Subgroups — #26109392

Order 12032.

494 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243516	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243388	C₂	1	2	yes	maximal
3	C₂×#15688874	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241272	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#436	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×#13093945	—	2	2	no	—
8	#15688993	C₂²	1	2	yes	—
9	#15688886	—	2	1	no	—
10	#15688886	—	2	1	no	—
11	#15688885	—	2	2	no	—
12	#15688874	—	2	2	no	—
13	C₂²×M₁₆×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#436	—	2	2	no	—
16	C₄₇×#436	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₄₇	—	2	1	no	—
19	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
26	#13093945	—	4	4	no	—
27	C₂²×D₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	—	2	1	no	—
29	C₂×M₁₆×C₄₇	—	2	4	no	—
30	C₂×M₁₆×C₄₇	—	4	4	no	—
31	C₂²×C₈×C₄₇	—	2	1	no	—
32	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄×D₄×C₄₇	—	2	1	no	—
35	C₄×D₄×C₄₇	—	2	1	no	—
36	C₄×D₄×C₄₇	C₂×C₄	1	2	yes	—
37	C₄×D₄×C₄₇	Q₈	1	2	yes	—
38	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	1	no	—
42	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
43	C₂²⋊C₈×C₄₇	—	2	4	no	—
44	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₄₇	D₄	1	2	yes	—
46	C₂⁴×C₄₇	—	2	1	no	—
47	C₂×D₄×C₄₇	—	2	1	no	—
48	C₂×D₄×C₄₇	—	2	2	no	—
49	C₂×D₄×C₄₇	—	2	2	no	—
50	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	—	2	2	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	—	4	2	no	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	4	2	no	—
60	M₁₆×C₄₇	—	4	8	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	2	2	no	—
64	C₂×C₈×C₄₇	—	4	2	no	—
65	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
69	C₂²⋊C₄×C₄₇	—	2	2	no	—
70	C₂²⋊C₄×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
74	C₄²×C₄₇	#41	1	1	yes	characteristic
75	C₄⋊Dic₄₇	—	4	2	no	—
76	C₄⋊Dic₄₇	—	4	2	no	—
77	C₄⋊Dic₄₇	—	4	2	no	—
78	C₄⋊Dic₄₇	—	4	2	no	—
79	C₄⋊Dic₄₇	—	8	2	no	—
80	C₂×C₄₇⋊C₈	—	4	2	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	4	2	no	—
85	C₂³×C₄₇	—	4	1	no	—
86	C₂³×C₄₇	—	4	1	no	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	#121	1	1	yes	characteristic
89	D₄×C₄₇	—	2	4	no	—
90	D₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	4	2	no	—
94	C₂×C₄×C₄₇	—	4	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	4	no	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	#135	1	2	yes	—
104	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×C₄×C₄₇	#137	1	2	yes	—
107	C₂×Dic₄₇	—	8	2	no	—
108	C₂×Dic₄₇	—	8	2	no	—
109	C₂×Dic₄₇	—	8	2	no	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	—	2	2	no	—
112	C₈×C₄₇	—	4	2	no	—
113	C₄₇⋊C₈	—	8	2	no	—
114	#8656	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	4	2	no	—
116	C₂²×C₄₇	—	4	2	no	—
117	C₂²×C₄₇	—	4	2	no	—
118	C₂²×C₄₇	—	8	2	no	—
119	C₂²×C₄₇	—	8	2	no	—
120	C₂²×C₄₇	#380	1	1	yes	characteristic
121	C₂²×C₄₇	#463	1	2	yes	—
122	C₂²×C₄₇	#469	1	1	yes	characteristic
123	C₂²×C₄₇	#470	1	1	yes	characteristic
124	C₂²×C₄₇	#471	1	2	yes	—
125	C₄×C₄₇	—	2	1	no	—
126	C₄×C₄₇	—	2	1	no	—
127	C₄×C₄₇	—	4	2	no	—
128	C₄×C₄₇	#416	1	1	yes	characteristic
129	C₄×C₄₇	#418	1	1	yes	characteristic
130	C₄×C₄₇	#428	1	2	yes	—
131	C₄×C₄₇	#468	1	2	yes	—
132	C₄×C₄₇	#478	1	2	yes	—
133	Dic₄₇	—	16	2	no	—
134	C₂×#436	D₄₇	1	1	yes	characteristic
135	#1834	—	47	1	no	—
136	#1802	—	47	2	no	—
137	C₂×#334	—	47	1	no	—
138	#1296	—	47	2	no	—
139	C₂×C₄₇	—	8	2	no	—
140	C₂×C₄₇	#1701	1	2	yes	—
141	C₂×C₄₇	#1703	1	2	yes	—
142	C₂×C₄₇	#1804	1	1	yes	characteristic
143	C₂×C₄₇	#1809	1	1	yes	characteristic
144	C₂×C₄₇	#1815	1	1	yes	characteristic
145	C₂²×M₁₆	—	2	1	no	—
146	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
147	#436	—	2	2	no	—
