Subgroups — #26225127

Order 12032.

654 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	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20252325	C₂	1	2	yes	maximal
3	#20244368	C₂	1	2	yes	maximal
4	#20244139	C₂	1	2	yes	maximal
5	#20244134	C₂	1	2	yes	maximal
6	#20243275	C₂	1	2	yes	maximal
7	#20243270	C₂	1	2	yes	maximal
8	#20243074	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1848	C₂	1	1	yes	maximal, characteristic
10	#15689680	C₂²	1	2	yes	—
11	#15689528	C₄	1	2	yes	complemented
12	#15689525	C₄	1	2	yes	complemented
13	#15689525	C₄	1	2	yes	complemented
14	#15689521	C₄	1	2	yes	complemented
15	#15689252	—	2	8	no	—
16	#15689161	C₂²	1	2	yes	—
17	#15689161	C₂²	1	2	yes	—
18	#15689151	C₂²	1	4	yes	—
19	#15689081	C₂²	1	4	yes	—
20	C₂×#13093935	C₂²	1	1	yes	characteristic
21	C₂×#13093934	C₂²	1	1	yes	characteristic
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688969	C₂²	1	1	yes	characteristic
26	#15688884	C₂²	1	4	yes	—
27	#15688884	C₂²	1	4	yes	—
28	C₂×#108×C₄₇	C₂²	1	2	yes	—
29	C₂×#108×C₄₇	C₂²	1	2	yes	—
30	C₄₇×#423	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
34	#13094024	C₂×C₄	1	4	yes	—
35	#13094016	C₂×C₄	1	4	yes	—
36	#13094003	—	2	4	no	—
37	C₄×Dic₉₄	—	2	4	no	—
38	#13093983	C₂³	1	4	yes	—
39	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
41	#13093958	C₂×C₄	1	4	yes	—
42	#13093958	C₂×C₄	1	4	yes	—
43	#13093956	C₂×C₄	1	4	yes	—
44	#13093956	C₂×C₄	1	4	yes	—
45	Dic₄₇⋊C₈	—	2	8	no	—
46	#13093935	D₄	1	4	yes	—
47	#13093934	—	2	2	no	—
48	#13093934	—	2	4	no	—
49	#13093934	Q₈	1	4	yes	complemented
50	C₈×Dic₄₇	—	2	2	no	—
51	C₈×Dic₄₇	—	2	4	no	—
52	C₈×Dic₄₇	Q₈	1	4	yes	complemented
53	C₄₇⋊#95	—	2	2	no	—
54	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#108×C₄₇	—	2	4	no	—
59	#108×C₄₇	—	2	4	no	—
60	C₄⋊C₈×C₄₇	—	2	4	no	—
61	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₄₇	—	2	1	no	—
68	C₂²×Dic₄₇	—	2	1	no	—
69	C₂×Dic₉₄	—	2	4	no	—
70	M₁₆×C₄₇	—	2	4	no	—
71	M₁₆×C₄₇	—	2	4	no	—
72	C₂×C₈×C₄₇	—	2	4	no	—
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	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₄₇	#41	1	2	yes	—
80	C₂×C₈×C₄₇	#41	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₇	C₂×D₄	1	2	yes	—
84	C₂²⋊Dic₄₇	—	2	4	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	C₂²×C₄	1	8	yes	—
87	Dic₄₇⋊C₄	—	2	8	no	—
88	C₄×Dic₄₇	—	2	4	no	—
89	C₄×Dic₄₇	—	2	1	no	—
90	C₄×Dic₄₇	—	2	1	no	—
91	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
92	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
93	C₂×C₄₇⋊C₈	—	2	1	no	—
94	C₂×C₄₇⋊C₈	—	2	1	no	—
95	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
97	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
98	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
99	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₄₇	—	2	8	no	—
110	C₂×Dic₄₇	—	2	2	no	—
111	C₂×Dic₄₇	—	2	2	no	—
112	C₂×Dic₄₇	—	4	2	no	—
113	C₂×Dic₄₇	—	4	2	no	—
114	Dic₉₄	—	8	4	no	—
115	C₈×C₄₇	—	4	4	no	—
116	C₈×C₄₇	C₄×Q₈	1	2	yes	—
117	C₈×C₄₇	C₄×Q₈	1	2	yes	—
118	C₈×C₄₇	C₄×Q₈	1	2	yes	—
119	C₈×C₄₇	C₄×Q₈	1	2	yes	—
120	C₄₇⋊C₈	—	2	2	no	—
121	C₄₇⋊C₈	C₄⋊Q₈	1	4	yes	—
122	#21733	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₄₇	#533	1	1	yes	characteristic
126	C₂²×C₄₇	#533	1	1	yes	characteristic
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
130	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₄×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₄×C₄₇	#532	1	1	yes	characteristic
133	Dic₄₇	—	4	2	no	—
134	Dic₄₇	—	4	2	no	—
135	Dic₄₇	—	4	8	no	—
136	#3160	—	47	1	no	—
137	#2961	—	47	2	no	—
138	#1995	—	47	2	no	—
139	#1946	—	47	2	no	—
140	#1945	—	47	2	no	—
141	#1848	D₄₇	1	1	yes	characteristic
142	#1773	—	47	2	no	—
143	#1772	—	47	2	no	—
144	#1726	—	47	1	no	—
145	C₂×C₄₇	—	2	1	no	—
146	C₂×C₄₇	—	2	1	no	—
147	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₄₇	#2957	1	2	yes	—
149	#533	—	47	2	no	—
