Subgroups — #26124995

Order 12032.

666 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689171	C₂	1	2	yes	maximal
2	#20251952	C₂	1	1	yes	maximal, characteristic
3	#20243127	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688888	C₂	1	2	yes	maximal
5	C₄₇×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×#10625329	C₂²	1	2	yes	—
8	#15689171	—	2	4	no	—
9	#15689171	—	2	4	no	—
10	C₂×#13093959	—	2	4	no	—
11	#15689150	—	2	4	no	—
12	#15689147	—	2	2	no	—
13	#15688892	C₄	1	4	yes	—
14	#15688888	—	2	4	no	—
15	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#117×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	C₂×C₄	1	4	yes	—
28	C₂×#10625329	D₄	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	—	2	2	no	—
33	#13093959	—	2	16	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	1	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	D₄	1	2	yes	—
40	C₂²×C₈×C₄₇	—	2	1	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	—	2	1	no	—
44	C₂²×C₈×C₄₇	—	2	2	no	—
45	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
46	#117×C₄₇	—	2	2	no	—
47	#117×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄²×C₄₇	D₄	1	2	yes	—
51	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	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×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	M₁₆×C₄₇	—	2	4	no	—
64	M₁₆×C₄₇	—	4	2	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	—	4	1	no	—
69	C₂×C₈×C₄₇	—	4	1	no	—
70	C₂×C₈×C₄₇	—	4	2	no	—
71	C₂×C₈×C₄₇	—	4	1	no	—
72	C₂×C₈×C₄₇	—	4	1	no	—
73	C₄⋊C₄×C₄₇	—	2	4	no	—
74	C₂²⋊C₄×C₄₇	—	2	4	no	—
75	C₄²×C₄₇	—	2	2	no	—
76	C₄²×C₄₇	—	2	2	no	—
77	#10625329	—	2	4	no	—
78	#10625329	—	2	4	no	—
79	#10625329	—	2	4	no	—
80	#10625329	—	2	4	no	—
81	#10625329	—	2	4	no	—
82	#10625329	—	4	2	no	—
83	#10625329	—	4	2	no	—
84	#10625329	C₄⋊C₄	1	8	yes	—
85	C₂×C₄₇⋊C₈	—	2	4	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	C₂³×C₄₇	—	4	2	no	—
92	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₄₇	#125	1	1	yes	characteristic
95	C₂³×C₄₇	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#120	1	2	yes	—
105	C₂×C₄×C₄₇	#121	1	2	yes	—
106	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
112	C₂×C₄×C₄₇	#123	1	2	yes	—
113	C₂×C₄×C₄₇	#125	1	2	yes	—
114	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
117	C₈×C₄₇	—	4	2	no	—
118	C₈×C₄₇	—	4	2	no	—
119	C₄₇⋊C₈	—	2	8	no	—
120	C₄₇⋊C₈	—	4	4	no	—
121	#11343	—	47	1	no	maximal, Sylow
122	C₂²×C₄₇	—	2	2	no	—
123	C₂²×C₄₇	—	2	4	no	—
124	C₂²×C₄₇	—	4	1	no	—
125	C₂²×C₄₇	—	4	2	no	—
126	C₂²×C₄₇	—	4	1	no	—
127	C₂²×C₄₇	#384	1	1	yes	characteristic
128	C₂²×C₄₇	#385	1	1	yes	characteristic
129	C₂²×C₄₇	#389	1	1	yes	characteristic
130	C₂²×C₄₇	#395	1	2	yes	—
131	C₂²×C₄₇	#398	1	2	yes	—
132	C₄×C₄₇	—	4	4	no	—
133	C₄×C₄₇	#385	1	2	yes	—
134	C₄×C₄₇	#389	1	2	yes	—
135	C₄×C₄₇	#392	1	1	yes	characteristic
136	C₄×C₄₇	#395	1	1	yes	characteristic
137	C₄×C₄₇	#398	1	1	yes	characteristic
138	C₄×C₄₇	#399	1	1	yes	characteristic
139	C₂×#430	—	47	2	no	—
140	#2905	—	47	1	no	—
141	#1742	—	47	1	no	—
142	C₂×#341	—	47	2	no	—
143	#1729	D₄₇	1	1	yes	characteristic
144	C₂×C₄₇	—	2	1	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	#1444	1	1	yes	characteristic
149	C₂×C₄₇	#1858	1	2	yes	—
150	C₂²×M₁₆	—	47	2	no	—
151	C₂²×M₁₆	—	47	2	no	—
152	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
153	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
155	#430	—	94	4	no	—
156	#430	—	94	4	no	—
