Subgroups — #26216649

Order 12032.

608 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251814	C₂	1	1	yes	maximal, characteristic
2	#20243536	C₂	1	2	yes	maximal, complemented
3	#20243509	C₂	1	2	yes	maximal, complemented
4	#20243493	C₂	1	1	yes	maximal, characteristic
5	#20243489	C₂	1	1	yes	maximal, characteristic
6	#20243436	C₂	1	1	yes	maximal, characteristic
7	#20243432	C₂	1	1	yes	maximal, characteristic
8	#20243273	C₂	1	2	yes	maximal, complemented
9	#20243188	C₂	1	1	yes	maximal, characteristic
10	#20243143	C₂	1	2	yes	maximal, complemented
11	C₄₇×#1757	C₂	1	1	yes	maximal, characteristic
12	C₂×#13094031	C₂²	1	1	yes	characteristic
13	#41×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×#13093959	C₂²	1	2	yes	—
15	#15689159	C₂²	1	2	yes	—
16	#15689107	C₂²	1	1	yes	characteristic
17	#15689097	C₂²	1	1	yes	characteristic
18	#15689081	C₄	1	2	yes	complemented
19	#15689081	C₂²	1	1	yes	characteristic
20	#15689078	C₄	1	2	yes	complemented
21	#15689078	C₂²	1	2	yes	—
22	#15689078	C₂²	1	1	yes	characteristic
23	M₁₆×Dic₄₇	C₄	1	2	yes	complemented
24	M₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂×#13093934	C₄	1	2	yes	complemented
26	C₂×#13093934	C₂²	1	1	yes	characteristic
27	#15688894	C₂²	1	2	yes	—
28	#15688884	C₂²	1	2	yes	—
29	#15688882	C₂²	1	2	yes	—
30	C₄₇⋊#338	C₂²	1	2	yes	—
31	#15688880	C₂²	1	2	yes	—
32	#15688879	C₂²	1	2	yes	—
33	#15688879	C₂²	1	2	yes	—
34	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
35	C₄₇×#413	C₂²	1	1	yes	characteristic
36	C₄₇×#410	C₂²	1	1	yes	characteristic
37	C₄₇×#405	C₂²	1	2	yes	—
38	C₄₇×#344	C₂²	1	2	yes	—
39	#13094031	—	2	2	no	—
40	#13094031	—	2	2	no	—
41	C₂×#10625329	C₂×C₄	1	2	yes	—
42	C₂×#10625329	C₂³	1	2	yes	—
43	C₂×#10625329	C₂³	1	1	yes	characteristic
44	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
46	Q₈×Dic₄₇	—	2	1	no	—
47	D₄×Dic₄₇	—	2	2	no	—
48	D₄×Dic₄₇	—	2	1	no	—
49	#13093983	C₂×C₄	1	2	yes	—
50	#13093983	C₂³	1	2	yes	—
51	#13093983	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
54	#13093959	—	2	2	no	—
55	#13093959	—	2	2	no	—
56	#13093957	—	2	2	no	—
57	#13093955	—	2	2	no	—
58	#13093943	—	2	1	no	—
59	#13093941	—	2	1	no	—
60	#13093939	—	2	2	no	—
61	#13093934	—	2	2	no	—
62	#13093934	C₂×C₄	1	1	yes	characteristic
63	#13093934	C₂×C₄	1	2	yes	—
64	#13093934	C₂×C₄	1	2	yes	—
65	#13093934	C₂×C₄	1	2	yes	—
66	#13093934	C₂×C₄	1	1	yes	characteristic
67	#13093934	C₂×C₄	1	1	yes	characteristic
68	#13093934	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₄₇	—	2	2	no	—
70	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
71	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
72	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
73	C₂×#41×C₄₇	C₂³	1	1	yes	characteristic
74	#131×C₄₇	—	2	2	no	—
75	#131×C₄₇	—	2	2	no	—
