Subgroups — #26216884

Order 12032.

810 conjugacy classes of subgroups.

The subgroup lattice has 306 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	C₂²×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243558	C₂	1	2	yes	maximal, complemented
4	#20243524	C₂	1	2	yes	maximal, complemented
5	#20243426	C₂	1	1	yes	maximal, characteristic
6	#20243424	C₂	1	2	yes	maximal
7	#108×Dic₄₇	C₂	1	2	yes	maximal, complemented
8	#20243289	C₂	1	2	yes	maximal, complemented
9	C₄₇×#1810	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₇⋊C₈	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	C₂²	1	4	yes	—
19	C₂×C₈×Dic₄₇	C₄	1	4	yes	—
20	C₂×C₈×Dic₄₇	C₄	1	4	yes	—
21	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₇	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	C₂³×C₈×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	C₂³	1	4	yes	—
33	C₂²×C₄₇⋊C₈	—	2	4	no	—
34	C₂²×C₄₇⋊C₈	C₂×C₄	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	2	yes	—
40	C₂×C₄×Dic₄₇	D₄	1	2	yes	—
41	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
42	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
44	#13093960	—	2	4	no	—
45	#13093959	—	2	8	no	—
46	Dic₄₇⋊C₈	—	2	8	no	—
47	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
48	#13093934	—	2	4	no	—
49	C₈×Dic₄₇	—	2	4	no	—
50	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
51	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
52	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
53	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
54	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
55	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₄₇	—	2	4	no	—
59	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	#108×C₄₇	—	2	4	no	—
63	C₂²⋊C₈×C₄₇	—	2	4	no	—
64	C₂⁴×C₄₇	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	—	2	2	no	—
66	C₂²×C₄×C₄₇	—	2	2	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	—	2	8	no	—
72	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
73	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
74	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
75	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
76	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
77	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
78	M₁₆×C₄₇	—	2	4	no	—
79	M₁₆×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	2	8	no	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	#41	1	4	yes	—
86	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
87	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
89	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
90	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
91	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
92	C₄×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
94	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
95	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
96	#10625329	—	2	4	no	—
97	#10625329	—	2	4	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	8	no	—
100	C₂×C₄₇⋊C₈	—	2	4	no	—
101	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
103	C₂×C₄₇⋊C₈	#41	1	4	yes	—
104	C₂³×C₄₇	—	2	2	no	—
105	C₂³×C₄₇	—	2	4	no	—
106	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	4	no	—
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	2	4	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	—	2	4	no	—
121	C₂×Dic₄₇	—	2	4	no	—
122	C₂×Dic₄₇	#108	1	8	yes	complemented
123	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	—	4	4	no	—
129	C₈×C₄₇	#117	1	4	yes	—
130	C₄₇⋊C₈	—	2	2	no	—
131	C₄₇⋊C₈	—	4	4	no	—
132	C₄₇⋊C₈	#117	1	4	yes	—
133	#20826	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	8	no	—
135	C₂²×C₄₇	—	2	2	no	—
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	C₂×#108	1	4	yes	—
139	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₂²×C₄₇	#533	1	2	yes	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
143	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
144	C₄×C₄₇	#532	1	2	yes	—
145	Dic₄₇	—	2	2	no	—
146	Dic₄₇	—	2	2	no	—
147	Dic₄₇	C₂×#108	1	4	yes	—
148	Dic₄₇	C₂×#108	1	4	yes	—
149	#2920	—	47	2	no	—
150	C₂²×C₄×C₈	—	47	1	no	—
151	#1848	—	47	2	no	—
