Subgroups — #26124475

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic
2	#20243105	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688883	C₂	1	2	yes	maximal
4	#20243084	C₂	1	2	yes	maximal
5	C₂×C₄₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂×#13093960	—	2	4	no	—
9	C₂×#13093960	—	2	2	no	—
10	#15688886	C₄	1	4	yes	complemented
11	#15688883	—	2	4	no	—
12	#15688877	—	2	4	no	—
13	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#336	—	2	2	no	—
17	C₂×#10625329	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	D₄	1	4	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	#13093983	—	2	2	no	—
25	#13093983	—	2	2	no	—
26	#13093983	—	2	2	no	—
27	#13093983	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	#13093960	—	2	8	no	—
32	C₂³×C₄×C₄₇	—	2	1	no	—
33	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₄₇	—	2	2	no	—
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	1	no	—
37	C₂²×C₈×C₄₇	—	2	1	no	—
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₄₇	D₄	1	2	yes	—
42	C₂²×C₈×C₄₇	D₄	1	2	yes	—
43	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
44	C₂×C₄²×C₄₇	—	2	2	no	—
45	C₂×C₄²×C₄₇	—	2	2	no	—
46	C₂×C₄²×C₄₇	—	2	1	no	—
47	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
48	C₂⁴×C₄₇	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	2	4	no	—
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	4	2	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
59	C₂²×C₄×C₄₇	#41	1	2	yes	—
60	C₂²×C₄×C₄₇	#41	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	2	no	—
66	C₂×C₈×C₄₇	—	2	4	no	—
67	C₂×C₈×C₄₇	—	2	2	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	4	1	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	4	2	no	—
73	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
74	C₄²×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	—	4	1	no	—
77	C₂²⋊Dic₄₇	—	4	2	no	—
78	C₂²⋊Dic₄₇	—	4	2	no	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄×Dic₄₇	—	4	2	no	—
82	C₄×Dic₄₇	—	4	2	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	2	4	no	—
85	#10625329	—	4	2	no	—
86	#10625329	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	4	2	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₄₇	#124	1	2	yes	—
95	C₂³×C₄₇	#125	1	1	yes	characteristic
96	C₂³×C₄₇	#126	1	2	yes	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	4	8	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#117	1	2	yes	—
103	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
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	#123	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#123	1	2	yes	—
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#125	1	2	yes	—
116	C₂×C₄×C₄₇	#125	1	2	yes	—
117	C₂×C₄×C₄₇	#126	1	2	yes	—
118	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	4	4	no	—
121	C₂×Dic₄₇	—	4	4	no	—
122	C₈×C₄₇	—	2	4	no	—
123	C₈×C₄₇	—	4	2	no	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#11258	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	1	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	1	no	—
132	C₂²×C₄₇	#388	1	2	yes	—
133	C₂²×C₄₇	#389	1	1	yes	characteristic
134	C₂²×C₄₇	#393	1	1	yes	characteristic
135	C₂²×C₄₇	#400	1	1	yes	characteristic
136	C₂²×C₄₇	#400	1	2	yes	—
137	C₂²×C₄₇	#443	1	2	yes	—
138	C₂²×C₄₇	#444	1	2	yes	—
139	C₄×C₄₇	—	4	4	no	—
140	C₄×C₄₇	#382	1	1	yes	characteristic
141	C₄×C₄₇	#385	1	1	yes	characteristic
142	C₄×C₄₇	#387	1	2	yes	—
143	C₄×C₄₇	#394	1	1	yes	characteristic
144	C₄×C₄₇	#397	1	1	yes	characteristic
145	C₄×C₄₇	#398	1	2	yes	—
146	Dic₄₇	—	8	4	no	—
147	C₂²×#108	—	47	1	no	—
148	#1735	—	47	1	no	—
149	C₂×#339	—	47	2	no	—
150	#1729	—	47	2	no	—
151	C₂×#336	D₄₇	1	1	yes	characteristic
152	C₂×C₄₇	—	2	1	no	—
153	C₂×C₄₇	—	2	1	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	#1442	1	1	yes	characteristic
156	C₂×C₄₇	#1929	1	4	yes	—
157	C₂×C₄₇	#1945	1	1	yes	characteristic
158	C₂×C₄₇	#1946	1	1	yes	characteristic
