Subgroups — #26216918

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252017	C₂	1	2	yes	maximal
2	C₂×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243558	C₂	1	2	yes	maximal, complemented
4	#20243524	C₂	1	2	yes	maximal, complemented
5	#20243430	C₂	1	1	yes	maximal, characteristic
6	#20243428	C₂	1	2	yes	maximal
7	#20243296	C₂	1	2	yes	maximal, complemented
8	#20243293	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1811	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	#15689177	—	2	4	no	—
13	#15689173	—	2	4	no	—
14	C₂×#13093960	C₂²	1	2	yes	—
15	C₂×#13093959	C₂²	1	4	yes	—
16	#15689146	C₂²	1	4	yes	—
17	#15689087	C₂²	1	2	yes	—
18	M₁₆×Dic₄₇	C₄	1	4	yes	—
19	M₁₆×Dic₄₇	C₄	1	4	yes	—
20	M₁₆×Dic₄₇	C₂²	1	4	yes	—
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688888	C₂²	1	2	yes	—
26	#15688887	C₂²	1	4	yes	—
27	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#108×C₄₇	C₂²	1	2	yes	—
29	C₄₇×#407	C₂²	1	2	yes	—
30	C₄₇×#341	C₂²	1	2	yes	—
31	C₂³×Dic₄₇	Q₈	1	2	yes	—
32	C₂×#10625329	—	2	4	no	—
33	C₂×#10625329	C₂×C₄	1	4	yes	—
34	C₂×#10625329	C₂³	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	D₄	1	4	yes	—
40	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
41	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	#13093960	—	2	4	no	—
44	#13093959	—	2	8	no	—
45	Dic₄₇⋊C₈	—	2	8	no	—
46	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
47	#13093934	—	2	4	no	—
48	#13093934	C₂×C₄	1	4	yes	—
49	#13093934	C₂×C₄	1	4	yes	—
50	C₈×Dic₄₇	—	2	4	no	—
51	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
52	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
53	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
54	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
55	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₄₇	—	2	4	no	—
57	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	#108×C₄₇	—	2	4	no	—
62	C₂²⋊C₈×C₄₇	—	2	4	no	—
63	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₇	—	4	4	no	—
71	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
72	C₂²×Dic₄₇	C₄⋊C₄	1	4	yes	—
73	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
74	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
75	M₁₆×C₄₇	—	2	8	no	—
76	M₁₆×C₄₇	—	2	4	no	—
77	M₁₆×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₄₇	#41	1	4	yes	—
83	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
84	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
85	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
86	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
89	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
90	#10625329	—	2	8	no	—
91	#10625329	—	2	4	no	—
92	#10625329	—	2	4	no	—
93	C₂×C₄₇⋊C₈	—	2	2	no	—
94	C₂×C₄₇⋊C₈	—	2	4	no	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄₇⋊C₈	#41	1	4	yes	—
98	C₂³×C₄₇	—	2	2	no	—
99	C₂³×C₄₇	—	4	2	no	—
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	4	no	—
106	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
111	C₂×Dic₄₇	—	2	8	no	—
112	C₂×Dic₄₇	—	2	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	2	no	—
115	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₈×C₄₇	—	2	2	no	—
119	C₈×C₄₇	—	4	4	no	—
120	C₈×C₄₇	#117	1	4	yes	—
121	C₄₇⋊C₈	—	2	2	no	—
122	C₄₇⋊C₈	—	4	4	no	—
123	C₄₇⋊C₈	#117	1	4	yes	—
124	#20831	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	—	4	4	no	—
129	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄₇	#533	1	2	yes	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
133	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
134	C₄×C₄₇	#532	1	2	yes	—
135	Dic₄₇	—	2	4	no	—
136	Dic₄₇	—	2	2	no	—
137	Dic₄₇	#430	1	4	yes	—
138	#2920	—	47	2	no	—
139	C₂×C₄×M₁₆	—	47	1	no	—
140	#1848	—	47	2	no	—
141	#1837	—	47	2	no	—
142	#1811	—	47	2	no	—
143	#1811	—	47	1	no	—
144	#1811	D₄₇	1	1	yes	characteristic
145	#1778	—	47	2	no	—
146	#1778	—	47	2	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	—	4	2	no	—
