Subgroups — #26188023

Order 12032.

652 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	#20252511	C₂	1	2	yes	maximal, complemented
2	C₄×#13094043	C₂	1	1	yes	maximal, characteristic
3	#20242298	C₂	1	1	yes	maximal, characteristic, complemented
4	#20242296	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242224	C₂	1	2	yes	maximal, complemented
6	#20242221	C₂	1	2	yes	maximal, complemented
7	#20242212	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242142	C₂	1	1	yes	maximal, characteristic, complemented
9	#20241824	C₂	1	2	yes	maximal, complemented
10	#20241674	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1580	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094043	C₄	1	1	yes	characteristic
13	C₄×#10625352	C₂²	1	2	yes	—
14	#15689253	C₂²	1	2	yes	—
15	#15689250	C₂²	1	2	yes	—
16	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
17	#15689217	C₄	1	1	yes	characteristic
18	#15689214	C₄	1	2	yes	—
19	C₄×#10625335	C₂²	1	2	yes	—
20	D₈×Dic₄₇	C₄	1	1	yes	characteristic
21	#15689194	—	2	2	no	—
22	#15689191	—	2	2	no	—
23	D₄×C₄₇⋊C₈	C₂²	1	2	yes	—
24	#15689177	—	2	2	no	—
25	#15689174	C₂²	1	1	yes	characteristic
26	#15689160	C₄	1	1	yes	characteristic
27	#15689145	C₂²	1	2	yes	—
28	C₂×Dic₄₇⋊C₈	—	2	2	no	—
29	#15689116	C₄	1	2	yes	—
30	C₄×#10625324	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
32	#15688874	C₂²	1	1	yes	characteristic
33	#15688873	C₂²	1	2	yes	—
34	#15688855	C₂²	1	2	yes	—
35	#15688853	C₂²	1	2	yes	—
36	D₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
37	#15688845	C₂²	1	1	yes	characteristic
38	#15688844	C₂²	1	1	yes	characteristic
39	#15688844	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₄₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₄₇	C₂²	1	2	yes	—
42	C₄₇×#334	C₂²	1	1	yes	characteristic
43	C₄₇×D₄⋊C₈	C₂²	1	2	yes	—
44	C₄₇×#322	C₂²	1	1	yes	characteristic
45	C₂×#10625352	C₂×C₄	1	2	yes	—
46	#13094043	—	2	4	no	—
47	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
48	C₂×#10625335	C₂×C₄	1	2	yes	—
49	C₂×#10625324	—	2	1	no	—
50	C₂×#10625324	—	2	2	no	—
51	C₂²×C₄₇⋊C₈	—	2	2	no	—
52	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₄₇	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₉₄	C₂³	1	2	yes	—
55	#13093989	C₂×C₄	1	2	yes	—
56	D₄×Dic₄₇	C₂×C₄	1	2	yes	—
57	C₄×C₂²⋊D₄₇	—	2	2	no	—
58	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
59	#13093982	C₂×C₄	1	2	yes	—
60	#13093980	—	2	2	no	—
61	C₂×C₄×Dic₄₇	—	2	2	no	—
62	C₄²×D₄₇	C₂³	1	1	yes	characteristic
63	#13093958	C₂×C₄	1	1	yes	characteristic
64	#13093955	C₂×C₄	1	2	yes	—
65	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
66	Dic₄₇⋊C₈	—	2	2	no	—
67	Dic₄₇⋊C₈	—	2	2	no	—
68	Dic₄₇⋊C₈	—	2	2	no	—
69	Dic₄₇⋊C₈	D₄	1	2	yes	—
70	#13093949	C₂×C₄	1	2	yes	—
71	#13093947	C₂×C₄	1	2	yes	—
72	#13093945	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
74	(C₂×D₄₇)⋊C₈	D₄	1	2	yes	—
75	#13093934	—	2	2	no	—
