Subgroups — #26233992

Order 12032.

766 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	#20246273	C₂	1	2	yes	maximal
2	#20244127	C₂	1	2	yes	maximal, complemented
3	#20243968	C₂	1	4	yes	maximal, complemented
4	#20243483	C₂	1	2	yes	maximal
5	#20243413	C₂	1	1	yes	maximal, characteristic
6	#20243287	C₂	1	2	yes	maximal, complemented
7	#20243164	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₇×#427	C₂	1	1	yes	maximal, characteristic
9	C₂×#13093956	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×#13093934	C₂²	1	2	yes	—
12	#15689022	C₂²	1	4	yes	—
13	#15689016	—	2	4	no	—
14	#15688989	—	2	2	no	—
15	#15688983	C₂²	1	2	yes	—
16	C₄⋊C₄×Dic₄₇	C₂²	1	4	yes	—
17	C₄×C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	#15688969	—	2	2	no	—
19	#15688886	C₂²	1	2	yes	—
20	#15688885	C₂²	1	4	yes	—
21	#15688885	C₂²	1	4	yes	—
22	#15688877	C₂²	1	2	yes	—
23	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₄₇	C₂²	1	2	yes	—
25	C₄₇×#427	C₄	1	8	yes	complemented
26	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×#106×C₄₇	C₂²	1	2	yes	—
29	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
36	#13093956	—	2	4	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
38	#13093934	—	2	4	no	—
39	C₄₇⋊#95	—	2	2	no	—
40	C₄₇⋊#95	—	2	4	no	—
41	C₄₇⋊#95	—	2	4	no	—
42	C₄₇⋊#95	D₄	1	2	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	—	2	4	no	—
45	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	#117×C₄₇	—	2	4	no	—
49	#117×C₄₇	C₂×C₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
52	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	#107×C₄₇	C₂×C₄	1	4	yes	—
55	#107×C₄₇	C₂×C₄	1	4	yes	—
56	#106×C₄₇	C₂×C₄	1	8	yes	—
57	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	—	2	2	no	—
59	C₂²×C₄×C₄₇	—	2	2	no	—
60	C₂²×C₄×C₄₇	—	2	2	no	—
61	C₂²×C₄×C₄₇	—	2	2	no	—
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	4	no	—
67	C₂²×Dic₄₇	C₂×D₄	1	4	yes	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	8	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₇	#41	1	4	yes	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	2	2	no	—
75	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
76	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
77	C₂²⋊C₄×C₄₇	—	2	8	no	—
78	C₄²×C₄₇	—	2	8	no	—
79	C₂²⋊Dic₄₇	—	4	2	no	—
80	C₂²⋊Dic₄₇	—	8	2	no	—
81	C₄⋊Dic₄₇	—	2	4	no	—
82	C₄⋊Dic₄₇	—	2	4	no	—
83	C₄⋊Dic₄₇	—	8	4	no	—
84	C₄×Dic₄₇	—	2	2	no	—
85	C₄×Dic₄₇	—	2	2	no	—
86	C₄×Dic₄₇	—	4	8	no	—
87	C₂×C₄₇⋊C₈	—	2	2	no	—
88	C₂×C₄₇⋊C₈	—	2	2	no	—
89	C₂×C₄₇⋊C₈	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	—	2	2	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	C₄×D₄	1	2	yes	—
94	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	4	no	—
100	C₂×C₄×C₄₇	—	2	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₇	2₋¹⁺⁴	1	2	yes	—
109	C₂×Dic₄₇	—	2	4	no	—
110	C₂×Dic₄₇	—	2	4	no	—
111	C₂×Dic₄₇	—	2	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₈×C₄₇	—	2	2	no	—
116	C₈×C₄₇	#117	1	4	yes	—
117	C₄₇⋊C₈	—	4	2	no	—
118	#22619	—	47	1	no	maximal, Sylow
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	2	2	no	—
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	—	4	4	no	—
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	#547	1	2	yes	—
125	C₂²×C₄₇	#575	1	2	yes	—
126	C₂²×C₄₇	#577	1	2	yes	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	—	4	4	no	—
129	C₄×C₄₇	—	4	4	no	—
130	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
131	C₄×C₄₇	#519	1	1	yes	characteristic
132	C₄×C₄₇	#552	1	2	yes	—
133	Dic₄₇	—	4	4	no	—
134	Dic₄₇	—	8	4	no	—
135	C₂×#427	D₄₇	1	1	yes	characteristic
136	#2346	—	47	2	no	—
137	#1943	—	47	2	no	—
138	#1919	—	47	4	no	—
139	#1825	—	47	2	no	—
140	#1807	—	47	1	no	—
141	#1776	—	47	2	no	—
142	#1752	—	47	1	no	—
143	C₂×C₄₇	—	4	2	no	—
144	C₂×C₄₇	#2344	1	1	yes	characteristic
145	C₂×C₄₇	#2943	1	1	yes	characteristic
