Subgroups — #26131723

Order 12032.

936 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688907	C₂	1	4	yes	maximal, complemented
2	#20244999	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	complemented
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
7	#15688907	—	2	8	no	—
8	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
10	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
11	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
12	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
13	C₂×#131×C₄₇	—	2	8	no	—
14	C₂×#131×C₄₇	—	2	1	no	—
15	C₂×#131×C₄₇	—	2	1	no	—
16	C₂×#131×C₄₇	C₄	1	2	yes	—
17	C₂×#131×C₄₇	C₂²	1	2	yes	—
18	C₂²×M₁₆×C₄₇	—	2	1	no	—
19	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
20	C₂³×C₈×C₄₇	—	2	1	no	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#13093964	—	2	4	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	4	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	C₂×#41×C₄₇	—	2	4	no	—
27	C₂×#41×C₄₇	—	2	4	no	—
28	C₂×#41×C₄₇	—	2	1	no	—
29	C₂×#41×C₄₇	—	2	1	no	—
30	C₂×#41×C₄₇	C₈	1	2	yes	—
31	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
33	C₂²×D₄×C₄₇	—	2	1	no	—
34	C₂²×D₄×C₄₇	C₈	1	1	yes	characteristic
35	C₂³×C₄×C₄₇	—	2	1	no	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	#131×C₄₇	—	2	8	no	—
38	#131×C₄₇	—	2	8	no	—
39	#131×C₄₇	—	2	4	no	—
40	#131×C₄₇	—	2	4	no	—
41	#131×C₄₇	—	2	4	no	—
42	#131×C₄₇	D₄	1	8	yes	—
43	C₂×M₁₆×C₄₇	—	2	8	no	—
44	C₂×M₁₆×C₄₇	—	2	1	no	—
45	C₂×M₁₆×C₄₇	—	2	4	no	—
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₄₇	—	2	8	no	—
50	C₂²×C₈×C₄₇	—	2	1	no	—
51	C₂²×C₈×C₄₇	—	2	2	no	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂²×C₈×C₄₇	—	2	4	no	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂⁴×C₄₇	—	2	1	no	—
61	C₂⁴×C₄₇	M₁₆	1	1	yes	characteristic
62	#41×C₄₇	—	2	4	no	—
63	#41×C₄₇	—	2	8	no	—
64	#41×C₄₇	—	2	4	no	—
65	#41×C₄₇	—	2	4	no	—
66	#41×C₄₇	—	2	4	no	—
67	#41×C₄₇	—	2	2	no	—
68	#41×C₄₇	—	2	2	no	—
69	#41×C₄₇	C₂²⋊C₄	1	4	yes	—
70	#41×C₄₇	M₁₆	1	4	yes	—
71	C₂×Q₈×C₄₇	—	2	4	no	—
72	C₂×Q₈×C₄₇	—	2	1	no	—
73	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
74	C₂×D₄×C₄₇	—	2	8	no	—
75	C₂×D₄×C₄₇	—	2	2	no	—
76	C₂×D₄×C₄₇	—	2	2	no	—
77	C₂×D₄×C₄₇	—	2	4	no	—
78	C₂×D₄×C₄₇	—	2	1	no	—
79	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₄₇	—	2	4	no	—
81	C₂²×C₄×C₄₇	—	2	4	no	—
82	C₂²×C₄×C₄₇	—	2	1	no	—
83	C₂²×C₄×C₄₇	—	2	2	no	—
84	C₂²×C₄×C₄₇	—	2	2	no	—
85	C₂²×C₄×C₄₇	—	2	2	no	—
86	C₂²×C₄×C₄₇	—	2	2	no	—
87	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
92	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	M₁₆×C₄₇	—	2	4	no	—
94	M₁₆×C₄₇	—	2	8	no	—
95	M₁₆×C₄₇	—	2	2	no	—
96	M₁₆×C₄₇	—	2	8	no	—
97	M₁₆×C₄₇	C₂²⋊C₄	1	4	yes	—
98	C₂×C₈×C₄₇	—	2	4	no	—
99	C₂×C₈×C₄₇	—	2	8	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	—	2	8	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	—	2	2	no	—
107	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄₇⋊C₁₆	—	4	2	no	—
113	C₄₇⋊C₁₆	—	4	2	no	—
114	C₂³×C₄₇	—	2	4	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	—	2	2	no	—
117	C₂³×C₄₇	—	2	2	no	—
118	C₂³×C₄₇	—	4	1	no	—
119	C₂³×C₄₇	—	4	2	no	—
120	C₂³×C₄₇	—	4	2	no	—
