Subgroups — #26369447

Order 12032.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#20252080	C₂	1	1	yes	maximal, characteristic
3	#20245211	C₂	1	4	yes	maximal, complemented
4	C₁₆×C₂²⋊D₄₇	C₂	1	4	yes	maximal, complemented
5	#20245148	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245002	C₂	1	1	yes	maximal, characteristic, complemented
7	#20244998	C₂	1	1	yes	maximal, characteristic, complemented
8	#20244984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
11	#15689630	C₄	1	1	yes	characteristic
12	C₂×#13093964	C₂²	1	1	yes	characteristic
13	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₄₇	—	2	4	no	—
15	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
16	C₂×C₁₆×D₄₇	C₂²	1	2	yes	—
17	#15689192	C₄	1	4	yes	—
18	C₈×C₂²⋊D₄₇	C₂²	1	4	yes	—
19	#15689151	C₄	1	1	yes	characteristic
20	#15689090	C₄	1	1	yes	characteristic
21	#15689086	C₄	1	1	yes	characteristic
22	#15689078	C₂²	1	1	yes	characteristic
23	Dic₄₇⋊C₁₆	C₂²	1	4	yes	complemented
24	C₂²⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
25	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
26	#15688898	C₂²	1	2	yes	complemented
27	C₁₆×Dic₄₇	C₂²	1	2	yes	complemented
28	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#504	—	2	4	no	—
30	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₇	C₂²	1	2	yes	complemented
32	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	complemented
34	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₄₇	—	2	2	no	—
38	C₂×C₈×D₄₇	—	2	2	no	—
39	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₇	C₂³	1	2	yes	—
41	#13094005	C₈	1	2	yes	—
42	#13094000	C₈	1	2	yes	—
43	#13093995	C₈	1	2	yes	—
44	#13093992	C₈	1	2	yes	—
45	C₄×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
46	#13093983	C₂×C₄	1	1	yes	characteristic
47	#13093964	—	2	2	no	—
48	C₂×C₄₇⋊C₁₆	—	2	2	no	—
49	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
50	C₂×C₄₇⋊C₁₆	C₂³	1	2	yes	—
51	C₁₆×D₄₇	—	2	4	no	—
52	C₁₆×D₄₇	—	2	4	no	—
53	Dic₄₇⋊C₈	C₂×C₄	1	4	yes	—
54	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
55	(C₂×D₄₇)⋊C₈	C₂×C₄	1	4	yes	—
56	#13093934	C₂×C₄	1	2	yes	—
57	C₈×Dic₄₇	C₂³	1	2	yes	—
58	C₂×#41×C₄₇	C₂×C₄	1	1	yes	characteristic
59	#131×C₄₇	—	2	2	no	—
60	#131×C₄₇	—	2	2	no	—
61	C₂×M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	M₃₂×C₄₇	—	2	2	no	—
66	M₃₂×C₄₇	—	2	4	no	—
67	C₂×C₁₆×C₄₇	—	2	4	no	—
68	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
69	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
70	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
72	C₂³×D₄₇	C₂×C₈	1	1	yes	characteristic
73	#41×C₄₇	—	2	2	no	—
74	#41×C₄₇	—	2	2	no	—
75	C₂×Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
77	C₂×D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₄₇	C₂×C₈	1	4	yes	—
81	C₂²×Dic₄₇	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₄₇	—	2	2	no	—
83	C₂×C₄×D₄₇	—	2	2	no	—
84	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
86	M₁₆×C₄₇	—	2	2	no	—
87	M₁₆×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₇	#41	1	2	yes	—
94	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
95	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₄₇	C₂×C₈	1	4	yes	—
97	(C₂×D₄₇)⋊C₄	C₂×C₈	1	4	yes	—
98	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
99	Dic₄₇⋊C₄	C₂×C₈	1	4	yes	—
100	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
101	#10625329	—	2	2	no	—
102	C₂×C₄₇⋊C₈	—	2	2	no	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
105	C₈×D₄₇	—	2	4	no	—