157	C₂×#108	—	94	4	no	—
158	#424	—	94	4	no	—
159	#423	—	94	2	no	—
160	#343	—	47	4	no	—
161	#341	—	94	4	no	—
162	#336	—	2	2	no	—
163	C₄₇	#11343	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	47	4	no	—
169	C₂×M₁₆	—	47	4	no	—
170	C₂×M₁₆	—	47	4	no	—
171	C₂×M₁₆	—	47	4	no	—
172	C₂×M₁₆	—	94	2	no	—
173	C₂×M₁₆	—	94	2	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	C₂×M₁₆	—	94	2	no	—
180	C₂×M₁₆	D₁₈₈	1	2	yes	—
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	—	2	1	no	—
186	C₂²×C₈	—	94	2	no	—
187	C₂²×C₈	—	94	2	no	—
188	C₂²×C₈	Dic₉₄	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
192	C₂×C₂²⋊C₄	Dic₉₄	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
194	#108	—	94	16	no	—
195	C₄⋊C₈	—	94	8	no	—
196	C₂⁴	C₂×Dic₉₄	1	1	yes	characteristic
197	C₂²×C₄	Dic₄₇⋊C₄	1	4	yes	—
198	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
201	C₂²×C₄	#10625350	1	1	yes	characteristic
202	C₂²×C₄	#10625350	1	1	yes	characteristic
203	C₂²×C₄	D₄×D₄₇	1	2	yes	—
204	C₂²×C₄	D₄×D₄₇	1	2	yes	—
205	C₂²×C₄	#10625352	1	2	yes	—
206	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
207	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	47	8	no	—
211	M₁₆	—	94	4	no	—
212	M₁₆	—	94	4	no	—
213	M₁₆	—	94	4	no	—
214	M₁₆	—	94	4	no	—
215	M₁₆	—	94	4	no	—
216	M₁₆	—	188	2	no	—
217	M₁₆	—	188	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	4	no	—
228	C₂×C₈	—	94	4	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	—	188	2	no	—
231	C₂×C₈	—	188	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
238	C₂³	#13094001	1	2	yes	—
239	C₂³	#13094016	1	1	yes	characteristic
240	C₂³	#13094023	1	2	yes	—
241	C₂³	#13094024	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
245	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
246	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
247	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
248	C₂×C₄	#13093992	1	1	yes	characteristic
249	C₂×C₄	#13093994	1	2	yes	—
250	C₂×C₄	#13093997	1	2	yes	—
251	C₂×C₄	#13093998	1	1	yes	characteristic
252	C₂×C₄	#13094001	1	2	yes	—
253	C₂×C₄	#13094002	1	2	yes	—
254	C₂×C₄	#13094003	1	1	yes	characteristic
255	C₂×C₄	#13094004	1	2	yes	—
256	C₂×C₄	#13094005	1	1	yes	characteristic
257	C₂×C₄	#13094006	1	2	yes	—
258	C₂×C₄	#13094014	1	2	yes	—
259	C₂×C₄	#13094020	1	1	yes	characteristic
260	C₂×C₄	#13094022	1	1	yes	characteristic
261	C₂×C₄	#13094023	1	2	yes	—
262	C₈	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	94	8	no	—
265	C₈	—	188	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	#15688992	1	1	yes	characteristic
272	C₂²	#15688995	1	1	yes	characteristic
273	C₂²	#15689021	1	1	yes	characteristic
274	C₂²	#15689044	1	2	yes	—
275	C₂²	#15689057	1	2	yes	—
276	C₄	—	4	4	no	—
277	C₄	#15688997	1	2	yes	—
278	C₄	#15689020	1	2	yes	—
279	C₄	#15689032	1	1	yes	characteristic
280	C₄	#15689043	1	1	yes	characteristic
281	C₄	#15689053	1	1	yes	characteristic
282	C₄	#15689059	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	#20241652	1	1	yes	characteristic
288	C₂	#20243607	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13093992
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1729	C₂
socle	C₂²×C₄₇	#385
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1729	D₄₇
hypocenter	C₄₇	#11343
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
163. C₄₇	#11343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689171	2
#20251952	2
#20243127	2
C₂×#15688888	2
C₄₇×#1729	2
#11343	47