Subgroups — #26109378

Order 12032.

434 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	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	#20243548	C₂	1	1	yes	maximal, characteristic
3	#20243394	C₂	1	2	yes	maximal
4	#20241272	C₂	1	2	yes	maximal
5	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	#15689253	—	2	2	no	—
8	#15689253	C₄	1	4	yes	complemented
9	C₂×#13093958	—	2	1	no	—
10	C₂×#13093958	—	2	1	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₇⋊C₈	—	2	1	no	—
13	C₂×#13093949	—	2	2	no	—
14	C₂×#13093934	—	2	1	no	—
15	#15688993	C₂²	1	2	yes	—
16	#15688886	—	2	2	no	—
17	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#334	—	2	2	no	—
20	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	1	no	—
22	C₄×Dic₉₄	—	2	1	no	—
23	C₄×Dic₉₄	—	2	1	no	—
24	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
25	C₄×Dic₉₄	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
30	C₂×Dic₄₇⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	#13093958	—	4	2	no	—
33	#13093958	—	4	2	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
36	Dic₄₇⋊C₈	—	2	4	no	—
37	#13093949	—	4	4	no	—
38	#13093934	—	2	4	no	—
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	—	2	2	no	—
44	C₄⋊C₈×C₄₇	—	2	2	no	—
45	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₇	D₄	1	2	yes	—
47	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
50	C₂²×Dic₄₇	—	2	1	no	—
51	C₂²×Dic₄₇	—	2	1	no	—
52	C₂²×Dic₄₇	—	4	2	no	—
53	C₂×Dic₉₄	—	2	2	no	—
54	C₂×Dic₉₄	—	2	2	no	—
55	C₂×Dic₉₄	—	2	1	no	—
56	C₂×Dic₉₄	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₄₇	—	2	1	no	—
58	C₂×C₈×C₄₇	—	2	1	no	—
59	C₂×C₈×C₄₇	—	4	2	no	—
60	C₂×C₈×C₄₇	—	4	2	no	—
61	C₂×C₈×C₄₇	—	4	2	no	—
62	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
65	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₄₇	#41	1	1	yes	characteristic
70	C₄⋊Dic₄₇	—	2	1	no	—
71	C₄⋊Dic₄₇	—	4	2	no	—
72	C₄⋊Dic₄₇	—	4	2	no	—
73	C₄⋊Dic₄₇	—	4	2	no	—
74	C₄⋊Dic₄₇	—	4	2	no	—
75	C₄⋊Dic₄₇	—	8	2	no	—
76	C₄⋊Dic₄₇	C₄⋊C₄	1	2	yes	—
77	Dic₄₇⋊C₄	—	2	2	no	—
78	Dic₄₇⋊C₄	—	2	2	no	—
79	C₄×Dic₄₇	—	2	2	no	—
80	C₄×Dic₄₇	—	2	2	no	—
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	4	1	no	—
83	C₂×C₄₇⋊C₈	—	4	1	no	—
84	C₂³×C₄₇	#121	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	1	no	—
86	C₂×C₄×C₄₇	—	2	1	no	—
87	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#137	1	2	yes	—
98	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	4	1	no	—
102	C₂×Dic₄₇	—	4	1	no	—
103	C₂×Dic₄₇	—	4	1	no	—
104	C₂×Dic₄₇	—	4	1	no	—
105	C₂×Dic₄₇	—	8	2	no	—
106	C₂×Dic₄₇	—	8	2	no	—
107	C₂×Dic₄₇	—	8	2	no	—
108	Dic₉₄	—	2	4	no	—
109	Dic₉₄	—	4	2	no	—
110	C₈×C₄₇	—	2	2	no	—
111	C₈×C₄₇	—	8	2	no	—
112	C₈×C₄₇	#103	1	4	yes	complemented
113	C₄₇⋊C₈	—	4	2	no	—
114	#8653	—	47	1	no	maximal, Sylow
115	C₂²×C₄₇	#380	1	1	yes	characteristic
116	C₂²×C₄₇	#460	1	1	yes	characteristic
117	C₂²×C₄₇	#462	1	1	yes	characteristic
118	C₂²×C₄₇	#469	1	1	yes	characteristic
119	C₂²×C₄₇	#470	1	1	yes	characteristic
120	C₂²×C₄₇	#472	1	1	yes	characteristic
121	C₂²×C₄₇	#473	1	1	yes	characteristic
122	C₄×C₄₇	—	2	1	no	—
123	C₄×C₄₇	—	2	1	no	—
124	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
125	C₄×C₄₇	#417	1	1	yes	characteristic
126	C₄×C₄₇	#428	1	2	yes	—
127	C₄×C₄₇	#468	1	2	yes	—
128	C₄×C₄₇	#475	1	1	yes	characteristic
129	C₄×C₄₇	#477	1	1	yes	characteristic
130	Dic₄₇	—	4	2	no	—
131	Dic₄₇	—	4	2	no	—
132	Dic₄₇	—	16	2	no	—
133	C₂×#446	—	47	1	no	—
134	#1845	—	47	1	no	—
135	#1803	—	47	2	no	—
136	C₂×#334	D₄₇	1	1	yes	characteristic
137	#1296	—	47	2	no	—
138	C₂×C₄₇	#1688	1	1	yes	characteristic
139	C₂×C₄₇	#1690	1	1	yes	characteristic
140	C₂×C₄₇	#1694	1	1	yes	characteristic
141	C₂×C₄₇	#1696	1	1	yes	characteristic
142	C₂×C₄₇	#1803	1	1	yes	characteristic
143	C₂×C₄₇	#1809	1	1	yes	characteristic
144	C₂×C₄₇	#1816	1	1	yes	characteristic
145	C₂×C₄×Q₈	—	47	1	no	—
146	#446	—	47	4	no	—
147	#446	—	94	2	no	—
148	C₂×#107	—	94	1	no	—
149	C₂×#107	—	94	1	no	—
150	C₂×C₄⋊C₈	—	47	1	no	—
151	C₂×C₄⋊C₈	—	94	1	no	—
152	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
153	C₂×#103	—	94	2	no	—
154	C₂×#97	—	94	1	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#384	—	47	2	no	—