106	C₈×D₄₇	—	2	2	no	—
107	C₈×D₄₇	—	2	2	no	—
108	C₁₆×C₄₇	—	2	2	no	—
109	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
110	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
111	C₄₇⋊C₁₆	—	2	2	no	—
112	C₄₇⋊C₁₆	—	2	2	no	—
113	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
114	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
115	C₂²×D₄₇	—	2	2	no	—
116	C₂²×D₄₇	—	2	4	no	—
117	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
118	Q₈×C₄₇	—	2	2	no	—
119	D₄×C₄₇	—	2	4	no	—
120	D₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	2	2	no	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#131	1	2	yes	—
127	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	2	2	no	—
130	C₂×Dic₄₇	C₂²×C₈	1	4	yes	—
131	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
132	C₄×D₄₇	—	2	2	no	—
133	C₄×D₄₇	—	2	2	no	—
134	C₄×D₄₇	—	2	4	no	—
135	C₈×C₄₇	—	2	2	no	—
136	C₈×C₄₇	C₄×D₄	1	2	yes	—
137	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₄₇⋊C₈	—	2	2	no	—
140	C₄₇⋊C₈	—	2	2	no	—
141	#33432	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	4	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	C₈×D₄	1	2	yes	—
146	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₄₇	—	2	8	no	—
148	C₂×D₄₇	—	2	2	no	—
149	C₂×D₄₇	—	2	2	no	—
150	C₂×D₄₇	#504	1	4	yes	complemented
151	C₄×C₄₇	—	2	2	no	—
152	C₄×C₄₇	C₈×D₄	1	2	yes	—
153	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
155	Dic₄₇	—	2	4	no	—
156	Dic₄₇	—	2	2	no	—
157	C₂×#504	D₄₇	1	1	yes	characteristic, complemented
158	C₂³×C₁₆	—	47	1	no	—
159	#2929	—	47	1	no	—
160	#2168	—	47	4	no	—
161	D₄×C₁₆	—	47	4	no	—
162	#2151	—	47	1	no	—
163	#2113	—	47	1	no	—
164	#2112	—	47	1	no	—
165	#2108	—	47	1	no	—
166	C₂×C₄₇	—	2	2	no	—
167	C₂×C₄₇	—	4	2	no	—
168	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#504	1	2	yes	—
170	D₄₇	—	2	4	no	—
171	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
172	C₂³×C₈	—	47	1	no	—
173	#525	—	47	1	no	—
174	#504	—	2	4	no	—
175	C₂×M₃₂	—	47	1	no	—
176	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
177	C₂×M₃₂	C₂×D₄₇	1	2	yes	complemented
178	C₂²×C₁₆	—	47	1	no	—
179	C₂²×C₁₆	—	47	2	no	—
180	C₂²×C₁₆	—	47	2	no	—
181	C₂²×C₁₆	—	94	4	no	—
182	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
183	C₂²×C₁₆	C₂×D₄₇	1	2	yes	complemented
184	#435	—	47	4	no	—
185	C₈×D₄	—	47	4	no	—
186	#424	—	47	1	no	—
187	#408	—	47	1	no	—
188	#407	—	47	1	no	—
189	#405	—	47	1	no	—
190	C₄⋊C₁₆	—	47	4	no	—
191	C₂²⋊C₁₆	—	47	4	no	—
192	C₂²⋊C₁₆	—	47	4	no	—
193	#346	—	47	2	no	—
194	C₄×C₁₆	—	47	2	no	—
195	C₄₇	#33432	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	C₄×D₄₇	1	1	yes	characteristic
197	C₂³×C₄	—	47	1	no	—
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	C₂×M₁₆	—	47	1	no	—
201	C₂×M₁₆	C₄×D₄₇	1	1	yes	characteristic
202	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
203	C₂²×C₈	—	47	1	no	—
204	C₂²×C₈	—	47	2	no	—
205	C₂²×C₈	—	47	2	no	—
206	C₂²×C₈	—	94	2	no	—
207	C₂²×C₈	—	94	2	no	—
208	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
209	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
210	#123	—	47	2	no	—
211	C₂²⋊Q₈	—	47	2	no	—
212	#121	—	47	2	no	—
213	#120	—	47	2	no	—
214	C₄×D₄	—	47	4	no	—
215	#117	—	47	1	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	2	4	no	—
218	M₃₂	—	94	2	no	—
219	C₂×C₁₆	—	2	4	no	—
220	C₂×C₁₆	—	47	2	no	—
221	C₂×C₁₆	—	47	2	no	—
222	C₂×C₁₆	—	94	2	no	—
223	C₂×C₁₆	—	94	4	no	—
224	C₂×C₁₆	—	94	4	no	—
225	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	complemented
