Subgroups — #26121531

Order 12032.

1518 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	C₂×#15688983	C₂	1	2	yes	maximal
2	C₂×#15688886	C₂	1	4	yes	maximal, complemented
3	C₂³×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
5	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
6	#15688983	—	2	4	no	—
7	#15688983	C₄	1	8	yes	complemented
8	C₂×C₄₇⋊#95	—	2	2	no	—
9	#15688886	—	2	8	no	—
10	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₄₇	—	2	4	no	—
12	C₂²×M₁₆×C₄₇	—	2	8	no	—
13	C₂²×M₁₆×C₄₇	C₄	1	4	yes	complemented
14	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂³×Dic₄₇	—	2	4	no	—
16	C₂×C₄⋊Dic₄₇	—	2	8	no	—
17	C₂×C₄⋊Dic₄₇	—	2	8	no	—
18	C₂×C₄⋊Dic₄₇	—	2	4	no	—
19	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
20	C₂×C₂²⋊Dic₄₇	—	2	8	no	—
21	C₂×C₂²⋊Dic₄₇	—	4	8	no	—
22	C₄₇⋊#95	—	2	8	no	—
23	C₂⁵×C₄₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₇	—	2	4	no	—
25	C₂³×C₄×C₄₇	—	2	4	no	—
26	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₇	D₄	1	4	yes	—
30	C₂³×C₄×C₄₇	Q₈	1	2	yes	—
31	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₄₇	—	2	32	no	—
33	C₂×M₁₆×C₄₇	—	2	8	no	—
34	C₂×M₁₆×C₄₇	—	2	16	no	—
35	C₂²×C₈×C₄₇	—	2	4	no	—
36	C₂²×C₈×C₄₇	—	2	4	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₄₇	D₄	1	8	yes	—
39	C₂⁴×C₄₇	—	2	4	no	—
40	C₂⁴×C₄₇	—	2	4	no	—
41	C₂⁴×C₄₇	—	2	4	no	—
42	C₂⁴×C₄₇	C₂²⋊C₄	1	4	yes	—
43	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
44	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	2	4	no	—
47	C₂²×C₄×C₄₇	—	2	4	no	—
48	C₂²×C₄×C₄₇	—	2	16	no	—
49	C₂²×C₄×C₄₇	—	2	8	no	—
50	C₂²×C₄×C₄₇	—	2	8	no	—
51	C₂²×C₄×C₄₇	—	2	8	no	—
52	C₂²×C₄×C₄₇	C₄²	1	4	yes	—
53	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	4	yes	—
63	C₂²×Dic₄₇	—	2	8	no	—
64	C₂²×Dic₄₇	—	4	8	no	—
65	C₂²×Dic₄₇	—	4	8	no	—
66	C₂²×Dic₄₇	—	4	4	no	—
67	C₂²×Dic₄₇	—	4	4	no	—
68	M₁₆×C₄₇	—	2	32	no	—
69	C₂×C₈×C₄₇	—	2	4	no	—
70	C₂×C₈×C₄₇	—	2	4	no	—
71	C₂×C₈×C₄₇	—	2	16	no	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	complemented
73	C₂²⋊Dic₄₇	—	4	16	no	—
74	C₄⋊Dic₄₇	—	2	16	no	—
75	C₄⋊Dic₄₇	—	4	4	no	—
76	C₄⋊Dic₄₇	—	4	4	no	—
77	C₂³×C₄₇	—	2	4	no	—
78	C₂³×C₄₇	—	2	8	no	—
79	C₂³×C₄₇	—	2	8	no	—
80	C₂³×C₄₇	—	2	4	no	—
81	C₂³×C₄₇	—	2	4	no	—
82	C₂³×C₄₇	—	2	8	no	—
83	C₂³×C₄₇	—	4	8	no	—
84	C₂³×C₄₇	—	4	4	no	—
85	C₂³×C₄₇	—	4	4	no	—
86	C₂³×C₄₇	#95	1	4	yes	—
87	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂³×C₄₇	#121	1	4	yes	—
90	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
91	C₂³×C₄₇	#137	1	4	yes	—
92	C₂×C₄×C₄₇	—	2	4	no	—
93	C₂×C₄×C₄₇	—	2	8	no	—
94	C₂×C₄×C₄₇	—	2	8	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	4	no	—
97	C₂×C₄×C₄₇	—	2	16	no	—