152	#1837	—	47	2	no	—
153	#1810	—	47	2	no	—
154	#1810	—	47	1	no	—
155	#1810	D₄₇	1	1	yes	characteristic
156	C₄×#108	—	47	2	no	—
157	C₄×#108	—	47	2	no	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	—	2	4	no	—
160	C₂×C₄₇	#2285	1	1	yes	characteristic
161	C₂×C₄₇	C₂²×#108	1	2	yes	—
162	C₂³×C₈	—	47	1	no	—
163	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
164	C₂²×C₄²	—	47	1	no	—
165	#432	—	94	4	no	—
166	#431	—	94	4	no	—
167	C₂×#108	—	47	4	no	—
168	C₂×#108	—	47	2	no	—
169	C₂×#108	C₂×D₄₇	1	2	yes	—
170	#423	—	47	4	no	—
171	#407	—	47	2	no	—
172	#407	C₂×D₄₇	1	2	yes	—
173	C₄×M₁₆	—	47	4	no	—
174	C₂×C₄×C₈	—	47	4	no	—
175	C₂×C₄×C₈	—	47	4	no	—
176	C₂×C₄×C₈	—	47	1	no	—
177	C₂×C₄×C₈	—	47	1	no	—
178	C₂×C₄×C₈	—	47	1	no	—
179	C₂×C₄×C₈	—	47	1	no	—
180	#341	—	47	4	no	—
181	#341	—	47	2	no	—
182	#341	C₂×D₄₇	1	2	yes	—
183	C₄₇	#20826	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	47	2	no	—
185	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
188	C₂²×C₈	—	2	4	no	—
189	C₂²×C₈	—	47	1	no	—
190	C₂²×C₈	—	47	4	no	—
191	C₂²×C₈	—	47	1	no	—
192	C₂²×C₈	—	94	4	no	—
193	C₂²×C₈	C₄×D₄₇	1	4	yes	—
194	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂×C₄²	—	47	2	no	—
197	C₂×C₄²	—	47	2	no	—
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	47	2	no	—
200	C₂×C₄²	—	47	2	no	—
201	C₂×C₄²	—	47	1	no	—
202	C₂×C₄²	—	47	1	no	—
203	#108	—	2	4	no	—
204	#108	—	94	8	no	—
205	#108	—	94	4	no	—
206	C₄⋊C₈	—	94	8	no	—
207	C₂²⋊C₈	—	2	4	no	—
208	C₂²⋊C₈	—	94	4	no	—
209	#97	—	94	4	no	—
210	C₄×C₈	—	47	4	no	—
211	C₄×C₈	—	47	2	no	—
212	C₄×C₈	—	47	2	no	—
213	C₄×C₈	—	47	2	no	—
214	C₄×C₈	—	47	2	no	—
215	C₄×C₈	—	47	4	no	—
216	C₄×C₈	—	94	4	no	—
217	C₂⁴	Q₈×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	47	2	no	—
223	C₂²×C₄	—	47	2	no	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	2	no	—
226	C₂²×C₄	—	94	8	no	—
227	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
228	C₂²×C₄	D₄×D₄₇	1	2	yes	—
229	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
231	M₁₆	—	2	4	no	—
232	M₁₆	—	2	4	no	—
233	M₁₆	—	94	4	no	—
234	M₁₆	—	94	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	47	2	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	—	94	8	no	—
242	C₂×C₈	—	94	2	no	—
243	C₂×C₈	—	94	4	no	—
244	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
246	C₂×C₈	#10625352	1	4	yes	—
247	C₄²	—	47	2	no	—
248	C₄²	—	47	2	no	—
249	C₄²	—	47	2	no	—
250	C₄²	—	47	2	no	—
251	C₄²	—	47	1	no	—
252	C₄²	—	47	2	no	—
253	C₄²	—	47	1	no	—
254	C₄²	—	47	1	no	—
255	C₄²	—	47	1	no	—
256	C₄²	—	47	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	4	no	—
259	C₂³	C₄⋊C₄×D₄₇	1	2	yes	—
260	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	47	2	no	—
266	C₂×C₄	—	47	2	no	—
267	C₂×C₄	—	47	8	no	—
268	C₂×C₄	—	47	2	no	—
269	C₂×C₄	—	47	2	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	94	4	no	—
274	C₂×C₄	—	94	4	no	—
275	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
278	C₂×C₄	#41×D₄₇	1	2	yes	—
279	C₂×C₄	C₂×#10625352	1	2	yes	—
280	C₈	—	2	2	no	—
281	C₈	—	4	4	no	—
282	C₈	—	47	4	no	—
283	C₈	—	94	2	no	—
284	C₈	—	188	4	no	—
285	C₈	#13093980	1	4	yes	—
286	C₂²	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	#108×D₄₇	1	4	yes	—
291	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
292	C₂²	#15689675	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	47	4	no	—
295	C₄	—	47	4	no	—
296	C₄	—	94	2	no	—
297	C₄	—	94	2	no	—
298	C₄	#117×D₄₇	1	1	yes	characteristic
299	C₄	C₂×#13093980	1	1	yes	characteristic
300	C₄	#15689668	1	2	yes	—
301	C₂	—	2	2	no	—
302	C₂	—	2	4	no	—
303	C₂	#20245790	1	1	yes	characteristic
304	C₂	C₂×#108×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×#1810	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1810	D₄₇
hypocenter	C₄₇	#20826
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1810

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. #108×Dic₄₇	C₂	semidirect
8. #20243289	C₂	semidirect
122. C₂×Dic₄₇	#108	semidirect
183. C₄₇	#20826	semidirect

Maximal subgroups

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

subgroup	index
#20252017	2
C₂²×C₈×Dic₄₇	2
#20243558	2
#20243524	2
#20243426	2
#20243424	2
#108×Dic₄₇	2
#20243289	2
C₄₇×#1810	2
#20826	47