Subgroups — #26370566

Order 12032.

806 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×#15689559	C₂	1	2	yes	maximal, complemented
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245425	C₂	1	4	yes	maximal, complemented
4	#20245420	C₂	1	4	yes	maximal, complemented
5	#20245144	C₂	1	1	yes	maximal, characteristic, complemented
6	#20245104	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094046	C₂²	1	1	yes	characteristic
9	C₄×#10625350	—	2	1	no	—
10	#15689559	—	2	8	no	—
11	C₂×#13093968	C₂²	1	4	yes	complemented
12	C₂×#13093965	C₂²	1	4	yes	complemented
13	C₂×#13093964	C₂²	1	2	yes	—
14	C₄×Dic₁₈₈	C₂²	1	2	yes	complemented
15	C₄×#10625325	—	2	2	no	—
16	C₄×D₃₇₆	C₂²	1	2	yes	complemented
17	#15689161	C₂²	1	1	yes	characteristic
18	#15689120	—	2	1	no	—
19	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	4	yes	complemented
20	#15688924	C₂²	1	8	yes	complemented
21	C₂²×D₈×C₄₇	—	2	1	no	—
22	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
23	C₂×#127×C₄₇	—	2	1	no	—
24	C₄₇×#493	—	2	2	no	—
25	C₄₇×#493	C₂²	1	4	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#10625350	—	2	1	no	—
28	#13094046	—	2	2	no	—
29	#13094046	D₄	1	4	yes	complemented
30	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
31	C₂×#10625325	—	2	1	no	—
32	C₂×D₃₇₆	C₂³	1	1	yes	characteristic
33	C₄×Dic₉₄	—	2	2	no	—
34	C₄×D₁₈₈	—	2	2	no	—
35	C₄×C₂²⋊D₄₇	—	4	2	no	—
36	#13093983	—	2	1	no	—
37	#13093981	—	4	1	no	—
38	C₄²×D₄₇	—	4	1	no	—
39	#13093968	—	2	8	no	—
40	#13093965	—	2	8	no	—
41	#13093964	—	2	4	no	—
42	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
43	#13093958	C₂³	1	2	yes	—
44	#13093956	—	2	1	no	—
45	#13093949	—	2	2	no	—
46	#13093946	—	2	2	no	—
47	C₂²×D₄×C₄₇	—	2	2	no	—
48	C₂²×D₄×C₄₇	—	4	2	no	—
49	C₂×D₈×C₄₇	—	2	4	no	—
50	C₂×D₈×C₄₇	—	2	2	no	—
51	C₂×D₈×C₄₇	—	2	4	no	—
52	C₂×D₈×C₄₇	—	2	4	no	—
53	C₂×D₈×C₄₇	—	4	4	no	—
54	C₂×D₈×C₄₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	#127×C₄₇	—	2	4	no	—
58	#127×C₄₇	—	4	1	no	—
59	#127×C₄₇	—	4	1	no	—
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	C₄×C₈×C₄₇	D₄	1	2	yes	—
62	C₄×C₈×C₄₇	C₂³	1	2	yes	—
63	C₂⁴×C₄₇	—	4	2	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	4	4	no	—
66	C₂×D₄×C₄₇	—	4	4	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	—	4	4	no	—
70	C₂×D₄×C₄₇	—	8	8	no	—
71	C₂²×C₄×C₄₇	D₈	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₄₇	—	4	1	no	—
74	#10625350	—	2	4	no	—
75	#10625350	—	4	2	no	—
76	C₂×D₁₈₈	—	2	1	no	—
77	C₂×C₄×D₄₇	—	4	1	no	—
78	C₂×C₄×D₄₇	—	4	2	no	—
79	C₂×Dic₉₄	—	2	1	no	—
80	D₈×C₄₇	—	2	4	no	—
81	D₈×C₄₇	—	2	4	no	—
82	D₈×C₄₇	—	4	8	no	—
83	D₈×C₄₇	—	4	4	no	—
84	C₂×C₈×C₄₇	—	2	2	no	—
85	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₇	#41	1	2	yes	—
89	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
90	C₄²×C₄₇	D₈	1	1	yes	characteristic
91	C₄²×C₄₇	D₈	1	1	yes	characteristic
92	C₄²×C₄₇	C₂×D₄	1	2	yes	—
93	C₂²⋊Dic₄₇	—	4	1	no	—
94	(C₂×D₄₇)⋊C₄	—	4	2	no	—
95	C₄⋊Dic₄₇	—	2	2	no	—
96	Dic₄₇⋊C₄	—	4	2	no	—
97	C₄×Dic₄₇	—	4	1	no	—
98	C₄×Dic₄₇	—	4	2	no	—
99	Dic₁₈₈	—	2	1	no	—
100	Dic₁₈₈	C₂×D₄	1	2	yes	complemented
101	D₃₇₆	—	2	1	no	—
102	D₃₇₆	C₂×D₄	1	2	yes	complemented
103	#10625325	—	2	4	no	—
