Subgroups — #26214388

Order 12032.

994 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689170	C₂	1	2	yes	maximal
2	#20251805	C₂	1	2	yes	maximal
3	C₂×#15689077	C₂	1	1	yes	maximal, characteristic
4	#20243296	C₂	1	2	yes	maximal, complemented
5	#20243295	C₂	1	2	yes	maximal, complemented
6	C₄×#13093960	C₂	1	1	yes	maximal, characteristic, complemented
7	#20243289	C₂	1	2	yes	maximal, complemented
8	#341×D₄₇	C₂	1	2	yes	maximal
9	C₄×#108×C₄₇	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	#15689170	C₄	1	16	yes	complemented
15	C₂×#13093960	C₂²	1	2	yes	—
16	C₂×#13093959	C₂²	1	4	yes	—
17	#15689080	—	2	4	no	—
18	#15689077	—	2	4	no	—
19	#15689077	C₄	1	8	yes	—
20	C₄×#10625329	C₂²	1	2	yes	—
21	M₁₆×Dic₄₇	C₂²	1	4	yes	—
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	C₂×#13093934	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×C₄×C₈×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	C₂×C₄	1	4	yes	—
35	C₂×#10625324	C₂×C₄	1	8	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	—	2	4	no	—
42	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
43	#13093981	C₂×C₄	1	4	yes	—
44	#13093981	D₄	1	2	yes	—
45	#13093981	D₄	1	2	yes	—
46	C₂×Dic₄₇⋊C₄	Q₈	1	2	yes	—
47	C₂×(C₂×D₄₇)⋊C₄	Q₈	1	2	yes	—
48	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
49	#13093960	C₂×C₄	1	8	yes	—
50	#13093959	C₂×C₄	1	16	yes	—
51	#13093935	—	2	2	no	—
52	#13093935	C₂×C₄	1	4	yes	—
53	#13093934	—	2	4	no	—
54	#13093934	C₂×C₄	1	4	yes	—
55	C₄×C₄₇⋊C₈	—	2	2	no	—
56	C₈×Dic₄₇	—	2	4	no	—
57	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
59	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
62	#108×C₄₇	C₂×C₄	1	8	yes	—
63	#97×C₄₇	—	2	2	no	—
64	C₄×C₈×C₄₇	—	2	2	no	—
65	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
66	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
73	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×D₄₇	#41	1	4	yes	—
75	M₁₆×C₄₇	—	2	4	no	—
76	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
85	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	4	yes	—
86	(C₂×D₄₇)⋊C₄	C₄⋊C₄	1	4	yes	—
87	Dic₄₇⋊C₄	C₄⋊C₄	1	4	yes	—
88	Dic₄₇⋊C₄	C₄⋊C₄	1	4	yes	—
89	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
90	C₄×Dic₄₇	C₂×D₄	1	2	yes	—
91	#10625329	—	2	4	no	—
92	#10625329	C₂²×C₄	1	8	yes	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	2	2	no	—
95	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
98	#10625324	—	2	16	no	—
99	#10625324	C₄²	1	16	yes	—
100	C₈×D₄₇	—	2	8	no	—
101	C₈×D₄₇	—	2	16	no	—
102	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×D₄₇	—	4	2	no	—
104	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂²×D₄₇	C₄×Q₈	1	4	yes	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
111	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
113	C₂×Dic₄₇	—	4	2	no	—
114	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×Dic₄₇	C₄×Q₈	1	4	yes	—
117	C₄×D₄₇	C₂×C₄²	1	4	yes	—
118	C₄×D₄₇	#117	1	4	yes	—
119	C₄×D₄₇	C₄×D₄	1	8	yes	—
120	C₈×C₄₇	—	2	2	no	—
121	C₈×C₄₇	—	2	8	no	—
122	C₈×C₄₇	C₂×C₄²	1	4	yes	—
123	C₄₇⋊C₈	—	2	2	no	—
124	C₄₇⋊C₈	—	2	8	no	—
125	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
126	#20459	—	47	1	no	maximal, Sylow
127	C₂²×C₄₇	—	2	2	no	—
128	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
130	C₂×D₄₇	—	2	4	no	—
131	C₂×D₄₇	—	4	4	no	—
132	C₂×D₄₇	C₄×C₄⋊C₄	1	4	yes	—
133	C₄×C₄₇	—	2	2	no	—
134	C₄×C₄₇	—	2	2	no	—
135	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
136	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
137	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
138	Dic₄₇	—	2	4	no	—
139	Dic₄₇	C₄×C₄⋊C₄	1	4	yes	—
140	C₂×#430	—	47	2	no	—
141	#2873	—	47	2	no	—
142	C₂×#405	—	47	1	no	—
143	#1778	—	47	2	no	—
144	#1778	—	47	2	no	—
145	C₄×#108	—	47	1	no	—
146	C₄×#108	—	47	2	no	—
147	C₄×#108	D₄₇	1	1	yes	characteristic, complemented
148	C₂×#341	—	47	2	no	—
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄₇	#2916	1	2	yes	—
152	D₄₇	—	2	4	no	—
153	C₂²×M₁₆	—	47	1	no	—
154	C₂²×M₁₆	—	47	4	no	—
155	C₂³×C₈	—	47	1	no	—
156	C₂×#117	—	47	1	no	—
