Subgroups — #26225050

Order 12032.

750 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20246137	C₂	1	2	yes	maximal
2	#20244362	C₂	1	2	yes	maximal
3	#20244132	C₂	1	2	yes	maximal
4	#20244117	C₂	1	2	yes	maximal
5	#20243552	C₂	1	1	yes	maximal, characteristic
6	#20243288	C₂	1	2	yes	maximal
7	#106×Dic₄₇	C₂	1	2	yes	maximal
8	#20243074	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄₇×#498	C₂	1	1	yes	maximal, characteristic
10	C₂×#13093958	C₂²	1	2	yes	—
11	C₂×#13093956	C₂²	1	2	yes	—
12	C₂×#13093935	C₂²	1	1	yes	characteristic
13	C₂×#13093934	C₂²	1	1	yes	characteristic
14	C₂×#13093934	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
17	#15689030	—	2	4	no	—
18	#15689022	C₂²	1	4	yes	—
19	#15689009	—	2	4	no	—
20	#15688989	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
22	#15688969	C₂²	1	1	yes	characteristic
23	#15688885	C₂²	1	4	yes	—
24	#15688885	C₂²	1	4	yes	—
25	C₂×C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
26	C₄₇×#498	C₄	1	8	yes	complemented
27	C₂×#106×C₄₇	C₂²	1	2	yes	—
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
37	#13093958	—	2	4	no	—
38	#13093956	—	2	4	no	—
39	#13093935	—	2	2	no	—
40	#13093934	—	2	2	no	—
41	#13093934	Q₈	1	4	yes	complemented
42	C₈×Dic₄₇	—	2	2	no	—
43	C₈×Dic₄₇	Q₈	1	4	yes	complemented
44	C₄₇⋊#95	—	2	2	no	—
45	C₄₇⋊#95	—	2	2	no	—
46	C₄₇⋊#95	—	2	4	no	—
47	C₄₇⋊#95	—	2	4	no	—
48	C₄₇⋊#95	D₄	1	4	yes	—
49	C₂²×Q₈×C₄₇	—	2	2	no	—
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₄₇	—	2	4	no	—
53	C₄⋊Q₈×C₄₇	C₂×C₄	1	8	yes	—
54	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
56	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
57	#106×C₄₇	C₂×C₄	1	8	yes	—
58	#106×C₄₇	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
60	C₂×Q₈×C₄₇	—	2	8	no	—
61	C₂×Q₈×C₄₇	—	8	4	no	—
62	C₂²×C₄×C₄₇	—	2	4	no	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
67	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₄₇	#41	1	2	yes	—
73	C₂×C₈×C₄₇	#41	1	2	yes	—
74	C₄⋊C₄×C₄₇	—	2	8	no	—
75	C₄⋊C₄×C₄₇	—	2	4	no	—
76	C₄⋊C₄×C₄₇	C₂²×C₄	1	8	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₇	#41	1	2	yes	—
79	C₄⋊Dic₄₇	—	2	4	no	—
80	C₄⋊Dic₄₇	—	2	4	no	—
81	C₄⋊Dic₄₇	—	8	4	no	—
82	C₄×Dic₄₇	—	2	1	no	—
83	C₄×Dic₄₇	—	2	1	no	—
84	C₄×Dic₄₇	—	8	4	no	—
85	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
86	C₄×Dic₄₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄₇⋊C₈	—	2	1	no	—
88	C₂×C₄₇⋊C₈	—	2	1	no	—
89	C₂×C₄₇⋊C₈	—	2	1	no	—
90	C₂×C₄₇⋊C₈	—	2	1	no	—
91	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂×Q₈	1	2	yes	—
93	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
94	Q₈×C₄₇	—	8	8	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	8	no	—
97	C₂×C₄×C₄₇	—	4	4	no	—
98	C₂×C₄×C₄₇	—	4	4	no	—
99	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
106	C₂×Dic₄₇	—	2	4	no	—
107	C₂×Dic₄₇	—	2	2	no	—
108	C₂×Dic₄₇	—	2	4	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	C₄⋊Q₈	1	4	yes	—
113	C₈×C₄₇	C₄×Q₈	1	4	yes	—
114	C₈×C₄₇	C₄×Q₈	1	4	yes	—
115	C₄₇⋊C₈	—	2	2	no	—
116	C₄₇⋊C₈	—	2	2	no	—
117	#21724	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂²×C₄₇	#532	1	1	yes	characteristic
121	C₂²×C₄₇	#577	1	4	yes	—
122	C₄×C₄₇	—	2	4	no	—
123	C₄×C₄₇	—	4	8	no	—
124	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
126	C₄×C₄₇	#533	1	1	yes	characteristic
127	C₄×C₄₇	#533	1	1	yes	characteristic
128	Dic₄₇	—	2	4	no	—
129	Dic₄₇	—	4	2	no	—
130	Dic₄₇	—	8	4	no	—
131	C₂×#498	D₄₇	1	1	yes	characteristic
132	#2331	—	47	2	no	—
133	#1993	—	47	2	no	—
134	#1944	—	47	2	no	—
135	#1941	—	47	2	no	—
136	#1846	—	47	1	no	—
137	#1776	—	47	2	no	—
138	C₄×#106	—	47	2	no	—
139	#1726	—	47	1	no	—
140	C₂×C₄₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₄₇	#2949	1	2	yes	—
142	C₂×C₄₇	#3163	1	2	yes	—
143	C₂×C₄₇	#3173	1	2	yes	—
144	C₂×C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
145	#498	Dic₄₇	1	8	yes	complemented
146	C₂×#107	—	47	2	no	—
147	C₂×#106	—	47	2	no	—
148	C₂×#106	C₂×D₄₇	1	2	yes	—
