Subgroups — #26214384

Order 12032.

1034 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₂×#15689075	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₄₇	C₂	1	2	yes	maximal
4	#20243295	C₂	1	2	yes	maximal, complemented
5	#20243294	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×Dic₄₇	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	C₂²×#10625324	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×D₄₇	C₂²	1	4	yes	—
13	C₂×C₄²×D₄₇	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	C₄×#10625329	C₂²	1	2	yes	—
18	#15689075	—	2	2	no	—
19	#15689075	C₄	1	8	yes	—
20	M₁₆×Dic₄₇	C₂²	1	4	yes	—
21	C₄×#10625324	—	2	2	no	—
22	C₂×#13093935	C₂²	1	1	yes	characteristic
23	#97×D₄₇	—	2	2	no	—
24	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₄₇	—	2	2	no	—
27	#15688888	C₂²	1	2	yes	—
28	#15688887	C₂²	1	4	yes	—
29	C₂×#108×C₄₇	C₂²	1	2	yes	—
30	C₄×M₁₆×C₄₇	C₂²	1	2	yes	—
31	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#341	C₂²	1	2	yes	—
33	C₂²×C₄×D₄₇	Q₈	1	2	yes	—
34	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
35	C₂×#10625329	C₂³	1	4	yes	—
36	C₂×#10625324	—	2	4	no	—
37	C₂×#10625324	—	2	4	no	—
38	C₂×#10625324	C₂×C₄	1	4	yes	—
39	C₂×#10625324	C₂×C₄	1	4	yes	—
40	C₂×#10625324	C₂×C₄	1	8	yes	—
41	M₁₆×D₄₇	—	2	8	no	—
42	M₁₆×D₄₇	C₂×C₄	1	16	yes	—
43	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₄₇	—	2	4	no	—
46	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
47	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₄²×D₄₇	C₂×C₄	1	4	yes	—
49	C₄²×D₄₇	D₄	1	2	yes	—
50	C₄²×D₄₇	D₄	1	2	yes	—
51	#13093960	C₂×C₄	1	8	yes	complemented
52	#13093959	C₂×C₄	1	16	yes	complemented
53	#13093935	—	2	2	no	—
54	#13093935	C₂×C₄	1	4	yes	—
55	#13093934	—	2	4	no	—
56	C₄×C₄₇⋊C₈	—	2	2	no	—
57	C₈×Dic₄₇	—	2	4	no	—
58	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
59	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
60	C₂×M₁₆×C₄₇	C₂³	1	4	yes	—
61	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
64	#108×C₄₇	C₂×C₄	1	8	yes	complemented
65	#97×C₄₇	—	2	2	no	—
66	#97×C₄₇	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₄₇	—	2	2	no	—
68	C₂³×D₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₂×Q₈	1	2	yes	—
72	C₂²×Dic₄₇	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×D₄₇	—	4	4	no	—
74	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
75	C₂×C₄×D₄₇	C₄⋊C₄	1	4	yes	—
76	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
77	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₄₇	C₂×D₄	1	2	yes	—
79	C₂×C₄×D₄₇	#41	1	4	yes	—
80	M₁₆×C₄₇	—	2	4	no	—
81	M₁₆×C₄₇	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₄₇	—	2	4	no	—
83	C₂×C₈×C₄₇	—	2	2	no	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
87	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₄₇	C₂×D₄	1	2	yes	—
89	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
90	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
91	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
92	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
93	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
