Subgroups — #26105901

Order 12032.

622 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689180	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688991	C₂	1	1	yes	maximal, characteristic
3	#20241230	C₂	1	4	yes	maximal
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	#15689180	—	2	2	no	—
6	#15689180	C₄	1	4	yes	complemented
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×#13093935	—	2	1	no	—
10	#15688991	—	2	1	no	—
11	#15688991	—	2	1	no	—
12	#15688991	—	2	1	no	—
13	#15688991	—	2	1	no	—
14	C₂×C₄₇⋊#95	C₄	1	2	yes	—
15	C₂×C₄₇⋊#95	C₄	1	2	yes	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#431	—	2	2	no	—
19	C₄₇×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄₇⋊C₈	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
27	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
28	#13093935	—	2	4	no	—
29	C₄₇⋊#95	—	2	2	no	—
30	C₄₇⋊#95	—	2	2	no	—
31	C₄₇⋊#95	—	2	2	no	—
32	C₄₇⋊#95	—	2	2	no	—
33	C₄₇⋊#95	—	2	2	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
47	C₄⋊C₈×C₄₇	—	2	4	no	—
48	C₂²⋊C₈×C₄₇	—	2	2	no	—
49	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₄₇	—	2	4	no	—
51	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	4	2	no	—
55	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
56	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	8	no	—
71	C₂²×Dic₄₇	—	4	4	no	—
72	C₂²×Dic₄₇	—	4	4	no	—
73	C₂×C₈×C₄₇	—	2	4	no	—
74	C₂×C₈×C₄₇	—	4	4	no	—
75	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
80	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
81	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	4	4	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	—	2	2	no	—
87	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₄₇	#131	1	4	yes	—
91	C₂×C₄×C₄₇	—	2	4	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	4	2	no	—
100	C₂×C₄×C₄₇	—	4	2	no	—
101	C₂×C₄×C₄₇	—	4	1	no	—
102	C₂×C₄×C₄₇	—	4	1	no	—
103	C₂×C₄×C₄₇	—	4	1	no	—
104	C₂×C₄×C₄₇	—	4	1	no	—
105	C₂×C₄×C₄₇	#95	1	2	yes	—
106	C₂×C₄×C₄₇	#95	1	2	yes	—
107	C₂×C₄×C₄₇	#95	1	2	yes	—
108	C₂×C₄×C₄₇	#95	1	2	yes	—
109	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	—	4	4	no	—
116	C₂×Dic₄₇	—	8	8	no	—
117	C₈×C₄₇	—	4	4	no	—
118	C₄₇⋊C₈	—	4	4	no	—
119	#7916	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	—	2	8	no	—
121	C₂²×C₄₇	—	2	1	no	—
122	C₂²×C₄₇	—	2	1	no	—
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
126	C₂²×C₄₇	#405	1	2	yes	—
127	C₂²×C₄₇	#408	1	2	yes	—
128	C₂²×C₄₇	#410	1	2	yes	—
129	C₂²×C₄₇	#413	1	2	yes	—
130	C₂²×C₄₇	#424	1	2	yes	—
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	2	no	—
133	C₄×C₄₇	—	2	1	no	—
134	C₄×C₄₇	—	2	1	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	Dic₄₇	—	8	4	no	—
140	C₂×#432	—	47	1	no	—
141	C₂×#431	D₄₇	1	1	yes	characteristic
142	C₂×#383	—	47	1	no	—
143	#1280	—	47	4	no	—
144	C₂×C₄₇	—	2	2	no	—
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	#1473	1	4	yes	—
147	C₂×C₄₇	#1729	1	1	yes	characteristic
148	C₂×C₄₇	#1740	1	1	yes	characteristic
149	C₂×C₄₇	#1742	1	1	yes	characteristic
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	#432	—	47	4	no	—
152	#432	—	94	2	no	—
153	#431	—	2	2	no	—
154	#431	Dic₄₇	1	4	yes	complemented
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	—	94	1	no	—
157	C₂×C₂²⋊C₈	—	47	2	no	—
158	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
159	C₂×#97	—	94	1	no	—
160	C₂×C₄×C₈	—	2	1	no	—
161	#383	—	94	1	no	—
162	#383	—	94	1	no	—
163	#383	—	94	1	no	—
164	#383	—	94	1	no	—
165	C₂×#95	—	47	2	no	—
166	C₂×#95	—	47	2	no	—
167	C₂×#95	—	47	2	no	—
168	C₄₇	#7916	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	—	47	4	no	—
170	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
171	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	94	2	no	—
175	C₂×C₄²	Dic₉₄	1	2	yes	—
176	C₂×C₄²	C₄×D₄₇	1	2	yes	—
177	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
178	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	94	4	no	—
185	C₂²⋊C₈	—	2	2	no	—
186	C₂²⋊C₈	—	47	4	no	—
187	C₂²⋊C₈	—	94	2	no	—
188	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
189	#97	—	94	4	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	—	94	2	no	—
192	#95	—	94	2	no	—
193	#95	—	94	2	no	—
194	#95	—	94	2	no	—
195	#95	—	94	2	no	—
196	#95	—	94	2	no	—
197	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	94	8	no	—
202	C₂²×C₄	—	94	4	no	—
203	C₂²×C₄	—	188	4	no	—
204	C₂²×C₄	—	188	4	no	—
205	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
206	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
207	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
208	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
209	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	4	no	—
221	C₂×C₈	—	94	4	no	—
222	C₂×C₈	—	188	4	no	—
223	C₄²	Dic₄₇⋊C₄	1	2	yes	—
224	C₄²	Dic₄₇⋊C₄	1	2	yes	—
225	C₄²	C₄⋊Dic₄₇	1	2	yes	—
226	C₄²	C₄⋊Dic₄₇	1	2	yes	—
227	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
228	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
229	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
230	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
234	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
235	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
236	C₂³	#13094030	1	4	yes	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	376	8	no	—
252	C₂×C₄	—	188	4	no	—
253	C₂×C₄	—	188	4	no	—
254	C₂×C₄	—	188	4	no	—
255	C₂×C₄	C₄₇⋊#95	1	2	yes	—
256	C₂×C₄	C₄₇⋊#95	1	2	yes	—
257	C₂×C₄	C₄₇⋊#95	1	2	yes	—
258	C₂×C₄	C₄₇⋊#95	1	2	yes	—
259	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	—	188	4	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
271	C₂²	#15689078	1	2	yes	—
272	C₂²	#15689089	1	2	yes	—
273	C₂²	#15689096	1	2	yes	—
274	C₂²	#15689105	1	2	yes	—
275	C₂²	#15689149	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	376	4	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	#20241755	1	4	yes	—
288	C₂	#20243084	1	1	yes	characteristic
289	C₂	#20243120	1	1	yes	characteristic
290	C₂	#20243126	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂³×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#431	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#431	D₄₇
hypocenter	C₄₇	#7916
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
6. #15689180	C₄	semidirect
19. C₄₇×#431	C₄	semidirect
154. #431	Dic₄₇	semidirect
168. C₄₇	#7916	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689180	2
C₂×#15688991	2
#20241230	2
C₂×C₄₇×#431	2
#7916	47