Subgroups — #26369637

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×D₄₇	C₂	1	1	yes	maximal, characteristic
2	#20252235	C₂	1	2	yes	maximal, complemented
3	#20245265	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245264	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245263	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093965	C₂	1	2	yes	maximal, complemented
7	#20245232	C₂	1	2	yes	maximal, complemented
8	#20245171	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245102	C₂	1	2	yes	maximal, complemented
10	#20244986	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#2177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₄₇	C₂²	1	2	yes	—
13	#15689595	—	2	2	no	—
14	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
15	C₂×#13093965	C₂²	1	2	yes	—
16	D₁₆×D₄₇	C₄	1	8	yes	complemented
17	C₂×#13093963	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
19	#15689233	—	2	2	no	—
20	#15689232	—	2	2	no	—
21	C₄×D₃₇₆	C₂²	1	2	yes	complemented
22	D₈×Dic₄₇	C₂²	1	2	yes	complemented
23	#15689205	C₂²	1	2	yes	complemented
24	#15689160	C₂²	1	2	yes	—
25	#15689126	—	2	2	no	—
26	#15689077	C₂²	1	1	yes	characteristic
27	#15688951	C₂²	1	1	yes	characteristic, complemented
28	#15688948	C₂²	1	2	yes	complemented
29	#15688925	C₂²	1	2	yes	complemented
30	#15688924	C₂²	1	2	yes	complemented
31	#15688923	C₂²	1	2	yes	complemented
32	#15688898	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₇	C₂²	1	2	yes	complemented
37	C₄₇×#367	C₂²	1	1	yes	characteristic, complemented
38	C₄₇×#357	C₂²	1	2	yes	complemented
39	C₄₇×#346	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₄₇	—	2	2	no	—
41	C₂×D₃₇₆	C₂³	1	2	yes	—
42	C₂×#10625334	—	2	2	no	—
43	D₈×D₄₇	—	2	4	no	—
44	D₈×D₄₇	C₂×C₄	1	8	yes	complemented
45	C₂×#10625324	D₄	1	1	yes	characteristic
46	C₂×C₈×D₄₇	C₂³	1	1	yes	characteristic
47	C₄×D₁₈₈	—	2	2	no	—
48	D₄×Dic₄₇	—	2	2	no	—
49	#13093986	—	2	2	no	—
50	C₄×C₂²⋊D₄₇	—	4	2	no	—
51	(C₂²⋊D₄₇)⋊C₄	—	4	2	no	—
52	#13093982	—	2	2	no	—
53	#13093981	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₇)⋊C₄	—	4	2	no	—
55	C₂²⋊C₄×D₄₇	—	4	2	no	—
56	D₇₅₂	C₂×C₄	1	4	yes	complemented
57	#13093965	C₂×C₄	1	8	yes	complemented
58	#13093963	—	2	2	no	—
59	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₁₆×D₄₇	C₂×C₄	1	4	yes	complemented
62	#13093958	C₂³	1	2	yes	—
63	#13093955	—	2	2	no	—
64	#13093946	—	2	2	no	—
65	#13093945	—	2	2	no	—
66	#13093944	—	2	2	no	—
67	#13093935	D₄	1	1	yes	characteristic
68	#13093934	D₄	1	1	yes	characteristic
69	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₄₇	C₂³	1	2	yes	—
71	C₄×D₄×C₄₇	—	2	2	no	—
72	D₁₆×C₄₇	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
75	#107×C₄₇	C₂³	1	2	yes	—
76	#102×C₄₇	—	2	2	no	—