94	#10625329	—	2	4	no	—
95	#10625329	C₂²×C₄	1	8	yes	—
96	C₂×C₄₇⋊C₈	—	2	4	no	—
97	C₂×C₄₇⋊C₈	—	2	2	no	—
98	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₄₇⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
101	#10625324	—	2	8	no	—
102	#10625324	—	2	16	no	—
103	#10625324	C₄²	1	16	yes	—
104	C₈×D₄₇	—	2	16	no	—
105	C₂³×C₄₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×D₄₇	—	4	2	no	—
107	C₂²×D₄₇	C₂×C₄⋊C₄	1	2	yes	—
108	C₂²×D₄₇	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
113	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
115	C₂×Dic₄₇	—	2	4	no	—
116	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×Dic₄₇	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₄₇	C₄×Q₈	1	4	yes	—
120	C₄×D₄₇	—	2	8	no	—
121	C₄×D₄₇	—	4	4	no	—
122	C₄×D₄₇	C₂×C₄²	1	4	yes	—
123	C₄×D₄₇	#117	1	4	yes	—
124	C₄×D₄₇	C₄×D₄	1	8	yes	—
125	C₈×C₄₇	—	2	2	no	—
126	C₈×C₄₇	—	2	8	no	—
127	C₈×C₄₇	C₂×C₄²	1	4	yes	—
128	C₄₇⋊C₈	—	2	2	no	—
129	C₄₇⋊C₈	—	2	8	no	—
130	C₄₇⋊C₈	C₂×C₄²	1	4	yes	—
131	C₄×#430	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	2	no	—
133	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₇	C₂×C₄×Q₈	1	2	yes	—
135	C₂×D₄₇	—	2	4	no	—
136	C₂×D₄₇	—	4	4	no	—
137	C₂×D₄₇	C₄×C₄⋊C₄	1	4	yes	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	C₂²×C₄²	1	1	yes	characteristic
140	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
141	C₄×C₄₇	C₂×C₄×D₄	1	2	yes	—
142	Dic₄₇	—	2	2	no	—
143	Dic₄₇	C₄×C₄⋊C₄	1	4	yes	—
144	Dic₄₇	#430	1	4	yes	complemented
145	C₂×#430	—	47	2	no	—
146	C₂×C₄×M₁₆	—	47	1	no	—
147	C₂×C₄×M₁₆	—	47	2	no	—
148	#1778	—	47	2	no	—
149	#1778	—	47	1	no	—
150	#1778	D₄₇	1	1	yes	characteristic, complemented
151	C₄×#108	—	47	2	no	—
152	C₄×#108	—	47	2	no	—
153	#1738	—	47	2	no	—
154	C₂×C₄₇	—	2	2	no	—
155	C₂×C₄₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄₇	C₂×#430	1	2	yes	—
157	D₄₇	—	2	4	no	—
158	C₂²×M₁₆	—	47	1	no	—
159	C₂²×M₁₆	—	47	1	no	—
160	C₂²×M₁₆	—	47	4	no	—
161	C₂²×C₄²	—	47	1	no	—
162	#430	—	47	16	no	—
163	C₂×#108	—	47	4	no	—
164	C₂×#108	—	47	2	no	—
165	C₂×#108	C₂×D₄₇	1	2	yes	—
166	C₄×M₁₆	—	47	8	no	—
167	C₄×M₁₆	—	47	4	no	—
168	C₄×M₁₆	—	47	2	no	—
169	C₄×M₁₆	—	94	2	no	—
170	C₄×M₁₆	—	94	2	no	—
171	C₄×M₁₆	C₂×D₄₇	1	2	yes	—
172	C₂×#97	—	47	1	no	—
173	C₂×#97	—	94	2	no	—
174	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
175	C₂×C₄×C₈	—	47	1	no	—
176	C₂×C₄×C₈	—	47	1	no	—
177	C₂×C₄×C₈	—	94	2	no	—
178	#341	—	47	4	no	—
179	#341	—	47	2	no	—
180	#341	C₂×D₄₇	1	2	yes	—
181	C₄₇	C₄×#430	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	47	1	no	—
183	C₂³×C₄	—	47	2	no	—
184	C₂×M₁₆	—	47	4	no	—
185	C₂×M₁₆	—	47	8	no	—
186	C₂×M₁₆	—	47	4	no	—
187	C₂×M₁₆	—	47	4	no	—
188	C₂×M₁₆	—	47	16	no	—
189	C₂×M₁₆	—	94	4	no	—
190	C₂×M₁₆	—	94	4	no	—
191	C₂×M₁₆	—	94	8	no	—
192	C₂×M₁₆	C₂²×D₄₇	1	4	yes	—
193	C₂²×C₈	—	47	1	no	—
194	C₂²×C₈	—	47	1	no	—
195	C₂²×C₈	—	94	4	no	—
196	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
198	C₂×C₄²	—	47	2	no	—
199	C₂×C₄²	—	47	1	no	—
200	C₂×C₄²	—	47	4	no	—
201	C₂×C₄²	—	47	2	no	—
202	C₂×C₄²	—	47	2	no	—
