—
156	D₁₈₈	#131	1	4	yes	—
157	C₄×D₄₇	—	4	2	no	—
158	C₄×D₄₇	—	4	2	no	—
159	C₄×D₄₇	—	8	2	no	—
160	Dic₉₄	—	2	2	no	—
161	Dic₉₄	—	2	2	no	—
162	Dic₉₄	—	8	1	no	—
163	C₈×C₄₇	—	2	2	no	—
164	C₈×C₄₇	—	2	2	no	—
165	C₈×C₄₇	—	4	2	no	—
166	C₈×C₄₇	—	4	2	no	—
167	C₄₇⋊C₈	—	2	2	no	—
168	C₄₇⋊C₈	—	2	2	no	—
169	C₄₇⋊C₈	—	2	2	no	—
170	C₄₇⋊C₈	—	4	1	no	—
171	#17924	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	1	no	—
173	C₂²×C₄₇	—	2	1	no	—
174	C₂²×C₄₇	—	2	1	no	—
175	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₄₇	—	2	4	no	—
177	C₂×D₄₇	—	4	2	no	—
178	C₂×D₄₇	—	4	2	no	—
179	C₂×D₄₇	—	8	4	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	—	2	1	no	—
183	C₄×C₄₇	—	2	1	no	—
184	C₄×C₄₇	C₂×#131	1	2	yes	—
185	C₄×C₄₇	#577	1	2	yes	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	—	4	2	no	—
188	Dic₄₇	—	4	2	no	—
189	#3101	—	47	1	no	—
190	#2985	—	47	1	no	—
191	#2972	—	47	1	no	—
192	#1585	—	47	2	no	—
193	#1582	—	47	2	no	—
194	SD₁₆⋊C₈	—	47	2	no	—
195	C₈×SD₁₆	—	47	2	no	—
196	#1572	D₄₇	1	1	yes	characteristic, complemented
197	#1490	—	47	1	no	—
198	#1485	—	47	1	no	—
199	#1453	—	47	1	no	—
200	C₂×C₄₇	—	4	1	no	—
201	C₂×C₄₇	#2929	1	1	yes	characteristic
202	C₂×C₄₇	#2949	1	1	yes	characteristic
203	C₂×C₄₇	#2956	1	1	yes	characteristic
204	D₄₇	—	4	4	no	—
205	#533	—	47	1	no	—
206	C₂×C₄×D₄	—	47	1	no	—
207	#483	—	47	2	no	—
208	#476	—	47	2	no	—
209	#461	—	47	2	no	—
210	#450	—	47	2	no	—
211	#446	—	47	2	no	—
212	#446	—	94	2	no	—
213	C₈×Q₈	—	47	2	no	—
214	C₄×SD₁₆	—	47	2	no	—
215	C₄×SD₁₆	—	47	2	no	—
216	C₈×D₄	—	47	2	no	—
217	C₈×D₄	—	47	2	no	—
218	C₈×D₄	—	94	2	no	—
219	#433	—	94	2	no	—
220	#432	C₂×D₄₇	1	1	yes	characteristic
221	#431	—	47	1	no	—
222	#431	C₂×D₄₇	1	1	yes	characteristic
223	#431	C₂×D₄₇	1	1	yes	characteristic
224	C₂×C₂²⋊C₈	—	94	2	no	—
225	C₄×M₁₆	—	47	1	no	—
226	#335	C₂×D₄₇	1	2	yes	complemented
227	#334	C₂×D₄₇	1	2	yes	complemented
228	Q₈⋊C₈	—	47	2	no	—
229	Q₈⋊C₈	—	47	2	no	—
230	D₄⋊C₈	—	47	2	no	—
231	D₄⋊C₈	—	47	2	no	—
232	#322	—	47	1	no	—
233	#322	—	47	1	no	—
234	#322	—	47	1	no	—
235	C₈²	—	47	1	no	—
236	C₄₇	#17924	1	1	yes	characteristic, complemented, Sylow
237	C₂²×D₄	—	47	1	no	—
238	C₂³×C₄	—	94	1	no	—
239	C₂×SD₁₆	—	47	2	no	—
240	C₂×SD₁₆	—	47	2	no	—
241	C₂×M₁₆	—	94	1	no	—
242	C₂²×C₈	—	94	2	no	—
243	C₂²×C₈	—	94	2	no	—
244	C₄⋊Q₈	—	47	1	no	—
245	C₄⋊Q₈	—	47	1	no	—
246	#125	—	94	1	no	—
247	C₂²⋊Q₈	—	47	2	no	—
248	C₂²⋊Q₈	—	94	1	no	—
249	C₄×Q₈	—	47	2	no	—
250	C₄×Q₈	—	94	1	no	—
251	C₄×D₄	—	47	2	no	—
252	C₄×D₄	—	47	2	no	—
253	C₄×D₄	—	94	1	no	—
254	C₄×D₄	—	94	1	no	—
255	#117	—	94	1	no	—
256	C₂×C₄⋊C₄	—	47	1	no	—
257	C₂×C₂²⋊C₄	—	94	1	no	—
258	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
259	#106	—	47	2	no	—
260	#106	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	94	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
265	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
266	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
267	#103	—	47	2	no	—
268	#103	—	47	2	no	—
269	#102	—	47	2	no	—
270	#102	—	47	2	no	—
271	C₂²⋊C₈	—	2	1	no	—
272	C₂²⋊C₈	—	2	1	no	—
273	C₂²⋊C₈	—	47	2	no	—
274	C₂²⋊C₈	—	94	2	no	—
275	C₂²⋊C₈	—	94	2	no	—
276	C₂²⋊C₈	C₄×D₄₇	1	2	yes	—
277	#97	—	2	1	no	—
278	#97	—	94	2	no	—
279	#97	—	94	1	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	—	47	1	no	—
284	C₄×C₈	—	47	1	no	—
285	C₄×C₈	—	94	2	no	—
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
288	C₂⁴	—	94	1	no	—
289	C₂×Q₈	—	47	2	no	—
290	C₂×Q₈	—	94	1	no	—
291	C₂×D₄	—	47	2	no	—
292	C₂×D₄	—	94	4	no	—
293	C₂×D₄	—	94	1	no	—
294	C₂²×C₄	—	94	1	no	—
295	C₂²×C₄	—	94	2	no	—
296	C₂²×C₄	—	94	2	no	—
297	C₂²×C₄	—	376	2	no	—
298	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
299	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
300	C₂²×C₄	#10625354	1	1	yes	characteristic
301	SD₁₆	—	94	4	no	—
302	SD₁₆	—	94	4	no	—
303	M₁₆	—	188	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	47	2	no	—
307	C₂×C₈	—	47	2	no	—
308	C₂×C₈	—	94	1	no	—
309	C₂×C₈	—	94	1	no	—
310	C₂×C₈	—	376	2	no	—
311	C₂×C₈	—	376	2	no	—
312	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
313	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
314	C₄⋊C₄	—	47	2	no	—
315	C₄⋊C₄	—	47	2	no	—
316	C₄⋊C₄	—	94	2	no	—
317	C₄⋊C₄	—	94	1	no	—
318	C₄⋊C₄	—	94	1	no	—
319	C₄⋊C₄	—	94	1	no	—
320	C₄⋊C₄	—	94	1	no	—
321	C₄⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₂²⋊C₄	—	94	2	no	—
324	C₂²⋊C₄	—	94	2	no	—
325	C₄²	—	94	2	no	—
326	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
327	C₄²	D₄×D₄₇	1	1	yes	characteristic
328	C₄²	#10625354	1	1	yes	characteristic
329	C₄²	C₂³×D₄₇	1	1	yes	characteristic
330	C₂³	—	94	4	no	—
331	C₂³	—	94	2	no	—
332	C₂³	—	376	1	no	—
333	C₂³	—	376	1	no	—
334	C₂³	#13093986	1	1	yes	characteristic
335	Q₈	—	94	2	no	—
336	Q₈	—	94	2	no	—
337	Q₈	—	376	1	no	—
338	D₄	—	47	4	no	—
339	D₄	—	94	4	no	—
340	D₄	—	94	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	2	no	—
346	C₂×C₄	—	94	2	no	—
347	C₂×C₄	—	376	2	no	—
348	C₂×C₄	—	376	1	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	#13093986	1	1	yes	characteristic
352	C₂×C₄	#13093986	1	1	yes	characteristic
353	C₂×C₄	#13093986	1	1	yes	characteristic
354	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
356	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	94	2	no	—
362	C₈	—	94	2	no	—
363	C₈	—	94	2	no	—
364	C₈	—	188	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	94	4	no	—
369	C₂²	—	376	4	no	—
370	C₂²	—	188	2	no	—
371	C₂²	—	188	2	no	—
372	C₂²	C₂×#13093986	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	188	2	no	—
378	C₄	—	188	2	no	—
379	C₄	—	188	2	no	—
380	C₄	#131×D₄₇	1	2	yes	—
381	C₄	#15689930	1	2	yes	—
382	C₂	—	4	1	no	—
383	C₂	—	188	4	no	—
384	C₂	#20252081	1	1	yes	characteristic
385	C₂	#20252246	1	1	yes	characteristic
386	C₂	#20252297	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1572	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1572	D₄₇
hypocenter	C₄₇	#17924
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1572

