Subgroups — #26220685

Order 12032.

722 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252039	C₂	1	2	yes	maximal, complemented
2	#20244277	C₂	1	2	yes	maximal
3	#20244137	C₂	1	2	yes	maximal, complemented
4	#20244101	C₂	1	4	yes	maximal, complemented
5	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
6	#20243198	C₂	1	2	yes	maximal, complemented
7	#20243074	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689184	—	2	4	no	—
10	#15689180	—	2	4	no	—
11	C₂×#13093960	C₂²	1	2	yes	—
12	#15689150	C₂²	1	4	yes	—
13	C₂×#13093951	C₂²	1	4	yes	—
14	#15689090	C₂²	1	4	yes	—
15	C₄×#10625329	C₂²	1	2	yes	—
16	C₂×#13093935	C₂²	1	1	yes	characteristic
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	#15688969	C₂²	1	1	yes	characteristic
20	#15688888	C₂²	1	2	yes	—
21	#15688882	C₂²	1	4	yes	—
22	C₂×#135×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×#41×C₄₇	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₄₇	C₄	1	2	yes	complemented
25	C₄×SD₁₆×C₄₇	C₄	1	4	yes	complemented
26	C₄×D₈×C₄₇	C₄	1	2	yes	complemented
27	C₄₇×#427	C₂²	1	1	yes	characteristic
28	C₄₇×#416	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#10625329	C₂³	1	4	yes	—
31	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
32	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
33	#13093960	—	2	4	no	—
34	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
35	#13093951	—	2	8	no	—
36	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
37	#13093935	—	2	2	no	—
38	#13093935	D₄	1	2	yes	—
39	#13093935	D₄	1	2	yes	—
40	#13093934	—	2	4	no	—
41	C₄×C₄₇⋊C₈	—	2	2	no	—
42	C₈×Dic₄₇	—	2	4	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₂×#41×C₄₇	C₂³	1	2	yes	—
45	#135×C₄₇	—	2	4	no	—
46	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₄₇	—	2	4	no	—
53	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
54	#117×C₄₇	—	2	2	no	—
55	#117×C₄₇	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₇	—	2	2	no	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	#107×C₄₇	C₂×C₄	1	2	yes	—
59	#106×C₄₇	C₂×C₄	1	2	yes	—
60	#103×C₄₇	C₂×C₄	1	4	yes	—
61	#102×C₄₇	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
64	#41×C₄₇	—	2	4	no	—
65	#41×C₄₇	—	2	4	no	—
66	C₂×Q₈×C₄₇	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₄₇	—	2	2	no	—
68	C₂×D₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	—	2	2	no	—
70	C₂²×C₄×C₄₇	—	2	4	no	—
71	C₂²×C₄×C₄₇	#41	1	2	yes	—
72	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₇	—	2	2	no	—
74	Q₁₆×C₄₇	—	2	2	no	—
75	SD₁₆×C₄₇	—	2	4	no	—
76	D₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₄₇	#41	1	4	yes	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	C₂²×C₄	1	4	yes	—
84	C₂²⋊C₄×C₄₇	—	2	2	no	—
85	C₂²⋊C₄×C₄₇	—	2	4	no	—
86	C₄²×C₄₇	—	2	4	no	—
87	C₄²×C₄₇	—	2	2	no	—
88	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₄₇	—	2	2	no	—
92	C₄×Dic₄₇	#41	1	4	yes	—
93	#10625329	—	2	4	no	—
94	#10625329	—	2	4	no	—
95	C₂×C₄₇⋊C₈	—	2	2	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	4	no	—
98	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
99	C₂³×C₄₇	—	2	2	no	—
100	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
101	Q₈×C₄₇	—	2	2	no	—
102	Q₈×C₄₇	—	2	2	no	—
103	D₄×C₄₇	—	2	2	no	—
104	D₄×C₄₇	—	2	2	no	—
105	D₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	8	4	no	—
111	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
116	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₂×Dic₄₇	—	2	4	no	—
118	C₂×Dic₄₇	—	4	4	no	—
119	C₈×C₄₇	—	2	2	no	—
120	C₈×C₄₇	#117	1	2	yes	—
121	C₈×C₄₇	#117	1	2	yes	—
122	C₄₇⋊C₈	—	2	4	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	#21294	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	2	no	—
126	C₂²×C₄₇	—	4	4	no	—
127	C₂²×C₄₇	—	4	2	no	—
128	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₇	#529	1	2	yes	—
130	C₄×C₄₇	—	2	2	no	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	4	4	no	—
133	C₄×C₄₇	—	4	2	no	—
134	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
135	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
136	C₄×C₄₇	#532	1	2	yes	—
137	Dic₄₇	—	4	4	no	—
138	C₄×#135	D₄₇	1	1	yes	characteristic
139	#2923	—	47	2	no	—
140	#1974	—	47	2	no	—
141	#1945	—	47	2	no	—
142	#1938	—	47	4	no	—
143	C₄×#108	—	47	1	no	—
