Subgroups — #26133174

Order 12032.

488 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093963	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252080	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688907	C₂	1	1	yes	maximal, characteristic, complemented
4	#20244998	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241478	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#350	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	#15689630	C₄	1	1	yes	characteristic
10	C₂×#13093964	C₂²	1	1	yes	characteristic
11	C₂×#13093963	—	2	2	no	—
12	C₂×#13093963	—	2	2	no	—
13	C₂×#13093963	—	2	2	no	—
14	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	#15689192	—	2	2	no	—
16	C₈×C₂²⋊D₄₇	—	2	2	no	—
17	#15689151	C₄	1	1	yes	characteristic
18	#15689090	C₄	1	1	yes	characteristic
19	#15689086	C₄	1	1	yes	characteristic
20	#15689078	C₂²	1	1	yes	characteristic
21	#15688907	—	2	2	no	—
22	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
23	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
26	C₄₇×#350	—	2	2	no	—
27	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₄₇	—	2	1	no	—
31	C₂×C₈×D₄₇	—	2	1	no	—
32	C₂×C₈×D₄₇	—	2	1	no	—
33	C₂×C₈×D₄₇	—	2	1	no	—
34	C₂×C₈×D₄₇	D₄	1	2	yes	—
35	C₂×C₈×D₄₇	D₄	1	2	yes	—
36	#13094005	—	2	1	no	—
37	#13094000	C₈	1	2	yes	—
38	#13093995	C₈	1	2	yes	—
39	#13093992	—	2	1	no	—
40	C₄×C₂²⋊D₄₇	—	2	2	no	—
41	#13093983	C₂×C₄	1	1	yes	characteristic
42	#13093964	—	2	1	no	—
43	#13093964	—	2	1	no	—
44	#13093963	—	2	8	no	—
45	C₂×C₄₇⋊C₁₆	—	2	1	no	—
46	C₂×C₄₇⋊C₁₆	—	2	1	no	—
47	C₂×C₄₇⋊C₁₆	—	2	1	no	—
48	C₂×C₄₇⋊C₁₆	D₄	1	2	yes	complemented
49	Dic₄₇⋊C₈	—	2	2	no	—
50	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
51	(C₂×D₄₇)⋊C₈	—	2	2	no	—
52	#13093934	—	2	1	no	—
53	C₈×Dic₄₇	—	2	1	no	—
54	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
55	#131×C₄₇	—	2	1	no	—
56	#131×C₄₇	—	2	1	no	—
57	#131×C₄₇	—	2	1	no	—
58	#131×C₄₇	—	2	1	no	—
59	C₂×M₁₆×C₄₇	—	2	1	no	—
60	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	—	2	1	no	—
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	M₃₂×C₄₇	—	2	1	no	—
64	M₃₂×C₄₇	—	2	1	no	—
65	C₂×C₁₆×C₄₇	—	2	1	no	—
66	C₂×C₁₆×C₄₇	—	2	1	no	—
67	C₂×C₁₆×C₄₇	—	2	1	no	—
68	C₂×C₁₆×C₄₇	D₄	1	2	yes	complemented
69	C₂³×D₄₇	M₁₆	1	1	yes	characteristic
70	#41×C₄₇	—	2	1	no	—
71	#41×C₄₇	—	2	1	no	—
72	#41×C₄₇	—	2	1	no	—
73	#41×C₄₇	—	2	1	no	—
74	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₄₇	—	2	1	no	—
76	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	—	2	1	no	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₇	—	2	2	no	—
80	C₂²×Dic₄₇	M₁₆	1	1	yes	characteristic
81	C₂×C₄×D₄₇	—	2	1	no	—
82	C₂×C₄×D₄₇	—	2	1	no	—
83	C₂×C₄×D₄₇	—	2	1	no	—
84	C₂×C₄×D₄₇	—	2	1	no	—
85	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
86	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
87	M₁₆×C₄₇	—	2	1	no	—
88	M₁₆×C₄₇	—	2	1	no	—
89	M₁₆×C₄₇	—	4	1	no	—
90	M₁₆×C₄₇	—	4	1	no	—
91	C₂×C₈×C₄₇	—	2	1	no	—
92	C₂×C₈×C₄₇	—	4	1	no	—
93	C₂×C₈×C₄₇	—	4	1	no	—
94	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
99	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
100	C₂²⋊Dic₄₇	—	2	1	no	—
101	C₂²⋊Dic₄₇	—	2	2	no	—
102	(C₂×D₄₇)⋊C₄	—	2	2	no	—
103	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
104	Dic₄₇⋊C₄	—	2	2	no	—
105	C₄×Dic₄₇	—	2	1	no	—
106	#10625329	—	2	1	no	—
107	#10625329	—	2	1	no	—
108	C₂×C₄₇⋊C₈	—	2	1	no	—
109	C₂×C₄₇⋊C₈	—	2	1	no	—
110	C₂×C₄₇⋊C₈	—	2	1	no	—
111	C₂×C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
112	C₈×D₄₇	—	2	2	no	—
