Subgroups — #26131571

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689246	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688934	C₂	1	1	yes	maximal, characteristic
3	#20245101	C₂	1	1	yes	maximal, characteristic
4	#20243285	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241497	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	C₂×#13094046	C₂²	1	1	yes	characteristic
9	C₂×#13094030	—	2	1	no	—
10	#15689246	—	2	2	no	—
11	#15689246	C₄	1	4	yes	complemented
12	C₂×#13093960	C₂²	1	1	yes	characteristic
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093956	C₂²	1	1	yes	characteristic
15	C₂×#13093952	—	2	1	no	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	#15688934	C₄	1	4	yes	complemented
18	#15688928	C₄	1	2	yes	complemented
19	#15688925	C₄	1	2	yes	complemented
20	#15688886	—	2	1	no	—
21	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#347	—	2	2	no	—
25	C₂×#10625350	—	2	1	no	—
26	#13094046	—	2	1	no	—
27	#13094046	—	2	1	no	—
28	#13094046	C₂×C₄	1	2	yes	—
29	#13094046	C₂×C₄	1	2	yes	—
30	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
31	C₂×#10625325	—	2	1	no	—
32	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
33	#13094030	—	2	4	no	—
34	#13094030	—	4	2	no	—
35	C₂×#10625329	—	2	1	no	—
36	C₂×#10625324	—	4	1	no	—
37	C₂×C₈×D₄₇	—	4	1	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄×Dic₄₇	—	4	1	no	—
41	#13093960	—	2	1	no	—
42	#13093960	C₂×C₄	1	2	yes	—
43	#13093958	—	2	1	no	—
44	#13093958	C₂×C₄	1	2	yes	—
45	#13093956	—	2	1	no	—
46	#13093956	C₂×C₄	1	2	yes	—
47	#13093952	—	2	4	no	—
48	C₄₇⋊#95	—	4	1	no	—
49	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
52	M₃₂×C₄₇	—	2	1	no	—
53	M₃₂×C₄₇	—	2	1	no	—
54	M₃₂×C₄₇	—	2	1	no	—
55	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
56	C₂×C₁₆×C₄₇	—	2	1	no	—
57	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
58	#97×C₄₇	C₂×C₄	1	2	yes	—
59	#97×C₄₇	D₄	1	2	yes	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₇	—	4	1	no	—
63	C₂²×Dic₄₇	—	4	1	no	—
64	C₂²×Dic₄₇	—	4	1	no	—
65	#10625350	—	2	2	no	—
66	#10625350	—	2	2	no	—
67	#10625350	—	4	1	no	—
68	#10625350	—	4	1	no	—
69	C₂×D₁₈₈	—	2	1	no	—
70	C₂×C₄×D₄₇	—	4	1	no	—
71	C₂×Dic₉₄	—	2	1	no	—
72	C₂×C₈×C₄₇	—	2	1	no	—
73	C₂×C₈×C₄₇	—	2	1	no	—
74	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
80	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
81	C₄²×C₄₇	—	2	1	no	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
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	C₄×Dic₄₇	—	8	2	no	—
88	#10625329	—	2	2	no	—
89	#10625329	—	4	1	no	—
90	C₂×C₄₇⋊C₈	—	4	1	no	—
91	Dic₁₈₈	—	2	1	no	—
92	Dic₁₈₈	C₄²	1	2	yes	complemented
93	D₃₇₆	—	2	1	no	—
94	D₃₇₆	C₄²	1	2	yes	complemented
95	#10625325	—	2	2	no	—
96	#10625325	—	2	2	no	—
97	#10625324	—	4	4	no	—
98	C₈×D₄₇	—	4	4	no	—
99	C₁₆×C₄₇	—	2	2	no	—
100	C₁₆×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×D₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	4	1	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
112	C₂²⋊D₄₇	—	4	2	no	—
113	C₂²⋊D₄₇	—	4	2	no	—
