Subgroups — #26220668

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	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
2	#20252030	C₂	1	2	yes	maximal, complemented
3	#20244276	C₂	1	2	yes	maximal
4	#20244136	C₂	1	2	yes	maximal, complemented
5	#20244092	C₂	1	4	yes	maximal, complemented
6	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#13093952	C₂	1	2	yes	maximal, complemented
8	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₂²	1	1	yes	characteristic
10	C₄×#10625350	C₂²	1	2	yes	—
11	C₄×Dic₁₈₈	C₄	1	2	yes	complemented
12	C₄×#10625325	C₄	1	4	yes	complemented
13	C₄×D₃₇₆	C₄	1	2	yes	complemented
14	#15689182	—	2	4	no	—
15	#15689174	—	2	4	no	—
16	C₂×#13093960	C₂²	1	2	yes	—
17	#15689161	C₂²	1	1	yes	characteristic
18	#15689149	C₂²	1	4	yes	—
19	C₂×#13093952	C₂²	1	4	yes	—
20	#15689120	C₂²	1	2	yes	—
21	#15689089	C₂²	1	4	yes	—
22	C₄×#10625329	C₂²	1	2	yes	—
23	#15688888	C₂²	1	2	yes	—
24	#15688883	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄³×C₄₇	C₂²	1	1	yes	characteristic
30	C₂×#10625350	C₂³	1	2	yes	—
31	#13094046	—	2	4	no	—
32	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
33	C₂×#10625325	C₂×C₄	1	2	yes	—
34	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
35	C₂×#10625329	C₂³	1	4	yes	—
36	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
37	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
38	C₄×C₂²⋊D₄₇	—	2	4	no	—
39	#13093983	C₂³	1	2	yes	—
40	#13093981	—	2	2	no	—
41	C₄²×D₄₇	—	2	2	no	—
42	#13093960	—	2	4	no	—
43	#13093958	C₂×C₄	1	2	yes	—
44	#13093956	C₂×C₄	1	2	yes	—
45	Dic₄₇⋊C₈	—	2	8	no	—
46	#13093952	—	2	8	no	—
47	#13093949	C₂×C₄	1	4	yes	—
48	#13093946	C₂×C₄	1	4	yes	—
49	(C₂×D₄₇)⋊C₈	—	2	8	no	—
50	#13093935	—	2	2	no	—
51	C₄×C₄₇⋊C₈	—	2	2	no	—
52	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₇	—	2	2	no	—
56	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₇	—	2	4	no	—
59	C₄×C₈×C₄₇	—	2	4	no	—
60	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₄₇	D₄	1	2	yes	—
63	C₄×C₈×C₄₇	D₄	1	2	yes	—
64	C₂²×C₄×C₄₇	—	2	2	no	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
67	C₂×C₂²⋊D₄₇	—	2	2	no	—
68	#10625350	—	2	4	no	—
69	#10625350	—	2	4	no	—
70	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₄₇	—	2	4	no	—
72	C₂×C₄×D₄₇	—	2	2	no	—
73	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	—	2	2	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₄₇	#41	1	4	yes	—
82	C₄²×C₄₇	—	2	8	no	—
83	C₄²×C₄₇	—	2	2	no	—
84	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₇	#41	1	4	yes	—
88	C₂²⋊Dic₄₇	—	2	2	no	—
89	(C₂×D₄₇)⋊C₄	—	2	4	no	—
90	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
91	Dic₄₇⋊C₄	—	2	4	no	—
92	C₄×Dic₄₇	—	2	4	no	—
93	C₄×Dic₄₇	—	2	2	no	—
94	#10625329	—	2	4	no	—
95	#10625329	—	2	4	no	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	Dic₁₈₈	—	2	2	no	—
98	D₃₇₆	—	2	2	no	—
99	#10625325	—	2	4	no	—
100	C₂³×C₄₇	C₂×#41	1	1	yes	characteristic
101	C₂²×D₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
110	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
111	C₂²⋊D₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	2	2	no	—
113	C₂×Dic₄₇	—	2	4	no	—
114	D₁₈₈	—	2	2	no	—
115	D₁₈₈	—	2	2	no	—
116	C₄×D₄₇	—	4	4	no	—
117	C₄×D₄₇	—	4	8	no	—
118	Dic₉₄	—	2	2	no	—
119	Dic₉₄	—	2	2	no	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	—	2	4	no	—
122	C₈×C₄₇	#117	1	2	yes	—
123	C₈×C₄₇	#117	1	2	yes	—
124	C₄₇⋊C₈	—	4	4	no	—
125	#21292	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₄₇	#529	1	2	yes	—
129	C₂×D₄₇	—	4	4	no	—
130	C₂×D₄₇	—	4	2	no	—
131	C₄×C₄₇	—	2	8	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	#443	1	4	yes	—
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	Dic₄₇	—	4	2	no	—
139	C₄×#135	—	47	1	no	—
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	D₄₇	1	1	yes	characteristic, complemented
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	#2293	1	1	yes	characteristic
149	C₂×C₄₇	C₂×#443	1	2	yes	—
150	D₄₇	—	8	2	no	—
151	C₂×#135	—	47	1	no	—
152	C₄×#41	—	47	2	no	—
153	C₄×Q₁₆	—	47	2	no	—
154	C₄×SD₁₆	—	47	4	no	—
155	C₄×D₈	—	47	2	no	—
