Subgroups — #26214396

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₄₇	C₂	1	2	yes	maximal
2	#20251805	C₂	1	2	yes	maximal
3	C₂×#15689080	C₂	1	1	yes	maximal, characteristic
4	#20243296	C₂	1	2	yes	maximal, complemented
5	#20243294	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093959	C₂	1	2	yes	maximal, complemented
7	#20243289	C₂	1	2	yes	maximal, complemented
8	#20243113	C₂	1	2	yes	maximal
9	C₄₇×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10625324	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₄₇	C₂²	1	4	yes	—
12	C₂²×C₈×D₄₇	C₂²	1	1	yes	characteristic
13	C₂×#13093981	C₂²	1	1	yes	characteristic
14	C₂×#13093960	C₂²	1	2	yes	—
15	C₂×#13093959	C₂²	1	4	yes	—
16	#108×D₄₇	C₄	1	16	yes	complemented
17	#15689080	—	2	4	no	—
18	#15689080	C₄	1	8	yes	—
19	#15689077	—	2	4	no	—
20	C₄×#10625329	C₂²	1	2	yes	—
21	M₁₆×Dic₄₇	C₂²	1	4	yes	—
22	C₂×#13093934	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	#15688888	C₂²	1	2	yes	—
26	#15688887	C₂²	1	4	yes	—
27	C₂×#108×C₄₇	C₂²	1	2	yes	—
28	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
29	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#341	C₂²	1	2	yes	—
31	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
32	C₂×#10625329	C₂³	1	4	yes	—
33	C₂×#10625324	—	2	4	no	—
34	C₂×#10625324	—	2	4	no	—
35	C₂×#10625324	C₂×C₄	1	4	yes	—
36	M₁₆×D₄₇	—	2	8	no	—
37	M₁₆×D₄₇	C₂×C₄	1	16	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₄₇	—	2	4	no	—
41	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
43	#13093981	C₂×C₄	1	4	yes	—
44	#13093981	D₄	1	4	yes	—
45	C₂×Dic₄₇⋊C₄	Q₈	1	2	yes	—
46	C₂×(C₂×D₄₇)⋊C₄	Q₈	1	2	yes	—
47	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
48	#13093960	C₂×C₄	1	8	yes	—
49	#13093959	C₂×C₄	1	16	yes	—
50	#13093935	—	2	2	no	—
51	#13093934	—	2	4	no	—
52	#13093934	C₂×C₄	1	4	yes	—
53	C₄×C₄₇⋊C₈	—	2	2	no	—
54	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
55	C₈×Dic₄₇	—	2	4	no	—
56	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
57	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
58	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
61	#108×C₄₇	C₂×C₄	1	8	yes	—
62	#97×C₄₇	—	2	2	no	—
63	#97×C₄₇	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₄₇	—	2	2	no	—
65	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
69	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
72	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₄×D₄₇	#41	1	4	yes	—
74	M₁₆×C₄₇	—	2	4	no	—
75	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂×D₄	1	2	yes	—
83	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	8	yes	—
84	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
85	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
86	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
87	#10625329	—	2	4	no	—
88	#10625329	C₂²×C₄	1	8	yes	—
89	C₂×C₄₇⋊C₈	—	2	4	no	—
90	C₂×C₄₇⋊C₈	—	2	2	no	—
91	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
94	#10625324	—	2	8	no	—
95	#10625324	—	2	16	no	—
96	C₈×D₄₇	—	2	16	no	—
97	C₈×D₄₇	C₄²	1	16	yes	—
98	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×D₄₇	—	4	2	no	—
100	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂²×D₄₇	C₄×Q₈	1	4	yes	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×Dic₄₇	—	4	2	no	—
109	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₄₇	C₄×Q₈	1	4	yes	—
112	C₄×D₄₇	C₂×C₄²	1	4	yes	—
113	C₄×D₄₇	#117	1	4	yes	—
114	C₄×D₄₇	C₄×D₄	1	4	yes	—
115	C₄×D₄₇	C₄×D₄	1	4	yes	—
116	C₈×C₄₇	—	2	2	no	—
117	C₈×C₄₇	—	2	8	no	—
118	C₈×C₄₇	C₂×C₄²	1	4	yes	—
119	C₄₇⋊C₈	—	2	2	no	—
120	C₄₇⋊C₈	—	2	8	no	—
121	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
122	#20460	—	47	1	no	maximal, Sylow
123	C₂²×C₄₇	—	2	2	no	—
124	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
126	C₂×D₄₇	—	2	4	no	—
127	C₂×D₄₇	—	4	4	no	—
128	C₂×D₄₇	C₄×C₄⋊C₄	1	4	yes	—
129	C₄×C₄₇	—	2	4	no	—
130	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
131	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
132	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
133	Dic₄₇	—	2	4	no	—
134	Dic₄₇	C₄×C₄⋊C₄	1	4	yes	—
135	C₂²×#108	—	47	2	no	—
136	#2873	—	47	2	no	—
137	C₂×#405	—	47	1	no	—
138	#1778	—	47	2	no	—
139	#1778	—	47	1	no	—
140	#1778	D₄₇	1	1	yes	characteristic, complemented
141	C₄×#108	—	47	2	no	—
142	C₄×#108	—	47	2	no	—
143	#1738	—	47	2	no	—
