Subgroups — #26125201

Order 12032.

1038 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688885	C₂	1	4	yes	maximal
2	C₂×#15688878	C₂	1	1	yes	maximal, characteristic
3	#20241618	C₂	1	1	yes	maximal, characteristic
4	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄₇⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂×C₄₇⋊#95	—	2	2	no	—
8	#15688885	—	2	8	no	—
9	#15688878	—	2	2	no	—
10	C₂×#125×C₄₇	—	2	8	no	—
11	C₂×#125×C₄₇	—	2	2	no	—
12	C₂×#125×C₄₇	C₄	1	4	yes	complemented
13	C₂²×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×Dic₄₇	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	C₂²×C₄₇⋊C₈	—	2	4	no	—
19	C₂²×C₄₇⋊C₈	—	2	2	no	—
20	C₂²×C₄₇⋊C₈	—	2	4	no	—
21	C₂²×C₄₇⋊C₈	Q₈	1	4	yes	—
22	C₂×C₄⋊Dic₄₇	—	2	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	Q₈	1	4	yes	—
26	C₄₇⋊#95	—	4	4	no	—
27	C₂³×C₄×C₄₇	—	2	2	no	—
28	C₂³×C₄×C₄₇	D₄	1	2	yes	—
29	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	#125×C₄₇	—	2	16	no	—
31	#125×C₄₇	—	2	16	no	—
32	C₂×C₄⋊C₄×C₄₇	—	2	16	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
34	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
38	C₂×C₄²×C₄₇	—	2	4	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	D₄	1	2	yes	—
41	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	—	2	4	no	—
43	C₂²×C₄×C₄₇	—	2	4	no	—
44	C₂²×C₄×C₄₇	—	2	8	no	—
45	C₂²×C₄×C₄₇	—	2	4	no	—
46	C₂²×C₄×C₄₇	—	2	2	no	—
47	C₂²×C₄×C₄₇	—	2	2	no	—
48	C₂²×C₄×C₄₇	—	4	4	no	—
49	C₂²×C₄×C₄₇	—	4	2	no	—
50	C₂²×C₄×C₄₇	—	4	2	no	—
51	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₄₇	#41	1	2	yes	—
58	C₂²×C₄×C₄₇	#41	1	2	yes	—
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	4	2	no	—
61	C₂²×Dic₄₇	—	4	2	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	C₂²×Dic₄₇	—	4	4	no	—
64	C₄⋊C₄×C₄₇	—	2	32	no	—
65	C₄⋊C₄×C₄₇	—	2	4	no	—
66	C₄⋊C₄×C₄₇	—	2	8	no	—
67	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
68	C₄²×C₄₇	—	2	4	no	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄⋊Dic₄₇	—	2	8	no	—
71	C₄⋊Dic₄₇	—	4	4	no	—
72	C₂×C₄₇⋊C₈	—	2	8	no	—
73	C₂×C₄₇⋊C₈	—	2	4	no	—
74	C₂×C₄₇⋊C₈	—	4	2	no	—
75	C₂×C₄₇⋊C₈	—	4	4	no	—
76	C₂×C₄₇⋊C₈	—	4	2	no	—
77	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₄₇	#120	1	2	yes	—
79	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₄₇	#124	1	1	yes	characteristic
81	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
82	C₂³×C₄₇	#135	1	4	yes	—
83	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
84	C₂³×C₄₇	#137	1	2	yes	—
85	C₂×C₄×C₄₇	—	2	4	no	—
86	C₂×C₄×C₄₇	—	2	8	no	—
87	C₂×C₄×C₄₇	—	2	4	no	—
88	C₂×C₄×C₄₇	—	2	4	no	—
89	C₂×C₄×C₄₇	—	4	4	no	—
90	C₂×C₄×C₄₇	—	4	4	no	—
91	C₂×C₄×C₄₇	—	4	4	no	—
92	C₂×C₄×C₄₇	—	4	8	no	—
93	C₂×C₄×C₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	4	8	no	—