104	C₄₇⋊C₁₆	—	2	4	no	—
105	C₂³×C₄₇	—	4	4	no	—
106	C₂³×C₄₇	—	8	8	no	—
107	C₂³×C₄₇	—	8	4	no	—
108	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
109	C₂²×D₄₇	—	4	1	no	—
110	D₄×C₄₇	—	4	4	no	—
111	D₄×C₄₇	—	4	4	no	—
112	D₄×C₄₇	—	8	8	no	—
113	D₄×C₄₇	—	8	4	no	—
114	D₄×C₄₇	—	8	8	no	—
115	C₂×C₄×C₄₇	—	2	4	no	—
116	C₂×C₄×C₄₇	#121	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
122	C₂²⋊D₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	1	no	—
124	C₂×Dic₄₇	—	4	2	no	—
125	D₁₈₈	—	2	2	no	—
126	D₁₈₈	—	4	1	no	—
127	C₄×D₄₇	—	4	4	no	—
128	C₄×D₄₇	—	8	4	no	—
129	Dic₉₄	—	2	2	no	—
130	Dic₉₄	—	4	1	no	—
131	C₈×C₄₇	—	2	2	no	—
132	C₈×C₄₇	#121	1	2	yes	—
133	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
134	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
135	#33582	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	8	8	no	—
138	C₂²×C₄₇	—	8	4	no	—
139	C₂²×C₄₇	—	16	8	no	—
140	C₂²×C₄₇	#459	1	2	yes	—
141	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
142	C₂×D₄₇	—	4	2	no	—
143	C₂×D₄₇	—	8	2	no	—
144	C₄×C₄₇	—	2	2	no	—
145	C₄×C₄₇	#459	1	2	yes	—
146	C₄×C₄₇	#509	1	4	yes	complemented
147	C₄×C₄₇	C₂×#121	1	1	yes	characteristic
148	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	Dic₄₇	—	4	2	no	—
150	Dic₄₇	—	8	2	no	—
151	C₂×#509	—	47	2	no	—
152	C₂×#493	D₄₇	1	1	yes	characteristic, complemented
153	C₄×#135	—	47	1	no	—
154	#2207	—	47	4	no	—
155	#2206	—	47	4	no	—
156	#2150	—	47	1	no	—
157	#2140	—	47	2	no	—
158	C₂×C₄₇	—	2	2	no	—
159	C₂×C₄₇	—	16	4	no	—
160	C₂×C₄₇	C₂×#459	1	1	yes	characteristic
161	C₂×C₄₇	C₂×#509	1	2	yes	—
162	D₄₇	—	8	2	no	—
163	C₂×#135	—	47	1	no	—
164	C₂²×D₈	—	2	1	no	—
165	C₂²×D₈	C₂×D₄₇	1	2	yes	—
166	C₂×#127	—	2	1	no	—
167	C₄×#41	—	94	1	no	—
168	#509	—	94	8	no	—
169	C₂×SD₃₂	—	47	4	no	—
170	C₂×D₁₆	—	47	4	no	—
171	C₂×M₃₂	—	47	2	no	—
172	#493	—	2	2	no	—
173	#493	C₂×D₄₇	1	4	yes	—
174	C₄×Q₁₆	—	47	2	no	—
175	C₄×SD₁₆	—	94	2	no	—
176	C₄×D₈	—	47	2	no	—
177	#427	—	47	1	no	—
178	#416	—	94	1	no	—
179	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
180	C₄⋊C₁₆	—	47	4	no	—
181	#357	—	47	8	no	—
182	C₄₇	#33582	1	1	yes	characteristic, complemented, Sylow
183	C₂×#41	—	94	1	no	—
184	C₂²×D₄	—	2	2	no	—
185	C₂²×D₄	—	4	2	no	—
186	#135	—	47	4	no	—
187	#135	—	94	2	no	—
188	C₂×Q₁₆	—	47	1	no	—
189	C₂×SD₁₆	—	94	1	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	4	no	—
192	C₂×D₈	—	2	4	no	—
193	C₂×D₈	—	2	2	no	—
194	C₂×D₈	—	4	4	no	—
195	C₂×D₈	—	47	1	no	—
196	C₂×D₈	C₂²×D₄₇	1	4	yes	—
197	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
199	#127	—	2	4	no	—
200	#127	—	4	1	no	—
201	#127	—	4	1	no	—
202	C₄×Q₈	—	94	2	no	—
203	C₄×D₄	—	94	2	no	—
204	C₄×D₄	—	188	2	no	—
205	#117	—	94	1	no	—
206	#117	—	188	1	no	—
207	C₂×C₄²	—	188	1	no	—
208	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
209	SD₃₂	—	94	8	no	—
210	D₁₆	—	94	8	no	—
211	M₃₂	—	94	4	no	—
212	C₂×C₁₆	—	47	4	no	—
213	#107	—	47	2	no	—
214	#106	—	94	1	no	—
215	#103	—	94	2	no	—
216	#102	—	94	2	no	—
217	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
218	C₄×C₈	C₂²×D₄₇	1	2	yes	—
219	C₂⁴	—	4	2	no	—
220	#41	—	94	4	no	—
221	#41	—	188	2	no	—
222	C₂×Q₈	—	94	1	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	8	8	no	—
