Subgroups — #26220669

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252032	C₂	1	2	yes	maximal, complemented
2	#20244277	C₂	1	2	yes	maximal
3	#20244137	C₂	1	2	yes	maximal, complemented
4	#20244094	C₂	1	4	yes	maximal, complemented
5	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093951	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
9	#15689184	—	2	4	no	—
10	#15689176	—	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×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
19	C₄²×Dic₄₇	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	—	2	2	no	—
33	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
34	#13093960	—	2	4	no	—
35	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
36	#13093951	—	2	8	no	—
37	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
38	#13093935	—	2	2	no	—
39	C₄×C₄₇⋊C₈	—	2	2	no	—
40	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
41	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
42	C₈×Dic₄₇	—	2	4	no	—
43	C₈×Dic₄₇	—	2	4	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	4	no	—
70	C₂²×C₄×C₄₇	—	2	2	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	8	no	—
92	C₄×Dic₄₇	—	2	2	no	—
93	C₄×Dic₄₇	#41	1	4	yes	—
94	#10625329	—	2	4	no	—
95	#10625329	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	2	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	—	2	4	no	—
99	C₂×C₄₇⋊C₈	C₂×D₄	1	4	yes	—
100	C₂³×C₄₇	—	2	2	no	—
101	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
102	Q₈×C₄₇	—	2	2	no	—
103	Q₈×C₄₇	—	2	2	no	—
104	D₄×C₄₇	—	2	2	no	—
105	D₄×C₄₇	—	2	2	no	—
106	D₄×C₄₇	—	4	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	—	4	8	no	—
112	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
118	C₂×Dic₄₇	—	2	8	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	#117	1	2	yes	—
122	C₈×C₄₇	#117	1	2	yes	—
123	C₄₇⋊C₈	—	2	4	no	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#21292	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₄₇	#529	1	2	yes	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	4	4	no	—
133	C₄×C₄₇	—	4	2	no	—
134	C₄×C₄₇	#443	1	4	yes	—
135	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
136	C₄×C₄₇	C₄×#41	1	1	yes	characteristic
137	C₄×C₄₇	#532	1	2	yes	—
138	Dic₄₇	—	2	8	no	—
139	C₄×#135	D₄₇	1	1	yes	characteristic
140	#2922	—	47	2	no	—
141	#1974	—	47	2	no	—
142	#1945	—	47	2	no	—
143	#1937	—	47	4	no	—
144	C₄×#108	—	47	1	no	—
145	C₄×#104	—	47	2	no	—
146	C₄²×C₈	—	47	1	no	—
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	—	8	2	no	—
149	C₂×C₄₇	#2293	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#443	1	2	yes	—
151	C₂×#135	C₂×D₄₇	1	1	yes	characteristic
152	C₄×#41	C₂×D₄₇	1	2	yes	—
153	C₄×Q₁₆	Dic₄₇	1	2	yes	complemented
154	C₄×SD₁₆	Dic₄₇	1	4	yes	complemented
155	C₄×D₈	Dic₄₇	1	2	yes	complemented
156	#433	—	94	4	no	—
157	#431	—	94	4	no	—
158	C₂×#108	—	47	2	no	—
159	#427	C₂×D₄₇	1	1	yes	characteristic
160	#424	—	47	4	no	—
161	C₂×#104	—	47	4	no	—
162	#416	C₂×D₄₇	1	2	yes	—
163	#408	—	47	4	no	—
164	C₄×M₁₆	—	47	2	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	C₂×C₄×C₈	—	47	2	no	—
167	C₂×C₄×C₈	—	47	2	no	—
168	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
169	C₄³	—	47	1	no	—
170	#341	—	47	2	no	—
171	#339	—	47	4	no	—
172	C₄₇	#21292	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	C₂²×D₄₇	1	2	yes	—
174	#135	—	2	4	no	—
175	C₂×Q₁₆	C₂×Dic₄₇	1	1	yes	characteristic
176	C₂×SD₁₆	C₂×Dic₄₇	1	2	yes	—