148	#436	Dic₄₇	1	4	yes	complemented
149	C₂×C₄⋊C₈	—	47	2	no	—
150	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
151	C₂×#102	—	94	2	no	—
152	C₂×C₂²⋊C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
154	#384	—	47	2	no	—
155	#340	—	94	1	no	—
156	#340	—	94	1	no	—
157	#340	—	94	2	no	—
158	#334	—	94	2	no	—
159	C₄₇	#8656	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
161	C₂³×C₄	—	2	1	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	4	4	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	94	2	no	—
166	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
167	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	Dic₉₄	1	2	yes	—
171	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
172	C₂×C₄⋊C₄	—	94	2	no	—
173	C₂×C₄⋊C₄	—	94	2	no	—
174	C₂×C₄⋊C₄	—	94	2	no	—
175	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	2	1	no	—
177	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
178	C₄⋊C₈	—	2	1	no	—
179	C₄⋊C₈	—	94	2	no	—
180	C₄⋊C₈	—	94	2	no	—
181	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
182	#102	—	188	4	no	—
183	C₂²⋊C₈	—	2	4	no	—
184	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
185	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
186	C₂⁴	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	C₄⋊Dic₄₇	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	188	2	no	—
197	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
198	C₂²×C₄	#10625350	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
200	M₁₆	—	4	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	188	2	no	—
206	C₂×C₈	—	188	2	no	—
207	C₂×C₈	—	188	2	no	—
208	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
209	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	376	2	no	—
212	C₄⋊C₄	—	188	2	no	—
213	C₄⋊C₄	—	188	2	no	—
214	C₄⋊C₄	—	188	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
220	C₄²	#10625350	1	1	yes	characteristic
221	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	#13093998	1	1	yes	characteristic
229	D₄	—	2	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	376	2	no	—
238	C₂×C₄	—	376	2	no	—
239	C₂×C₄	—	376	2	no	—
240	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
242	C₂×C₄	#13093983	1	1	yes	characteristic
243	C₂×C₄	#13093998	1	1	yes	characteristic
244	C₂×C₄	#13094003	1	1	yes	characteristic
245	C₂×C₄	#13094003	1	1	yes	characteristic
246	C₂×C₄	#13094044	1	2	yes	—
247	C₂×C₄	#13094050	1	2	yes	—
248	C₂×C₄	#13094052	1	2	yes	—
249	C₂×C₄	#13094057	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	4	2	no	—
253	C₈	—	376	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	#15688983	1	1	yes	characteristic
260	C₂²	#15689338	1	2	yes	—
261	C₂²	#15689376	1	1	yes	characteristic
262	C₂²	#15689381	1	1	yes	characteristic
263	C₂²	#15689386	1	2	yes	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	752	2	no	—
268	C₄	#15689119	1	1	yes	characteristic
269	C₄	#15689128	1	1	yes	characteristic
270	C₄	#15689165	1	2	yes	—
271	C₄	#15689369	1	2	yes	—
272	C₄	#15689423	1	2	yes	—
273	C₂	—	8	2	no	—
274	C₂	#20242949	1	2	yes	—
275	C₂	#20242963	1	2	yes	—
276	C₂	#20243398	1	1	yes	characteristic
277	C₂	#20243421	1	1	yes	characteristic
278	C₂	#20243447	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#436	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₄×C₄₇	D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#436	D₄₇
hypocenter	C₄₇	#8656
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #20243516	C₂	semidirect
3. C₂×#15688874	C₂	semidirect
4. #20241272	C₂	semidirect
16. C₄₇×#436	C₄	semidirect
148. #436	Dic₄₇	semidirect
159. C₄₇	#8656	semidirect

Maximal subgroups

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

subgroup	index
#20243516	2
#20243388	2
C₂×#15688874	2
#20241272	2
C₂×C₄₇×#436	2
#8656	47