149	C₂×C₄₇	#2285	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#430	1	2	yes	—
151	C₂²×M₁₆	—	47	1	no	—
152	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
153	C₂²×C₄²	—	47	1	no	—
154	#432	—	94	4	no	—
155	#431	—	94	4	no	—
156	C₂×#108	—	47	4	no	—
157	C₂×#108	—	47	2	no	—
158	C₂×#108	C₂×D₄₇	1	2	yes	—
159	#423	—	47	4	no	—
160	#407	—	47	2	no	—
161	#407	C₂×D₄₇	1	2	yes	—
162	C₄×M₁₆	—	47	4	no	—
163	C₄×M₁₆	—	47	4	no	—
164	C₄×M₁₆	—	47	4	no	—
165	C₂×#97	—	47	1	no	—
166	C₂×#97	—	47	1	no	—
167	C₂×C₄×C₈	—	47	1	no	—
168	C₂×C₄×C₈	—	47	1	no	—
169	#341	—	47	4	no	—
170	#341	—	47	2	no	—
171	#341	C₂×D₄₇	1	2	yes	—
172	C₄₇	#20831	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	47	2	no	—
174	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	—	47	4	no	—
177	C₂×M₁₆	—	47	4	no	—
178	C₂×M₁₆	—	94	4	no	—
179	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
180	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
181	C₂²×C₈	—	47	1	no	—
182	C₂²×C₈	—	47	1	no	—
183	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	47	2	no	—
187	C₂×C₄²	—	47	4	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	47	1	no	—
190	C₂×C₄²	—	47	1	no	—
191	#108	—	2	4	no	—
192	#108	—	94	8	no	—
193	#108	—	94	4	no	—
194	C₄⋊C₈	—	94	8	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	C₂²⋊C₈	—	94	4	no	—
197	#97	—	47	4	no	—
198	#97	—	47	4	no	—
199	#97	—	94	4	no	—
200	C₄×C₈	—	47	2	no	—
201	C₄×C₈	—	47	2	no	—
202	C₄×C₈	—	47	2	no	—
203	C₄×C₈	—	47	2	no	—
204	C₄×C₈	—	94	4	no	—
205	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	47	4	no	—
209	C₂²×C₄	—	47	2	no	—
210	C₂²×C₄	—	47	4	no	—
211	C₂²×C₄	—	47	2	no	—
212	C₂²×C₄	—	188	4	no	—
213	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
214	C₂²×C₄	D₄×D₄₇	1	2	yes	—
215	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	M₁₆	—	2	8	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	94	8	no	—
221	M₁₆	—	94	4	no	—
222	M₁₆	—	94	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	47	2	no	—
226	C₂×C₈	—	47	4	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	—	94	2	no	—
229	C₂×C₈	—	94	4	no	—
230	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
232	C₂×C₈	#10625352	1	4	yes	—
233	C₄²	—	47	4	no	—
234	C₄²	—	47	4	no	—
235	C₄²	—	47	1	no	—
236	C₄²	—	47	1	no	—
237	C₄²	—	47	2	no	—
238	C₄²	—	47	2	no	—
239	C₄²	—	47	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
243	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	47	2	no	—
249	C₂×C₄	—	47	2	no	—
250	C₂×C₄	—	47	4	no	—
251	C₂×C₄	—	94	4	no	—
252	C₂×C₄	—	94	8	no	—
253	C₂×C₄	—	188	2	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
256	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
258	C₂×C₄	#41×D₄₇	1	2	yes	—
259	C₂×C₄	C₂×#10625352	1	2	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	4	4	no	—
262	C₈	—	47	4	no	—
263	C₈	—	94	2	no	—
264	C₈	—	188	4	no	—
265	C₈	#13093980	1	4	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
271	C₂²	#15689675	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	47	4	no	—
274	C₄	—	94	2	no	—
275	C₄	—	94	4	no	—
276	C₄	#117×D₄₇	1	1	yes	characteristic
277	C₄	C₂×#13093980	1	1	yes	characteristic
278	C₄	#15689668	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	#20245790	1	1	yes	characteristic
282	C₂	#430×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1811	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1811	D₄₇
hypocenter	C₄₇	#20831
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1811

Complements

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

N	H ≅ G/N	type
3. #20243558	C₂	semidirect
4. #20243524	C₂	semidirect
7. #20243296	C₂	semidirect
8. #20243293	C₂	semidirect
172. C₄₇	#20831	semidirect

Maximal subgroups

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

subgroup	index
#20252017	2
C₂×M₁₆×Dic₄₇	2
#20243558	2
#20243524	2
#20243430	2
#20243428	2
#20243296	2
#20243293	2
C₄₇×#1811	2
#20831	47