177	C₂×D₈	C₂×Dic₄₇	1	1	yes	characteristic
178	C₂×M₁₆	—	47	4	no	—
179	C₂²×C₈	—	47	2	no	—
180	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₄×Q₈	C₂×Dic₄₇	1	4	yes	—
183	C₄×D₄	—	2	4	no	—
184	C₄×D₄	C₂×Dic₄₇	1	4	yes	—
185	#117	—	2	2	no	—
186	#117	C₂²×D₄₇	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	47	2	no	—
189	C₂×C₄²	—	94	2	no	—
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	#108	—	94	4	no	—
192	#107	C₂×Dic₄₇	1	2	yes	—
193	#106	C₂×Dic₄₇	1	2	yes	—
194	C₄⋊C₈	—	94	8	no	—
195	#104	—	94	8	no	—
196	#103	C₂×Dic₄₇	1	4	yes	—
197	#102	C₂×Dic₄₇	1	4	yes	—
198	C₂²⋊C₈	—	94	8	no	—
199	#97	—	94	2	no	—
200	C₄×C₈	—	47	2	no	—
201	C₄×C₈	—	47	2	no	—
202	C₄×C₈	—	94	4	no	—
203	C₄×C₈	—	94	4	no	—
204	C₄×C₈	—	94	2	no	—
205	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
206	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	C₂²×Dic₄₇	1	2	yes	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×Dic₄₇	1	2	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	94	2	no	—
215	C₂²×C₄	#10625352	1	2	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	2	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	94	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	47	4	no	—
224	C₂×C₈	—	94	2	no	—
225	C₂×C₈	—	94	2	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	#10625350	1	4	yes	—
228	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
229	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
230	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×Dic₄₇	1	4	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	47	4	no	—
238	C₄²	—	94	8	no	—
239	C₄²	—	94	2	no	—
240	C₄²	D₄×D₄₇	1	1	yes	characteristic
241	C₄²	D₄×D₄₇	1	1	yes	characteristic
242	C₄²	C₂²×Dic₄₇	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×#10625352	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	94	8	no	—
256	C₂×C₄	—	94	4	no	—
257	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
258	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
259	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
261	C₂×C₄	#41×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×#10625350	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	94	4	no	—
265	C₈	—	188	4	no	—
266	C₈	#13093983	1	2	yes	—
267	C₈	#13093983	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×D₄×Dic₄₇	1	1	yes	characteristic
272	C₂²	#15689655	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	94	8	no	—
277	C₄	#15689240	1	4	yes	—
278	C₄	C₂×#13093983	1	1	yes	characteristic
279	C₄	#41×Dic₄₇	1	1	yes	characteristic
280	C₄	#15689670	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	#20245853	1	1	yes	characteristic
284	C₂	C₂×#15689240	1	2	yes	—

	subgroup	G/H
center	C₄²	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	#21292
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20252032	C₂	semidirect
3. #20244137	C₂	semidirect
4. #20244094	C₂	semidirect
5. C₄×#13093960	C₂	semidirect
6. C₄×#13093951	C₂	semidirect
7. C₄×C₈×Dic₄₇	C₂	semidirect
24. C₄×Q₁₆×C₄₇	C₄	semidirect
25. C₄×SD₁₆×C₄₇	C₄	semidirect
26. C₄×D₈×C₄₇	C₄	semidirect
153. C₄×Q₁₆	Dic₄₇	semidirect
154. C₄×SD₁₆	Dic₄₇	semidirect
155. C₄×D₈	Dic₄₇	semidirect
172. C₄₇	#21292	semidirect

Maximal subgroups

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

subgroup	index
#20252032	2
#20244277	2
#20244137	2
#20244094	2
C₄×#13093960	2
C₄×#13093951	2
C₄×C₈×Dic₄₇	2
C₄×#135×C₄₇	2
#21292	47