226	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
227	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
228	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
229	C₄⋊C₈	—	47	4	no	—
230	C₂²⋊C₈	—	47	4	no	—
231	C₂²⋊C₈	—	47	4	no	—
232	#97	—	47	2	no	—
233	C₄×C₈	—	47	2	no	—
234	C₂⁴	—	47	1	no	—
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₈×D₄₇	1	1	yes	characteristic
238	C₂×D₄	—	47	4	no	—
239	C₂×D₄	C₈×D₄₇	1	2	yes	—
240	C₂×D₄	C₈×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	—	47	1	no	—
242	C₂²×C₄	—	47	2	no	—
243	C₂²×C₄	—	47	2	no	—
244	C₂²×C₄	—	94	2	no	—
245	C₂²×C₄	—	94	2	no	—
246	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	94	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	47	2	no	—
254	C₂×C₈	—	47	2	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	—	94	4	no	—
257	C₂×C₈	—	94	2	no	—
258	C₂×C₈	—	94	2	no	—
259	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
261	C₂×C₈	#10625350	1	2	yes	—
262	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
263	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
264	C₄⋊C₄	—	47	4	no	—
265	C₄⋊C₄	—	47	2	no	—
266	C₂²⋊C₄	—	47	2	no	—
267	C₂²⋊C₄	—	47	4	no	—
268	C₂²⋊C₄	—	47	4	no	—
269	C₄²	—	47	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	94	2	no	—
272	C₁₆	—	94	2	no	—
273	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
274	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
275	C₂³	—	47	2	no	—
276	C₂³	—	94	4	no	—
277	C₂³	—	94	2	no	—
278	C₂³	C₂×C₈×D₄₇	1	2	yes	—
279	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	2	no	—
283	D₄	—	94	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	47	4	no	—
287	C₂×C₄	—	47	2	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	2	no	—
291	C₂×C₄	—	94	4	no	—
292	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
295	C₂×C₄	#13094030	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	94	2	no	—
299	C₈	—	94	2	no	—
300	C₈	C₄×C₂²⋊D₄₇	1	2	yes	—
301	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
302	C₈	C₂×#10625350	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	47	4	no	—
307	C₂²	—	94	8	no	—
308	C₂²	—	94	2	no	—
309	C₂²	—	94	2	no	—
310	C₂²	C₈×C₂²⋊D₄₇	1	2	yes	—
311	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	94	2	no	—
314	C₄	—	94	4	no	—
315	C₄	C₈×C₂²⋊D₄₇	1	2	yes	—
316	C₄	C₂×C₄×C₂²⋊D₄₇	1	1	yes	characteristic
317	C₄	C₂×#13094030	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	94	4	no	—
321	C₂	C₂×C₈×C₂²⋊D₄₇	1	1	yes	characteristic
322	C₂	#504×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#504	C₂
prime radical	#15689630	C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×#504	D₄₇
hypocenter	C₄₇	#33432
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₆×D₄₇	C₂	semidirect
3. #20245211	C₂	semidirect
4. C₁₆×C₂²⋊D₄₇	C₂	semidirect
5. #20245148	C₂	semidirect
6. #20245002	C₂	semidirect
7. #20244998	C₂	semidirect
8. #20244984	C₂	semidirect
9. C₂×C₄₇×#504	C₂	semidirect
23. Dic₄₇⋊C₁₆	C₂²	semidirect
24. C₂²⋊(C₄₇⋊C₁₆)	C₂²	semidirect
26. #15688898	C₂²	semidirect
27. C₁₆×Dic₄₇	C₂²	semidirect
31. C₂×M₃₂×C₄₇	C₂²	semidirect
33. C₂²×C₁₆×C₄₇	C₂²	semidirect
68. C₂×C₁₆×C₄₇	D₄	semidirect
150. C₂×D₄₇	#504	semidirect
157. C₂×#504	D₄₇	semidirect
177. C₂×M₃₂	C₂×D₄₇	semidirect
183. C₂²×C₁₆	C₂×D₄₇	semidirect
195. C₄₇	#33432	semidirect
225. 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×C₁₆×D₄₇	2
#20252080	2
#20245211	2
C₁₆×C₂²⋊D₄₇	2
#20245148	2
#20245002	2
#20244998	2
#20244984	2
C₂×C₄₇×#504	2
#33432	47