76	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
77	C₂×M₁₆×C₄₇	C₂³	1	2	yes	—
78	C₂×M₁₆×C₄₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
80	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
81	#117×C₄₇	C₂³	1	2	yes	—
82	#101×C₄₇	—	2	1	no	—
83	#100×C₄₇	—	2	1	no	—
84	#99×C₄₇	—	2	2	no	—
85	#97×C₄₇	—	2	2	no	—
86	C₄×C₈×C₄₇	—	2	2	no	—
87	#41×C₄₇	—	2	2	no	—
88	#41×C₄₇	—	2	2	no	—
89	C₂×Q₈×C₄₇	#41	1	1	yes	characteristic
90	C₂×D₄×C₄₇	#41	1	2	yes	—
91	C₂×D₄×C₄₇	#41	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₄₇	—	2	2	no	—
95	C₂²×Dic₄₇	—	2	1	no	—
96	M₁₆×C₄₇	—	2	2	no	—
97	M₁₆×C₄₇	—	2	4	no	—
98	M₁₆×C₄₇	—	2	2	no	—
99	M₁₆×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	#41	1	2	yes	—
107	C₂×C₈×C₄₇	#41	1	2	yes	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₂²⋊C₄×C₄₇	—	2	2	no	—
110	C₄²×C₄₇	—	2	2	no	—
111	C₂²⋊Dic₄₇	—	2	2	no	—
112	C₂²⋊Dic₄₇	—	2	1	no	—
113	C₂²⋊Dic₄₇	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	2	no	—
115	C₄⋊Dic₄₇	—	2	1	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄×Dic₄₇	—	2	2	no	—
118	C₄×Dic₄₇	—	2	1	no	—
119	C₄×Dic₄₇	—	2	2	no	—
120	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
121	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
123	#10625329	—	2	2	no	—
124	#10625329	—	2	4	no	—
125	#10625329	—	2	2	no	—
126	#10625329	—	2	2	no	—
127	C₂×C₄₇⋊C₈	—	2	2	no	—
128	C₂×C₄₇⋊C₈	—	2	2	no	—
129	C₂×C₄₇⋊C₈	—	2	2	no	—
130	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
131	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄₇⋊C₈	#41	1	2	yes	—
134	C₂×C₄₇⋊C₈	#41	1	2	yes	—
135	C₂³×C₄₇	#117	1	2	yes	—
136	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
137	Q₈×C₄₇	—	2	2	no	—
138	D₄×C₄₇	—	2	4	no	—
139	D₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	#117	1	2	yes	—
144	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
146	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
147	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₄₇	—	2	1	no	—
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	2	1	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₂×Dic₄₇	—	8	1	no	—
153	C₂×Dic₄₇	—	8	2	no	—
154	C₈×C₄₇	—	2	2	no	—
155	C₈×C₄₇	—	4	2	no	—
156	C₈×C₄₇	#117	1	1	yes	characteristic
157	C₈×C₄₇	#117	1	1	yes	characteristic
158	C₈×C₄₇	#117	1	2	yes	—
159	C₄₇⋊C₈	—	2	2	no	—
160	C₄₇⋊C₈	—	4	2	no	—
161	C₄₇⋊C₈	#117	1	1	yes	characteristic
162	C₄₇⋊C₈	#117	1	1	yes	characteristic
163	C₄₇⋊C₈	#117	1	2	yes	—
164	#20788	—	47	1	no	maximal, Sylow
165	C₂²×C₄₇	—	2	2	no	—
166	C₂²×C₄₇	—	4	1	no	—
167	C₂²×C₄₇	—	4	2	no	—
168	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
169	C₂²×C₄₇	#529	1	2	yes	—
170	C₄×C₄₇	—	2	2	no	—
171	C₄×C₄₇	—	8	2	no	—
172	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
173	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