95	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₄₇	#120	1	2	yes	—
99	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
100	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
101	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₄₇	#124	1	4	yes	—
103	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×Dic₄₇	—	4	4	no	—
107	C₂×Dic₄₇	—	4	8	no	—
108	C₂×Dic₄₇	—	8	4	no	—
109	C₂×Dic₄₇	—	8	2	no	—
110	C₂×Dic₄₇	—	8	2	no	—
111	C₄₇⋊C₈	—	4	4	no	—
112	#11385	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	#379	1	1	yes	characteristic
114	C₂²×C₄₇	#391	1	1	yes	characteristic
115	C₂²×C₄₇	#391	1	1	yes	characteristic
116	C₂²×C₄₇	#393	1	2	yes	—
117	C₂²×C₄₇	#395	1	2	yes	—
118	C₂²×C₄₇	#416	1	2	yes	—
119	C₂²×C₄₇	#417	1	2	yes	—
120	C₂²×C₄₇	#449	1	4	yes	—
121	C₂²×C₄₇	#452	1	4	yes	—
122	C₂²×C₄₇	#478	1	4	yes	—
123	C₂²×C₄₇	#479	1	4	yes	—
124	C₂²×C₄₇	#488	1	2	yes	—
125	C₂²×C₄₇	#490	1	1	yes	characteristic
126	C₂²×C₄₇	#491	1	1	yes	characteristic
127	C₂²×C₄₇	#499	1	2	yes	—
128	C₂²×C₄₇	#501	1	2	yes	—
129	C₄×C₄₇	—	4	4	no	—
130	C₄×C₄₇	—	4	8	no	—
131	C₄×C₄₇	#391	1	4	yes	—
132	C₄×C₄₇	#393	1	2	yes	—
133	C₄×C₄₇	#395	1	2	yes	—
134	Dic₄₇	—	8	4	no	—
135	C₂²×#125	D₄₇	1	1	yes	characteristic
136	C₂×#340	—	47	4	no	—
137	C₂×#336	—	47	1	no	—
138	C₄⋊#95	—	47	1	no	—
139	C₂×C₄₇	#1446	1	1	yes	characteristic
140	C₂×C₄₇	#1788	1	2	yes	—
141	C₂×C₄₇	#1989	1	2	yes	—
142	C₂×C₄₇	#1991	1	1	yes	characteristic
143	C₂×C₄₇	#1992	1	1	yes	characteristic
144	C₂×C₄₇	#2025	1	2	yes	—
145	C₂×C₄₇	#2026	1	2	yes	—
146	C₂×C₄₇	#2059	1	4	yes	—
147	C₂³×C₈	—	47	2	no	—
148	C₂×#125	—	2	8	no	—
149	C₂×#125	—	2	2	no	—
150	C₂×#125	Dic₄₇	1	4	yes	complemented
151	C₂²×C₄⋊C₄	—	2	2	no	—
152	C₂²×C₄⋊C₄	—	47	2	no	—
153	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
154	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
155	C₂×#95	—	94	2	no	—
156	#340	—	94	8	no	—
157	#336	—	94	2	no	—
158	C₄₇	#11385	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	2	2	no	—
160	C₂³×C₄	—	94	2	no	—
161	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂²×C₈	—	47	4	no	—
164	C₂²×C₈	—	94	2	no	—
165	C₂²×C₈	—	94	4	no	—
166	C₂²×C₈	—	94	2	no	—
167	C₂²×C₈	—	94	4	no	—
168	#125	—	2	16	no	—
169	#125	—	2	16	no	—
170	C₂×C₄⋊C₄	—	2	16	no	—
171	C₂×C₄⋊C₄	—	2	8	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	47	4	no	—
175	C₂×C₄⋊C₄	—	94	4	no	—
176	C₂×C₄⋊C₄	—	94	4	no	—
177	C₂×C₄⋊C₄	—	94	2	no	—
178	C₂×C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
179	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	8	yes	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
182	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
183	#95	—	188	4	no	—
184	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
185	C₂²×C₄	—	2	4	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	94	4	no	—
