Subgroups — #26186995

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253947	C₂	1	1	yes	maximal, characteristic
2	#20252322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20252101	C₂	1	1	yes	maximal, characteristic
4	#20242233	C₂	1	4	yes	maximal, complemented
5	#20242226	C₂	1	4	yes	maximal, complemented
6	#20242144	C₂	1	1	yes	maximal, characteristic, complemented
7	#20241844	C₂	1	1	yes	maximal, characteristic, complemented
8	#20241840	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13094030	—	2	1	no	—
11	#15689680	C₂²	1	1	yes	characteristic
12	#15689673	C₂²	1	1	yes	characteristic
13	#15689521	C₄	1	1	yes	characteristic
14	#15689496	C₄	1	1	yes	characteristic
15	#15689474	C₄	1	1	yes	characteristic
16	#15689470	C₄	1	1	yes	characteristic
17	#15689352	C₄	1	4	yes	—
18	#15689253	—	2	1	no	—
19	#15689253	C₂²	1	2	yes	—
20	C₈×Dic₉₄	—	2	1	no	—
21	C₈×Dic₉₄	C₂²	1	2	yes	—
22	C₄×#10625325	C₂²	1	4	yes	—
23	#15689202	—	2	1	no	—
24	#15689202	C₂²	1	2	yes	—
25	C₈×D₁₈₈	—	2	1	no	—
26	C₈×D₁₈₈	C₂²	1	2	yes	—
27	#15689149	—	2	1	no	—
28	#15689147	C₂²	1	1	yes	characteristic
29	#15689147	C₂²	1	1	yes	characteristic
30	#15689089	—	2	1	no	—
31	#15689078	—	2	1	no	—
32	#15688874	C₂²	1	4	yes	—
33	#15688855	C₂²	1	4	yes	—
34	#15688852	C₂²	1	4	yes	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#322	C₂²	1	2	yes	—
39	C₄₇×#322	C₂²	1	2	yes	—
40	C₂×#10625350	—	2	1	no	—
41	C₂×#10625325	C₂×C₄	1	4	yes	—
42	#13094030	—	4	4	no	—
43	C₂×#10625329	—	2	1	no	—
44	C₂×#10625329	—	2	1	no	—
45	C₂×#10625324	—	2	2	no	—
46	C₂×C₈×D₄₇	—	2	2	no	—
47	#13094024	C₂×C₄	1	1	yes	characteristic
48	#13094024	C₂×C₄	1	1	yes	characteristic
49	#13094024	C₂×C₄	1	1	yes	characteristic
50	#13094021	C₂×C₄	1	1	yes	characteristic
51	#13094016	—	2	1	no	—
52	#13094012	—	2	1	no	—
53	#13094003	—	2	1	no	—
54	#13094003	C₂×C₄	1	2	yes	—
55	#13093998	—	2	1	no	—
56	#13093998	C₂×C₄	1	2	yes	—
57	C₄×Dic₉₄	—	2	1	no	—
58	C₄×Dic₉₄	C₂³	1	2	yes	—
59	C₄×D₁₈₈	—	2	1	no	—
60	C₄×D₁₈₈	C₂³	1	2	yes	—
61	#13093983	—	2	1	no	—
62	#13093956	C₂×C₄	1	4	yes	—
63	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
64	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	#13093949	C₂×C₄	1	4	yes	—
70	#13093946	C₂×C₄	1	4	yes	—
71	(C₂×D₄₇)⋊C₈	—	2	2	no	—
72	(C₂×D₄₇)⋊C₈	—	2	2	no	—
73	#13093934	—	2	2	no	—
74	C₈×Dic₄₇	—	2	2	no	—
75	C₂²×C₈×C₄₇	—	2	1	no	—
76	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₄₇	—	2	1	no	—
81	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
84	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
85	C₄×C₈×C₄₇	C₂³	1	2	yes	—
86	C₄×C₈×C₄₇	C₂³	1	2	yes	—
87	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
92	C₂×C₂²⋊D₄₇	—	2	2	no	—
93	#10625350	—	4	2	no	—
94	#10625350	—	4	2	no	—
95	C₂×D₁₈₈	—	2	1	no	—
96	C₂×D₁₈₈	—	2	1	no	—
97	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₄₇	—	2	2	no	—
99	C₂×Dic₉₄	—	2	1	no	—
100	C₂×Dic₉₄	—	2	1	no	—
101	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	4	2	no	—
105	C₂×C₈×C₄₇	—	4	2	no	—
106	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
107	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₇	#41	1	1	yes	characteristic
116	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
117	C₂²⋊Dic₄₇	—	2	2	no	—
118	(C₂×D₄₇)⋊C₄	—	2	2	no	—
119	(C₂×D₄₇)⋊C₄	—	2	2	no	—
120	C₄⋊Dic₄₇	—	2	1	no	—
121	C₄⋊Dic₄₇	—	2	1	no	—
122	C₄⋊Dic₄₇	—	2	1	no	—
123	C₄⋊Dic₄₇	—	2	1	no	—
124	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
125	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
126	Dic₄₇⋊C₄	—	2	2	no	—
127	Dic₄₇⋊C₄	—	2	2	no	—
128	C₄×Dic₄₇	—	2	2	no	—
129	#10625329	—	4	2	no	—
130	#10625329	—	4	2	no	—
131	C₂×C₄₇⋊C₈	—	2	2	no	—
132	C₂×C₄₇⋊C₈	—	2	2	no	—