76	#13093934	—	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	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₄₇	C₂×C₄	1	1	yes	characteristic
83	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₄₇	—	2	2	no	—
85	C₄×D₄×C₄₇	C₂³	1	2	yes	—
86	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
88	#102×C₄₇	C₂×C₄	1	2	yes	—
89	C₂²⋊C₈×C₄₇	—	2	2	no	—
90	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₄₇	—	2	2	no	—
95	C₂×C₂²⋊D₄₇	—	2	2	no	—
96	C₂²×Dic₄₇	—	2	2	no	—
97	C₂²×Dic₄₇	—	2	2	no	—
98	#10625352	—	2	4	no	—
99	#10625352	—	4	2	no	—
100	C₂×C₄×D₄₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₄₇	#41	1	1	yes	characteristic
103	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
104	D₈×C₄₇	M₁₆	1	4	yes	complemented
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	—	2	1	no	—
107	C₂×C₈×C₄₇	—	8	2	no	—
108	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
111	C₄⋊C₄×C₄₇	C₂²×C₄	1	2	yes	—
112	C₂²⋊C₄×C₄₇	—	2	2	no	—
113	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
114	C₂²⋊Dic₄₇	—	2	2	no	—
115	#10625335	—	2	4	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
118	Dic₄₇⋊C₄	—	2	2	no	—
119	Dic₄₇⋊C₄	—	2	2	no	—
120	C₄×Dic₄₇	—	2	2	no	—
121	C₄×Dic₄₇	—	2	2	no	—
122	C₄×Dic₄₇	—	2	2	no	—
123	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₄₇	#41	1	1	yes	characteristic
126	C₂×C₄₇⋊C₈	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	1	no	—
128	C₂×C₄₇⋊C₈	—	8	2	no	—
129	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
132	Dic₁₈₈	—	2	2	no	—
133	#10625324	—	4	2	no	—
134	#10625324	—	8	2	no	—
135	C₈×D₄₇	—	2	2	no	—
136	C₈×D₄₇	—	4	1	no	—
137	C₂³×C₄₇	—	2	2	no	—
138	C₂²×D₄₇	C₄×D₄	1	1	yes	characteristic
139	D₄×C₄₇	—	2	2	no	—
140	D₄×C₄₇	C₂×M₁₆	1	4	yes	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	2	no	—
144	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₄₇	—	4	4	no	—
148	C₂×Dic₄₇	—	2	2	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	4	4	no	—
151	C₂×Dic₄₇	—	4	4	no	—
152	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
153	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₄₇	C₄×D₄	1	1	yes	characteristic
155	C₄×D₄₇	—	2	1	no	—
156	C₄×D₄₇	—	2	1	no	—
157	C₄×D₄₇	—	2	2	no	—
158	C₄×D₄₇	—	2	2	no	—
159	Dic₉₄	—	2	2	no	—
160	Dic₉₄	—	2	2	no	—
161	C₈×C₄₇	—	2	1	no	—
162	C₈×C₄₇	—	2	2	no	—
163	C₈×C₄₇	—	4	1	no	—
164	C₈×C₄₇	—	4	2	no	—
165	C₈×C₄₇	C₂×M₁₆	1	2	yes	—
166	C₄₇⋊C₈	—	2	1	no	—
167	C₄₇⋊C₈	—	2	1	no	—
168	C₄₇⋊C₈	—	2	2	no	—
169	C₄₇⋊C₈	—	4	1	no	—
170	C₄₇⋊C₈	—	4	2	no	—
171	#18065	—	47	1	no	maximal, Sylow
172	C₂²×C₄₇	—	2	4	no	—
173	C₂²×C₄₇	—	4	2	no	—
174	C₂²×C₄₇	—	4	2	no	—
175	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₄₇	—	2	1	no	—
177	C₂×D₄₇	—	2	1	no	—
178	C₂×D₄₇	—	2	1	no	—
179	C₄×C₄₇	—	2	1	no	—
180	C₄×C₄₇	—	4	2	no	—
181	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
182	C₄×C₄₇	#568	1	1	yes	characteristic
183	C₄×C₄₇	#577	1	2	yes	—
184	Dic₄₇	—	2	1	no	—
185	Dic₄₇	—	2	1	no	—
186	Dic₄₇	—	4	2	no	—