174	C₄×C₄₇	#529	1	2	yes	—
175	Dic₄₇	—	4	2	no	—
176	Dic₄₇	—	4	1	no	—
177	Dic₄₇	—	4	1	no	—
178	Dic₄₇	—	8	2	no	—
179	#2875	—	47	1	no	—
180	#1841	—	47	2	no	—
181	#1833	—	47	2	no	—
182	#1827	—	47	1	no	—
183	#1827	—	47	1	no	—
184	#1813	—	47	1	no	—
185	#1812	—	47	1	no	—
186	#1773	—	47	2	no	—
187	#1757	—	47	1	no	—
188	#1757	D₄₇	1	1	yes	characteristic
189	#1747	—	47	2	no	—
190	C₂×C₄₇	—	2	1	no	—
191	C₂×C₄₇	—	2	2	no	—
192	C₂×C₄₇	—	4	2	no	—
193	C₂×C₄₇	#2283	1	1	yes	characteristic
194	C₂×#131	—	47	1	no	—
195	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
196	C₄×#41	—	47	1	no	—
197	C₂×#108	—	47	2	no	—
198	#427	—	47	2	no	—
199	#413	—	47	1	no	—
200	#413	C₂×D₄₇	1	1	yes	characteristic
201	#410	—	47	1	no	—
202	#410	C₂×D₄₇	1	1	yes	characteristic
203	#405	—	47	2	no	—
204	#405	—	47	1	no	—
205	#405	—	47	2	no	—
206	#405	—	47	1	no	—
207	#405	—	47	2	no	—
208	#405	C₂×D₄₇	1	2	yes	—
209	C₄×M₁₆	—	47	2	no	—
210	C₄×M₁₆	—	47	1	no	—
211	C₂×#97	—	47	2	no	—
212	C₂×#97	—	47	1	no	—
213	#344	—	47	2	no	—
214	#344	C₂×D₄₇	1	2	yes	—
215	#339	—	47	2	no	—
216	#339	—	47	2	no	—
217	#338	—	47	2	no	—
218	H(3,ℤ/4)	—	47	2	no	—
219	H(3,ℤ/4)	—	47	2	no	—
220	H(3,ℤ/4)	—	47	2	no	—
221	C₄₇	#20788	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	C₂²×D₄₇	1	1	yes	characteristic
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	94	2	no	—
226	#131	—	94	2	no	—
227	C₂×M₁₆	—	47	2	no	—
228	C₂×M₁₆	—	47	1	no	—
229	C₂×M₁₆	—	47	2	no	—
230	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
231	C₂×M₁₆	C₂²×D₄₇	1	2	yes	—
232	C₂×M₁₆	C₂²×D₄₇	1	1	yes	characteristic
233	C₂²×C₈	—	47	1	no	—
234	C₂²×C₈	—	47	2	no	—
235	C₂²×C₈	C₄×D₄₇	1	2	yes	—
236	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
237	C₄×Q₈	—	94	1	no	—
238	C₄×D₄	—	94	2	no	—
239	C₄×D₄	—	94	1	no	—
240	#117	—	47	2	no	—
241	#117	—	47	1	no	—
242	#117	—	47	2	no	—
243	#117	C₂²×D₄₇	1	2	yes	—
244	C₂×C₄²	—	47	2	no	—
245	C₂×C₄²	—	47	1	no	—
246	#108	—	94	2	no	—
247	#108	—	94	2	no	—
248	#107	—	94	2	no	—
249	#106	—	94	2	no	—
250	#101	—	2	1	no	—
251	#101	—	94	1	no	—
252	#100	—	2	1	no	—
253	#100	—	94	1	no	—
254	#99	—	2	2	no	—
255	#99	—	94	2	no	—
256	#97	—	2	2	no	—
257	#97	—	47	1	no	—
258	#97	—	47	1	no	—
259	#97	—	47	2	no	—
260	#97	—	47	1	no	—
261	#97	—	47	2	no	—
262	#97	—	47	1	no	—
263	#97	—	47	2	no	—
264	#97	—	94	2	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	47	2	no	—
267	C₄×C₈	—	47	2	no	—
268	C₄×C₈	—	47	2	no	—
269	C₄×C₈	—	94	2	no	—
270	#41	—	2	2	no	—
271	#41	—	2	2	no	—
272	C₂×Q₈	#10625350	1	1	yes	characteristic
273	C₂×D₄	#10625350	1	2	yes	—
274	C₂×D₄	#10625350	1	1	yes	characteristic
275	C₂²×C₄	—	94	2	no	—
276	C₂²×C₄	—	94	1	no	—
277	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
278	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	94	4	no	—
284	M₁₆	—	94	2	no	—
