Subgroups — #26369392

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094030	C₂	1	1	yes	maximal, characteristic
2	#20245194	C₂	1	4	yes	maximal, complemented
3	#20245190	C₂	1	2	yes	maximal, complemented
4	#20245186	C₂	1	2	yes	maximal, complemented
5	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245001	C₂	1	2	yes	maximal
7	#20244984	C₂	1	2	yes	maximal, complemented
8	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094030	C₄	1	1	yes	characteristic
10	C₂×#13094030	C₂²	1	1	yes	characteristic
11	C₄×#10625350	C₄	1	1	yes	characteristic
12	C₂×#13093964	C₂²	1	2	yes	—
13	#15689078	C₄	1	1	yes	characteristic
14	#15689078	C₂²	1	1	yes	characteristic
15	#15689077	C₄	1	2	yes	—
16	#15689077	C₂²	1	2	yes	—
17	C₄×#10625329	C₄	1	1	yes	characteristic
18	C₄×#10625324	C₄	1	2	yes	—
19	C₄×C₈×D₄₇	C₂²	1	2	yes	—
20	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
21	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	2	yes	—
22	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
23	(C₂×D₄₇)⋊C₁₆	C₂²	1	4	yes	—
24	#15688898	C₂²	1	4	yes	—
25	C₁₆×Dic₄₇	C₂²	1	4	yes	—
26	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₄₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
32	#13094030	—	2	2	no	—
33	#13094030	—	2	2	no	—
34	#13094030	—	2	4	no	—
35	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
36	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
37	C₂×#10625324	C₂×C₄	1	2	yes	—
38	C₂×#10625324	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₇	C₂³	1	2	yes	—
41	C₄×Dic₉₄	C₈	1	1	yes	characteristic
42	C₄×Dic₉₄	C₈	1	1	yes	characteristic
43	C₄×D₁₈₈	C₈	1	1	yes	characteristic
44	C₄×D₁₈₈	C₈	1	1	yes	characteristic
45	C₄×C₂²⋊D₄₇	C₈	1	2	yes	—
46	C₄×C₂²⋊D₄₇	C₈	1	2	yes	—
47	#13093983	C₂×C₄	1	1	yes	characteristic
48	#13093981	C₂×C₄	1	2	yes	—
49	C₄²×D₄₇	C₂×C₄	1	2	yes	—
50	#13093964	—	2	4	no	—
51	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
52	#13093935	C₂×C₄	1	2	yes	—
53	#13093934	C₂×C₄	1	2	yes	—
54	#13093934	C₂×C₄	1	2	yes	—
55	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
56	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
57	C₈×Dic₄₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₁₆×C₄₇	—	2	4	no	—
62	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
63	C₂×C₁₆×C₄₇	C₂³	1	2	yes	—
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₄₇	C₂×C₈	1	2	yes	—
72	#10625350	—	2	2	no	—
73	#10625350	—	2	2	no	—
74	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₄₇	#41	1	2	yes	—
84	C₂×C₈×C₄₇	#41	1	2	yes	—
85	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
88	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₄₇	C₂×C₈	1	2	yes	—
91	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
92	(C₂×D₄₇)⋊C₄	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₄₇	C₂×C₈	1	1	yes	characteristic
95	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
96	Dic₄₇⋊C₄	C₂×C₈	1	2	yes	—
97	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
98	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
99	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
100	#10625329	—	2	2	no	—
101	#10625329	M₁₆	1	4	yes	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
