Subgroups — #26105742

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689145	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241222	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	—	2	2	no	—
7	C₂²×C₈×D₄₇	C₄	1	2	yes	—
8	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
9	#15689145	—	2	4	no	—
10	#15689145	C₄	1	8	yes	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₈	—	2	2	no	—
13	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
15	C₄×C₈×D₄₇	—	2	4	no	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
21	C₂×#10625324	—	2	8	no	—
22	C₂×#10625324	—	4	4	no	—
23	C₂×#10625324	—	4	4	no	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₈×D₄₇	—	2	4	no	—
27	C₂×C₈×D₄₇	—	2	4	no	—
28	C₂×C₈×D₄₇	—	2	4	no	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
32	C₄²×D₄₇	C₂×C₄	1	4	yes	—
33	C₄²×D₄₇	D₄	1	4	yes	—
34	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
36	Dic₄₇⋊C₈	—	2	8	no	—
37	C₄×C₄₇⋊C₈	—	2	2	no	—
38	C₈×Dic₄₇	—	2	4	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₇	—	2	2	no	—
43	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₄₇	—	2	2	no	—
45	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×D₄₇	—	2	2	no	—
53	C₂×C₄×D₄₇	—	2	2	no	—
54	C₂×C₄×D₄₇	—	2	4	no	—
55	C₂×C₄×D₄₇	C₄²	1	2	yes	—
56	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×D₄₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
59	C₂×C₄×D₄₇	D₈	1	2	yes	—
60	C₂×C₄×D₄₇	SD₁₆	1	4	yes	—
61	C₂×C₄×D₄₇	Q₁₆	1	2	yes	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	4	2	no	—
66	C₂×C₈×C₄₇	—	4	2	no	—
67	C₂×C₈×C₄₇	M₁₆	1	4	yes	complemented
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂×D₄	1	2	yes	—
70	C₄×Dic₄₇	C₄²	1	2	yes	—
71	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
72	C₄×Dic₄₇	C₂²⋊C₄	1	4	yes	—
73	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
74	C₂×C₄₇⋊C₈	—	2	2	no	—
75	C₂×C₄₇⋊C₈	—	2	2	no	—
76	C₂×C₄₇⋊C₈	—	2	4	no	—
77	C₂×C₄₇⋊C₈	—	4	2	no	—
78	C₂×C₄₇⋊C₈	—	4	2	no	—
79	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	complemented
80	#10625324	—	4	16	no	—
81	C₈×D₄₇	—	2	16	no	—
82	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×D₄₇	—	2	2	no	—
84	C₂²×D₄₇	—	2	2	no	—
85	C₂²×D₄₇	#95	1	2	yes	—
86	C₂²×D₄₇	#106	1	2	yes	—
87	C₂²×D₄₇	#107	1	2	yes	—
88	C₂×C₄×C₄₇	—	2	2	no	—
89	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
94	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	#95	1	2	yes	—
100	C₂×Dic₄₇	#95	1	2	yes	—
101	C₂×Dic₄₇	#95	1	2	yes	—
102	C₂×Dic₄₇	#106	1	2	yes	—
103	C₂×Dic₄₇	#107	1	2	yes	—
104	C₂×Dic₄₇	#108	1	4	yes	—
105	C₄×D₄₇	—	2	2	no	—
106	C₄×D₄₇	—	2	2	no	—
107	C₄×D₄₇	—	2	4	no	—
108	C₄×D₄₇	—	2	8	no	—
109	C₄×D₄₇	#102	1	8	yes	—
110	C₄×D₄₇	#103	1	8	yes	—
111	C₈×C₄₇	—	2	4	no	—
112	C₈×C₄₇	—	4	4	no	—
113	C₄₇⋊C₈	—	2	4	no	—
114	C₄₇⋊C₈	—	4	4	no	—
115	#7874	—	47	1	no	maximal, Sylow
116	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
117	C₂²×C₄₇	#423	1	2	yes	—
118	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#108	1	2	yes	—
121	C₂×D₄₇	—	2	4	no	—
122	C₂×D₄₇	—	2	4	no	—
123	C₂×D₄₇	—	2	2	no	—
124	C₂×D₄₇	—	2	2	no	—
125	C₂×D₄₇	#340	1	4	yes	—
126	C₄×C₄₇	—	2	2	no	—
127	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
128	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
129	C₄×C₄₇	#407	1	2	yes	—
130	C₄×C₄₇	C₂×#102	1	2	yes	—
131	C₄×C₄₇	C₂×#103	1	2	yes	—
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	#340	1	4	yes	—
134	Dic₄₇	#341	1	4	yes	—
135	C₂×#423	—	47	2	no	—
136	C₂²×C₄×C₈	—	47	1	no	—
137	#1277	—	47	2	no	—
138	#1277	—	47	1	no	—
139	#1277	D₄₇	1	1	yes	characteristic, complemented
140	C₂×C₄₇	#1566	1	2	yes	—
141	C₂×C₄₇	#1567	1	2	yes	—
142	C₂×C₄₇	#1728	1	1	yes	characteristic
143	C₂×C₄₇	C₂×#340	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#341	1	1	yes	characteristic
145	D₄₇	—	2	4	no	—
146	C₂²×M₁₆	—	94	2	no	—
147	C₂³×C₈	—	47	2	no	—
148	C₂²×C₄²	—	47	1	no	—