195	C₂²×C₄	—	188	2	no	—
196	C₂²×C₄	—	188	2	no	—
197	C₂²×C₄	—	188	4	no	—
198	C₂²×C₄	—	188	4	no	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
200	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
201	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
202	C₂²×C₄	#10625354	1	2	yes	—
203	C₂²×C₄	#10625354	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
207	C₂×C₈	—	94	8	no	—
208	C₂×C₈	—	94	4	no	—
209	C₂×C₈	—	188	2	no	—
210	C₂×C₈	—	188	4	no	—
211	C₂×C₈	—	188	2	no	—
212	C₄⋊C₄	—	2	32	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	94	8	no	—
216	C₄⋊C₄	—	188	4	no	—
217	C₄⋊C₄	C₂²⋊Dic₄₇	1	8	yes	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
221	C₂³	#13093993	1	2	yes	—
222	C₂³	#13094000	1	2	yes	—
223	C₂³	#13094013	1	1	yes	characteristic
224	C₂³	#13094022	1	1	yes	characteristic
225	C₂³	#13094044	1	4	yes	—
226	C₂³	#13094049	1	2	yes	—
227	C₂³	#13094054	1	2	yes	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	376	2	no	—
239	C₂×C₄	—	376	2	no	—
240	C₂×C₄	—	376	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	8	no	—
243	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
244	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
245	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
246	C₂×C₄	#13093993	1	2	yes	—
247	C₂×C₄	#13094000	1	4	yes	—
248	C₂×C₄	#13094000	1	4	yes	—
249	C₂×C₄	#13094000	1	2	yes	—
250	C₂×C₄	#13094013	1	4	yes	—
251	C₂×C₄	#13094013	1	1	yes	characteristic
252	C₂×C₄	#13094022	1	4	yes	—
253	C₂×C₄	#13094022	1	1	yes	characteristic
254	C₈	—	188	4	no	—
255	C₂²	#15688979	1	1	yes	characteristic
256	C₂²	#15689030	1	1	yes	characteristic
257	C₂²	#15689030	1	1	yes	characteristic
258	C₂²	#15689034	1	2	yes	—
259	C₂²	#15689042	1	2	yes	—
260	C₂²	#15689119	1	2	yes	—
261	C₂²	#15689124	1	2	yes	—
262	C₂²	#15689268	1	4	yes	—
263	C₂²	#15689285	1	4	yes	—
264	C₂²	#15689420	1	4	yes	—
265	C₂²	#15689427	1	4	yes	—
266	C₂²	#15689480	1	2	yes	—
267	C₂²	#15689489	1	1	yes	characteristic
268	C₂²	#15689493	1	1	yes	characteristic
269	C₂²	#15689522	1	2	yes	—
270	C₂²	#15689533	1	2	yes	—
271	C₄	—	4	4	no	—
272	C₄	—	4	8	no	—
273	C₄	—	376	4	no	—
274	C₄	#15689030	1	4	yes	—
275	C₄	#15689034	1	2	yes	—
276	C₄	#15689042	1	2	yes	—
277	C₂	#20241659	1	1	yes	characteristic
278	C₂	#20243328	1	2	yes	—
279	C₂	#20244346	1	2	yes	—
280	C₂	#20244354	1	1	yes	characteristic
281	C₂	#20244358	1	1	yes	characteristic
282	C₂	#20244531	1	2	yes	—
283	C₂	#20244536	1	2	yes	—
284	C₂	#20244723	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#11385
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
12. C₂×#125×C₄₇	C₄	semidirect
150. C₂×#125	Dic₄₇	semidirect
158. C₄₇	#11385	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688885	2
C₂×#15688878	2
#20241618	2
C₂²×#125×C₄₇	2
#11385	47