Subgroups — #26105874

Order 12032.

802 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689176	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241230	C₂	1	4	yes	maximal
4	C₂×C₄₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	C₄	1	2	yes	—
6	C₂²×C₄×Dic₄₇	C₄	1	2	yes	—
7	#15689176	—	2	2	no	—
8	#15689176	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
12	C₄×C₄⋊Dic₄₇	—	2	1	no	—
13	C₄×C₄⋊Dic₄₇	—	2	1	no	—
14	C₄×C₄⋊Dic₄₇	—	2	1	no	—
15	C₄×C₄⋊Dic₄₇	—	2	1	no	—
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	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	4	no	—
28	C₂×C₄⋊Dic₄₇	C₈	1	8	yes	complemented
29	C₂×C₄×Dic₄₇	—	2	2	no	—
30	C₂×C₄×Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	4	4	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
35	C₄×C₄₇⋊C₈	—	2	4	no	—
36	C₄₇⋊#95	—	2	2	no	—
37	C₄₇⋊#95	—	2	2	no	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
50	C₄⋊C₈×C₄₇	—	2	4	no	—
51	C₂²⋊C₈×C₄₇	—	2	2	no	—
52	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₄₇	—	2	4	no	—
54	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
58	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₄₇	M₁₆	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	2	4	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂²×Dic₄₇	—	4	4	no	—
78	C₂²×Dic₄₇	C₂×C₈	1	8	yes	—
79	C₂×C₈×C₄₇	—	2	4	no	—
80	C₂×C₈×C₄₇	—	4	4	no	—
81	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
88	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
89	C₄⋊Dic₄₇	—	2	8	no	—
90	C₄⋊Dic₄₇	—	2	8	no	—
91	C₄×Dic₄₇	—	4	4	no	—
92	C₄×Dic₄₇	—	4	4	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	4	no	—
95	C₂³×C₄₇	—	2	2	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₄₇	C₂²×C₈	1	2	yes	—
102	C₂³×C₄₇	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	#95	1	2	yes	—
114	C₂×C₄×C₄₇	#95	1	2	yes	—
115	C₂×C₄×C₄₇	#95	1	2	yes	—
116	C₂×C₄×C₄₇	#95	1	2	yes	—
117	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
118	C₂×C₄×C₄₇	#97	1	4	yes	—
119	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
120	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
121	C₂×C₄×C₄₇	#100	1	2	yes	—
122	C₂×C₄×C₄₇	#101	1	2	yes	—
123	C₂×C₄×C₄₇	C₄⋊C₈	1	8	yes	—
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	8	no	—
129	C₂×Dic₄₇	—	4	8	no	—
130	C₂×Dic₄₇	—	4	4	no	—
131	C₂×Dic₄₇	—	4	4	no	—
132	C₂×Dic₄₇	—	4	8	no	—
133	C₈×C₄₇	—	4	4	no	—
134	C₄₇⋊C₈	—	4	4	no	—
135	#7908	—	47	1	no	maximal, Sylow
136	C₂²×C₄₇	—	2	2	no	—
137	C₂²×C₄₇	—	2	2	no	—
138	C₂²×C₄₇	—	2	2	no	—
139	C₂²×C₄₇	—	2	2	no	—
140	C₂²×C₄₇	—	2	1	no	—
141	C₂²×C₄₇	—	2	1	no	—
142	C₂²×C₄₇	—	2	1	no	—
143	C₂²×C₄₇	—	2	1	no	—
144	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
145	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
147	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
148	C₂²×C₄₇	#410	1	2	yes	—
149	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
150	C₂²×C₄₇	C₂×#101	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
152	C₄×C₄₇	—	2	1	no	—
153	C₄×C₄₇	—	2	1	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	#336	1	4	yes	—
158	C₄×C₄₇	#336	1	4	yes	—
159	C₄×C₄₇	#343	1	4	yes	—
160	Dic₄₇	—	4	8	no	—
161	C₂×#431	—	47	1	no	—
162	C₂×#431	D₄₇	1	1	yes	characteristic
163	C₂×C₄×C₄⋊C₄	—	47	1	no	—
164	#1280	—	47	4	no	—
165	C₂×C₄₇	—	2	1	no	—
166	C₂×C₄₇	—	2	1	no	—
167	C₂×C₄₇	—	2	2	no	—
168	C₂×C₄₇	#1460	1	2	yes	—
169	C₂×C₄₇	#1461	1	2	yes	—
170	C₂×C₄₇	C₂×#336	1	1	yes	characteristic
171	C₂×C₄₇	#1740	1	1	yes	characteristic
172	C₂×C₄₇	C₂×#343	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	—	47	2	no	—
174	C₂²×C₄²	—	47	2	no	—
175	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
176	#431	—	2	2	no	—
177	#431	—	47	4	no	—
178	#431	—	94	2	no	—
179	#431	Dic₄₇	1	4	yes	complemented
180	C₂×C₄⋊C₈	—	2	1	no	—
181	C₂×C₄⋊C₈	—	94	1	no	—
182	C₂×C₂²⋊C₈	—	47	2	no	—
183	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
184	C₂×C₄×C₈	—	2	1	no	—
185	C₂×C₄×C₈	—	94	1	no	—
186	C₄×C₄⋊C₄	—	94	1	no	—