187	Dic₄₇	—	4	2	no	—
188	Dic₄₇	#444	1	2	yes	—
189	#2990	—	47	2	no	—
190	C₄×#135	—	47	1	no	—
191	#1593	—	47	1	no	—
192	#1593	—	47	1	no	—
193	#1582	—	47	2	no	—
194	#1582	—	47	2	no	—
195	#1580	—	47	1	no	—
196	#1580	D₄₇	1	1	yes	characteristic, complemented
197	#1566	—	47	1	no	—
198	#1489	—	47	2	no	—
199	#1451	—	47	1	no	—
200	C₂×C₄₇	—	4	4	no	—
201	C₂×C₄₇	D₄×M₁₆	1	1	yes	characteristic
202	C₂×C₄₇	#2952	1	1	yes	characteristic
203	C₂×C₄₇	C₂×#444	1	1	yes	characteristic
204	D₄₇	—	4	1	no	—
205	C₂×#135	—	47	1	no	—
206	C₄×#41	—	47	2	no	—
207	#446	—	47	2	no	—
208	#446	—	47	2	no	—
209	C₄×Q₁₆	—	47	1	no	—
210	C₄×Q₁₆	—	47	1	no	—
211	C₄×SD₁₆	—	47	2	no	—
212	C₄×SD₁₆	—	47	2	no	—
213	C₄×D₈	—	47	1	no	—
214	C₄×D₈	C₂×D₄₇	1	1	yes	characteristic
215	#435	—	94	2	no	—
216	#435	—	94	2	no	—
217	C₈×D₄	—	47	2	no	—
218	C₈×D₄	C₂×D₄₇	1	2	yes	—
219	#432	—	94	2	no	—
220	#431	—	47	1	no	—
221	#427	—	47	1	no	—
222	#423	—	47	2	no	—
223	C₂×C₄⋊C₈	—	94	2	no	—
224	#416	—	47	2	no	—
225	C₄×M₁₆	—	47	1	no	—
226	C₂×C₄×C₈	—	47	1	no	—
227	#334	—	47	2	no	—
228	#334	—	47	1	no	—
229	#334	C₂×D₄₇	1	1	yes	characteristic
230	Q₈⋊C₈	—	47	2	no	—
231	Q₈⋊C₈	—	47	2	no	—
232	D₄⋊C₈	—	47	2	no	—
233	D₄⋊C₈	C₂×D₄₇	1	2	yes	—
234	#322	—	47	1	no	—
235	#322	—	47	1	no	—
236	#322	—	47	1	no	—
237	#322	C₂×D₄₇	1	1	yes	characteristic
238	C₄₇	#18065	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	47	2	no	—
240	#135	—	94	4	no	—
241	C₂×Q₁₆	—	47	1	no	—
242	C₂×SD₁₆	—	47	2	no	—
243	C₂×D₈	C₄×D₄₇	1	1	yes	characteristic
244	C₂×M₁₆	—	94	2	no	—
245	C₂×M₁₆	—	94	1	no	—
246	C₂²×C₈	—	2	2	no	—
247	C₂²×C₈	—	47	1	no	—
248	C₂²×C₈	—	47	1	no	—
249	C₂²×C₈	—	94	2	no	—
250	C₄×Q₈	—	47	2	no	—
251	C₄×Q₈	—	47	2	no	—
252	C₄×D₄	—	47	2	no	—
253	C₄×D₄	—	94	2	no	—
254	C₄×D₄	—	94	2	no	—
255	C₄×D₄	C₂²×D₄₇	1	2	yes	—
256	#117	—	47	2	no	—
257	#117	—	94	2	no	—
258	C₂×C₄²	—	47	1	no	—
259	C₂×C₄²	—	94	2	no	—
260	#107	—	47	1	no	—
261	#107	C₄×D₄₇	1	1	yes	characteristic
262	#106	—	47	2	no	—
263	C₄⋊C₈	—	47	2	no	—
264	C₄⋊C₈	—	47	2	no	—
265	C₄⋊C₈	—	94	2	no	—
266	C₄⋊C₈	—	94	2	no	—
267	C₄⋊C₈	—	94	2	no	—
268	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
269	#103	—	47	2	no	—
270	#103	—	47	2	no	—
271	#102	—	47	2	no	—
272	#102	C₄×D₄₇	1	2	yes	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	—	47	2	no	—
275	C₂²⋊C₈	—	94	2	no	—
276	#97	—	94	1	no	—
277	#97	—	94	2	no	—
278	#97	—	94	2	no	—
279	C₄×C₈	—	47	1	no	—
280	C₄×C₈	—	47	1	no	—
281	C₄×C₈	—	47	1	no	—
282	C₄×C₈	—	47	1	no	—
283	C₄×C₈	—	47	1	no	—
284	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
287	#41	—	94	4	no	—
288	#41	—	188	2	no	—
289	C₂×Q₈	—	47	2	no	—
290	C₂×D₄	—	94	2	no	—
291	C₂×D₄	C₂×C₄×D₄₇	1	2	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	47	1	no	—
294	C₂²×C₄	—	47	1	no	—
295	C₂²×C₄	—	47	1	no	—
296	C₂²×C₄	—	94	2	no	—
297	C₂²×C₄	—	94	2	no	—
298	Q₁₆	—	94	2	no	—
299	SD₁₆	—	94	4	no	—
300	D₈	#10625324	1	4	yes	complemented
301	M₁₆	—	376	2	no	—
