Subgroups — #26131550

Order 12032.

486 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689236	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688937	C₂	1	1	yes	maximal, characteristic
3	#20245105	C₂	1	1	yes	maximal, characteristic
4	#20243272	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241494	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₇₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₈₈	—	2	1	no	—
10	#15689236	—	2	2	no	—
11	#15689236	C₄	1	4	yes	complemented
12	C₂×#13093960	C₂²	1	1	yes	characteristic
13	#15689161	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₂²	1	1	yes	characteristic
15	C₂×#13093946	—	2	1	no	—
16	#15689078	—	2	1	no	—
17	#15688937	C₄	1	4	yes	complemented
18	#15688925	C₄	1	4	yes	complemented
19	#15688883	—	2	1	no	—
20	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
23	C₄₇×#347	—	2	2	no	—
24	C₂²×D₁₈₈	—	2	1	no	—
25	C₂²×C₄×D₄₇	—	4	1	no	—
26	C₂×D₃₇₆	—	2	1	no	—
27	C₂×D₃₇₆	—	2	1	no	—
28	C₂×D₃₇₆	—	2	1	no	—
29	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
30	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
31	C₂×D₃₇₆	C₂×C₄	1	2	yes	—
32	C₂×#10625329	—	2	1	no	—
33	C₄×D₁₈₈	—	2	4	no	—
34	C₄×D₁₈₈	—	4	2	no	—
35	#13093983	—	2	1	no	—
36	C₂×C₄⋊Dic₄₇	—	2	1	no	—
37	C₂×(C₂×D₄₇)⋊C₄	—	4	1	no	—
38	#13093960	—	2	1	no	—
39	#13093960	C₂×C₄	1	2	yes	complemented
40	#13093958	C₂×C₄	1	2	yes	complemented
41	#13093956	—	2	1	no	—
42	#13093956	—	2	1	no	—
43	#13093956	C₂×C₄	1	2	yes	—
44	#13093946	—	2	4	no	—
45	#13093934	—	4	1	no	—
46	C₈×Dic₄₇	—	4	1	no	—
47	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
50	M₃₂×C₄₇	—	2	1	no	—
51	M₃₂×C₄₇	—	2	1	no	—
52	M₃₂×C₄₇	—	2	1	no	—
53	M₃₂×C₄₇	C₂×C₄	1	2	yes	complemented
54	C₂×C₁₆×C₄₇	—	2	1	no	—
55	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
56	#97×C₄₇	C₂×C₄	1	2	yes	—
57	#97×C₄₇	D₄	1	2	yes	—
58	C₂³×D₄₇	—	4	1	no	—
59	C₂²×C₄×C₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₄₇	—	4	1	no	—
62	C₂×D₁₈₈	—	2	2	no	—
63	C₂×D₁₈₈	—	2	2	no	—
64	C₂×D₁₈₈	—	2	2	no	—
65	C₂×D₁₈₈	—	4	1	no	—
66	C₂×D₁₈₈	—	4	1	no	—
67	C₂×D₁₈₈	—	4	1	no	—
68	C₂×C₄×D₄₇	—	4	4	no	—
69	C₂×C₄×D₄₇	—	8	4	no	—
70	C₂×C₈×C₄₇	—	2	1	no	—
71	C₂×C₈×C₄₇	—	2	1	no	—
72	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
78	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
79	C₄²×C₄₇	—	2	1	no	—
80	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
81	C₂²⋊Dic₄₇	—	4	1	no	—
82	(C₂×D₄₇)⋊C₄	—	4	4	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	—	4	1	no	—
86	C₄×Dic₄₇	—	4	1	no	—
87	#10625329	—	2	2	no	—
88	#10625329	—	4	1	no	—
89	C₂×C₄₇⋊C₈	—	4	1	no	—
90	D₃₇₆	—	2	2	no	—
91	D₃₇₆	—	2	2	no	—
92	D₃₇₆	—	2	2	no	—
93	D₃₇₆	C₄²	1	4	yes	—
94	C₁₆×C₄₇	—	2	2	no	—
95	C₁₆×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂²×D₄₇	—	4	2	no	—
98	C₂²×D₄₇	—	4	2	no	—
99	C₂²×D₄₇	—	4	2	no	—
100	C₂²×D₄₇	—	8	2	no	—
101	C₂²×D₄₇	—	8	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