187	C₄×C₄⋊C₄	—	94	1	no	—
188	C₄×C₄⋊C₄	—	94	1	no	—
189	C₄×C₄⋊C₄	—	94	1	no	—
190	C₂×#95	—	47	2	no	—
191	C₄₇	#7908	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	47	4	no	—
193	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
194	C₂³×C₄	C₄×D₄₇	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	94	2	no	—
198	C₂×C₄⋊C₄	—	47	8	no	—
199	C₂×C₄⋊C₄	—	94	4	no	—
200	C₂×C₄⋊C₄	—	94	2	no	—
201	C₂×C₄⋊C₄	—	94	2	no	—
202	C₂×C₄²	—	94	2	no	—
203	C₂×C₄²	—	94	2	no	—
204	C₂×C₄²	—	188	4	no	—
205	C₂×C₄²	Dic₉₄	1	2	yes	—
206	C₂×C₄²	C₄×D₄₇	1	2	yes	—
207	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
208	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	94	4	no	—
215	C₂²⋊C₈	—	2	2	no	—
216	C₂²⋊C₈	—	47	4	no	—
217	C₂²⋊C₈	—	94	2	no	—
218	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
219	C₄×C₈	—	2	4	no	—
220	C₄×C₈	—	94	4	no	—
221	#95	—	94	2	no	—
222	#95	—	94	2	no	—
223	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	47	8	no	—
228	C₂²×C₄	—	94	4	no	—
229	C₂²×C₄	—	94	4	no	—
230	C₂²×C₄	—	188	4	no	—
231	C₂²×C₄	—	188	4	no	—
232	C₂²×C₄	C₈×D₄₇	1	4	yes	—
233	C₂²×C₄	#10625324	1	4	yes	—
234	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
235	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
236	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
237	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
238	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	4	4	no	—
250	C₂×C₈	—	94	4	no	—
251	C₂×C₈	—	188	4	no	—
252	C₄⋊C₄	—	94	8	no	—
253	C₄⋊C₄	—	94	8	no	—
254	C₄²	—	188	4	no	—
255	C₄²	—	188	4	no	—
256	C₄²	Dic₄₇⋊C₄	1	2	yes	—
257	C₄²	Dic₄₇⋊C₄	1	2	yes	—
258	C₄²	C₄⋊Dic₄₇	1	2	yes	—
259	C₄²	C₄⋊Dic₄₇	1	2	yes	—
260	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
261	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
262	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
263	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
268	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
269	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
270	C₂³	C₂×C₈×D₄₇	1	2	yes	—
271	C₂³	C₂×#10625324	1	2	yes	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	94	8	no	—
283	C₂×C₄	—	188	8	no	—
284	C₂×C₄	—	188	4	no	—
285	C₂×C₄	—	188	4	no	—
286	C₂×C₄	—	188	8	no	—
287	C₂×C₄	C₄₇⋊#95	1	2	yes	—
288	C₂×C₄	C₄₇⋊#95	1	2	yes	—
289	C₂×C₄	C₄₇⋊#95	1	2	yes	—
290	C₂×C₄	C₄₇⋊#95	1	2	yes	—
291	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
292	C₂×C₄	#13093934	1	4	yes	—
293	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
294	C₂×C₄	(C₂×D₄₇)⋊C₈	1	4	yes	—
295	C₂×C₄	#13093940	1	2	yes	—
296	C₂×C₄	#13093942	1	2	yes	—
297	C₂×C₄	Dic₄₇⋊C₈	1	8	yes	—
298	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
300	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
302	C₈	—	4	4	no	—
303	C₈	—	188	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
313	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
314	C₂²	C₂×#13093934	1	1	yes	characteristic
315	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	2	yes	—
316	C₂²	#15689096	1	2	yes	—
317	C₂²	C₂×#13093940	1	1	yes	characteristic
318	C₂²	C₂×#13093942	1	1	yes	characteristic
319	C₂²	C₂×Dic₄₇⋊C₈	1	2	yes	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	188	8	no	—
326	C₄	#15688877	1	4	yes	—
327	C₄	#15688877	1	4	yes	—
328	C₄	#15688891	1	4	yes	—
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	2	2	no	—
332	C₂	#20241708	1	2	yes	—
333	C₂	#20241711	1	2	yes	—
334	C₂	C₂×#15688877	1	1	yes	characteristic
335	C₂	#20243120	1	1	yes	characteristic
336	C₂	C₂×#15688891	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄₇	C₂×C₄×C₈
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	#7908
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
8. #15689176	C₄	semidirect
19. C₄₇×#431	C₄	semidirect
28. C₂×C₄⋊Dic₄₇	C₈	semidirect
179. #431	Dic₄₇	semidirect
191. C₄₇	#7908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689176	2
C₂×C₄×C₄⋊Dic₄₇	2
#20241230	2
C₂×C₄₇×#431	2
#7908	47