149	C₂×#106	C₂×D₄₇	1	2	yes	—
150	C₂×#97	—	47	1	no	—
151	C₂×#97	—	47	1	no	—
152	C₂×#97	—	47	1	no	—
153	C₂×C₄×C₈	—	47	1	no	—
154	C₂×C₄×C₈	—	47	1	no	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	#391	—	94	4	no	—
157	#389	—	47	4	no	—
158	#387	—	94	4	no	—
159	#382	—	47	2	no	—
160	C₄×C₄⋊C₄	—	47	4	no	—
161	#376	—	47	1	no	—
162	#340	—	47	4	no	—
163	#340	—	47	4	no	—
164	C₄₇	#21724	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	2	2	no	—
166	C₂²×C₈	—	47	1	no	—
167	C₂²×C₈	—	47	1	no	—
168	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	C₂×Dic₄₇	1	8	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	47	4	no	—
174	C₂×C₄⋊C₄	—	94	4	no	—
175	C₂×C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
176	C₂×C₄²	—	47	1	no	—
177	C₂×C₄²	—	47	1	no	—
178	C₂×C₄²	—	94	4	no	—
179	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
180	#107	—	94	4	no	—
181	#106	—	94	4	no	—
182	#106	C₂×Dic₄₇	1	8	yes	—
183	#106	C₂×Dic₄₇	1	8	yes	—
184	#97	—	47	4	no	—
185	#97	—	94	2	no	—
186	#97	—	94	2	no	—
187	C₄×C₈	—	47	4	no	—
188	C₄×C₈	—	94	2	no	—
189	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
190	#95	—	47	4	no	—
191	#95	—	94	4	no	—
192	#95	—	94	4	no	—
193	#95	—	94	2	no	—
194	#95	—	94	2	no	—
195	C₂×Q₈	—	2	8	no	—
196	C₂×Q₈	—	8	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	47	2	no	—
199	C₂²×C₄	—	47	2	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	D₄×D₄₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	C₂×C₈	—	47	2	no	—
204	C₂×C₈	—	47	2	no	—
205	C₂×C₈	—	94	1	no	—
206	C₂×C₈	—	94	1	no	—
207	C₂×C₈	—	94	1	no	—
208	C₂×C₈	—	94	1	no	—
209	C₂×C₈	Q₈×D₄₇	1	2	yes	—
210	C₂×C₈	Q₈×D₄₇	1	2	yes	—
211	C₂×C₈	#10625354	1	2	yes	—
212	C₂×C₈	#10625354	1	2	yes	—
213	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
214	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	94	4	no	—
218	C₄⋊C₄	—	94	4	no	—
219	C₄⋊C₄	—	376	4	no	—
220	C₄⋊C₄	C₂²×Dic₄₇	1	8	yes	—
221	C₄²	—	47	2	no	—
222	C₄²	—	47	2	no	—
223	C₄²	—	94	1	no	—
224	C₄²	—	94	1	no	—
225	C₄²	—	376	4	no	—
226	C₄²	#10625352	1	2	yes	—
227	C₄²	C₂²×Dic₄₇	1	2	yes	—
228	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
229	Q₈	—	8	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	47	4	no	—
235	C₂×C₄	—	94	4	no	—
236	C₂×C₄	—	94	4	no	—
237	C₂×C₄	—	94	2	no	—
238	C₂×C₄	—	188	4	no	—
239	C₂×C₄	—	188	4	no	—
240	C₂×C₄	—	188	4	no	—
241	C₂×C₄	D₄×Dic₄₇	1	4	yes	—
242	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
243	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
244	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
246	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
247	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
248	C₈	—	94	2	no	—
249	C₈	—	94	2	no	—
250	C₈	Q₈×Dic₄₇	1	4	yes	—
251	C₈	Q₈×Dic₄₇	1	4	yes	—
252	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
253	C₂²	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
254	C₂²	#15689674	1	1	yes	characteristic
255	C₂²	#15689929	1	4	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	4	8	no	—
258	C₄	—	94	4	no	—
259	C₄	—	376	4	no	—
260	C₄	—	188	2	no	—
261	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
262	C₄	C₂×Q₈×Dic₄₇	1	1	yes	characteristic
263	C₄	#15689678	1	1	yes	characteristic
264	C₄	#15689678	1	1	yes	characteristic
265	C₂	C₄⋊Q₈×Dic₄₇	1	1	yes	characteristic
266	C₂	#20252248	1	2	yes	—
267	C₂	#20253913	1	2	yes	—
268	C₂	#20254007	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#498	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#498	D₄₇
hypocenter	C₄₇	#21724
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
26. C₄₇×#498	C₄	semidirect
41. #13093934	Q₈	semidirect
43. C₈×Dic₄₇	Q₈	semidirect
145. #498	Dic₄₇	semidirect
164. C₄₇	#21724	semidirect

Maximal subgroups

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

subgroup	index
#20246137	2
#20244362	2
#20244132	2
#20244117	2
#20243552	2
#20243288	2
#106×Dic₄₇	2
#20243074	2
C₂×C₄₇×#498	2
#21724	47