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂³×D₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	2	2	no	—
80	C₂²×C₄×C₄₇	—	4	2	no	—
81	C₂×C₂²⋊D₄₇	—	4	2	no	—
82	C₂²×Dic₄₇	—	4	2	no	—
83	D₄×D₄₇	—	2	8	no	—
84	D₄×D₄₇	—	4	4	no	—
85	C₂×D₁₈₈	—	2	2	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×C₄×D₄₇	—	4	2	no	—
88	C₂×C₄×D₄₇	C₂×D₄	1	1	yes	characteristic
89	D₈×C₄₇	—	2	2	no	—
90	D₈×C₄₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₂²⋊C₄×C₄₇	—	4	2	no	—
95	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₄₇	—	4	2	no	—
97	#10625334	—	2	8	no	—
98	(C₂×D₄₇)⋊C₄	—	4	2	no	—
99	(C₂×D₄₇)⋊C₄	—	4	2	no	—
100	(C₂×D₄₇)⋊C₄	—	4	2	no	—
101	(C₂×D₄₇)⋊C₄	D₈	1	2	yes	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	Dic₄₇⋊C₄	D₈	1	2	yes	—
104	C₄×Dic₄₇	—	4	2	no	—
105	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
108	D₃₇₆	—	2	2	no	—
109	D₃₇₆	C₂²×C₄	1	4	yes	—
110	#10625324	—	2	2	no	—
111	C₈×D₄₇	C₂²×C₄	1	2	yes	—
112	C₈×D₄₇	#41	1	2	yes	—
113	C₁₆×C₄₇	—	2	1	no	—
114	C₁₆×C₄₇	C₂²×C₄	1	2	yes	—
115	C₄₇⋊C₁₆	—	2	1	no	—
116	C₄₇⋊C₁₆	C₂²×C₄	1	2	yes	—
117	C₂³×C₄₇	—	4	2	no	—
118	C₂²×D₄₇	—	4	2	no	—
119	C₂²×D₄₇	—	4	8	no	—
120	C₂²×D₄₇	—	16	2	no	—
121	C₂²×D₄₇	—	16	2	no	—
122	C₂²×D₄₇	C₂×D₈	1	1	yes	characteristic
123	D₄×C₄₇	—	2	4	no	—
124	D₄×C₄₇	—	4	2	no	—
125	C₂×C₄×C₄₇	—	4	2	no	—
126	C₂×C₄×C₄₇	—	16	2	no	—
127	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₄₇	—	4	8	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	16	2	no	—
132	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
133	C₂×Dic₄₇	C₂×D₈	1	1	yes	characteristic
134	D₁₈₈	—	2	4	no	—
135	D₁₈₈	—	4	2	no	—
136	C₄×D₄₇	—	16	2	no	—
137	C₄×D₄₇	—	16	2	no	—
138	C₄×D₄₇	C₄×D₄	1	2	yes	—
139	C₄×D₄₇	#135	1	2	yes	—
140	C₈×C₄₇	—	2	1	no	—
141	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
142	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
143	C₄₇⋊C₈	—	2	1	no	—
144	C₄₇⋊C₈	C₄×D₄	1	2	yes	—
145	#33456	—	47	1	no	maximal, Sylow
146	C₂²×C₄₇	—	4	4	no	—
147	C₂²×C₄₇	—	8	2	no	—
148	C₂²×C₄₇	—	8	2	no	—
149	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₄₇	—	2	2	no	—
151	C₂×D₄₇	—	4	4	no	—
152	C₂×D₄₇	—	8	2	no	—
153	C₂×D₄₇	—	8	2	no	—
154	C₂×D₄₇	—	16	8	no	—
155	C₂×D₄₇	C₄×D₈	1	2	yes	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	8	2	no	—
158	C₄×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
160	Dic₄₇	—	2	2	no	—
161	Dic₄₇	—	8	2	no	—
162	Dic₄₇	C₄×D₈	1	2	yes	—
163	C₂²×D₁₆	—	47	1	no	—
164	#2948	—	47	2	no	—
165	#2177	—	47	1	no	—
166	#2177	—	47	1	no	—
167	#2177	—	47	1	no	—
168	#2177	D₄₇	1	1	yes	characteristic, complemented
169	C₄×D₁₆	—	47	2	no	—
170	C₄×D₁₆	—	47	2	no	—
171	#2158	—	47	1	no	—
172	#2140	—	47	2	no	—
173	#2108	—	47	1	no	—
174	C₂×C₄₇	—	8	4	no	—
175	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
176	C₂×C₄₇	#3415	1	2	yes	—