285	M₁₆	—	94	2	no	—
286	M₁₆	—	94	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	47	1	no	—
291	C₂×C₈	—	47	1	no	—
292	C₂×C₈	—	47	2	no	—
293	C₂×C₈	—	47	2	no	—
294	C₂×C₈	—	47	2	no	—
295	C₂×C₈	—	94	2	no	—
296	C₂×C₈	—	94	2	no	—
297	C₂×C₈	—	94	2	no	—
298	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
299	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
301	C₂×C₈	#10625352	1	2	yes	—
302	C₂×C₈	#10625352	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	94	2	no	—
305	C₄⋊C₄	—	94	1	no	—
306	C₄⋊C₄	—	94	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	94	2	no	—
309	C₂²⋊C₄	—	94	1	no	—
310	C₂²⋊C₄	—	94	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	47	2	no	—
313	C₄²	—	47	1	no	—
314	C₄²	—	47	1	no	—
315	C₄²	—	94	2	no	—
316	C₄²	—	94	1	no	—
317	C₄²	—	94	2	no	—
318	C₂³	#13093981	1	2	yes	—
319	C₂³	C₂×#10625350	1	1	yes	characteristic
320	Q₈	—	2	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	94	1	no	—
327	C₂×C₄	—	94	2	no	—
328	C₂×C₄	—	94	1	no	—
329	C₂×C₄	—	376	1	no	—
330	C₂×C₄	—	376	2	no	—
331	C₂×C₄	—	188	2	no	—
332	C₂×C₄	#13093981	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
334	C₂×C₄	#41×D₄₇	1	2	yes	—
335	C₂×C₄	C₂×#10625352	1	2	yes	—
336	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	47	1	no	—
340	C₈	—	47	1	no	—
341	C₈	—	47	2	no	—
342	C₈	—	94	2	no	—
343	C₈	—	188	2	no	—
344	C₈	#13093980	1	1	yes	characteristic
345	C₈	#13093980	1	1	yes	characteristic
346	C₈	#13093980	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×#13093981	1	1	yes	characteristic
351	C₂²	#15689653	1	2	yes	—
352	C₄	—	2	2	no	—
353	C₄	—	8	2	no	—
354	C₄	—	376	2	no	—
355	C₄	—	188	1	no	—
356	C₄	—	188	2	no	—
357	C₄	—	188	1	no	—
358	C₄	#117×D₄₇	1	1	yes	characteristic
359	C₄	C₂×#13093980	1	1	yes	characteristic
360	C₄	#15689653	1	2	yes	—
361	C₂	—	2	2	no	—
362	C₂	—	2	1	no	—
363	C₂	—	4	2	no	—
364	C₂	#20245782	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13093980
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1757	C₂
socle	C₂×C₄₇	#2283
prime radical	C₂×#41×C₄₇	C₂³
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1757	D₄₇
hypocenter	C₄₇	#20788
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1757

Complements

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

N	H ≅ G/N	type
2. #20243536	C₂	semidirect
3. #20243509	C₂	semidirect
8. #20243273	C₂	semidirect
10. #20243143	C₂	semidirect
18. #15689081	C₄	semidirect
20. #15689078	C₄	semidirect
23. M₁₆×Dic₄₇	C₄	semidirect
25. C₂×#13093934	C₄	semidirect
221. C₄₇	#20788	semidirect

Maximal subgroups

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

subgroup	index
#20251814	2
#20243536	2
#20243509	2
#20243493	2
#20243489	2
#20243436	2
#20243432	2
#20243273	2
#20243188	2
#20243143	2
C₄₇×#1757	2
#20788	47