156	#433	—	94	4	no	—
157	#431	—	94	4	no	—
158	C₂×#108	—	47	2	no	—
159	#427	—	47	1	no	—
160	#424	—	47	4	no	—
161	C₂×#104	—	47	4	no	—
162	#416	—	47	2	no	—
163	#408	—	47	4	no	—
164	C₄×M₁₆	—	47	2	no	—
165	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
166	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
169	C₄³	C₂×D₄₇	1	1	yes	characteristic
170	#341	—	47	2	no	—
171	#339	—	47	4	no	—
172	C₄₇	#21292	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	47	2	no	—
174	#135	—	94	4	no	—
175	C₂×Q₁₆	—	47	1	no	—
176	C₂×SD₁₆	—	47	2	no	—
177	C₂×D₈	—	47	1	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
181	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
182	C₄×Q₈	—	47	4	no	—
183	C₄×D₄	—	47	4	no	—
184	C₄×D₄	—	94	4	no	—
185	#117	—	47	2	no	—
186	#117	—	94	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	94	2	no	—
189	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	#108	—	94	4	no	—
192	#107	—	47	2	no	—
193	#106	—	47	2	no	—
194	C₄⋊C₈	—	94	8	no	—
195	#104	—	94	8	no	—
196	#103	—	47	4	no	—
197	#102	—	47	4	no	—
198	C₂²⋊C₈	—	94	8	no	—
199	#97	—	94	2	no	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	—	94	2	no	—
203	C₄×C₈	C₄×D₄₇	1	2	yes	—
204	C₄×C₈	C₄×D₄₇	1	2	yes	—
205	C₄×C₈	D₁₈₈	1	2	yes	—
206	C₄×C₈	D₁₈₈	1	2	yes	—
207	#41	—	94	4	no	—
208	#41	—	94	4	no	—
209	C₂×Q₈	—	47	2	no	—
210	C₂×D₄	—	47	2	no	—
211	C₂×D₄	—	94	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	94	4	no	—
214	C₂²×C₄	—	94	2	no	—
215	C₂²×C₄	#10625350	1	2	yes	—
216	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
217	Q₁₆	—	94	2	no	—
218	SD₁₆	—	94	4	no	—
219	D₈	—	94	2	no	—
220	M₁₆	—	94	4	no	—
221	M₁₆	—	94	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	94	4	no	—
226	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
229	C₂×C₈	C₂×D₁₈₈	1	4	yes	—
230	C₂×C₈	#10625350	1	4	yes	—
231	C₄⋊C₄	—	47	4	no	—
232	C₄⋊C₄	—	94	4	no	—
233	C₂²⋊C₄	—	94	2	no	—
234	C₂²⋊C₄	—	94	4	no	—
235	C₄²	—	2	8	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	94	4	no	—
238	C₄²	—	94	2	no	—
239	C₄²	C₂×C₄×D₄₇	1	2	yes	—
240	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
241	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
242	C₄²	#10625350	1	4	yes	—
243	C₂³	—	94	2	no	—
244	C₂³	C₂×#10625350	1	1	yes	characteristic
245	Q₈	—	94	2	no	—
246	Q₈	—	94	2	no	—
247	D₄	—	94	2	no	—
248	D₄	—	94	2	no	—
249	D₄	—	188	4	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	94	4	no	—
254	C₂×C₄	—	94	2	no	—
255	C₂×C₄	—	188	8	no	—
256	C₂×C₄	—	188	4	no	—
257	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
258	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
259	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
261	C₂×C₄	C₂×#10625350	1	2	yes	—
262	C₂×C₄	C₂×#10625350	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	—	188	4	no	—
266	C₈	#13093981	1	2	yes	—
267	C₈	#13093981	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	188	2	no	—
270	C₂²	—	188	4	no	—
271	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
272	C₂²	#15689657	1	2	yes	—
273	C₄	—	2	8	no	—
274	C₄	—	2	2	no	—
275	C₄	—	188	2	no	—
276	C₄	—	188	4	no	—
277	C₄	#15689241	1	4	yes	—
278	C₄	C₂×#13093981	1	1	yes	characteristic
279	C₄	C₄×#10625350	1	1	yes	characteristic
280	C₄	#15689673	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	376	2	no	—
283	C₂	#20245852	1	1	yes	characteristic
284	C₂	C₂×#15689241	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄²×C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₂²
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#21292
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
2. #20252030	C₂	semidirect
4. #20244136	C₂	semidirect
5. #20244092	C₂	semidirect
6. C₄×#13093960	C₂	semidirect
7. C₄×#13093952	C₂	semidirect
8. C₄²×C₈×C₄₇	C₂	semidirect
11. C₄×Dic₁₈₈	C₄	semidirect
12. C₄×#10625325	C₄	semidirect
13. C₄×D₃₇₆	C₄	semidirect
146. C₄²×C₈	D₄₇	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
C₄×#13094046	2
#20252030	2
#20244276	2
#20244136	2
#20244092	2
C₄×#13093960	2
C₄×#13093952	2
C₄²×C₈×C₄₇	2
#21292	47