110	C₂×Dic₄₇	—	4	2	no	—
111	C₂×Dic₄₇	—	4	2	no	—
112	C₂×Dic₄₇	—	8	2	no	—
113	D₁₈₈	—	2	4	no	—
114	D₁₈₈	—	4	2	no	—
115	D₁₈₈	—	4	2	no	—
116	D₁₈₈	—	4	2	no	—
117	C₄×D₄₇	—	8	8	no	—
118	C₈×C₄₇	—	2	2	no	—
119	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
120	C₈×C₄₇	#117	1	1	yes	characteristic
121	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
122	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
123	C₄₇⋊C₈	—	4	2	no	—
124	#12647	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	#418	1	1	yes	characteristic
127	C₂²×C₄₇	#440	1	1	yes	characteristic
128	C₂²×C₄₇	#442	1	1	yes	characteristic
129	C₂×D₄₇	—	4	4	no	—
130	C₂×D₄₇	—	8	2	no	—
131	C₂×D₄₇	—	8	2	no	—
132	C₂×D₄₇	—	8	4	no	—
133	C₂×D₄₇	—	8	4	no	—
134	C₄×C₄₇	—	4	2	no	—
135	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₄₇	#417	1	1	yes	characteristic
137	C₄×C₄₇	#440	1	1	yes	characteristic
138	C₄×C₄₇	#442	1	1	yes	characteristic
139	Dic₄₇	—	8	2	no	—
140	Dic₄₇	—	8	2	no	—
141	C₂×#442	—	47	1	no	—
142	C₂×#361	—	47	1	no	—
143	#2140	—	47	1	no	—
144	C₂×#347	D₄₇	1	1	yes	characteristic, complemented
145	#1773	—	47	1	no	—
146	#1386	—	47	1	no	—
147	#1383	—	47	1	no	—
148	C₂×C₄₇	—	2	1	no	—
149	C₂×C₄₇	—	2	1	no	—
150	C₂×C₄₇	#1762	1	1	yes	characteristic
151	C₂×C₄₇	AGL(1,ℤ/16)	1	2	yes	—
152	D₄₇	—	8	4	no	—
153	C₂²×D₈	—	47	1	no	—
154	C₂×C₄×D₄	—	94	1	no	—
155	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
156	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
157	#442	—	47	4	no	—
158	#442	—	94	2	no	—
159	C₂×#108	—	47	1	no	—
160	#427	—	47	1	no	—
161	C₂×#106	—	47	1	no	—
162	C₂×#102	—	94	1	no	—
163	#405	—	94	1	no	—
164	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
165	#361	—	47	4	no	—
166	#357	—	47	4	no	—
167	#347	—	2	2	no	—
168	#339	—	94	1	no	—
169	C₄₇	#12647	1	1	yes	characteristic, complemented, Sylow
170	C₂²×D₄	—	94	1	no	—
171	C₂³×C₄	—	188	1	no	—
172	C₂×D₈	—	47	2	no	—
173	C₂×D₈	—	47	2	no	—
174	C₂×D₈	—	47	2	no	—
175	C₂×D₈	—	94	1	no	—
176	C₂×D₈	—	94	1	no	—
177	C₂×D₈	—	94	1	no	—
178	C₂×M₁₆	—	94	1	no	—
179	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₄×D₄	—	94	4	no	—
182	C₄×D₄	—	188	2	no	—
183	#117	—	94	1	no	—
184	C₂×C₄⋊C₄	—	94	1	no	—
185	C₂×C₂²⋊C₄	—	188	1	no	—
186	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₄×D₄₇	1	2	yes	complemented
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	C₄×D₄₇	1	2	yes	complemented
193	#108	—	47	2	no	—
194	#108	—	94	1	no	—
195	#107	—	47	2	no	—
196	#106	—	47	2	no	—
197	#106	—	94	1	no	—
198	#106	—	94	1	no	—
199	#102	—	94	4	no	—
200	#97	—	188	1	no	—
201	#97	C₄×D₄₇	1	2	yes	—
202	#97	C₂²⋊D₄₇	1	2	yes	—
203	C₄×C₈	—	188	1	no	—
204	C₂⁴	—	188	1	no	—
205	C₂×D₄	—	94	2	no	—
206	C₂×D₄	—	94	2	no	—
207	C₂×D₄	—	94	2	no	—
208	C₂×D₄	—	188	1	no	—
209	C₂×D₄	—	188	1	no	—
210	C₂×D₄	—	188	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	376	4	no	—
213	C₂²×C₄	—	188	4	no	—
214	C₂²×C₄	—	188	1	no	—