177	D₄₇	—	2	2	no	—
178	D₄₇	—	8	4	no	—
179	C₂²×D₈	—	47	2	no	—
180	#518	—	94	2	no	—
181	C₂×D₁₆	—	47	2	no	—
182	C₂×D₁₆	—	47	8	no	—
183	C₂×D₁₆	—	47	1	no	—
184	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
185	C₂×M₃₂	—	47	1	no	—
186	C₂²×C₁₆	—	47	1	no	—
187	#442	—	94	2	no	—
188	#442	—	94	2	no	—
189	C₄×D₈	—	47	2	no	—
190	C₄×D₈	—	47	2	no	—
191	C₄×D₈	—	47	2	no	—
192	C₄×D₈	C₂×D₄₇	1	2	yes	complemented
193	#427	—	47	2	no	—
194	#418	—	94	2	no	—
195	#405	—	47	1	no	—
196	#367	—	47	1	no	—
197	#367	C₂×D₄₇	1	1	yes	characteristic, complemented
198	#366	—	47	2	no	—
199	#357	—	47	2	no	—
200	#357	—	47	2	no	—
201	#357	—	47	2	no	—
202	#357	C₂×D₄₇	1	2	yes	complemented
203	#346	—	47	1	no	—
204	#346	C₂×D₄₇	1	1	yes	characteristic, complemented
205	C₄×C₁₆	—	47	1	no	—
206	C₄×C₁₆	—	47	1	no	—
207	C₄₇	#33456	1	1	yes	characteristic, complemented, Sylow
208	C₂²×D₄	—	94	2	no	—
209	C₂×D₈	—	47	8	no	—
210	C₂×D₈	—	47	2	no	—
211	C₂×D₈	—	94	4	no	—
212	C₂×D₈	—	94	2	no	—
213	C₂×D₈	C₂²×D₄₇	1	2	yes	—
214	C₂×M₁₆	—	47	1	no	—
215	C₂²×C₈	—	47	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	94	2	no	—
218	C₄×D₄	—	94	2	no	—
219	C₄×D₄	—	94	2	no	—
220	C₄×D₄	—	188	2	no	—
221	C₄×D₄	—	188	2	no	—
222	#117	—	47	1	no	—
223	#117	—	94	2	no	—
224	C₂×C₂²⋊C₄	—	188	2	no	—
225	C₂×C₂²⋊C₄	—	188	2	no	—
226	D₁₆	—	47	8	no	—
227	D₁₆	—	47	4	no	—
228	D₁₆	C₄×D₄₇	1	4	yes	complemented
229	M₃₂	—	94	2	no	—
230	C₂×C₁₆	—	47	1	no	—
231	C₂×C₁₆	—	47	4	no	—
232	C₂×C₁₆	—	47	1	no	—
233	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
235	#107	—	47	2	no	—
236	#107	C₂²×D₄₇	1	2	yes	—
237	#106	—	94	2	no	—
238	#102	—	2	2	no	—
239	#102	—	94	2	no	—
240	#102	—	94	2	no	—
241	#102	—	94	2	no	—
242	#97	—	47	1	no	—
243	#97	—	47	1	no	—
244	C₄×C₈	—	47	1	no	—
245	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
246	C₂⁴	—	188	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	94	8	no	—
249	C₂×D₄	—	94	2	no	—
250	C₂×D₄	—	188	4	no	—
251	C₂×D₄	—	188	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	47	1	no	—
254	C₂²×C₄	—	188	2	no	—
255	C₂²×C₄	—	188	2	no	—
256	C₂²×C₄	—	188	2	no	—
257	D₈	—	2	2	no	—
258	D₈	—	47	4	no	—
259	D₈	—	94	8	no	—
260	D₈	—	94	2	no	—
261	D₈	C₂×C₄×D₄₇	1	4	yes	—
262	M₁₆	—	94	2	no	—
263	C₂×C₈	—	47	1	no	—
264	C₂×C₈	—	47	2	no	—
265	C₂×C₈	—	47	2	no	—
266	C₂×C₈	—	47	1	no	—
267	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	47	2	no	—
271	C₄⋊C₄	—	94	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	47	2	no	—
274	C₂²⋊C₄	—	188	2	no	—
275	C₂²⋊C₄	—	188	2	no	—
276	C₂²⋊C₄	—	188	2	no	—
277	C₂²⋊C₄	—	188	2	no	—
278	C₄²	—	47	1	no	—
279	C₄²	—	188	2	no	—
280	C₄²	D₄×D₄₇	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	47	2	no	—
283	C₁₆	—	94	1	no	—
284	C₁₆	C₂×C₄×D₄₇	1	2	yes	—
285	C₂³	—	4	2	no	—
286	C₂³	—	47	1	no	—