230	C₂×D₄	—	94	1	no	—
231	C₂×D₄	—	188	1	no	—
232	C₂²×C₄	—	188	2	no	—
233	C₂²×C₄	—	188	1	no	—
234	C₂²×C₄	#10625334	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
236	Q₁₆	—	47	2	no	—
237	Q₁₆	—	94	1	no	—
238	SD₁₆	—	94	4	no	—
239	D₈	—	2	4	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	8	no	—
242	D₈	—	4	4	no	—
243	D₈	—	47	2	no	—
244	D₈	—	94	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	D₄×D₄₇	1	2	yes	—
247	C₂×C₈	#10625352	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
249	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
251	C₄⋊C₄	—	94	2	no	—
252	C₄⋊C₄	—	188	2	no	—
253	C₂²⋊C₄	—	188	1	no	—
254	C₂²⋊C₄	—	188	2	no	—
255	C₄²	—	188	1	no	—
256	C₄²	—	188	2	no	—
257	C₄²	#10625334	1	1	yes	characteristic
258	C₄²	#10625334	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
260	C₁₆	—	94	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	8	8	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	188	1	no	—
265	C₂³	C₂×#10625334	1	1	yes	characteristic
266	Q₈	—	94	2	no	—
267	Q₈	—	188	1	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	8	no	—
272	D₄	—	8	4	no	—
273	D₄	—	94	2	no	—
274	D₄	—	188	4	no	—
275	D₄	—	188	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	376	4	no	—
278	C₂×C₄	—	188	2	no	—
279	C₂×C₄	—	188	1	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	#13093995	1	2	yes	—
282	C₂×C₄	C₂×#10625334	1	2	yes	—
283	C₂×C₄	C₂×#10625334	1	2	yes	—
284	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
285	C₂×C₄	#13094050	1	2	yes	—
286	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	#13093995	1	2	yes	—
289	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
290	C₈	C₂×#10625352	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	8	no	—
294	C₂²	—	16	8	no	—
295	C₂²	—	376	2	no	—
296	C₂²	—	188	2	no	—
297	C₂²	#15689315	1	2	yes	—
298	C₂²	C₂²×#10625334	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	376	2	no	—
301	C₄	—	188	2	no	—
302	C₄	#15689315	1	2	yes	—
303	C₄	#15689559	1	4	yes	complemented
304	C₄	C₂×#13093995	1	1	yes	characteristic
305	C₄	C₂×#13094050	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	—	376	2	no	—
309	C₂	C₂×#15689315	1	1	yes	characteristic
310	C₂	C₂×#15689559	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#493	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#493	D₄₇
hypocenter	C₄₇	#33582
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#15689559	C₂	semidirect
2. C₄×#13094046	C₂	semidirect
3. #20245425	C₂	semidirect
4. #20245420	C₂	semidirect
5. #20245144	C₂	semidirect
6. #20245104	C₂	semidirect
7. C₂×C₄₇×#493	C₂	semidirect
11. C₂×#13093968	C₂²	semidirect
12. C₂×#13093965	C₂²	semidirect
14. C₄×Dic₁₈₈	C₂²	semidirect
16. C₄×D₃₇₆	C₂²	semidirect
19. C₄⋊(C₄₇⋊C₁₆)	C₂²	semidirect
20. #15688924	C₂²	semidirect
29. #13094046	D₄	semidirect
100. Dic₁₈₈	C₂×D₄	semidirect
102. D₃₇₆	C₂×D₄	semidirect
146. C₄×C₄₇	#509	semidirect
152. C₂×#493	D₄₇	semidirect
182. C₄₇	#33582	semidirect
303. C₄	#15689559	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689559	2
C₄×#13094046	2
#20245425	2
#20245420	2
#20245144	2
#20245104	2
C₂×C₄₇×#493	2
#33582	47