121	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
122	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
123	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
124	Q₈×C₄₇	—	2	4	no	—
125	Q₈×C₄₇	—	2	2	no	—
126	Q₈×C₄₇	C₂²⋊C₈	1	4	yes	—
127	D₄×C₄₇	—	2	8	no	—
128	D₄×C₄₇	—	2	4	no	—
129	D₄×C₄₇	—	2	4	no	—
130	D₄×C₄₇	—	2	4	no	—
131	D₄×C₄₇	—	2	2	no	—
132	D₄×C₄₇	C₂²⋊C₈	1	4	yes	—
133	C₂×C₄×C₄₇	—	2	4	no	—
134	C₂×C₄×C₄₇	—	2	8	no	—
135	C₂×C₄×C₄₇	—	2	2	no	—
136	C₂×C₄×C₄₇	—	2	2	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	4	no	—
139	C₂×C₄×C₄₇	—	2	1	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	2	2	no	—
142	C₂×C₄×C₄₇	—	2	1	no	—
143	C₂×C₄×C₄₇	—	2	4	no	—
144	C₂×C₄×C₄₇	—	2	1	no	—
145	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
151	C₈×C₄₇	—	2	1	no	—
152	C₈×C₄₇	—	2	4	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	#120	1	2	yes	—
155	C₈×C₄₇	#120	1	2	yes	—
156	#12683	—	47	1	no	maximal, Sylow
157	C₂²×C₄₇	—	2	1	no	—
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	4	no	—
165	C₂²×C₄₇	—	4	4	no	—
166	C₂²×C₄₇	—	4	8	no	—
167	C₂²×C₄₇	—	4	2	no	—
168	C₂²×C₄₇	—	4	2	no	—
169	C₂²×C₄₇	#350	1	2	yes	complemented
170	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂²×C₄₇	#407	1	2	yes	—
172	C₄×C₄₇	—	2	1	no	—
173	C₄×C₄₇	—	2	4	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	#379	1	1	yes	characteristic
176	C₄×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
177	C₄×C₄₇	#407	1	1	yes	characteristic
178	C₂²×#131	D₄₇	1	1	yes	characteristic
179	C₂×#350	—	47	4	no	—
180	#2112	—	47	1	no	—
181	C₂×C₂²⋊C₁₆	—	47	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	—	4	1	no	—
185	C₂×C₄₇	—	4	4	no	—
186	C₂×C₄₇	#1779	1	1	yes	characteristic
187	C₂×C₄₇	C₂×#350	1	1	yes	characteristic
188	C₂×C₄₇	#2117	1	1	yes	characteristic
189	C₂²×#41	Dic₄₇	1	1	yes	characteristic
190	C₂×#131	—	2	8	no	—
191	C₂×#131	—	2	1	no	—
192	C₂×#131	—	2	1	no	—
193	C₂×#131	Dic₄₇	1	2	yes	—
194	C₂×#131	C₂×D₄₇	1	2	yes	—
195	C₂²×M₁₆	—	2	1	no	—
196	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
197	C₂³×C₈	—	2	1	no	—
198	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
199	C₂×M₃₂	—	47	2	no	—
200	C₂²×C₁₆	—	47	2	no	—
201	#350	—	94	8	no	—
202	C₂²⋊C₁₆	—	94	1	no	—
203	C₂²⋊C₁₆	—	94	1	no	—
204	C₂²⋊C₁₆	—	94	1	no	—
205	C₂²⋊C₁₆	—	94	1	no	—
206	C₄₇	#12683	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	—	2	4	no	—
208	C₂×#41	—	2	4	no	—
209	C₂×#41	—	2	1	no	—
210	C₂×#41	—	2	1	no	—
211	C₂×#41	C₄₇⋊C₈	1	2	yes	—
212	C₂×#41	C₂×Dic₄₇	1	2	yes	—
213	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
214	C₂²×D₄	—	2	1	no	—
215	C₂²×D₄	C₄₇⋊C₈	1	1	yes	characteristic
216	C₂³×C₄	—	2	1	no	—
217	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
218	#131	—	2	8	no	—
219	#131	—	2	8	no	—
220	#131	—	2	4	no	—
221	#131	—	2	4	no	—
222	#131	—	2	4	no	—
223	#131	C₂²⋊D₄₇	1	8	yes	—
224	C₂×M₁₆	—	2	8	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	2	4	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
230	C₂²×C₈	—	2	8	no	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	—	2	4	no	—
234	C₂²×C₈	—	2	2	no	—
235	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
236	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
