Subgroups — #26124915

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689151	C₂	1	2	yes	maximal
2	#20243105	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688884	C₂	1	2	yes	maximal
4	C₂×#15688877	C₂	1	1	yes	maximal, characteristic
5	C₂²×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15689151	—	2	4	no	—
10	#15689151	—	2	4	no	—
11	C₂×Dic₄₇⋊C₈	—	2	4	no	—
12	#15688886	C₄	1	4	yes	complemented
13	#15688884	—	2	4	no	—
14	#15688877	—	2	2	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#108×C₄₇	—	2	4	no	—
18	C₂×#108×C₄₇	—	2	2	no	—
19	C₂³×Dic₄₇	—	2	1	no	—
20	C₂²×C₄₇⋊C₈	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	#13093983	D₄	1	4	yes	—
28	#13093983	Q₈	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
36	Dic₄₇⋊C₈	—	2	16	no	—
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	2	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	—	2	2	no	—
43	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	—	2	2	no	—
46	C₂²×C₈×C₄₇	—	2	2	no	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₇	D₄	1	2	yes	—
49	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
50	#108×C₄₇	—	2	8	no	—
51	C₂⁴×C₄₇	#41	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂²×Dic₄₇	—	4	2	no	—
66	C₂²×Dic₄₇	—	4	2	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	2	4	no	—
69	M₁₆×C₄₇	—	4	2	no	—
70	M₁₆×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	2	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	4	no	—
76	C₂×C₈×C₄₇	—	4	2	no	—
77	C₂×C₈×C₄₇	—	4	2	no	—
78	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
79	C₂²⋊Dic₄₇	—	2	4	no	—
80	C₂²⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	C₄⋊C₄	1	8	yes	—
83	C₄×Dic₄₇	—	2	4	no	—
84	C₄×Dic₄₇	—	2	4	no	—
85	C₄×Dic₄₇	—	2	4	no	—
86	C₄×Dic₄₇	—	4	1	no	—
87	C₄×Dic₄₇	—	4	1	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂×C₄₇⋊C₈	—	4	1	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	1	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
97	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₄₇	#123	1	2	yes	—
99	C₂³×C₄₇	#125	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#120	1	2	yes	—
105	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
106	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#121	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#125	1	2	yes	—
113	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	8	no	—
120	C₂×Dic₄₇	—	4	8	no	—
121	C₂×Dic₄₇	—	4	2	no	—
122	C₂×Dic₄₇	—	4	2	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₈×C₄₇	—	2	4	no	—
126	C₈×C₄₇	—	4	4	no	—
127	C₄₇⋊C₈	—	4	2	no	—
128	#11329	—	47	1	no	maximal, Sylow
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	4	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	#385	1	2	yes	—
136	C₂²×C₄₇	#389	1	1	yes	characteristic
137	C₂²×C₄₇	#395	1	1	yes	characteristic
138	C₂²×C₄₇	#398	1	2	yes	—
139	C₂²×C₄₇	#399	1	1	yes	characteristic
140	C₂²×C₄₇	#443	1	2	yes	—
141	C₂²×C₄₇	#444	1	2	yes	—
142	C₄×C₄₇	#384	1	1	yes	characteristic
143	C₄×C₄₇	#385	1	1	yes	characteristic
144	C₄×C₄₇	#389	1	2	yes	—
145	C₄×C₄₇	#392	1	1	yes	characteristic
146	C₄×C₄₇	#395	1	2	yes	—
147	C₄×C₄₇	#398	1	1	yes	characteristic
148	Dic₄₇	—	4	4	no	—
149	Dic₄₇	—	4	8	no	—
150	C₂²×#108	D₄₇	1	1	yes	characteristic
151	C₂×#424	—	47	2	no	—
152	#1735	—	47	1	no	—
153	C₂×#339	—	47	2	no	—
154	C₂×#336	—	47	1	no	—
155	C₂×C₄₇	—	2	1	no	—
156	C₂×C₄₇	—	2	1	no	—
157	C₂×C₄₇	—	2	2	no	—
158	C₂×C₄₇	#1444	1	1	yes	characteristic
159	C₂×C₄₇	#1866	1	4	yes	—
160	C₂×C₄₇	#1945	1	1	yes	characteristic
161	C₂×C₄₇	#1946	1	1	yes	characteristic
162	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
163	C₂³×C₈	—	47	1	no	—
164	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂×#117	—	47	2	no	—
166	C₂²×C₄²	—	47	1	no	—
167	C₂×#108	—	2	4	no	—
168	C₂×#108	—	2	2	no	—
169	#424	—	94	4	no	—
170	#424	—	94	4	no	—