113	C₈×D₄₇	—	2	2	no	—
114	C₈×D₄₇	—	2	2	no	—
115	C₈×D₄₇	—	2	2	no	—
116	C₁₆×C₄₇	—	2	2	no	—
117	C₁₆×C₄₇	—	4	1	no	—
118	C₄₇⋊C₁₆	—	2	2	no	—
119	C₄₇⋊C₁₆	—	4	1	no	—
120	C₂³×C₄₇	—	2	1	no	—
121	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₄₇	—	2	1	no	—
123	C₂²×D₄₇	—	2	1	no	—
124	C₂²×D₄₇	—	4	2	no	—
125	C₂²×D₄₇	C₂²⋊C₈	1	2	yes	—
126	Q₈×C₄₇	—	2	1	no	—
127	Q₈×C₄₇	—	2	1	no	—
128	D₄×C₄₇	—	2	1	no	—
129	D₄×C₄₇	—	2	1	no	—
130	D₄×C₄₇	—	4	1	no	—
131	D₄×C₄₇	—	4	1	no	—
132	C₂×C₄×C₄₇	—	2	1	no	—
133	C₂×C₄×C₄₇	—	4	1	no	—
134	C₂×C₄×C₄₇	—	4	1	no	—
135	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
141	C₂²⋊D₄₇	—	4	4	no	—
142	C₂×Dic₄₇	—	2	1	no	—
143	C₂×Dic₄₇	—	2	1	no	—
144	C₂×Dic₄₇	—	2	2	no	—
145	C₂×Dic₄₇	C₂²⋊C₈	1	2	yes	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	2	2	no	—
150	C₈×C₄₇	—	4	1	no	—
151	C₈×C₄₇	#120	1	1	yes	characteristic
152	C₈×C₄₇	#121	1	1	yes	characteristic
153	C₈×C₄₇	#121	1	1	yes	characteristic
154	C₈×C₄₇	#123	1	1	yes	characteristic
155	C₄₇⋊C₈	—	2	2	no	—
156	C₄₇⋊C₈	—	4	1	no	—
157	#12984	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	4	2	no	—
160	C₂²×C₄₇	—	4	1	no	—
161	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₄₇	#435	1	1	yes	characteristic
163	C₂²×C₄₇	#435	1	1	yes	characteristic
164	C₂×D₄₇	—	2	2	no	—
165	C₂×D₄₇	—	2	2	no	—
166	C₂×D₄₇	—	2	2	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₂×D₄₇	—	4	2	no	—
169	C₄×C₄₇	—	4	1	no	—
170	C₄×C₄₇	#386	1	1	yes	characteristic
171	C₄×C₄₇	#408	1	1	yes	characteristic
172	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
173	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
174	Dic₄₇	—	2	2	no	—
175	Dic₄₇	—	4	2	no	—
176	C₂²×M₃₂	—	47	1	no	—
177	#2929	—	47	1	no	—
178	C₂×#350	—	47	1	no	—
179	C₂×#350	D₄₇	1	1	yes	characteristic, complemented
180	#2112	—	47	1	no	—
181	#1379	—	47	1	no	—
182	#1379	—	47	1	no	—
183	C₂×C₄₇	—	2	1	no	—
184	C₂×C₄₇	—	2	1	no	—
185	C₂×C₄₇	—	8	1	no	—
186	C₂×C₄₇	#1869	1	1	yes	characteristic
187	C₂×C₄₇	#2117	1	2	yes	—
188	D₄₇	—	4	2	no	—
189	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
190	C₂³×C₈	—	47	1	no	—
191	#525	—	47	1	no	—
192	C₂×M₃₂	—	47	1	no	—
193	C₂×M₃₂	—	94	2	no	—
194	C₂×M₃₂	—	94	2	no	—
195	C₂×M₃₂	—	94	2	no	—
196	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
197	C₂²×C₁₆	—	47	1	no	—
198	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
199	#435	—	94	2	no	—
200	C₈×D₄	—	94	2	no	—
201	#424	—	47	1	no	—
202	#408	—	47	1	no	—
203	#407	—	47	1	no	—
204	#405	—	47	1	no	—
205	#350	—	2	2	no	—
206	#350	—	94	2	no	—
207	C₂²⋊C₁₆	—	94	2	no	—
208	C₄₇	#12984	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
210	C₂³×C₄	—	47	1	no	—
211	#131	—	2	1	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	47	1	no	—
217	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	—	47	1	no	—
220	C₂²×C₈	—	47	2	no	—
221	C₂²×C₈	—	47	2	no	—
222	C₂²×C₈	—	94	1	no	—
223	C₂²×C₈	—	94	1	no	—
224	C₂²×C₈	—	94	1	no	—
225	C₂²×C₈	—	94	1	no	—
226	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
227	#123	—	94	1	no	—
228	C₂²⋊Q₈	—	47	2	no	—
229	#121	—	47	2	no	—
230	#120	—	94	1	no	—
231	C₄×D₄	—	94	2	no	—
232	#117	—	47	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	94	8	no	—
236	M₃₂	—	94	1	no	—
237	M₃₂	—	94	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	47	2	no	—
242	C₂×C₁₆	—	94	1	no	—
243	C₂×C₁₆	—	94	1	no	—
244	C₂×C₁₆	—	94	1	no	—
245	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	complemented
246	C₄⋊C₈	—	94	2	no	—
247	C₂²⋊C₈	—	94	2	no	—
248	C₂²⋊C₈	—	94	2	no	—
249	#97	—	94	1	no	—