149	#423	—	47	8	no	—
150	#423	—	94	4	no	—
151	C₂×C₄⋊C₈	—	47	2	no	—
152	C₂×C₄⋊C₈	—	94	2	no	—
153	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
154	C₂×C₄×C₈	—	47	2	no	—
155	C₂×C₄×C₈	—	47	1	no	—
156	C₂×C₄×C₈	—	94	4	no	—
157	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
158	C₄₇	#7874	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	47	1	no	—
160	C₂³×C₄	—	47	1	no	—
161	C₂³×C₄	—	47	1	no	—
162	C₂×M₁₆	—	94	8	no	—
163	C₂×M₁₆	—	188	4	no	—
164	C₂×M₁₆	—	188	4	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	47	2	no	—
167	C₂²×C₈	—	94	4	no	—
168	C₂²×C₈	—	94	4	no	—
169	C₂²×C₈	—	94	4	no	—
170	C₂²×C₈	—	94	2	no	—
171	C₂²×C₈	C₄×D₄₇	1	2	yes	—
172	C₂×C₄²	—	47	1	no	—
173	C₂×C₄²	—	47	1	no	—
174	C₂×C₄²	—	47	1	no	—
175	C₂×C₄²	—	47	4	no	—
176	C₂×C₄²	—	47	4	no	—
177	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	47	4	no	—
180	C₄⋊C₈	—	94	8	no	—
181	C₄⋊C₈	—	94	2	no	—
182	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	94	4	no	—
185	C₄×C₈	—	94	2	no	—
186	C₂⁴	—	47	1	no	—
187	C₂²×C₄	—	47	1	no	—
188	C₂²×C₄	—	47	1	no	—
189	C₂²×C₄	—	47	4	no	—
190	C₂²×C₄	—	47	2	no	—
191	C₂²×C₄	—	47	4	no	—
192	C₂²×C₄	—	47	2	no	—
193	C₂²×C₄	—	47	1	no	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	47	2	no	—
196	C₂²×C₄	—	47	4	no	—
197	C₂²×C₄	—	94	4	no	—
198	C₂²×C₄	—	94	2	no	—
199	C₂²×C₄	—	94	2	no	—
200	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
201	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
203	M₁₆	—	188	16	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	47	4	no	—
210	C₂×C₈	—	94	16	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	—	94	4	no	—
214	C₂×C₈	—	188	2	no	—
215	C₂×C₈	—	188	2	no	—
216	C₂×C₈	#10625324	1	4	yes	complemented
217	C₄²	—	47	2	no	—
218	C₄²	—	47	2	no	—
219	C₄²	—	47	4	no	—
220	C₄²	—	47	4	no	—
221	C₄²	C₂×C₄×D₄₇	1	2	yes	—
222	C₄²	D₄×D₄₇	1	2	yes	—
223	C₂³	—	47	2	no	—
224	C₂³	—	47	2	no	—
225	C₂³	—	47	2	no	—
226	C₂³	—	94	2	no	—
227	C₂³	—	94	2	no	—
228	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	47	2	no	—
231	C₂×C₄	—	47	2	no	—
232	C₂×C₄	—	47	2	no	—
233	C₂×C₄	—	47	2	no	—
234	C₂×C₄	—	47	4	no	—
235	C₂×C₄	—	47	8	no	—
236	C₂×C₄	—	47	8	no	—
237	C₂×C₄	—	47	2	no	—
238	C₂×C₄	—	94	8	no	—
239	C₂×C₄	—	94	2	no	—
240	C₂×C₄	—	94	2	no	—
241	C₂×C₄	—	94	2	no	—
242	C₂×C₄	—	94	4	no	—
243	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
246	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
247	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
248	C₂×C₄	C₂×#10625324	1	2	yes	—
249	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
250	C₂×C₄	SD₁₆×D₄₇	1	2	yes	—
251	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	4	4	no	—
254	C₈	—	94	4	no	—
255	C₈	—	188	4	no	—
256	C₂²	—	47	4	no	—
257	C₂²	—	94	2	no	—
258	C₂²	—	94	2	no	—
259	C₂²	—	94	4	no	—
260	C₂²	—	94	4	no	—
261	C₂²	#95×D₄₇	1	1	yes	characteristic
262	C₂²	#15689145	1	2	yes	—
263	C₂²	#106×D₄₇	1	1	yes	characteristic
264	C₂²	#107×D₄₇	1	1	yes	characteristic
265	C₂²	#108×D₄₇	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	47	4	no	—
268	C₄	—	47	4	no	—
269	C₄	—	94	2	no	—
270	C₄	C₄×#10625324	1	1	yes	characteristic
271	C₄	#15689075	1	1	yes	characteristic
272	C₄	#15689085	1	2	yes	—
273	C₄	#102×D₄₇	1	2	yes	—
274	C₄	#103×D₄₇	1	2	yes	—
275	C₂	—	94	4	no	—
276	C₂	#20242142	1	2	yes	—
277	C₂	#20242145	1	2	yes	—
278	C₂	#20243080	1	1	yes	characteristic
279	C₂	#340×D₄₇	1	1	yes	characteristic
280	C₂	#341×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#7874
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
3. #20241223	C₂	semidirect
4. #20241222	C₂	semidirect
5. C₄₇×#1277	C₂	semidirect
67. C₂×C₈×C₄₇	M₁₆	semidirect
79. C₂×C₄₇⋊C₈	M₁₆	semidirect
139. #1277	D₄₇	semidirect
158. C₄₇	#7874	semidirect
216. C₂×C₈	#10625324	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689145	2
C₂×C₄×C₈×D₄₇	2
#20241223	2
#20241222	2
C₄₇×#1277	2
#7874	47