133	#10625325	—	4	4	no	—
134	#10625324	—	4	4	no	—
135	C₈×D₄₇	—	4	4	no	—
136	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₄₇	#135	1	2	yes	—
145	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	4	4	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	2	no	—
151	C₂×Dic₄₇	—	2	2	no	—
152	D₁₈₈	—	2	4	no	—
153	D₁₈₈	—	4	2	no	—
154	D₁₈₈	—	4	2	no	—
155	C₄×D₄₇	—	4	2	no	—
156	C₄×D₄₇	—	4	2	no	—
157	Dic₉₄	—	2	4	no	—
158	Dic₉₄	—	4	2	no	—
159	Dic₉₄	—	4	2	no	—
160	C₈×C₄₇	—	2	2	no	—
161	C₈×C₄₇	—	2	2	no	—
162	C₈×C₄₇	—	2	1	no	—
163	C₈×C₄₇	—	2	1	no	—
164	C₈×C₄₇	—	4	2	no	—
165	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
166	C₈×C₄₇	C₂×SD₁₆	1	2	yes	—
167	C₄₇⋊C₈	—	4	2	no	—
168	C₄₇⋊C₈	—	4	2	no	—
169	#17976	—	47	1	no	maximal, Sylow
170	C₂²×C₄₇	—	2	1	no	—
171	C₂²×C₄₇	—	2	1	no	—
172	C₂²×C₄₇	C₄×SD₁₆	1	2	yes	—
173	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₄₇	—	4	2	no	—
175	C₂×D₄₇	—	4	2	no	—
176	C₂×D₄₇	—	4	2	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
180	C₄×C₄₇	#444	1	2	yes	—
181	C₄×C₄₇	#568	1	2	yes	—
182	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
184	Dic₄₇	—	4	2	no	—
185	Dic₄₇	—	4	2	no	—
186	Dic₄₇	—	4	2	no	—
187	#3167	—	47	1	no	—
188	#2960	—	47	1	no	—
189	#2931	—	47	1	no	—
190	#1583	—	47	4	no	—
191	#1582	—	47	4	no	—
192	#1566	—	47	1	no	—
193	#1493	—	47	1	no	—
194	#1492	—	47	1	no	—
195	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
196	C₂×C₄₇	—	2	2	no	—
197	C₂×C₄₇	#2932	1	1	yes	characteristic
198	C₂×C₄₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
199	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
200	D₄₇	—	8	2	no	—
201	C₂×#131	—	94	1	no	—
202	#533	—	47	1	no	—
203	#532	—	47	1	no	—
204	#498	—	47	1	no	—
205	#492	—	47	1	no	—
206	#487	—	47	1	no	—
207	#486	—	47	1	no	—
208	#465	—	47	4	no	—
209	#446	—	47	2	no	—
210	#446	—	94	1	no	—
211	C₈×Q₈	—	47	2	no	—
212	C₈×Q₈	—	94	1	no	—
213	C₄×SD₁₆	—	47	4	no	—
214	#436	—	47	2	no	—
215	#436	—	94	1	no	—
216	C₈×D₄	—	47	2	no	—
217	C₈×D₄	—	94	1	no	—
218	#424	—	94	1	no	—
219	#423	—	47	1	no	—
220	#423	—	47	1	no	—
221	#408	—	94	1	no	—
222	#405	—	94	1	no	—
223	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
226	#334	—	47	4	no	—
227	Q₈⋊C₈	—	47	4	no	—
228	D₄⋊C₈	—	47	4	no	—
229	#322	C₂×D₄₇	1	2	yes	—
230	#322	C₂×D₄₇	1	2	yes	—
231	C₄₇	#17976	1	1	yes	characteristic, complemented, Sylow
232	C₂×#41	—	94	1	no	—
233	C₂×SD₁₆	—	47	4	no	—
234	#131	—	188	4	no	—
235	C₂×M₁₆	—	94	1	no	—
236	C₂×M₁₆	—	94	1	no	—
237	C₂×M₁₆	—	94	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	94	2	no	—
240	C₂²×C₈	C₄×D₄₇	1	2	yes	—
241	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
242	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
243	C₄⋊Q₈	—	47	1	no	—
244	C₄⋊Q₈	—	47	1	no	—
245	C₄⋊Q₈	—	47	1	no	—
246	#127	—	47	1	no	—
247	#125	—	94	1	no	—
248	#124	—	94	1	no	—
249	C₂²⋊Q₈	—	47	2	no	—
250	C₂²⋊Q₈	—	94	1	no	—
251	#121	—	47	2	no	—
252	#121	—	94	1	no	—
253	C₄×Q₈	—	47	2	no	—
254	C₄×Q₈	—	94	1	no	—
255	C₄×D₄	—	47	2	no	—
256	C₄×D₄	—	94	1	no	—
257	#117	—	94	1	no	—
258	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
259	#106	—	47	4	no	—
260	C₄⋊C₈	—	47	2	no	—
261	C₄⋊C₈	—	47	2	no	—
262	C₄⋊C₈	—	94	2	no	—
263	C₄⋊C₈	—	94	2	no	—
264	C₄⋊C₈	—	94	1	no	—
265	C₄⋊C₈	—	94	1	no	—
266	#103	—	47	4	no	—
267	#102	—	47	4	no	—
268	C₂²⋊C₈	—	94	2	no	—
269	C₂²⋊C₈	—	94	2	no	—
270	#97	—	94	2	no	—
271	C₄×C₈	—	2	1	no	—
272	C₄×C₈	—	94	2	no	—
273	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
274	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
275	C₄×C₈	D₁₈₈	1	1	yes	characteristic
276	C₄×C₈	D₁₈₈	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₄₇	1	2	yes	—
278	C₄×C₈	C₂²×D₄₇	1	2	yes	—