98	C₂×C₄×C₄₇	#95	1	4	yes	—
99	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₇	#117	1	2	yes	—
104	C₂×C₄×C₄₇	C₄×D₄	1	8	yes	—
105	C₂×C₄×C₄₇	#120	1	4	yes	—
106	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
107	C₂×C₄×C₄₇	#123	1	4	yes	—
108	C₂×Dic₄₇	—	4	8	no	—
109	C₂×Dic₄₇	—	4	8	no	—
110	C₂×Dic₄₇	—	4	8	no	—
111	C₂×Dic₄₇	—	8	8	no	—
112	C₂×Dic₄₇	—	8	8	no	—
113	C₈×C₄₇	—	2	8	no	—
114	#10722	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	2	8	no	—
117	C₂²×C₄₇	—	4	8	no	—
118	C₂²×C₄₇	—	4	8	no	—
119	C₂²×C₄₇	—	4	8	no	—
120	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
121	C₂²×C₄₇	#380	1	2	yes	—
122	C₂²×C₄₇	#386	1	4	yes	—
123	C₂²×C₄₇	#417	1	4	yes	—
124	C₂²×C₄₇	#418	1	2	yes	—
125	C₂²×C₄₇	#419	1	2	yes	—
126	C₂²×C₄₇	#428	1	4	yes	—
127	C₂²×C₄₇	#468	1	8	yes	—
128	C₂²×C₄₇	#469	1	4	yes	—
129	C₂²×C₄₇	#470	1	4	yes	—
130	C₄×C₄₇	—	2	4	no	—
131	C₄×C₄₇	C₄×C₂²⋊C₄	1	2	yes	—
132	C₄×C₄₇	#379	1	1	yes	characteristic
133	C₄×C₄₇	#381	1	1	yes	characteristic
134	C₄×C₄₇	#385	1	4	yes	—
135	Dic₄₇	—	8	8	no	—
136	C₂³×M₁₆	D₄₇	1	1	yes	characteristic
137	C₂×#380	—	47	2	no	—
138	C₂×#340	—	47	4	no	—
139	C₂×C₄₇	—	4	4	no	—
140	C₂×C₄₇	C₂²⋊#95	1	1	yes	characteristic
141	C₂×C₄₇	#1736	1	2	yes	—
142	C₂×C₄₇	#1809	1	4	yes	—
143	C₂×C₄₇	#1894	1	4	yes	—
144	C₂×C₄₇	#1895	1	4	yes	—
145	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
146	C₂²×M₁₆	—	2	4	no	—
147	C₂²×M₁₆	—	2	8	no	—
148	C₂²×M₁₆	Dic₄₇	1	4	yes	complemented
149	C₂³×C₈	C₂×D₄₇	1	2	yes	—
150	C₂²×C₄⋊C₄	—	47	4	no	—
151	C₂²×C₂²⋊C₄	—	94	2	no	—
152	#380	—	47	8	no	—
153	#380	—	94	4	no	—
154	C₂×#95	—	94	2	no	—
155	#340	—	94	8	no	—
156	C₄₇	#10722	1	1	yes	characteristic, complemented, Sylow
157	C₂⁵	D₁₈₈	1	1	yes	characteristic
158	C₂³×C₄	—	2	4	no	—
159	C₂³×C₄	—	2	4	no	—
160	C₂³×C₄	—	94	4	no	—
161	C₂³×C₄	Dic₉₄	1	2	yes	—
162	C₂³×C₄	C₄×D₄₇	1	4	yes	—
163	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
165	C₂³×C₄	C₂²⋊D₄₇	1	4	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
167	C₂×M₁₆	—	2	32	no	—
168	C₂×M₁₆	—	2	8	no	—
169	C₂×M₁₆	—	2	16	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
173	C₂²×C₈	C₂²⋊D₄₇	1	8	yes	—
174	C₂×C₄⋊C₄	—	47	8	no	—
175	C₂×C₄⋊C₄	—	94	8	no	—
176	C₂×C₄⋊C₄	—	94	8	no	—
177	C₂×C₄⋊C₄	—	94	4	no	—
178	C₂×C₂²⋊C₄	—	94	8	no	—
179	C₂×C₂²⋊C₄	—	188	8	no	—
180	#95	—	94	8	no	—
181	C₂⁴	—	2	4	no	—
182	C₂⁴	—	2	4	no	—
183	C₂⁴	—	2	4	no	—
184	C₂⁴	C₄⋊Dic₄₇	1	2	yes	—
185	C₂⁴	(C₂×D₄₇)⋊C₄	1	4	yes	—
186	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	94	8	no	—
195	C₂²×C₄	—	188	8	no	—
196	C₂²×C₄	—	188	8	no	—
197	C₂²×C₄	—	188	4	no	—
198	C₂²×C₄	—	188	4	no	—
199	C₂²×C₄	C₄×Dic₄₇	1	4	yes	—
200	C₂²×C₄	Dic₄₇⋊C₄	1	8	yes	—