250	C₄×C₈	—	94	1	no	—
251	C₂⁴	—	47	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	—	2	1	no	—
255	#41	—	2	1	no	—
256	C₂×Q₈	C₈×D₄₇	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	94	2	no	—
259	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	47	1	no	—
262	C₂²×C₄	—	47	2	no	—
263	C₂²×C₄	—	47	2	no	—
264	C₂²×C₄	—	94	1	no	—
265	C₂²×C₄	—	94	1	no	—
266	C₂²×C₄	—	94	1	no	—
267	C₂²×C₄	—	94	1	no	—
268	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	4	1	no	—
272	M₁₆	—	4	1	no	—
273	M₁₆	—	94	1	no	—
274	M₁₆	—	94	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	47	2	no	—
279	C₂×C₈	—	94	1	no	—
280	C₂×C₈	—	94	1	no	—
281	C₂×C₈	—	94	2	no	—
282	C₂×C₈	—	94	2	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	94	1	no	—
286	C₂×C₈	#10625350	1	1	yes	characteristic
287	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
288	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
289	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
290	C₂×C₈	#10625352	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₄⋊C₄	—	47	2	no	—
293	C₄⋊C₄	—	94	2	no	—
294	C₂²⋊C₄	—	94	2	no	—
295	C₂²⋊C₄	—	94	2	no	—
296	C₂²⋊C₄	—	94	1	no	—
297	C₄²	—	94	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	4	1	no	—
300	C₁₆	—	94	2	no	—
301	C₁₆	—	188	1	no	—
302	C₂³	—	2	1	no	—
303	C₂³	—	47	2	no	—
304	C₂³	—	94	1	no	—
305	C₂³	—	94	1	no	—
306	C₂³	—	188	2	no	—
307	C₂³	M₁₆×D₄₇	1	1	yes	characteristic
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	1	no	—
314	D₄	—	188	4	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	47	2	no	—
319	C₂×C₄	—	94	1	no	—
320	C₂×C₄	—	94	1	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	94	2	no	—
324	C₂×C₄	—	94	2	no	—
325	C₂×C₄	—	94	2	no	—
326	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
327	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
329	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
330	C₂×C₄	#13094030	1	1	yes	characteristic
331	C₂×C₄	#13094032	1	1	yes	characteristic
332	C₈	—	4	1	no	—
333	C₈	—	94	2	no	—
334	C₈	—	188	1	no	—
335	C₈	#13093992	1	1	yes	characteristic
336	C₈	#13093995	1	1	yes	characteristic
337	C₈	#13093997	1	1	yes	characteristic
338	C₈	#13094006	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	94	2	no	—
343	C₂²	—	94	2	no	—
344	C₂²	—	94	2	no	—
345	C₂²	—	188	2	no	—
346	C₂²	—	188	2	no	—
347	C₂²	C₂²⋊C₈×D₄₇	1	1	yes	characteristic
348	C₂²	#15689192	1	1	yes	characteristic
349	C₂²	#15689197	1	1	yes	characteristic
350	C₄	—	4	1	no	—
351	C₄	—	94	2	no	—
352	C₄	—	188	2	no	—
353	C₄	#15688999	1	1	yes	characteristic
354	C₄	#15689088	1	1	yes	characteristic
355	C₄	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
356	C₄	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	8	1	no	—
360	C₂	—	188	2	no	—
361	C₂	#20243677	1	1	yes	characteristic
362	C₂	#20245014	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#350	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	#15689630	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#350	D₄₇
hypocenter	C₄₇	#12984
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093963	C₂	semidirect
3. C₂×#15688907	C₂	semidirect
4. #20244998	C₂	semidirect
5. #20241478	C₂	semidirect
6. #20241476	C₂	semidirect
7. C₂×C₄₇×#350	C₂	semidirect
48. C₂×C₄₇⋊C₁₆	D₄	semidirect
68. C₂×C₁₆×C₄₇	D₄	semidirect
179. C₂×#350	D₄₇	semidirect
208. C₄₇	#12984	semidirect
245. C₂×C₁₆	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093963	2
#20252080	2
C₂×#15688907	2
#20244998	2
#20241478	2
#20241476	2
C₂×C₄₇×#350	2
#12984	47