Subgroups — #26124963

Order 12032.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689172	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689171	C₂	1	2	yes	maximal
3	C₂×C₄₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	#20243082	C₂	1	2	yes	maximal
5	C₄₇×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689172	—	2	4	no	—
9	#15689171	—	2	4	no	—
10	#15689171	—	2	4	no	—
11	C₂×#13093960	—	2	2	no	—
12	C₂×#13093959	—	2	4	no	—
13	C₄₇⋊#338	C₄	1	4	yes	complemented
14	#15688878	—	2	4	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	—	2	2	no	—
19	C₂×#10625329	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	—	2	2	no	—
27	C₂×#10625329	—	2	2	no	—
28	C₂×#10625329	C₂×C₄	1	4	yes	—
29	C₂×#10625329	D₄	1	4	yes	—
30	C₂×#10625329	Q₈	1	4	yes	—
31	C₂²×C₄₇⋊C₈	—	2	2	no	—
32	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
33	#13093960	—	2	8	no	—
34	#13093959	—	2	16	no	—
35	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	—	2	1	no	—
40	C₂×M₁₆×C₄₇	—	2	2	no	—
41	C₂×M₁₆×C₄₇	—	2	2	no	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
44	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
45	C₂²×C₈×C₄₇	—	2	1	no	—
46	C₂²×C₈×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	—	2	1	no	—
48	C₂×C₄²×C₄₇	—	2	2	no	—
49	C₂×C₄²×C₄₇	—	2	1	no	—
50	C₂×C₄²×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	#41	1	2	yes	—
64	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
65	M₁₆×C₄₇	—	2	4	no	—
66	M₁₆×C₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	4	no	—
68	M₁₆×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	—	2	2	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₂×C₈×C₄₇	—	4	1	no	—
74	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	—	4	2	no	—
77	C₄²×C₄₇	—	4	1	no	—
78	#10625329	—	2	4	no	—
79	#10625329	—	2	4	no	—
80	#10625329	—	2	4	no	—
81	#10625329	—	2	4	no	—
82	#10625329	—	2	4	no	—
83	#10625329	—	2	4	no	—
84	#10625329	—	4	2	no	—
85	#10625329	—	4	2	no	—
86	C₂×C₄₇⋊C₈	—	2	4	no	—
87	C₂×C₄₇⋊C₈	—	2	4	no	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	4	2	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
94	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
108	C₂×C₄×C₄₇	#120	1	2	yes	—
109	C₂×C₄×C₄₇	#121	1	2	yes	—
110	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#123	1	2	yes	—
116	C₂×C₄×C₄₇	#125	1	2	yes	—
117	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
118	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
121	C₈×C₄₇	—	2	4	no	—
122	C₈×C₄₇	—	4	2	no	—
123	C₄₇⋊C₈	—	2	8	no	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#11338	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	4	1	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	—	4	1	no	—
131	C₂²×C₄₇	#389	1	2	yes	—
132	C₂²×C₄₇	#389	1	1	yes	characteristic
133	C₂²×C₄₇	#395	1	1	yes	characteristic
134	C₂²×C₄₇	#395	1	2	yes	—
135	C₂²×C₄₇	#399	1	1	yes	characteristic
136	C₄×C₄₇	—	4	4	no	—
137	C₄×C₄₇	#384	1	1	yes	characteristic
138	C₄×C₄₇	#385	1	1	yes	characteristic
139	C₄×C₄₇	#385	1	2	yes	—
140	C₄×C₄₇	#392	1	1	yes	characteristic
141	C₄×C₄₇	#398	1	1	yes	characteristic
142	C₄×C₄₇	#398	1	2	yes	—
143	C₂×#430	—	47	2	no	—
144	C₂×#430	—	47	1	no	—
145	C₂×#338	—	47	1	no	—
146	#1728	—	47	2	no	—
147	#1728	D₄₇	1	1	yes	characteristic
148	C₂×C₄₇	—	2	2	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	—	2	1	no	—
151	C₂×C₄₇	—	2	2	no	—
152	C₂×C₄₇	#1444	1	1	yes	characteristic
153	C₂×C₄₇	#2091	1	2	yes	—
154	C₂²×M₁₆	—	47	2	no	—
155	C₂²×M₁₆	—	47	2	no	—
156	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