201	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
202	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
203	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
204	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
206	C₂²×C₄	#10625350	1	4	yes	—
207	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
209	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
210	M₁₆	—	2	32	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	complemented
215	C₄⋊C₄	—	94	16	no	—
216	C₄⋊C₄	—	188	4	no	—
217	C₄⋊C₄	—	188	4	no	—
218	C₂²⋊C₄	—	188	16	no	—
219	C₂³	—	2	4	no	—
220	C₂³	—	2	8	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	4	8	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₄₇⋊#95	1	4	yes	—
229	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
230	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
231	C₂³	#13093998	1	4	yes	—
232	C₂³	#13094052	1	4	yes	—
233	C₂³	#13094057	1	4	yes	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	16	no	—
240	C₂×C₄	—	376	8	no	—
241	C₂×C₄	—	376	8	no	—
242	C₂×C₄	—	188	8	no	—
243	C₂×C₄	—	188	8	no	—
244	C₂×C₄	—	188	8	no	—
245	C₂×C₄	C₄₇⋊#95	1	4	yes	—
246	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
250	C₂×C₄	#13093983	1	2	yes	—
251	C₂×C₄	C₄×C₂²⋊D₄₇	1	8	yes	—
252	C₂×C₄	#13093993	1	4	yes	—
253	C₂×C₄	#13094003	1	4	yes	—
254	C₂×C₄	#13094007	1	4	yes	—
255	C₈	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	8	no	—
259	C₂²	—	4	8	no	—
260	C₂²	—	4	8	no	—
261	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
262	C₂²	#15688983	1	2	yes	—
263	C₂²	#15689002	1	4	yes	—
264	C₂²	#15689125	1	4	yes	—
265	C₂²	#15689129	1	2	yes	—
266	C₂²	#15689133	1	2	yes	—
267	C₂²	#15689165	1	4	yes	—
268	C₂²	#15689369	1	8	yes	—
269	C₂²	#15689376	1	4	yes	—
270	C₂²	#15689381	1	4	yes	—
271	C₄	—	2	4	no	—
272	C₄	—	376	8	no	—
273	C₄	C₄×C₂²⋊Dic₄₇	1	2	yes	—
274	C₄	#15688979	1	1	yes	characteristic
275	C₄	#15688986	1	1	yes	characteristic
276	C₄	#15688996	1	4	yes	—
277	C₂	—	4	4	no	—
278	C₂	#20241611	1	1	yes	characteristic
279	C₂	#20243108	1	2	yes	—
280	C₂	#20243421	1	4	yes	—
281	C₂	#20243821	1	4	yes	—
282	C₂	#20243826	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂³×M₁₆×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
prime radical	C₂⁵×C₄₇	D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂³×M₁₆	D₄₇
hypocenter	C₄₇	#10722
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#15688886	C₂	semidirect
7. #15688983	C₄	semidirect
13. C₂²×M₁₆×C₄₇	C₄	semidirect
72. C₂×C₈×C₄₇	C₂²⋊C₄	semidirect
148. C₂²×M₁₆	Dic₄₇	semidirect
156. C₄₇	#10722	semidirect
214. C₂×C₈	C₂²⋊Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688983	2
C₂×#15688886	2
C₂³×M₁₆×C₄₇	2
#10722	47