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄₇	#2916	1	2	yes	—
147	D₄₇	—	2	4	no	—
148	C₂²×M₁₆	—	47	1	no	—
149	C₂²×M₁₆	—	47	4	no	—
150	C₂³×C₈	—	47	1	no	—
151	C₂×#117	—	47	1	no	—
152	C₂×#108	—	47	4	no	—
153	C₂×#108	—	47	2	no	—
154	C₂×#108	—	47	16	no	—
155	C₂×#108	C₂×D₄₇	1	2	yes	—
156	#405	—	47	8	no	—
157	#405	—	94	4	no	—
158	#405	—	94	4	no	—
159	C₄×M₁₆	—	47	4	no	—
160	C₄×M₁₆	—	47	2	no	—
161	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
162	C₂×#97	—	47	1	no	—
163	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
164	C₂×C₄×C₈	—	47	1	no	—
165	C₂×C₄×C₈	—	47	1	no	—
166	#341	—	47	4	no	—
167	#341	—	47	2	no	—
168	#341	C₂×D₄₇	1	2	yes	—
169	C₄₇	#20460	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	—	47	1	no	—
171	C₂×M₁₆	—	47	4	no	—
172	C₂×M₁₆	—	47	4	no	—
173	C₂×M₁₆	—	47	16	no	—
174	C₂×M₁₆	—	94	4	no	—
175	C₂×M₁₆	—	94	4	no	—
176	C₂×M₁₆	—	94	8	no	—
177	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
178	C₂²×C₈	—	47	1	no	—
179	C₂²×C₈	—	47	4	no	—
180	C₂²×C₈	—	47	8	no	—
181	C₂²×C₈	—	47	1	no	—
182	C₂²×C₈	—	94	4	no	—
183	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
185	#117	—	47	4	no	—
186	#117	—	47	4	no	—
187	C₂×C₄⋊C₄	—	47	2	no	—
188	C₂×C₂²⋊C₄	—	47	2	no	—
189	C₂×C₄²	—	47	1	no	—
190	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
191	#108	—	47	16	no	—
192	#108	—	47	8	no	—
193	#108	C₄×D₄₇	1	8	yes	—
194	#97	—	2	2	no	—
195	#97	—	47	4	no	—
196	#97	—	94	4	no	—
197	#97	—	94	2	no	—
198	#97	C₄×D₄₇	1	4	yes	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	47	4	no	—
201	C₄×C₈	—	47	4	no	—
202	C₄×C₈	—	94	4	no	—
203	C₄×C₈	—	94	2	no	—
204	C₂⁴	—	47	1	no	—
205	C₂²×C₄	—	47	1	no	—
206	C₂²×C₄	—	47	2	no	—
207	C₂²×C₄	—	47	4	no	—
208	C₂²×C₄	—	47	2	no	—
209	C₂²×C₄	—	47	4	no	—
210	C₂²×C₄	—	47	2	no	—
211	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
212	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
213	M₁₆	—	2	4	no	—
214	M₁₆	—	47	8	no	—
215	M₁₆	—	94	8	no	—
216	M₁₆	—	94	16	no	—
217	M₁₆	—	94	4	no	—
218	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	47	2	no	—
222	C₂×C₈	—	47	4	no	—
223	C₂×C₈	—	47	16	no	—
224	C₂×C₈	—	47	2	no	—
225	C₂×C₈	—	94	16	no	—
226	C₂×C₈	—	94	4	no	—
227	C₂×C₈	—	94	2	no	—
228	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
230	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
231	C₄⋊C₄	—	47	8	no	—
232	C₂²⋊C₄	—	47	8	no	—
233	C₄²	—	47	2	no	—
234	C₄²	—	47	2	no	—
235	C₄²	C₂×C₄×D₄₇	1	2	yes	—
236	C₄²	D₄×D₄₇	1	2	yes	—
237	C₂³	—	47	2	no	—
238	C₂³	—	47	4	no	—
239	C₂³	—	188	2	no	—
240	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	47	2	no	—
243	C₂×C₄	—	47	4	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	47	4	no	—
246	C₂×C₄	—	47	4	no	—
247	C₂×C₄	—	47	4	no	—
248	C₂×C₄	—	47	4	no	—
249	C₂×C₄	—	188	2	no	—
250	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
251	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
252	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
253	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
254	C₂×C₄	#41×D₄₇	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	8	no	—
257	C₈	—	47	4	no	—
258	C₈	—	94	2	no	—
259	C₈	—	94	8	no	—
260	C₈	C₄²×D₄₇	1	4	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	47	4	no	—
263	C₂²	—	94	4	no	—
264	C₂²	—	188	4	no	—
265	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
266	C₂²	C₄×Q₈×D₄₇	1	2	yes	—
267	C₄	—	2	4	no	—
268	C₄	—	47	4	no	—
269	C₄	—	94	4	no	—
270	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
271	C₄	#117×D₄₇	1	1	yes	characteristic
272	C₄	C₄×D₄×D₄₇	1	2	yes	—
273	C₂	—	2	2	no	—
274	C₂	—	94	4	no	—
275	C₂	C₄×C₄⋊C₄×D₄₇	1	1	yes	characteristic
276	C₂	#20251998	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1778	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1778	D₄₇
hypocenter	C₄₇	#20460
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
4. #20243296	C₂	semidirect
5. #20243294	C₂	semidirect
6. C₄×#13093959	C₂	semidirect
7. #20243289	C₂	semidirect
9. C₄₇×#1778	C₂	semidirect
16. #108×D₄₇	C₄	semidirect
140. #1778	D₄₇	semidirect
169. C₄₇	#20460	semidirect

Maximal subgroups

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

subgroup	index
C₂×#108×D₄₇	2
#20251805	2
C₂×#15689080	2
#20243296	2
#20243294	2
C₄×#13093959	2
#20243289	2
#20243113	2
C₄₇×#1778	2
#20460	47