157	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
159	#430	—	94	4	no	—
160	#430	—	94	4	no	—
161	#430	—	94	4	no	—
162	C₂×#108	—	94	4	no	—
163	C₂×#108	—	94	2	no	—
164	#338	—	47	4	no	—
165	#336	—	2	2	no	—
166	#336	—	94	4	no	—
167	C₄₇	#11338	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	Dic₉₄	1	2	yes	—
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	47	4	no	—
176	C₂×M₁₆	—	47	4	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	94	2	no	—
179	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	—	94	2	no	—
181	C₂×M₁₆	—	94	2	no	—
182	C₂×M₁₆	—	94	2	no	—
183	C₂×M₁₆	—	94	2	no	—
184	C₂×M₁₆	—	94	2	no	—
185	C₂×M₁₆	—	94	2	no	—
186	C₂×M₁₆	—	94	2	no	—
187	C₂×M₁₆	Dic₉₄	1	2	yes	—
188	C₂×M₁₆	C₄×D₄₇	1	2	yes	—
189	C₂×M₁₆	D₁₈₈	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	47	4	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	#108	—	94	8	no	—
199	#108	—	94	16	no	—
200	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
207	C₂²×C₄	#10625350	1	1	yes	characteristic
208	C₂²×C₄	#10625350	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₄₇	1	2	yes	—
210	C₂²×C₄	D₄×D₄₇	1	2	yes	—
211	C₂²×C₄	#10625352	1	2	yes	—
212	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	94	4	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	94	4	no	—
222	M₁₆	—	94	4	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	—	188	2	no	—
225	M₁₆	—	188	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	47	8	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	94	4	no	—
234	C₂×C₈	—	94	4	no	—
235	C₂×C₈	—	188	2	no	—
236	C₂×C₈	—	188	2	no	—
237	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₄×Dic₉₄	1	1	yes	characteristic
243	C₂³	#13094001	1	2	yes	—
244	C₂³	#13094003	1	1	yes	characteristic
245	C₂³	#13094023	1	2	yes	—
246	C₂³	#13094024	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
253	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
254	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
255	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
256	C₂×C₄	#13093992	1	1	yes	characteristic
257	C₂×C₄	#13093994	1	2	yes	—
258	C₂×C₄	#13093997	1	2	yes	—
259	C₂×C₄	#13093998	1	1	yes	characteristic
260	C₂×C₄	#13094001	1	2	yes	—
261	C₂×C₄	#13094002	1	2	yes	—
262	C₂×C₄	#13094004	1	2	yes	—
263	C₂×C₄	#13094005	1	1	yes	characteristic
264	C₂×C₄	#13094006	1	2	yes	—
265	C₂×C₄	#13094014	1	2	yes	—
266	C₂×C₄	#13094016	1	1	yes	characteristic
267	C₂×C₄	#13094020	1	1	yes	characteristic
268	C₂×C₄	#13094022	1	1	yes	characteristic
269	C₂×C₄	#13094023	1	2	yes	—
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	94	8	no	—
273	C₈	—	188	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	#15689020	1	2	yes	—
280	C₂²	#15689021	1	1	yes	characteristic
281	C₂²	#15689043	1	1	yes	characteristic
282	C₂²	#15689044	1	2	yes	—
283	C₂²	#15689059	1	1	yes	characteristic
284	C₄	—	4	4	no	—
285	C₄	#15688992	1	1	yes	characteristic
286	C₄	#15688995	1	1	yes	characteristic
287	C₄	#15688997	1	2	yes	—
288	C₄	#15689032	1	1	yes	characteristic
289	C₄	#15689053	1	1	yes	characteristic
290	C₄	#15689057	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#20241652	1	1	yes	characteristic
296	C₂	#20244904	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1728	C₂
socle	C₂²×C₄₇	#399
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1728	D₄₇
hypocenter	C₄₇	#11338
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
13. C₄₇⋊#338	C₄	semidirect
167. C₄₇	#11338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689172	2
C₂×#15689171	2
C₂×C₄₇⋊#338	2
#20243082	2
C₄₇×#1728	2
#11338	47