287	C₂³	—	188	8	no	—
288	C₂³	—	188	2	no	—
289	C₂³	—	752	2	no	—
290	C₂³	—	752	2	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	94	4	no	—
294	D₄	—	188	8	no	—
295	D₄	—	188	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	—	47	1	no	—
299	C₂×C₄	—	47	2	no	—
300	C₂×C₄	—	47	2	no	—
301	C₂×C₄	—	47	2	no	—
302	C₂×C₄	—	188	2	no	—
303	C₂×C₄	—	752	2	no	—
304	C₂×C₄	—	752	2	no	—
305	C₂×C₄	—	752	2	no	—
306	C₂×C₄	D₈×D₄₇	1	2	yes	—
307	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	47	2	no	—
310	C₈	—	94	1	no	—
311	C₈	C₂²×C₄×D₄₇	1	1	yes	characteristic
312	C₈	#41×D₄₇	1	1	yes	characteristic
313	C₂²	—	4	4	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	47	2	no	—
317	C₂²	—	94	2	no	—
318	C₂²	—	376	2	no	—
319	C₂²	—	376	2	no	—
320	C₂²	—	188	4	no	—
321	C₂²	—	752	8	no	—
322	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	47	2	no	—
326	C₄	—	94	2	no	—
327	C₄	—	376	2	no	—
328	C₄	C₄×D₄×D₄₇	1	1	yes	characteristic
329	C₄	#135×D₄₇	1	1	yes	characteristic
330	C₂	—	8	4	no	—
331	C₂	—	94	2	no	—
332	C₂	—	376	4	no	—
333	C₂	C₄×D₈×D₄₇	1	1	yes	characteristic
334	C₂	#20256528	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₇
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#2177	C₂
socle	C₂²×C₄₇	C₂²×D₈
prime radical	C₂×D₁₆×D₄₇	C₂
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	#2177	D₄₇
hypocenter	C₄₇	#33456
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

Complements

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

N	H ≅ G/N	type
2. #20252235	C₂	semidirect
3. #20245265	C₂	semidirect
4. #20245264	C₂	semidirect
5. #20245263	C₂	semidirect
6. C₄×#13093965	C₂	semidirect
7. #20245232	C₂	semidirect
8. #20245171	C₂	semidirect
9. #20245102	C₂	semidirect
10. #20244986	C₂	semidirect
11. C₄₇×#2177	C₂	semidirect
16. D₁₆×D₄₇	C₄	semidirect
21. C₄×D₃₇₆	C₂²	semidirect
22. D₈×Dic₄₇	C₂²	semidirect
23. #15689205	C₂²	semidirect
27. #15688951	C₂²	semidirect
28. #15688948	C₂²	semidirect
29. #15688925	C₂²	semidirect
30. #15688924	C₂²	semidirect
31. #15688923	C₂²	semidirect
32. #15688898	C₂²	semidirect
33. C₄×C₄₇⋊C₁₆	C₂²	semidirect
34. C₁₆×Dic₄₇	C₂²	semidirect
36. C₄×D₈×C₄₇	C₂²	semidirect
37. C₄₇×#367	C₂²	semidirect
38. C₄₇×#357	C₂²	semidirect
39. C₄₇×#346	C₂²	semidirect
44. D₈×D₄₇	C₂×C₄	semidirect
56. D₇₅₂	C₂×C₄	semidirect
57. #13093965	C₂×C₄	semidirect
61. C₁₆×D₄₇	C₂×C₄	semidirect
72. D₁₆×C₄₇	C₂×C₄	semidirect
168. #2177	D₄₇	semidirect
192. C₄×D₈	C₂×D₄₇	semidirect
197. #367	C₂×D₄₇	semidirect
202. #357	C₂×D₄₇	semidirect
204. #346	C₂×D₄₇	semidirect
207. C₄₇	#33456	semidirect
228. D₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₄₇	2
#20252235	2
#20245265	2
#20245264	2
#20245263	2
C₄×#13093965	2
#20245232	2
#20245171	2
#20245102	2
#20244986	2
C₄₇×#2177	2
#33456	47