104	#10625324	—	2	4	no	—
105	#10625324	—	2	4	no	—
106	C₈×D₄₇	—	2	2	no	—
107	C₈×D₄₇	—	2	2	no	—
108	C₈×D₄₇	—	2	4	no	—
109	C₁₆×C₄₇	—	2	2	no	—
110	C₁₆×C₄₇	—	2	2	no	—
111	C₄₇⋊C₁₆	—	2	4	no	—
112	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₄₇	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₄₇	—	2	2	no	—
115	C₂×C₄×C₄₇	—	4	1	no	—
116	C₂×C₄×C₄₇	#117	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₂²⋊D₄₇	—	2	4	no	—
125	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
126	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
127	C₂×Dic₄₇	C₂²×C₈	1	2	yes	—
128	D₁₈₈	—	2	2	no	—
129	C₄×D₄₇	—	2	2	no	—
130	C₄×D₄₇	—	2	2	no	—
131	C₄×D₄₇	—	2	4	no	—
132	C₄×D₄₇	—	4	2	no	—
133	Dic₉₄	—	2	2	no	—
134	C₈×C₄₇	—	2	1	no	—
135	C₈×C₄₇	—	2	1	no	—
136	C₈×C₄₇	#117	1	2	yes	—
137	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
138	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
139	C₄₇⋊C₈	—	2	2	no	—
140	C₄₇⋊C₈	C₂×M₁₆	1	4	yes	—
141	#33423	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	#431	1	2	yes	—
145	C₂²×C₄₇	C₂³×C₈	1	1	yes	characteristic
146	C₂×D₄₇	—	2	2	no	—
147	C₂×D₄₇	—	2	2	no	—
148	C₂×D₄₇	—	2	2	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	#431	1	2	yes	—
153	C₄×C₄₇	#504	1	2	yes	—
154	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
155	C₄×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
156	Dic₄₇	—	2	2	no	—
157	Dic₄₇	—	2	2	no	—
158	Dic₄₇	—	2	2	no	—
159	C₄×#131	—	47	1	no	—
160	#2164	—	47	4	no	—
161	#2163	—	47	2	no	—
162	#2162	—	47	2	no	—
163	#2150	—	47	1	no	—
164	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
165	#2113	—	47	2	no	—
166	#2108	—	47	2	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	C₂×#431	1	1	yes	characteristic
169	C₂×C₄₇	C₂×#504	1	1	yes	characteristic
170	C₂×C₄₇	#3407	1	1	yes	characteristic
171	D₄₇	—	4	2	no	—
172	C₂×#131	—	47	1	no	—
173	C₂×#131	—	47	1	no	—
174	C₄×#41	—	47	1	no	—
175	C₂×M₃₂	—	47	2	no	—
176	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
177	#405	—	47	1	no	—
178	#405	—	47	1	no	—
179	#405	—	47	2	no	—
180	#405	—	47	2	no	—
181	C₄×M₁₆	—	47	1	no	—
182	C₄×M₁₆	—	47	2	no	—
183	C₂×C₄×C₈	—	47	2	no	—
184	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
185	C₄⋊C₁₆	—	47	2	no	—
186	C₄⋊C₁₆	—	47	2	no	—
187	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
188	C₄⋊C₁₆	C₂×D₄₇	1	1	yes	characteristic
189	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
190	C₂²⋊C₁₆	—	47	4	no	—
191	C₂²⋊C₁₆	—	47	4	no	—
192	#346	—	47	4	no	—
193	C₄×C₁₆	—	47	4	no	—
194	C₄₇	#33423	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	—	47	1	no	—
196	#131	—	94	4	no	—
197	#131	—	94	2	no	—
198	#131	—	94	2	no	—
199	C₂×M₁₆	—	47	1	no	—
200	C₂×M₁₆	—	47	1	no	—
201	C₂×M₁₆	—	47	2	no	—
202	C₂×M₁₆	—	47	2	no	—
203	C₂²×C₈	—	47	2	no	—
204	C₂²×C₈	—	47	2	no	—
205	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
207	C₄×Q₈	—	47	1	no	—
208	C₄×Q₈	—	47	1	no	—
209	C₄×D₄	—	47	2	no	—
210	C₄×D₄	—	47	2	no	—
211	C₄×D₄	—	47	1	no	—
212	C₄×D₄	—	47	1	no	—
213	#117	—	47	1	no	—
214	#117	—	47	2	no	—
215	C₂×C₄²	—	47	2	no	—
216	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
217	M₃₂	—	94	4	no	—
218	C₂×C₁₆	—	2	4	no	—
219	C₂×C₁₆	—	47	4	no	—
