Subgroups — #26181904

Order 12032.

874 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252016	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241681	C₂	1	2	yes	maximal, complemented
4	#20241674	C₂	1	4	yes	maximal, complemented
5	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#10625324	C₂	1	4	yes	maximal, complemented
7	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₇	C₄	1	2	yes	—
9	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
10	#15689179	C₄	1	4	yes	—
11	#15689177	C₄	1	4	yes	—
12	#15689176	C₂²	1	2	yes	—
13	#15689174	C₂²	1	4	yes	—
14	#15689145	C₄	1	4	yes	—
15	#15689085	C₄	1	4	yes	—
16	C₄×#10625324	C₂²	1	4	yes	—
17	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
19	C₄×C₈×D₄₇	C₄	1	4	yes	—
20	C₄×C₈×D₄₇	C₂²	1	2	yes	—
21	C₄×C₈×D₄₇	C₂²	1	2	yes	—
22	#15688845	C₂²	1	4	yes	complemented
23	#15688844	C₂²	1	4	yes	complemented
24	#15688844	C₂²	1	2	yes	complemented
25	C₈×C₄₇⋊C₈	C₂²	1	2	yes	complemented
26	C₄₇×#431	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄₇×#322	C₂²	1	2	yes	complemented
29	C₈²×C₄₇	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
31	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×#10625324	C₂×C₄	1	8	yes	—
33	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₄₇	—	2	8	no	—
35	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
39	C₄²×D₄₇	C₂×C₄	1	2	yes	—
40	C₄²×D₄₇	C₂×C₄	1	2	yes	—
41	C₄²×D₄₇	C₂×C₄	1	2	yes	—
42	C₄²×D₄₇	C₂³	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
45	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
46	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
47	#13093934	C₂×C₄	1	8	yes	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
50	C₄×C₄₇⋊C₈	C₂³	1	4	yes	—
51	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
56	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₄₇	C₂³	1	4	yes	—
60	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂³×D₄₇	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₄₇	C₄²	1	1	yes	characteristic
65	C₂²×Dic₄₇	C₄²	1	2	yes	—
66	C₂×C₄×D₄₇	—	2	4	no	—
67	C₂×C₄×D₄₇	—	2	8	no	—
68	C₂×C₄×D₄₇	C₄²	1	2	yes	—
69	C₂×C₄×D₄₇	C₄²	1	2	yes	—
70	C₂×C₄×D₄₇	C₄²	1	2	yes	—
71	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	M₁₆	1	8	yes	complemented
75	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₄×Dic₄₇	C₄²	1	2	yes	—
82	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
83	C₄×Dic₄₇	C₄²	1	2	yes	—
84	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
85	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
87	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
91	#10625324	—	2	16	no	—
92	C₈×D₄₇	—	2	8	no	—
93	C₈×D₄₇	—	2	8	no	—
94	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
95	C₂²×D₄₇	—	2	2	no	—
96	C₂²×D₄₇	—	2	4	no	—
97	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
107	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
108	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
109	C₄×D₄₇	—	2	4	no	—
110	C₄×D₄₇	—	2	4	no	—
111	C₄×D₄₇	—	2	4	no	—
112	C₄×D₄₇	—	2	4	no	—
113	C₄×D₄₇	—	2	4	no	—
114	C₄×D₄₇	#131	1	8	yes	—
115	C₈×C₄₇	—	2	8	no	—
116	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
117	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
118	C₄₇⋊C₈	—	2	4	no	—
119	C₄₇⋊C₈	—	2	8	no	—
120	#17323	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
123	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
124	C₂×D₄₇	—	2	8	no	—
125	C₂×D₄₇	—	2	2	no	—
126	C₂×D₄₇	—	2	2	no	—
127	C₂×D₄₇	#405	1	4	yes	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	C₄×M₁₆	1	2	yes	—
130	C₄×C₄₇	C₂²×M₁₆	1	2	yes	—
131	C₄×C₄₇	C₂×#131	1	4	yes	—
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	—	2	2	no	—
134	Dic₄₇	#405	1	4	yes	—
135	#2920	—	47	2	no	—
136	C₂²×C₄×C₈	—	47	1	no	—
137	#1452	—	47	2	no	—
138	#1451	—	47	4	no	—
139	#1450	—	47	1	no	—
140	#1450	D₄₇	1	1	yes	characteristic, complemented
141	C₈×M₁₆	—	47	4	no	—
142	C₂×C₄₇	—	2	2	no	—
143	C₂×C₄₇	C₂×C₄×M₁₆	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#405	1	2	yes	—
145	D₄₇	—	2	4	no	—
146	C₂³×C₈	—	47	2	no	—
147	C₂²×C₄²	—	47	1	no	—
148	#432	—	47	4	no	—
149	#432	—	47	4	no	—
150	#431	—	47	2	no	—
151	#431	—	47	4	no	—
152	#431	C₂×D₄₇	1	2	yes	—