171	C₂×C₄⋊C₈	—	94	4	no	—
172	#340	—	47	4	no	—
173	#339	—	94	4	no	—
174	#336	—	94	2	no	—
175	C₄₇	#11329	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	94	1	no	—
177	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	47	2	no	—
188	C₂²×C₈	—	94	2	no	—
189	C₂²×C₈	—	94	1	no	—
190	C₂²×C₈	—	94	1	no	—
191	C₂²×C₈	—	94	2	no	—
192	C₂²×C₈	Dic₉₄	1	2	yes	—
193	C₂²×C₈	C₄×D₄₇	1	2	yes	—
194	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
195	#117	—	47	4	no	—
196	#117	—	47	4	no	—
197	#117	—	94	2	no	—
198	#117	—	94	2	no	—
199	C₂×C₄⋊C₄	—	47	4	no	—
200	C₂×C₂²⋊C₄	—	94	2	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	94	2	no	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	—	94	1	no	—
206	#108	—	2	8	no	—
207	C₄⋊C₈	—	94	16	no	—
208	C₂⁴	#10625354	1	1	yes	characteristic
209	C₂²×C₄	—	94	4	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	—	94	2	no	—
212	C₂²×C₄	—	188	2	no	—
213	C₂²×C₄	—	188	2	no	—
214	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	#10625350	1	2	yes	—
217	C₂²×C₄	D₄×D₄₇	1	2	yes	—
218	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
219	C₂²×C₄	#10625352	1	1	yes	characteristic
220	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
221	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	94	4	no	—
236	C₂×C₈	—	94	2	no	—
237	C₂×C₈	—	188	1	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	—	188	1	no	—
240	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
241	C₄⋊C₄	—	47	8	no	—
242	C₄⋊C₄	—	94	4	no	—
243	C₂²⋊C₄	—	94	4	no	—
244	C₂²⋊C₄	—	94	4	no	—
245	C₄²	—	94	4	no	—
246	C₄²	—	94	4	no	—
247	C₄²	—	94	4	no	—
248	C₄²	—	188	1	no	—
249	C₄²	—	188	1	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	#13093986	1	1	yes	characteristic
253	C₂³	#13094000	1	2	yes	—
254	C₂³	#13094000	1	1	yes	characteristic
255	C₂³	#13094009	1	2	yes	—
256	C₂³	#13094015	1	1	yes	characteristic
257	C₂×C₄	—	94	4	no	—
258	C₂×C₄	—	94	8	no	—
259	C₂×C₄	—	188	2	no	—
260	C₂×C₄	—	188	8	no	—
261	C₂×C₄	—	188	4	no	—
262	C₂×C₄	—	188	4	no	—
263	C₂×C₄	—	188	2	no	—
264	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
265	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
267	C₂×C₄	#13093989	1	1	yes	characteristic
268	C₂×C₄	#13093992	1	2	yes	—
269	C₂×C₄	#13093993	1	1	yes	characteristic
270	C₂×C₄	#13093995	1	1	yes	characteristic
271	C₂×C₄	#13093996	1	2	yes	—
272	C₂×C₄	#13094002	1	2	yes	—
273	C₂×C₄	#13094002	1	2	yes	—
274	C₂×C₄	#13094003	1	2	yes	—
275	C₂×C₄	#13094007	1	1	yes	characteristic
276	C₂×C₄	#13094014	1	2	yes	—
277	C₂×C₄	#13094020	1	1	yes	characteristic
278	C₂×C₄	#13094022	1	1	yes	characteristic
279	C₂×C₄	#13094022	1	2	yes	—
280	C₂×C₄	#13094023	1	2	yes	—
281	C₂×C₄	#13094023	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₈	—	188	2	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	#15688994	1	2	yes	—
292	C₂²	#15689019	1	1	yes	characteristic
293	C₂²	#15689042	1	1	yes	characteristic
294	C₂²	#15689056	1	2	yes	—
295	C₂²	#15689058	1	1	yes	characteristic
296	C₂²	#15689239	1	2	yes	—
297	C₂²	#15689244	1	2	yes	—
298	C₄	—	188	4	no	—
299	C₄	—	188	8	no	—
300	C₄	#15688992	1	1	yes	characteristic
301	C₄	#15688996	1	1	yes	characteristic
302	C₄	#15689018	1	2	yes	—
303	C₄	#15689031	1	1	yes	characteristic
304	C₄	#15689043	1	2	yes	—
305	C₄	#15689054	1	1	yes	characteristic
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	#20241651	1	1	yes	characteristic
310	C₂	#20243660	1	4	yes	—
311	C₂	#20244134	1	1	yes	characteristic
312	C₂	#20244139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#108×C₄₇	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#108	D₄₇
hypocenter	C₄₇	#11329
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
12. #15688886	C₄	semidirect
175. C₄₇	#11329	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689151	2
#20243105	2
C₂×#15688884	2
C₂×#15688877	2
C₂²×#108×C₄₇	2
#11329	47