Complements

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

N	H ≅ G/N	type
2. #20252485	C₂	semidirect
4. #20242244	C₂	semidirect
5. #20242223	C₂	semidirect
6. #20242199	C₂	semidirect
7. #20242187	C₂	semidirect
9. #20241808	C₂	semidirect
10. #20241683	C₂	semidirect
11. C₄₇×#1572	C₂	semidirect
19. #15689250	C₂²	semidirect
20. #15689247	C₂²	semidirect
30. #15688853	C₂²	semidirect
31. #15688853	C₂²	semidirect
34. #15688845	C₂²	semidirect
35. #15688845	C₂²	semidirect
36. #15688844	C₂²	semidirect
37. C₈×C₄₇⋊C₈	C₂²	semidirect
41. C₄₇×#335	C₂²	semidirect
42. C₄₇×#334	C₂²	semidirect
196. #1572	D₄₇	semidirect
226. #335	C₂×D₄₇	semidirect
227. #334	C₂×D₄₇	semidirect
236. C₄₇	#17924	semidirect

Maximal subgroups

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

subgroup	index
#20253397	2
#20252485	2
#20252411	2
#20242244	2
#20242223	2
#20242199	2
#20242187	2
#20241830	2
#20241808	2
#20241683	2
C₄₇×#1572	2
#17924	47