153	#423	—	47	4	no	—
154	#407	—	47	4	no	—
155	C₄×M₁₆	—	47	4	no	—
156	C₂×C₄×C₈	—	47	2	no	—
157	C₂×C₄×C₈	—	47	1	no	—
158	C₂×C₄×C₈	—	47	4	no	—
159	C₂×C₄×C₈	—	47	2	no	—
160	C₂×C₄×C₈	—	47	2	no	—
161	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
162	#322	—	47	4	no	—
163	#322	—	47	2	no	—
164	#322	—	47	4	no	—
165	#322	C₂×D₄₇	1	2	yes	complemented
166	C₈²	—	47	2	no	—
167	C₈²	C₂×D₄₇	1	2	yes	complemented
168	C₄₇	#17323	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	47	1	no	—
170	C₂³×C₄	—	47	2	no	—
171	C₂×M₁₆	—	47	8	no	—
172	C₂²×C₈	—	47	2	no	—
173	C₂²×C₈	—	47	4	no	—
174	C₂²×C₈	—	47	4	no	—
175	C₂²×C₈	—	94	8	no	—
176	C₂²×C₈	C₄×D₄₇	1	2	yes	—
177	C₂×C₄²	—	47	1	no	—
178	C₂×C₄²	—	47	2	no	—
179	C₂×C₄²	—	47	2	no	—
180	C₂×C₄²	—	47	2	no	—
181	C₂×C₄²	—	47	2	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	C₄⋊C₈	—	47	8	no	—
185	C₄⋊C₈	—	47	4	no	—
186	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
187	C₂²⋊C₈	—	47	4	no	—
188	C₂²⋊C₈	—	47	8	no	—
189	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
190	#97	—	47	8	no	—
191	C₄×C₈	—	47	4	no	—
192	C₄×C₈	—	47	4	no	—
193	C₄×C₈	—	47	2	no	—
194	C₄×C₈	—	47	4	no	—
195	C₄×C₈	—	47	2	no	—
196	C₄×C₈	C₄×D₄₇	1	2	yes	—
197	C₄×C₈	C₂²×D₄₇	1	4	yes	—
198	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
200	C₂⁴	—	47	1	no	—
201	C₂²×C₄	—	47	2	no	—
202	C₂²×C₄	—	47	2	no	—
203	C₂²×C₄	—	47	1	no	—
204	C₂²×C₄	—	47	2	no	—
205	C₂²×C₄	—	47	2	no	—
206	C₂²×C₄	—	47	2	no	—
207	C₂²×C₄	—	47	2	no	—
208	C₂²×C₄	—	47	2	no	—
209	C₂²×C₄	—	94	8	no	—
210	C₂²×C₄	—	94	4	no	—
211	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
213	M₁₆	—	94	16	no	—
214	C₂×C₈	—	47	4	no	—
215	C₂×C₈	—	47	8	no	—
216	C₂×C₈	—	94	4	no	—
217	C₂×C₈	—	94	8	no	—
218	C₂×C₈	—	94	8	no	—
219	C₂×C₈	#10625324	1	8	yes	complemented
220	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
221	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
222	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
223	C₄²	—	47	1	no	—
224	C₄²	—	47	2	no	—
225	C₄²	—	47	2	no	—
226	C₄²	—	47	1	no	—
227	C₄²	—	47	2	no	—
228	C₄²	—	47	2	no	—
229	C₄²	—	47	2	no	—
230	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₄₇	1	2	yes	—
232	C₄²	C₂³×D₄₇	1	1	yes	characteristic
233	C₂³	—	47	2	no	—
234	C₂³	—	94	4	no	—
235	C₂³	—	94	2	no	—
236	C₂³	C₄²×D₄₇	1	1	yes	characteristic
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	47	2	no	—
239	C₂×C₄	—	47	8	no	—
240	C₂×C₄	—	47	2	no	—
241	C₂×C₄	—	47	2	no	—
242	C₂×C₄	—	94	4	no	—
243	C₂×C₄	—	94	4	no	—
244	C₂×C₄	—	94	4	no	—
245	C₂×C₄	—	94	2	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	—	94	4	no	—
249	C₂×C₄	C₄²×D₄₇	1	2	yes	—
250	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625324	1	4	yes	—
252	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
254	C₈	—	2	8	no	—
255	C₈	—	94	4	no	—
256	C₈	—	94	8	no	—
257	C₈	C₂×#10625324	1	4	yes	—
258	C₈	C₂×#10625324	1	4	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	47	4	no	—
261	C₂²	—	94	8	no	—
262	C₂²	—	94	2	no	—
263	C₂²	—	94	2	no	—
264	C₂²	C₄×#10625324	1	2	yes	—
265	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	47	4	no	—
268	C₄	—	94	2	no	—
269	C₄	—	94	2	no	—
270	C₄	C₄×#10625324	1	2	yes	—
271	C₄	C₂²×#10625324	1	2	yes	—
272	C₄	#131×D₄₇	1	4	yes	—
273	C₂	—	2	2	no	—
274	C₂	—	94	4	no	—
275	C₂	C₂×C₄×#10625324	1	1	yes	characteristic
276	C₂	#405×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1450	C₂
prime radical	C₂³×D₄₇	C₄²
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1450	D₄₇
hypocenter	C₄₇	#17323
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20241681	C₂	semidirect
4. #20241674	C₂	semidirect
5. #20241671	C₂	semidirect
6. C₈×#10625324	C₂	semidirect
7. C₄₇×#1450	C₂	semidirect
22. #15688845	C₂²	semidirect
23. #15688844	C₂²	semidirect
24. #15688844	C₂²	semidirect
25. C₈×C₄₇⋊C₈	C₂²	semidirect
28. C₄₇×#322	C₂²	semidirect
29. C₈²×C₄₇	C₂²	semidirect
74. C₂×C₈×C₄₇	M₁₆	semidirect
140. #1450	D₄₇	semidirect
165. #322	C₂×D₄₇	semidirect
167. C₈²	C₂×D₄₇	semidirect
168. C₄₇	#17323	semidirect
219. 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
#20252016	2
C₂×C₄×C₈×D₄₇	2
#20241681	2
#20241674	2
#20241671	2
C₈×#10625324	2
C₄₇×#1450	2
#17323	47