237	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	M₃₂	—	94	4	no	—
242	C₂×C₁₆	—	94	2	no	—
243	C₂×C₁₆	—	94	4	no	—
244	C₂×C₁₆	—	94	2	no	—
245	C₂⁴	—	2	1	no	—
246	C₂⁴	#10625329	1	1	yes	characteristic
247	#41	—	2	4	no	—
248	#41	—	2	8	no	—
249	#41	—	2	4	no	—
250	#41	—	2	4	no	—
251	#41	—	2	4	no	—
252	#41	—	2	2	no	—
253	#41	—	2	2	no	—
254	#41	#10625329	1	4	yes	—
255	#41	C₂²⋊Dic₄₇	1	4	yes	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
259	C₂×D₄	—	2	8	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	4	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
274	C₂²×C₄	#10625329	1	1	yes	characteristic
275	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
276	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	8	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	8	no	—
282	M₁₆	C₂²⋊Dic₄₇	1	4	yes	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	8	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	8	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
294	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
295	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
296	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
297	C₁₆	—	188	2	no	—
298	C₁₆	—	188	2	no	—
299	C₂³	—	2	4	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
307	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
308	C₂³	C₂×#10625329	1	1	yes	characteristic
309	Q₈	—	2	4	no	—
310	Q₈	—	2	2	no	—
311	Q₈	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
312	D₄	—	2	8	no	—
313	D₄	—	2	4	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	D₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
331	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
332	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
333	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
335	C₂×C₄	C₂×#10625329	1	2	yes	—
336	C₈	—	2	1	no	—
337	C₈	—	2	4	no	—
338	C₈	—	2	1	no	—
339	C₈	#13093993	1	2	yes	—
340	C₈	#13093993	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	4	4	no	—
350	C₂²	—	4	8	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	#15688907	1	2	yes	complemented
354	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
355	C₂²	#15689087	1	2	yes	—
356	C₄	—	2	1	no	—
357	C₄	—	2	4	no	—
358	C₄	—	2	1	no	—
359	C₄	#15688979	1	1	yes	characteristic
360	C₄	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
361	C₄	#15689087	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	2	2	no	—
364	C₂	—	4	1	no	—
365	C₂	—	4	4	no	—
366	C₂	#20243298	1	1	yes	characteristic
367	C₂	C₂×#15688907	1	1	yes	characteristic
368	C₂	#20245016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂²×#131×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#12683
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#15688907	C₂	semidirect
2. #20244999	C₂	semidirect
3. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
5. C₂×#13093964	C₂²	semidirect
6. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
169. C₂²×C₄₇	#350	semidirect
206. C₄₇	#12683	semidirect
353. C₂²	#15688907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688907	2
#20244999	2
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂²×#131×C₄₇	2
#12683	47