279	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
281	#41	—	188	2	no	—
282	#41	—	188	2	no	—
283	C₂×Q₈	—	47	2	no	—
284	C₂×Q₈	—	94	1	no	—
285	C₂×Q₈	—	94	1	no	—
286	C₂×D₄	—	47	2	no	—
287	C₂×D₄	—	94	2	no	—
288	C₂×D₄	—	94	1	no	—
289	C₂×D₄	—	94	1	no	—
290	C₂²×C₄	—	94	2	no	—
291	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
292	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
293	C₂²×C₄	#10625350	1	1	yes	characteristic
294	SD₁₆	—	188	4	no	—
295	M₁₆	—	188	2	no	—
296	M₁₆	—	188	2	no	—
297	M₁₆	—	188	4	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	94	2	no	—
303	C₂×C₈	—	94	2	no	—
304	C₂×C₈	—	188	4	no	—
305	C₂×C₈	#10625325	1	4	yes	complemented
306	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
307	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
308	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
309	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
310	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
311	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
312	C₄⋊C₄	—	47	2	no	—
313	C₄⋊C₄	—	47	2	no	—
314	C₄⋊C₄	—	94	2	no	—
315	C₄⋊C₄	—	94	1	no	—
316	C₄⋊C₄	—	94	1	no	—
317	C₄⋊C₄	—	94	1	no	—
318	C₄⋊C₄	—	94	1	no	—
319	C₄⋊C₄	—	94	2	no	—
320	C₂²⋊C₄	—	94	2	no	—
321	C₂²⋊C₄	—	94	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₄²	—	94	2	no	—
324	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
325	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
326	C₄²	#10625350	1	1	yes	characteristic
327	C₄²	C₂³×D₄₇	1	1	yes	characteristic
328	C₂³	—	94	2	no	—
329	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
330	Q₈	—	94	4	no	—
331	Q₈	—	188	2	no	—
332	Q₈	—	188	2	no	—
333	D₄	—	94	4	no	—
334	D₄	—	188	4	no	—
335	D₄	—	188	2	no	—
336	D₄	—	188	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	94	2	no	—
340	C₂×C₄	—	94	2	no	—
341	C₂×C₄	—	94	2	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	—	188	2	no	—
344	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
345	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
346	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
347	C₂×C₄	C₂×#10625325	1	2	yes	—
348	C₂×C₄	#13094046	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
351	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	2	2	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	188	2	no	—
358	C₈	—	188	2	no	—
359	C₈	C₂×#10625325	1	2	yes	—
360	C₈	C₂×#10625325	1	2	yes	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	188	2	no	—
364	C₂²	—	188	2	no	—
365	C₂²	—	188	2	no	—
366	C₂²	C₄×#10625325	1	2	yes	—
367	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	188	2	no	—
371	C₄	—	188	2	no	—
372	C₄	—	188	2	no	—
373	C₄	C₄×#10625325	1	2	yes	—
374	C₄	#15689246	1	2	yes	—
375	C₄	#15689889	1	2	yes	—
376	C₄	C₂²×#10625325	1	1	yes	characteristic
377	C₄	C₂×#13094046	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	376	2	no	—
380	C₂	#20252107	1	1	yes	characteristic
381	C₂	C₂×C₄×#10625325	1	1	yes	characteristic
382	C₂	C₂×#15689246	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	#15689673	C₂²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#17976
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #20252322	C₂	semidirect
4. #20242233	C₂	semidirect
5. #20242226	C₂	semidirect
6. #20242144	C₂	semidirect
7. #20241844	C₂	semidirect
9. C₂×C₄₇×#322	C₂	semidirect
106. C₂×C₈×C₄₇	SD₁₆	semidirect
195. C₂×#322	D₄₇	semidirect
231. C₄₇	#17976	semidirect
305. C₂×C₈	#10625325	semidirect

Maximal subgroups

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

subgroup	index
#20253947	2
#20252322	2
#20252101	2
#20242233	2
#20242226	2
#20242144	2
#20241844	2
#20241840	2
C₂×C₄₇×#322	2
#17976	47