114	C₂×Dic₄₇	—	4	1	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	—	4	2	no	—
117	C₂×Dic₄₇	—	8	1	no	—
118	C₂×Dic₄₇	—	8	1	no	—
119	C₂×Dic₄₇	—	8	1	no	—
120	C₂×Dic₄₇	—	8	1	no	—
121	D₁₈₈	—	2	2	no	—
122	D₁₈₈	—	4	1	no	—
123	C₄×D₄₇	—	4	2	no	—
124	C₄×D₄₇	—	4	2	no	—
125	Dic₉₄	—	2	2	no	—
126	Dic₉₄	—	4	1	no	—
127	C₈×C₄₇	—	2	2	no	—
128	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
129	C₈×C₄₇	#117	1	1	yes	characteristic
130	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
131	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
132	C₄₇⋊C₈	—	4	2	no	—
133	#12650	—	47	1	no	maximal, Sylow
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	#417	1	1	yes	characteristic
137	C₂²×C₄₇	#442	1	1	yes	characteristic
138	C₂²×C₄₇	#442	1	1	yes	characteristic
139	C₂×D₄₇	—	4	2	no	—
140	C₂×D₄₇	—	8	1	no	—
141	C₂×D₄₇	—	8	1	no	—
142	C₄×C₄₇	—	4	2	no	—
143	C₄×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₄₇	#418	1	1	yes	characteristic
145	C₄×C₄₇	#440	1	1	yes	characteristic
146	C₄×C₄₇	#440	1	1	yes	characteristic
147	Dic₄₇	—	4	2	no	—
148	Dic₄₇	—	8	2	no	—
149	Dic₄₇	—	8	2	no	—
150	C₂×#444	—	47	1	no	—
151	C₂×#360	—	47	1	no	—
152	#2139	—	47	1	no	—
153	C₂×#347	D₄₇	1	1	yes	characteristic, complemented
154	#1776	—	47	1	no	—
155	#1387	—	47	1	no	—
156	#1383	—	47	1	no	—
157	C₂×C₄₇	—	2	1	no	—
158	C₂×C₄₇	—	2	1	no	—
159	C₂×C₄₇	#1762	1	1	yes	characteristic
160	C₂×C₄₇	#2180	1	1	yes	characteristic
161	C₂×C₄₇	AGL(1,ℤ/16)	1	1	yes	characteristic
162	D₄₇	—	8	2	no	—
163	C₂×#135	—	47	1	no	—
164	C₂×#131	—	94	1	no	—
165	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
166	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
167	#444	—	47	4	no	—
168	#444	—	94	2	no	—
169	C₂×#108	—	47	1	no	—
170	C₂×#107	—	47	1	no	—
171	C₂×#106	—	47	1	no	—
172	C₂×#104	—	94	1	no	—
173	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
174	C₄×C₄⋊C₄	—	94	1	no	—
175	#360	—	47	4	no	—
176	#358	—	47	2	no	—
177	#357	—	47	2	no	—
178	#347	—	2	2	no	—
179	#340	—	94	1	no	—
180	C₄₇	#12650	1	1	yes	characteristic, complemented, Sylow
181	C₂×#41	—	94	1	no	—
182	#135	—	47	2	no	—
183	#135	—	47	2	no	—
184	#135	—	94	1	no	—
185	#135	—	94	1	no	—
186	C₂×Q₁₆	—	47	1	no	—
187	C₂×SD₁₆	—	94	1	no	—
188	C₂×D₈	—	47	1	no	—
189	#131	—	94	4	no	—
190	#131	—	188	2	no	—
191	C₂×M₁₆	—	94	1	no	—
192	C₂×M₁₆	—	188	1	no	—
193	C₂²×C₈	—	188	1	no	—
194	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	94	1	no	—
197	C₂×C₄⋊C₄	—	94	1	no	—
198	C₂×C₄²	—	188	1	no	—
199	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	C₄×D₄₇	1	2	yes	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
206	#108	—	47	2	no	—
207	#108	—	94	1	no	—
208	#107	—	47	2	no	—
209	#107	—	94	1	no	—
210	#106	—	47	2	no	—
211	#106	—	94	1	no	—
212	#104	—	94	4	no	—
213	#97	C₄×D₄₇	1	2	yes	—
214	#97	C₂²⋊D₄₇	1	2	yes	—
215	#95	—	188	1	no	—
216	#41	—	94	2	no	—
217	#41	—	94	2	no	—
218	#41	—	188	1	no	—
219	#41	—	188	1	no	—
220	C₂×Q₈	—	94	1	no	—
221	C₂×D₄	—	94	1	no	—
222	C₂×D₄	—	188	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	188	1	no	—