203	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
204	#108	—	47	16	no	—
205	#108	—	47	8	no	—
206	#108	C₄×D₄₇	1	8	yes	complemented
207	#97	—	2	2	no	—
208	#97	—	47	4	no	—
209	#97	—	94	4	no	—
210	#97	—	94	2	no	—
211	#97	C₄×D₄₇	1	4	yes	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	47	4	no	—
214	C₄×C₈	—	47	4	no	—
215	C₄×C₈	—	94	4	no	—
216	C₄×C₈	—	94	2	no	—
217	C₂⁴	—	47	1	no	—
218	C₂²×C₄	—	47	1	no	—
219	C₂²×C₄	—	47	2	no	—
220	C₂²×C₄	—	47	4	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	47	4	no	—
223	C₂²×C₄	—	47	2	no	—
224	C₂²×C₄	—	47	4	no	—
225	C₂²×C₄	—	47	4	no	—
226	C₂²×C₄	—	188	4	no	—
227	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
228	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
229	M₁₆	—	2	4	no	—
230	M₁₆	—	47	16	no	—
231	M₁₆	—	47	8	no	—
232	M₁₆	—	94	8	no	—
233	M₁₆	—	94	16	no	—
234	M₁₆	—	94	4	no	—
235	M₁₆	C₂×C₄×D₄₇	1	8	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	47	2	no	—
239	C₂×C₈	—	47	4	no	—
240	C₂×C₈	—	47	2	no	—
241	C₂×C₈	—	94	16	no	—
242	C₂×C₈	—	94	4	no	—
243	C₂×C₈	—	94	2	no	—
244	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
247	C₄²	—	47	4	no	—
248	C₄²	—	47	4	no	—
249	C₄²	—	47	2	no	—
250	C₄²	—	47	1	no	—
251	C₄²	—	47	1	no	—
252	C₄²	C₂×C₄×D₄₇	1	2	yes	—
253	C₄²	D₄×D₄₇	1	2	yes	—
254	C₂³	—	47	2	no	—
255	C₂³	—	47	4	no	—
256	C₂³	—	188	2	no	—
257	C₂³	C₂×Q₈×D₄₇	1	1	yes	characteristic
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	47	2	no	—
260	C₂×C₄	—	47	2	no	—
261	C₂×C₄	—	47	4	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	47	4	no	—
264	C₂×C₄	—	47	8	no	—
265	C₂×C₄	—	47	2	no	—
266	C₂×C₄	—	94	8	no	—
267	C₂×C₄	—	94	4	no	—
268	C₂×C₄	—	188	4	no	—
269	C₂×C₄	C₄⋊C₄×D₄₇	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
273	C₂×C₄	#41×D₄₇	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	2	8	no	—
276	C₈	—	47	4	no	—
277	C₈	—	94	2	no	—
278	C₈	—	94	8	no	—
279	C₈	C₄²×D₄₇	1	4	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	47	4	no	—
282	C₂²	—	94	4	no	—
283	C₂²	—	188	4	no	—
284	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
285	C₂²	C₄×Q₈×D₄₇	1	2	yes	—
286	C₄	—	2	4	no	—
287	C₄	—	47	4	no	—
288	C₄	—	47	4	no	—
289	C₄	—	94	2	no	—
290	C₄	C₂×C₄²×D₄₇	1	1	yes	characteristic
291	C₄	#117×D₄₇	1	1	yes	characteristic
292	C₄	C₄×D₄×D₄₇	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	94	4	no	—
295	C₂	C₄×C₄⋊C₄×D₄₇	1	1	yes	characteristic
296	C₂	#430×D₄₇	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	C₄×#430
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. #20243295	C₂	semidirect
5. #20243294	C₂	semidirect
6. #108×Dic₄₇	C₂	semidirect
7. #20243289	C₂	semidirect
9. C₄₇×#1778	C₂	semidirect
14. #15689170	C₄	semidirect
51. #13093960	C₂×C₄	semidirect
52. #13093959	C₂×C₄	semidirect
64. #108×C₄₇	C₂×C₄	semidirect
144. Dic₄₇	#430	semidirect
150. #1778	D₄₇	semidirect
181. C₄₇	C₄×#430	semidirect
206. #108	C₄×D₄₇	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
C₂×#15689075	2
C₄×M₁₆×D₄₇	2
#20243295	2
#20243294	2
#108×Dic₄₇	2
#20243289	2
#20243113	2
C₄₇×#1778	2
C₄×#430	47