146	C₂×C₄₇	#2952	1	1	yes	characteristic
147	C₂×C₄₇	#3348	1	4	yes	—
148	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
149	C₂×#117	C₂×D₄₇	1	2	yes	—
150	#427	Dic₄₇	1	8	yes	complemented
151	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
152	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
153	C₂×#106	—	47	2	no	—
154	C₂×#106	C₂×D₄₇	1	2	yes	—
155	C₂×C₂²⋊C₈	—	47	1	no	—
156	C₂×#97	—	47	2	no	—
157	#389	—	47	4	no	—
158	#388	—	94	4	no	—
159	#382	—	94	2	no	—
160	#380	—	47	2	no	—
161	C₄×C₄⋊C₄	—	47	4	no	—
162	C₄×C₂²⋊C₄	—	47	1	no	—
163	#376	—	94	2	no	—
164	#340	—	47	2	no	—
165	#340	—	47	4	no	—
166	#340	—	47	4	no	—
167	#336	—	47	2	no	—
168	C₄₇	#22619	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	47	2	no	—
172	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
173	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
175	#117	—	2	4	no	—
176	#117	C₂×Dic₄₇	1	8	yes	—
177	C₂×C₄⋊C₄	—	47	4	no	—
178	C₂×C₄⋊C₄	—	94	4	no	—
179	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
180	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
181	C₂×C₂²⋊C₄	—	2	2	no	—
182	C₂×C₂²⋊C₄	—	47	2	no	—
183	C₂×C₂²⋊C₄	—	94	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	47	2	no	—
186	C₂×C₄²	—	94	4	no	—
187	#107	C₂×Dic₄₇	1	4	yes	—
188	#107	C₂×Dic₄₇	1	4	yes	—
189	#106	—	94	4	no	—
190	#106	C₂×Dic₄₇	1	8	yes	—
191	C₂²⋊C₈	—	94	2	no	—
192	#97	—	94	4	no	—
193	#95	—	47	2	no	—
194	#95	—	94	4	no	—
195	#95	—	94	4	no	—
196	#95	—	94	2	no	—
197	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	47	4	no	—
203	C₂²×C₄	—	94	4	no	—
204	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	#10625352	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	94	2	no	—
211	C₂×C₈	—	94	2	no	—
212	C₂×C₈	—	94	2	no	—
213	C₂×C₈	#10625350	1	4	yes	—
214	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
215	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	94	4	no	—
219	C₄⋊C₄	—	94	4	no	—
220	C₄⋊C₄	—	376	4	no	—
221	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
222	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
223	C₂²⋊C₄	—	2	8	no	—
224	C₂²⋊C₄	—	376	2	no	—
225	C₂²⋊C₄	—	188	2	no	—
226	C₄²	—	2	8	no	—
227	C₄²	—	94	2	no	—
228	C₄²	—	94	2	no	—
229	C₄²	—	188	8	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	D₄×Dic₄₇	1	2	yes	—
234	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	94	4	no	—
245	C₂×C₄	—	94	4	no	—
246	C₂×C₄	—	94	4	no	—
247	C₂×C₄	—	188	4	no	—
248	C₂×C₄	—	188	4	no	—
249	C₂×C₄	—	188	4	no	—
250	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
251	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
253	C₂×C₄	C₂×#10625350	1	2	yes	—
254	C₂×C₄	#13094072	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	188	2	no	—
257	C₈	#13093983	1	4	yes	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
263	C₂²	#15689774	1	2	yes	—
264	C₂²	#15689920	1	2	yes	—
265	C₂²	#15689930	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	4	no	—
269	C₄	—	376	4	no	—
270	C₄	—	188	4	no	—
271	C₄	C₂×#13093983	1	1	yes	characteristic
272	C₄	#15689601	1	1	yes	characteristic
273	C₄	#15689808	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	#20246257	1	1	yes	characteristic
276	C₂	#20252189	1	1	yes	characteristic
277	C₂	#20252272	1	1	yes	characteristic
278	C₂	#20255899	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄₇×#427	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#427	D₄₇
hypocenter	C₄₇	#22619
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #20244127	C₂	semidirect
3. #20243968	C₂	semidirect
6. #20243287	C₂	semidirect
25. C₄₇×#427	C₄	semidirect
150. #427	Dic₄₇	semidirect
168. C₄₇	#22619	semidirect

Maximal subgroups

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

subgroup	index
#20246273	2
#20244127	2
#20243968	2
#20243483	2
#20243413	2
#20243287	2
#20243164	2
C₂×C₄₇×#427	2
#22619	47