220	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
221	C₂×C₁₆	C₂²×D₄₇	1	2	yes	—
222	#97	—	47	2	no	—
223	#97	—	47	2	no	—
224	#97	—	47	2	no	—
225	C₄×C₈	—	47	2	no	—
226	C₄×C₈	—	47	2	no	—
227	C₄×C₈	—	47	2	no	—
228	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
229	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
232	#41	—	94	2	no	—
233	#41	—	94	2	no	—
234	C₂×Q₈	—	47	1	no	—
235	C₂×D₄	—	47	2	no	—
236	C₂×D₄	—	47	1	no	—
237	C₂²×C₄	—	47	2	no	—
238	C₂²×C₄	—	47	2	no	—
239	C₂²×C₄	—	47	2	no	—
240	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
241	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
243	M₁₆	—	47	4	no	—
244	M₁₆	—	94	4	no	—
245	M₁₆	—	94	2	no	—
246	M₁₆	—	94	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	47	2	no	—
249	C₂×C₈	—	47	2	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	94	2	no	—
252	C₂×C₈	—	94	2	no	—
253	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂×C₈	#10625352	1	2	yes	—
257	C₂×C₈	#10625354	1	2	yes	—
258	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
259	C₄⋊C₄	—	47	2	no	—
260	C₄⋊C₄	—	47	2	no	—
261	C₄⋊C₄	—	47	1	no	—
262	C₄⋊C₄	—	47	1	no	—
263	C₂²⋊C₄	—	47	2	no	—
264	C₂²⋊C₄	—	47	2	no	—
265	C₂²⋊C₄	—	47	2	no	—
266	C₄²	—	47	2	no	—
267	C₄²	—	47	2	no	—
268	C₄²	—	47	2	no	—
269	C₄²	C₈×D₄₇	1	1	yes	characteristic
270	C₄²	C₈×D₄₇	1	1	yes	characteristic
271	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₁₆	—	2	2	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	—	94	4	no	—
276	C₂³	—	47	2	no	—
277	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
278	Q₈	—	94	2	no	—
279	D₄	—	94	4	no	—
280	D₄	—	94	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	47	2	no	—
284	C₂×C₄	—	47	2	no	—
285	C₂×C₄	—	47	2	no	—
286	C₂×C₄	—	94	4	no	—
287	C₂×C₄	—	94	2	no	—
288	C₂×C₄	—	94	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	#13093982	1	2	yes	—
291	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
292	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	47	4	no	—
301	C₈	—	94	2	no	—
302	C₈	#13093982	1	2	yes	—
303	C₈	C₂×#10625352	1	1	yes	characteristic
304	C₈	C₂×#10625354	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	94	2	no	—
308	C₂²	—	94	2	no	—
309	C₂²	—	94	2	no	—
310	C₂²	#15689175	1	2	yes	—
311	C₂²	C₂²×C₈×D₄₇	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	94	2	no	—
316	C₄	—	94	2	no	—
317	C₄	—	94	2	no	—
318	C₄	#15689175	1	2	yes	—
319	C₄	#15689539	1	2	yes	—
320	C₄	C₂×#13093982	1	1	yes	characteristic
321	C₄	C₂×M₁₆×D₄₇	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	188	2	no	—
324	C₂	C₂×#15689175	1	1	yes	characteristic
325	C₂	C₂×#15689539	1	1	yes	characteristic
326	C₂	#20256490	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₈
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#33423
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #20245194	C₂	semidirect
3. #20245190	C₂	semidirect
4. #20245186	C₂	semidirect
5. #20245144	C₂	semidirect
7. #20244984	C₂	semidirect
8. C₂×C₄₇×C₄⋊C₁₆	C₂	semidirect
164. C₂×C₄⋊C₁₆	D₄₇	semidirect
194. C₄₇	#33423	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094030	2
#20245194	2
#20245190	2
#20245186	2
#20245144	2
#20245001	2
#20244984	2
C₂×C₄₇×C₄⋊C₁₆	2
#33423	47