144	#1759	—	47	2	no	—
145	#1726	—	47	1	no	—
146	C₂×C₄₇	—	2	2	no	—
147	C₂×C₄₇	—	8	2	no	—
148	C₂×C₄₇	#2293	1	1	yes	characteristic
149	C₂×C₄₇	#2957	1	2	yes	—
150	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
151	C₄×#41	C₂×D₄₇	1	2	yes	—
152	C₄×Q₁₆	Dic₄₇	1	2	yes	complemented
153	C₄×SD₁₆	Dic₄₇	1	4	yes	complemented
154	C₄×D₈	Dic₄₇	1	2	yes	complemented
155	#433	—	94	4	no	—
156	#432	—	94	4	no	—
157	C₂×#108	—	47	2	no	—
158	#427	C₂×D₄₇	1	1	yes	characteristic
159	#424	—	47	4	no	—
160	C₂×#104	—	47	4	no	—
161	#416	C₂×D₄₇	1	2	yes	—
162	#408	—	47	4	no	—
163	C₄×M₁₆	—	47	2	no	—
164	C₂×#97	—	47	1	no	—
165	C₂×#97	—	47	2	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
168	#376	—	47	1	no	—
169	#341	—	47	2	no	—
170	#339	—	47	4	no	—
171	C₄₇	#21294	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	C₂²×D₄₇	1	2	yes	—
173	#135	—	2	4	no	—
174	C₂×Q₁₆	C₂×Dic₄₇	1	1	yes	characteristic
175	C₂×SD₁₆	C₂×Dic₄₇	1	2	yes	—
176	C₂×D₈	C₂×Dic₄₇	1	1	yes	characteristic
177	C₂×M₁₆	—	47	4	no	—
178	C₂²×C₈	—	47	2	no	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×Dic₄₇	1	4	yes	—
184	#117	—	2	2	no	—
185	#117	C₂²×D₄₇	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	47	2	no	—
188	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
189	#108	—	94	4	no	—
190	#107	C₂×Dic₄₇	1	2	yes	—
191	#106	C₂×Dic₄₇	1	2	yes	—
192	C₄⋊C₈	—	94	8	no	—
193	#104	—	94	8	no	—
194	#103	C₂×Dic₄₇	1	4	yes	—
195	#102	C₂×Dic₄₇	1	4	yes	—
196	C₂²⋊C₈	—	94	8	no	—
197	#97	—	47	2	no	—
198	#97	—	47	2	no	—
199	#97	—	94	4	no	—
200	#97	—	94	2	no	—
201	C₄×C₈	—	94	4	no	—
202	C₄×C₈	—	94	2	no	—
203	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
204	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
205	#95	—	94	2	no	—
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	C₂×Q₈	C₂²×Dic₄₇	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×Dic₄₇	1	2	yes	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	94	2	no	—
214	C₂²×C₄	#10625352	1	2	yes	—
215	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	2	no	—
219	M₁₆	—	94	4	no	—
220	M₁₆	—	94	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	94	2	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	#10625350	1	4	yes	—
227	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
228	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	47	4	no	—
237	C₄²	—	94	2	no	—
238	C₄²	D₄×D₄₇	1	1	yes	characteristic
239	C₄²	D₄×D₄₇	1	1	yes	characteristic
240	C₄²	C₂²×Dic₄₇	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	C₂×#10625352	1	1	yes	characteristic
243	Q₈	—	2	2	no	—
244	Q₈	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	94	4	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
256	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
257	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
259	C₂×C₄	#41×D₄₇	1	2	yes	—
260	C₂×C₄	C₂×#10625350	1	2	yes	—
261	C₈	—	2	2	no	—
262	C₈	—	94	4	no	—
263	C₈	—	188	4	no	—
264	C₈	#13093983	1	2	yes	—
265	C₈	#13093983	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	4	no	—
269	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
270	C₂²	#15689655	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	188	4	no	—
276	C₄	C₂×#13093983	1	1	yes	characteristic
277	C₄	#41×Dic₄₇	1	1	yes	characteristic
278	C₄	#15689670	1	2	yes	—
279	C₂	—	2	2	no	—
280	C₂	—	8	2	no	—
281	C₂	#20245853	1	1	yes	characteristic
282	C₂	#20252305	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#135×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#135×C₄₇	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#135	D₄₇
hypocenter	C₄₇	#21294
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20252039	C₂	semidirect
3. #20244137	C₂	semidirect
4. #20244101	C₂	semidirect
5. C₄×#13093960	C₂	semidirect
6. #20243198	C₂	semidirect
7. #20243074	C₂	semidirect
24. C₄×Q₁₆×C₄₇	C₄	semidirect
25. C₄×SD₁₆×C₄₇	C₄	semidirect
26. C₄×D₈×C₄₇	C₄	semidirect
152. C₄×Q₁₆	Dic₄₇	semidirect
153. C₄×SD₁₆	Dic₄₇	semidirect
154. C₄×D₈	Dic₄₇	semidirect
171. C₄₇	#21294	semidirect

Maximal subgroups

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

subgroup	index
#20252039	2
#20244277	2
#20244137	2
#20244101	2
C₄×#13093960	2
#20243198	2
#20243074	2
C₄×#135×C₄₇	2
#21294	47