150	#500	—	47	2	no	—
151	#499	—	47	2	no	—
152	#499	—	47	2	no	—
153	#498	—	47	2	no	—
154	#446	—	94	8	no	—
155	C₂×#108	C₂×D₄₇	1	2	yes	—
156	C₂×#108	C₂×D₄₇	1	2	yes	—
157	#427	—	47	2	no	—
158	#427	—	47	2	no	—
159	#424	—	47	4	no	—
160	#423	C₂×D₄₇	1	2	yes	—
161	#405	—	47	4	no	—
162	C₂×#97	—	47	1	no	—
163	C₂×#97	—	47	1	no	—
164	C₂×#97	—	47	1	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	1	no	—
167	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
168	#376	—	47	1	no	—
169	#339	—	47	4	no	—
170	#339	—	47	4	no	—
171	C₄₇	#21733	1	1	yes	characteristic, complemented, Sylow
172	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
173	C₂²×C₈	—	47	1	no	—
174	C₂²×C₈	—	47	1	no	—
175	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
177	C₄⋊Q₈	—	47	4	no	—
178	#125	—	47	4	no	—
179	C₂²⋊Q₈	—	94	4	no	—
180	C₄×Q₈	—	94	4	no	—
181	#117	—	47	4	no	—
182	C₂×C₄²	—	47	1	no	—
183	C₂×C₄²	—	47	1	no	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	#108	—	2	4	no	—
186	#108	—	2	4	no	—
187	#107	—	47	4	no	—
188	#107	—	47	4	no	—
189	#106	—	47	4	no	—
190	#106	—	47	4	no	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	94	8	no	—
193	#97	—	47	4	no	—
194	#97	—	47	4	no	—
195	#97	—	94	4	no	—
196	#97	—	94	2	no	—
197	C₄×C₈	—	47	4	no	—
198	C₄×C₈	—	94	4	no	—
199	C₄×C₈	—	94	2	no	—
200	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
201	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
202	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
203	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
204	#95	—	94	2	no	—
205	C₂×Q₈	—	94	4	no	—
206	C₂²×C₄	—	94	1	no	—
207	C₂²×C₄	—	94	1	no	—
208	C₂²×C₄	#10625354	1	2	yes	—
209	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	47	2	no	—
214	C₂×C₈	—	47	2	no	—
215	C₂×C₈	—	47	2	no	—
216	C₂×C₈	—	47	2	no	—
217	C₂×C₈	—	94	1	no	—
218	C₂×C₈	—	94	1	no	—
219	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
223	C₂×C₈	#10625352	1	2	yes	—
224	C₂×C₈	#10625352	1	2	yes	—
225	C₂×C₈	Q₈×D₄₇	1	2	yes	—
226	C₂×C₈	Q₈×D₄₇	1	2	yes	—
227	C₄⋊C₄	—	47	8	no	—
228	C₄⋊C₄	—	94	8	no	—
229	C₄⋊C₄	—	94	4	no	—
230	C₂²⋊C₄	—	94	4	no	—
231	C₄²	—	47	2	no	—
232	C₄²	—	47	2	no	—
233	C₄²	—	94	4	no	—
234	C₄²	—	94	1	no	—
235	C₄²	—	94	1	no	—
236	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
238	C₄²	D₄×D₄₇	1	2	yes	—
239	C₂³	C₂×#10625354	1	1	yes	characteristic
240	Q₈	—	376	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	94	8	no	—
243	C₂×C₄	—	94	2	no	—
244	C₂×C₄	—	94	2	no	—
245	C₂×C₄	—	188	2	no	—
246	C₂×C₄	—	188	2	no	—
247	C₂×C₄	#13093986	1	2	yes	—
248	C₂×C₄	#13093986	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
255	C₈	—	4	4	no	—
256	C₈	—	47	4	no	—
257	C₈	—	94	2	no	—
258	C₈	#13093989	1	2	yes	—
259	C₈	#13093989	1	2	yes	—
260	C₈	#13093989	1	2	yes	—
261	C₈	#13093989	1	2	yes	—
262	C₂²	—	2	2	no	—
263	C₂²	C₂×#13093986	1	1	yes	characteristic
264	C₂²	#15689678	1	1	yes	characteristic
265	C₂²	#15689678	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	188	2	no	—
269	C₄	—	188	2	no	—
270	C₄	—	188	8	no	—
271	C₄	C₂×#13093989	1	1	yes	characteristic
272	C₄	C₂×#13093989	1	1	yes	characteristic
273	C₄	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
274	C₄	#15689671	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	#20245955	1	1	yes	characteristic
278	C₂	#20252304	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂³×C₄₇	C₂×#41
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#21733
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
11. #15689528	C₄	semidirect
12. #15689525	C₄	semidirect
13. #15689525	C₄	semidirect
14. #15689521	C₄	semidirect
49. #13093934	Q₈	semidirect
52. C₈×Dic₄₇	Q₈	semidirect
171. C₄₇	#21733	semidirect

Maximal subgroups

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

subgroup	index
#20253902	2
#20252325	2
#20244368	2
#20244139	2
#20244134	2
#20243275	2
#20243270	2
#20243074	2
C₄₇×#1848	2
#21733	47