302	M₁₆	—	188	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	8	2	no	—
306	C₂×C₈	—	47	1	no	—
307	C₂×C₈	—	47	1	no	—
308	C₂×C₈	—	47	2	no	—
309	C₂×C₈	—	94	2	no	—
310	C₂×C₈	—	94	2	no	—
311	C₂×C₈	—	94	1	no	—
312	C₂×C₈	—	376	2	no	—
313	C₂×C₈	—	188	1	no	—
314	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
316	C₂×C₈	D₄×D₄₇	1	2	yes	—
317	C₄⋊C₄	—	47	2	no	—
318	C₄⋊C₄	—	94	2	no	—
319	C₄⋊C₄	—	94	2	no	—
320	C₄⋊C₄	C₂×C₄×D₄₇	1	2	yes	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	94	2	no	—
323	C₂²⋊C₄	—	94	2	no	—
324	C₄²	—	47	1	no	—
325	C₄²	—	47	1	no	—
326	C₄²	—	47	1	no	—
327	C₄²	—	94	2	no	—
328	C₄²	—	94	2	no	—
329	C₄²	—	94	2	no	—
330	C₄²	C₂³×D₄₇	1	1	yes	characteristic
331	C₂³	—	2	2	no	—
332	C₂³	—	47	1	no	—
333	Q₈	—	94	2	no	—
334	Q₈	—	94	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	188	4	no	—
337	D₄	C₂×#10625324	1	4	yes	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	47	1	no	—
342	C₂×C₄	—	47	1	no	—
343	C₂×C₄	—	47	1	no	—
344	C₂×C₄	—	94	2	no	—
345	C₂×C₄	—	94	1	no	—
346	C₂×C₄	—	94	1	no	—
347	C₂×C₄	—	94	2	no	—
348	C₂×C₄	—	94	2	no	—
349	C₂×C₄	—	94	2	no	—
350	C₂×C₄	—	188	4	no	—
351	C₂×C₄	—	188	4	no	—
352	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
354	C₂×C₄	#41×D₄₇	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	2	no	—
359	C₈	—	94	1	no	—
360	C₈	—	94	1	no	—
361	C₈	—	94	2	no	—
362	C₈	—	188	1	no	—
363	C₈	—	188	2	no	—
364	C₈	C₂×#10625324	1	2	yes	—
365	C₂²	—	2	4	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	94	1	no	—
369	C₂²	—	94	1	no	—
370	C₂²	—	94	1	no	—
371	C₂²	C₄×D₄×D₄₇	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	4	2	no	—
374	C₄	—	47	2	no	—
375	C₄	—	94	1	no	—
376	C₄	—	94	1	no	—
377	C₄	—	188	2	no	—
378	C₄	—	188	2	no	—
379	C₄	C₂²×#10625324	1	1	yes	characteristic
380	C₄	#15689888	1	1	yes	characteristic
381	C₄	#15689933	1	2	yes	—
382	C₂	—	4	4	no	—
383	C₂	—	188	1	no	—
384	C₂	D₄×#10625324	1	1	yes	characteristic
385	C₂	#20252270	1	1	yes	characteristic
386	C₂	#444×D₄₇	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×#1580	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094043	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1580	D₄₇
hypocenter	C₄₇	#18065
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1580

Complements

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

N	H ≅ G/N	type
1. #20252511	C₂	semidirect
3. #20242298	C₂	semidirect
4. #20242296	C₂	semidirect
5. #20242224	C₂	semidirect
6. #20242221	C₂	semidirect
7. #20242212	C₂	semidirect
8. #20242142	C₂	semidirect
9. #20241824	C₂	semidirect
10. #20241674	C₂	semidirect
11. C₄₇×#1580	C₂	semidirect
104. D₈×C₄₇	M₁₆	semidirect
196. #1580	D₄₇	semidirect
238. C₄₇	#18065	semidirect
300. D₈	#10625324	semidirect

Maximal subgroups

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

subgroup	index
#20252511	2
C₄×#13094043	2
#20242298	2
#20242296	2
#20242224	2
#20242221	2
#20242212	2
#20242142	2
#20241824	2
#20241674	2
C₄₇×#1580	2
#18065	47