225	C₂²×C₄	—	188	1	no	—
226	C₂²×C₄	—	188	1	no	—
227	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
228	Q₁₆	—	47	2	no	—
229	Q₁₆	—	94	1	no	—
230	SD₁₆	—	94	2	no	—
231	SD₁₆	—	94	2	no	—
232	D₈	—	47	2	no	—
233	D₈	—	94	1	no	—
234	M₁₆	—	94	2	no	—
235	M₁₆	—	188	4	no	—
236	M₁₆	—	188	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	188	4	no	—
240	C₂×C₈	—	188	1	no	—
241	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
242	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
245	C₂×C₈	#10625350	1	1	yes	characteristic
246	C₂×C₈	#10625350	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₄⋊C₄	—	94	2	no	—
249	C₄⋊C₄	—	94	2	no	—
250	C₄⋊C₄	—	188	1	no	—
251	C₄⋊C₄	—	188	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	376	2	no	—
254	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	188	1	no	—
258	C₂³	#13094052	1	1	yes	characteristic
259	Q₈	—	94	2	no	—
260	Q₈	—	188	1	no	—
261	D₄	—	94	2	no	—
262	D₄	—	188	2	no	—
263	D₄	—	188	2	no	—
264	D₄	—	188	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	376	1	no	—
269	C₂×C₄	—	376	1	no	—
270	C₂×C₄	—	376	1	no	—
271	C₂×C₄	—	376	1	no	—
272	C₂×C₄	—	188	2	no	—
273	C₂×C₄	—	188	2	no	—
274	C₂×C₄	—	188	1	no	—
275	C₂×C₄	—	188	2	no	—
276	C₂×C₄	—	188	2	no	—
277	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
279	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
280	C₂×C₄	#13094052	1	1	yes	characteristic
281	C₂×C₄	#13094052	1	1	yes	characteristic
282	C₂×C₄	#13094057	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	188	2	no	—
285	C₈	C₄²×D₄₇	1	1	yes	characteristic
286	C₈	#13093981	1	1	yes	characteristic
287	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
288	C₈	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	376	1	no	—
292	C₂²	—	376	1	no	—
293	C₂²	—	188	2	no	—
294	C₂²	#15689125	1	1	yes	characteristic
295	C₂²	#15689236	1	1	yes	characteristic
296	C₂²	#15689236	1	1	yes	characteristic
297	C₄	—	4	2	no	—
298	C₄	—	376	2	no	—
299	C₄	—	376	2	no	—
300	C₄	—	188	2	no	—
301	C₄	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
302	C₄	#15689129	1	1	yes	characteristic
303	C₄	#15689225	1	1	yes	characteristic
304	C₄	#15689225	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	376	2	no	—
308	C₂	#20243214	1	1	yes	characteristic
309	C₂	#20245281	1	1	yes	characteristic
310	C₂	#20245288	1	1	yes	characteristic

	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	C₂×#13094046	C₂²
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#347	D₄₇
hypocenter	C₄₇	#12650
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. #20243285	C₂	semidirect
5. #20241497	C₂	semidirect
6. #20241487	C₂	semidirect
7. C₂×C₄₇×#347	C₂	semidirect
11. #15689246	C₄	semidirect
17. #15688934	C₄	semidirect
18. #15688928	C₄	semidirect
19. #15688925	C₄	semidirect
92. Dic₁₈₈	C₄²	semidirect
94. D₃₇₆	C₄²	semidirect
153. C₂×#347	D₄₇	semidirect
180. C₄₇	#12650	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689246	2
C₂×#15688934	2
#20245101	2
#20243285	2
#20241497	2
#20241487	2
C₂×C₄₇×#347	2
#12650	47