157	#340	—	94	2	no	—
158	#334	—	2	2	no	—
159	C₄₇	#8653	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	47	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	94	1	no	—
163	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
164	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
165	C₄×Q₈	—	47	2	no	—
166	C₄×Q₈	—	47	2	no	—
167	C₄×Q₈	—	94	1	no	—
168	C₄×Q₈	—	94	1	no	—
169	C₂×C₄⋊C₄	—	47	1	no	—
170	C₂×C₄⋊C₄	—	94	1	no	—
171	C₂×C₄⋊C₄	—	94	2	no	—
172	C₂×C₄⋊C₄	—	94	2	no	—
173	C₂×C₄⋊C₄	—	94	2	no	—
174	C₂×C₄²	—	94	1	no	—
175	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
176	#107	—	188	2	no	—
177	#107	—	188	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	47	2	no	—
181	C₄⋊C₈	—	94	4	no	—
182	C₄⋊C₈	—	94	1	no	—
183	#103	—	188	4	no	—
184	#97	—	94	4	no	—
185	C₄×C₈	C₄×D₄₇	1	2	yes	—
186	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
187	C₂×Q₈	—	47	2	no	—
188	C₂×Q₈	—	94	1	no	—
189	C₂×Q₈	—	94	2	no	—
190	C₂×Q₈	—	94	2	no	—
191	C₂²×C₄	—	94	1	no	—
192	C₂²×C₄	—	94	1	no	—
193	C₂²×C₄	—	188	2	no	—
194	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
195	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
196	C₂²×C₄	#10625354	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	94	2	no	—
203	C₂×C₈	—	188	1	no	—
204	C₂×C₈	—	188	1	no	—
205	C₂×C₈	#10625325	1	2	yes	—
206	C₂×C₈	Dic₁₈₈	1	2	yes	—
207	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
208	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
209	C₄⋊C₄	—	47	2	no	—
210	C₄⋊C₄	—	94	2	no	—
211	C₄⋊C₄	—	94	1	no	—
212	C₄⋊C₄	—	94	2	no	—
213	C₄⋊C₄	—	376	2	no	—
214	C₄⋊C₄	—	188	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₄⋊C₄	—	188	2	no	—
217	C₄⋊C₄	—	188	2	no	—
218	C₄²	—	94	2	no	—
219	C₄²	—	94	2	no	—
220	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₄²	#10625352	1	1	yes	characteristic
222	C₄²	Q₈×D₄₇	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	C₂³	#13093999	1	1	yes	characteristic
225	Q₈	—	94	4	no	—
226	Q₈	—	188	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	94	2	no	—
230	C₂×C₄	—	94	2	no	—
231	C₂×C₄	—	376	2	no	—
232	C₂×C₄	—	376	2	no	—
233	C₂×C₄	—	376	2	no	—
234	C₂×C₄	—	188	1	no	—
235	C₂×C₄	—	188	1	no	—
236	C₂×C₄	—	188	1	no	—
237	C₂×C₄	—	188	1	no	—
238	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
240	C₂×C₄	#13093982	1	1	yes	characteristic
241	C₂×C₄	#13093995	1	1	yes	characteristic
242	C₂×C₄	#13094000	1	1	yes	characteristic
243	C₂×C₄	#13094004	1	1	yes	characteristic
244	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
245	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
246	C₂×C₄	#13094051	1	2	yes	—
247	C₂×C₄	#13094052	1	1	yes	characteristic
248	C₂×C₄	#13094053	1	2	yes	—
249	C₂×C₄	#13094057	1	1	yes	characteristic
250	C₈	—	2	2	no	—
251	C₈	—	8	2	no	—
252	C₈	—	188	2	no	—
253	C₈	#13093949	1	4	yes	complemented
254	C₂²	#15688980	1	1	yes	characteristic
255	C₂²	#15689324	1	1	yes	characteristic
256	C₂²	#15689334	1	1	yes	characteristic
257	C₂²	#15689377	1	1	yes	characteristic
258	C₂²	#15689382	1	1	yes	characteristic
259	C₂²	#15689394	1	1	yes	characteristic
260	C₂²	#15689399	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	188	2	no	—
264	C₄	—	188	2	no	—
265	C₄	—	752	2	no	—
266	C₄	C₂×#13093949	1	1	yes	characteristic
267	C₄	#15689125	1	1	yes	characteristic
268	C₄	#15689162	1	2	yes	—
269	C₄	#15689366	1	2	yes	—
270	C₄	#15689409	1	1	yes	characteristic
271	C₄	#15689419	1	1	yes	characteristic
272	C₂	#20242873	1	1	yes	characteristic
273	C₂	#20242883	1	1	yes	characteristic
274	C₂	#20242907	1	1	yes	characteristic
275	C₂	#20242917	1	1	yes	characteristic
276	C₂	#20243394	1	1	yes	characteristic
277	C₂	#20243418	1	1	yes	characteristic
278	C₂	#20243449	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle, prime radical	C₂³×C₄₇	#121
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#8653
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #15689253	C₄	semidirect
112. C₈×C₄₇	#103	semidirect
159. C₄₇	#8653	semidirect
253. C₈	#13093949	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
#20243548	2
#20243394	2
#20241272	2
C₂×C₄₇×#334	2
#8653	47