157	#430	—	47	16	no	—
158	C₂×#108	—	47	4	no	—
159	C₂×#108	—	47	2	no	—
160	C₂×#108	C₂×D₄₇	1	2	yes	—
161	#405	—	47	8	no	—
162	#405	—	94	4	no	—
163	#405	—	94	4	no	—
164	C₄×M₁₆	—	47	4	no	—
165	C₄×M₁₆	—	47	2	no	—
166	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
167	C₂×#97	—	47	1	no	—
168	C₂×#97	—	47	1	no	—
169	C₂×C₄×C₈	—	47	1	no	—
170	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
171	#341	—	47	4	no	—
172	#341	—	47	2	no	—
173	#341	C₂×D₄₇	1	2	yes	—
174	C₄₇	#20459	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₄	—	47	1	no	—
176	C₂×M₁₆	—	47	4	no	—
177	C₂×M₁₆	—	47	8	no	—
178	C₂×M₁₆	—	47	4	no	—
179	C₂×M₁₆	—	47	16	no	—
180	C₂×M₁₆	—	94	4	no	—
181	C₂×M₁₆	—	94	8	no	—
182	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
183	C₂²×C₈	—	47	1	no	—
184	C₂²×C₈	—	47	1	no	—
185	C₂²×C₈	—	47	4	no	—
186	C₂²×C₈	—	94	4	no	—
187	C₂²×C₈	—	94	4	no	—
188	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	#117	—	47	4	no	—
191	#117	—	47	2	no	—
192	#117	—	47	2	no	—
193	C₂×C₄⋊C₄	—	47	2	no	—
194	C₂×C₂²⋊C₄	—	47	2	no	—
195	C₂×C₄²	—	47	1	no	—
196	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
197	#108	—	47	16	no	—
198	#108	—	47	8	no	—
199	#108	C₄×D₄₇	1	8	yes	—
200	#97	—	2	2	no	—
201	#97	—	47	4	no	—
202	#97	—	47	4	no	—
203	#97	—	94	4	no	—
204	#97	—	94	2	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	—	47	4	no	—
207	C₄×C₈	—	94	4	no	—
208	C₄×C₈	—	94	2	no	—
209	C₄×C₈	C₄×D₄₇	1	4	yes	—
210	C₂⁴	—	47	1	no	—
211	C₂²×C₄	—	47	1	no	—
212	C₂²×C₄	—	47	2	no	—
213	C₂²×C₄	—	47	4	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	47	4	no	—
216	C₂²×C₄	—	47	2	no	—
217	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
218	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	47	16	no	—
221	M₁₆	—	47	8	no	—
222	M₁₆	—	94	16	no	—
223	M₁₆	—	94	4	no	—
224	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	47	2	no	—
228	C₂×C₈	—	47	4	no	—
229	C₂×C₈	—	47	2	no	—
230	C₂×C₈	—	94	8	no	—
231	C₂×C₈	—	94	16	no	—
232	C₂×C₈	—	94	4	no	—
233	C₂×C₈	—	94	2	no	—
234	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
237	C₄⋊C₄	—	47	4	no	—
238	C₄⋊C₄	—	47	4	no	—
239	C₂²⋊C₄	—	47	4	no	—
240	C₂²⋊C₄	—	47	4	no	—
241	C₄²	—	47	2	no	—
242	C₄²	—	47	2	no	—
243	C₄²	C₂×C₄×D₄₇	1	2	yes	—
244	C₄²	D₄×D₄₇	1	1	yes	characteristic
245	C₄²	D₄×D₄₇	1	1	yes	characteristic
246	C₂³	—	47	2	no	—
247	C₂³	—	47	4	no	—
248	C₂³	—	188	2	no	—
249	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	47	2	no	—
252	C₂×C₄	—	47	4	no	—
253	C₂×C₄	—	47	4	no	—
254	C₂×C₄	—	47	4	no	—
255	C₂×C₄	—	47	8	no	—
256	C₂×C₄	—	47	4	no	—
257	C₂×C₄	—	188	2	no	—
258	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
259	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
260	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
262	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
263	C₂×C₄	#41×D₄₇	1	2	yes	—
264	C₈	—	2	2	no	—
265	C₈	—	2	8	no	—
266	C₈	—	47	4	no	—
267	C₈	—	94	2	no	—
268	C₈	—	94	8	no	—
269	C₈	C₄²×D₄₇	1	4	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	47	4	no	—
272	C₂²	—	94	4	no	—
273	C₂²	—	188	4	no	—
274	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
275	C₂²	C₄×Q₈×D₄₇	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	47	4	no	—
279	C₄	—	94	4	no	—
280	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
281	C₄	#117×D₄₇	1	1	yes	characteristic
282	C₄	C₄×D₄×D₄₇	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	94	4	no	—
285	C₂	C₄×C₄⋊C₄×D₄₇	1	1	yes	characteristic
286	C₂	#20251996	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₇	C₂×Q₈
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#20459
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #20243296	C₂	semidirect
5. #20243295	C₂	semidirect
6. C₄×#13093960	C₂	semidirect
7. #20243289	C₂	semidirect
9. C₄×#108×C₄₇	C₂	semidirect
14. #15689170	C₄	semidirect
147. C₄×#108	D₄₇	semidirect
174. C₄₇	#20459	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689170	2
#20251805	2
C₂×#15689077	2
#20243296	2
#20243295	2
C₄×#13093960	2
#20243289	2
#341×D₄₇	2
C₄×#108×C₄₇	2
#20459	47