215	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
216	D₈	—	47	4	no	—
217	D₈	—	94	2	no	—
218	D₈	—	94	2	no	—
219	D₈	—	94	2	no	—
220	M₁₆	—	94	2	no	—
221	M₁₆	—	188	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	188	1	no	—
225	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₈	#10625350	1	1	yes	characteristic
230	C₂×C₈	#10625350	1	1	yes	characteristic
231	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
232	C₄⋊C₄	—	94	2	no	—
233	C₄⋊C₄	—	94	2	no	—
234	C₄⋊C₄	—	188	1	no	—
235	C₂²⋊C₄	—	188	4	no	—
236	C₂²⋊C₄	—	188	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	188	1	no	—
239	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₂³	—	376	2	no	—
243	C₂³	—	376	2	no	—
244	C₂³	—	188	2	no	—
245	C₂³	—	188	2	no	—
246	C₂³	—	188	2	no	—
247	C₂³	#13094052	1	1	yes	characteristic
248	D₄	—	94	4	no	—
249	D₄	—	188	2	no	—
250	D₄	—	188	2	no	—
251	D₄	—	188	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	376	8	no	—
255	C₂×C₄	—	376	2	no	—
256	C₂×C₄	—	188	2	no	—
257	C₂×C₄	—	188	2	no	—
258	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
260	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
261	C₂×C₄	#13094052	1	1	yes	characteristic
262	C₂×C₄	#13094052	1	1	yes	characteristic
263	C₂×C₄	#13094057	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	188	2	no	—
266	C₈	C₄²×D₄₇	1	1	yes	characteristic
267	C₈	#13093981	1	1	yes	characteristic
268	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	376	2	no	—
272	C₂²	—	376	2	no	—
273	C₂²	—	376	4	no	—
274	C₂²	—	376	4	no	—
275	C₂²	—	188	4	no	—
276	C₂²	#15689129	1	1	yes	characteristic
277	C₂²	#15689225	1	1	yes	characteristic
278	C₂²	#15689236	1	1	yes	characteristic
279	C₄	—	4	2	no	—
280	C₄	—	376	2	no	—
281	C₄	—	376	2	no	—
282	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
283	C₄	#15689125	1	1	yes	characteristic
284	C₄	#15689225	1	1	yes	characteristic
285	C₄	#15689236	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	376	4	no	—
289	C₂	#20243214	1	1	yes	characteristic
290	C₂	#20245288	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689236
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#347	C₂
socle	C₂²×C₄₇	#442
prime radical	C₂²×D₃₇₆	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#347	D₄₇
hypocenter	C₄₇	#12647
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
4. #20243272	C₂	semidirect
5. #20241494	C₂	semidirect
6. #20241487	C₂	semidirect
7. C₂×C₄₇×#347	C₂	semidirect
11. #15689236	C₄	semidirect
17. #15688937	C₄	semidirect
18. #15688925	C₄	semidirect
39. #13093960	C₂×C₄	semidirect
40. #13093958	C₂×C₄	semidirect
53. M₃₂×C₄₇	C₂×C₄	semidirect
55. C₂×C₁₆×C₄₇	C₂×C₄	semidirect
144. C₂×#347	D₄₇	semidirect
169. C₄₇	#12647	semidirect
190. M₃₂	C₄×D₄₇	semidirect
192. C₂×C₁₆	C₄×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689236	2
C₂×#15688937	2
#20245105	2
#20243272	2
#20241494	2
#20241487	2
C₂×C₄₇×#347	2
#12647	47