159	C₂²×M₁₆	—	47	2	no	—
160	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
161	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
162	C₂×#117	—	47	2	no	—
163	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
164	C₂×#108	—	94	4	no	—
165	C₂×#108	—	94	2	no	—
166	#340	—	47	4	no	—
167	#339	—	94	4	no	—
168	#336	—	2	2	no	—
169	#336	—	94	4	no	—
170	C₄₇	#11258	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂×M₁₆	—	47	4	no	—
174	C₂×M₁₆	—	94	2	no	—
175	C₂×M₁₆	—	94	2	no	—
176	C₂×M₁₆	—	94	2	no	—
177	C₂×M₁₆	—	94	2	no	—
178	C₂×M₁₆	—	94	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	94	2	no	—
186	C₂²×C₈	Dic₉₄	1	2	yes	—
187	C₂²×C₈	C₄×D₄₇	1	2	yes	—
188	C₂²×C₈	D₁₈₈	1	2	yes	—
189	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
190	#117	—	94	2	no	—
191	#117	—	94	2	no	—
192	#117	—	94	2	no	—
193	#117	—	94	2	no	—
194	C₂×C₄⋊C₄	—	47	4	no	—
195	C₂×C₂²⋊C₄	—	94	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	94	2	no	—
200	C₂×C₄²	Dic₉₄	1	2	yes	—
201	#108	—	94	8	no	—
202	C₂⁴	#10625350	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	94	4	no	—
208	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
212	C₂²×C₄	#10625350	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₄₇	1	2	yes	—
214	C₂²×C₄	#10625352	1	2	yes	—
215	C₂²×C₄	#10625352	1	2	yes	—
216	C₂²×C₄	#10625354	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	M₁₆	—	94	4	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	188	2	no	—
221	M₁₆	—	188	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	94	4	no	—
232	C₂×C₈	—	188	2	no	—
233	C₂×C₈	—	188	2	no	—
234	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
235	C₄⋊C₄	—	94	4	no	—
236	C₄⋊C₄	—	94	4	no	—
237	C₂²⋊C₄	—	188	2	no	—
238	C₂²⋊C₄	—	188	2	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	188	2	no	—
243	C₄²	—	188	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
247	C₂³	#13094003	1	1	yes	characteristic
248	C₂³	#13094010	1	2	yes	—
249	C₂³	#13094016	1	1	yes	characteristic
250	C₂³	#13094019	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	188	4	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
259	C₂×C₄	#13093982	1	2	yes	—
260	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
261	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
262	C₂×C₄	#13093992	1	1	yes	characteristic
263	C₂×C₄	#13093998	1	1	yes	characteristic
264	C₂×C₄	#13093999	1	2	yes	—
265	C₂×C₄	#13094001	1	2	yes	—
266	C₂×C₄	#13094001	1	2	yes	—
267	C₂×C₄	#13094005	1	1	yes	characteristic
268	C₂×C₄	#13094008	1	2	yes	—
269	C₂×C₄	#13094009	1	2	yes	—
270	C₂×C₄	#13094011	1	1	yes	characteristic
271	C₂×C₄	#13094013	1	1	yes	characteristic
272	C₂×C₄	#13094015	1	2	yes	—
273	C₂×C₄	#13094015	1	2	yes	—
274	C₂×C₄	#13094017	1	2	yes	—
275	C₂×C₄	#13094024	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	188	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	#15689015	1	2	yes	—
286	C₂²	#15689021	1	1	yes	characteristic
287	C₂²	#15689035	1	1	yes	characteristic
288	C₂²	#15689062	1	1	yes	characteristic
289	C₂²	#15689065	1	2	yes	—
290	C₂²	#15689241	1	2	yes	—
291	C₂²	#15689246	1	2	yes	—
292	C₄	—	4	4	no	—
293	C₄	—	376	4	no	—
294	C₄	#15688988	1	1	yes	characteristic
295	C₄	#15688995	1	1	yes	characteristic
296	C₄	#15689010	1	2	yes	—
297	C₄	#15689040	1	1	yes	characteristic
298	C₄	#15689048	1	1	yes	characteristic
299	C₄	#15689055	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	#20241644	1	1	yes	characteristic
304	C₂	#20244037	1	4	yes	—
305	C₂	#20244136	1	1	yes	characteristic
306	C₂	#20244141	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#336	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#336	D₄₇
hypocenter	C₄₇	#11258
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
10. #15688886	C₄	semidirect
170. C₄₇	#11258	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093960	2
#20243105	2
C₂×#15688883	2
#20243084	2
C₂×C₄₇×#336	2
#11258	47