Subgroups — #26109342

Order 12032.

526 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	#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×C₈×D₄×C₄₇	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	C₂³×C₈×C₄₇	—	2	1	no	—
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₄₇	—	2	2	no	—
16	C₈×D₄×C₄₇	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	C₂²×C₈×C₄₇	—	2	1	no	—
30	C₂²×C₈×C₄₇	—	2	4	no	—
31	C₂²×C₈×C₄₇	—	4	4	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	C₄⋊C₄	1	2	yes	—
50	C₂×D₄×C₄₇	SD₁₆	1	4	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	C₂×C₈×C₄₇	—	2	2	no	—
61	C₂×C₈×C₄₇	—	2	2	no	—
62	C₂×C₈×C₄₇	—	2	2	no	—
63	C₂×C₈×C₄₇	—	4	8	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	—	2	4	no	—
85	C₂³×C₄₇	—	4	2	no	—
86	C₂³×C₄₇	—	4	1	no	—
87	C₂³×C₄₇	—	4	1	no	—
88	C₂³×C₄₇	#121	1	1	yes	characteristic
89	D₄×C₄₇	—	2	2	no	—
90	D₄×C₄₇	—	2	4	no	—
91	D₄×C₄₇	#106	1	4	yes	—
92	C₂×C₄×C₄₇	—	2	1	no	—
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	4	2	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
106	C₂×C₄×C₄₇	#135	1	2	yes	—
107	C₂×C₄×C₄₇	#135	1	2	yes	—
108	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×Dic₄₇	—	8	2	no	—
110	C₂×Dic₄₇	—	8	2	no	—
111	C₂×Dic₄₇	—	8	2	no	—
112	C₈×C₄₇	—	2	2	no	—
113	C₈×C₄₇	—	2	2	no	—
114	C₈×C₄₇	—	4	2	no	—
115	C₄₇⋊C₈	—	8	2	no	—
116	#8646	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	—	4	2	no	—
119	C₂²×C₄₇	—	4	2	no	—
120	C₂²×C₄₇	—	4	4	no	—
121	C₂²×C₄₇	—	4	4	no	—
122	C₂²×C₄₇	#380	1	1	yes	characteristic
123	C₂²×C₄₇	#463	1	2	yes	—
124	C₂²×C₄₇	#465	1	2	yes	—
125	C₂²×C₄₇	#472	1	1	yes	characteristic
126	C₂²×C₄₇	#473	1	1	yes	characteristic
127	C₄×C₄₇	—	2	1	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	4	2	no	—
130	C₄×C₄₇	#416	1	1	yes	characteristic
131	C₄×C₄₇	#418	1	1	yes	characteristic
132	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
133	C₄×C₄₇	#427	1	1	yes	characteristic
134	C₄×C₄₇	#465	1	2	yes	—
135	C₄×C₄₇	#478	1	2	yes	—
136	Dic₄₇	—	16	2	no	—
137	C₂×C₈×D₄	D₄₇	1	1	yes	characteristic
138	#1834	—	47	1	no	—
139	#1802	—	47	2	no	—
140	C₂×#334	—	47	1	no	—
141	#1296	—	47	2	no	—
142	C₂×C₄₇	—	4	4	no	—
143	C₂×C₄₇	#1677	1	2	yes	—
144	C₂×C₄₇	#1679	1	2	yes	—
145	C₂×C₄₇	#1804	1	1	yes	characteristic
146	C₂×C₄₇	#1807	1	1	yes	characteristic
147	C₂×C₄₇	#1816	1	1	yes	characteristic
148	C₂³×C₈	—	2	1	no	—
149	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
150	C₈×D₄	—	2	2	no	—
151	C₈×D₄	Dic₄₇	1	4	yes	complemented
152	C₂×C₄⋊C₈	—	47	2	no	—
153	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
154	C₂×#102	—	94	2	no	—
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	#384	—	47	2	no	—
158	#340	—	94	1	no	—
159	#340	—	94	1	no	—
160	#340	—	94	2	no	—
161	#334	—	94	2	no	—
162	C₄₇	#8646	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₄	D₁₈₈	1	1	yes	characteristic
164	C₂³×C₄	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	—	4	4	no	—
168	C₂²×C₈	—	94	2	no	—
169	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
170	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	Dic₉₄	1	2	yes	—
174	C₄×D₄	C₂×Dic₄₇	1	2	yes	—
175	C₂×C₄⋊C₄	—	94	2	no	—
176	C₂×C₄⋊C₄	—	94	2	no	—
177	C₂×C₄⋊C₄	—	94	2	no	—
178	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
181	C₄⋊C₈	—	2	1	no	—
182	C₄⋊C₈	—	94	2	no	—
183	C₄⋊C₈	—	94	2	no	—
184	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
185	#102	—	188	4	no	—
186	C₂²⋊C₈	—	2	4	no	—
187	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
188	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
189	C₂⁴	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	#10625325	1	4	yes	—
193	C₂×D₄	C₄⋊Dic₄₇	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	188	2	no	—
200	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
201	C₂²×C₄	#10625350	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	8	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	188	2	no	—
209	C₂×C₈	—	188	2	no	—
210	C₂×C₈	—	188	2	no	—
211	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
212	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	376	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₄⋊C₄	—	188	2	no	—
217	C₄⋊C₄	—	188	2	no	—
218	C₄⋊C₄	—	188	2	no	—
219	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
223	C₄²	#10625350	1	1	yes	characteristic
224	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	2	4	no	—
228	C₂³	—	4	2	no	—
229	C₂³	—	4	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	#13093998	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	#13093956	1	4	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	376	2	no	—
243	C₂×C₄	—	376	2	no	—
244	C₂×C₄	—	376	2	no	—
245	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
247	C₂×C₄	#13093983	1	1	yes	characteristic
248	C₂×C₄	#13093998	1	1	yes	characteristic
249	C₂×C₄	#13094003	1	1	yes	characteristic
250	C₂×C₄	#13094003	1	1	yes	characteristic
251	C₂×C₄	C₂×#10625325	1	2	yes	—
252	C₂×C₄	#13094044	1	2	yes	—
253	C₂×C₄	#13094046	1	2	yes	—
254	C₂×C₄	#13094050	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	376	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	#15688983	1	1	yes	characteristic
265	C₂²	#15689338	1	2	yes	—
266	C₂²	#15689352	1	2	yes	—
267	C₂²	#15689393	1	1	yes	characteristic
268	C₂²	#15689398	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	752	2	no	—
273	C₄	#15689119	1	1	yes	characteristic
274	C₄	#15689128	1	1	yes	characteristic
275	C₄	C₂×#13093956	1	1	yes	characteristic
276	C₄	#15689161	1	1	yes	characteristic
277	C₄	#15689352	1	2	yes	—
278	C₄	#15689423	1	2	yes	—
279	C₂	—	4	4	no	—
280	C₂	#20242797	1	2	yes	—
281	C₂	#20242811	1	2	yes	—
282	C₂	#20243398	1	1	yes	characteristic
283	C₂	#20243413	1	1	yes	characteristic
284	C₂	#20243452	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈×D₄×C₄₇	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₄×C₄₇	D₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈×D₄	D₄₇
hypocenter	C₄₇	#8646
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

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. C₈×D₄×C₄₇	C₄	semidirect
151. C₈×D₄	Dic₄₇	semidirect
162. C₄₇	#8646	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×C₈×D₄×C₄₇	2
#8646	47