Subgroups — #26106303

Order 12032.

786 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245786	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241268	C₂	1	4	yes	maximal
4	C₂×C₄₇×#432	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₇	—	2	1	no	—
6	C₂²×C₄×Dic₄₇	—	2	1	no	—
7	#15688991	—	2	1	no	—
8	#15688989	C₄	1	2	yes	complemented
9	#15688986	—	2	1	no	—
10	#15688983	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₄₇	—	2	2	no	—
12	C₄×C₄⋊Dic₄₇	C₄	1	2	yes	complemented
13	C₄×C₄⋊Dic₄₇	C₄	1	4	yes	complemented
14	C₄×C₂²⋊Dic₄₇	—	2	1	no	—
15	#15688969	—	2	1	no	—
16	C₂×C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#432	—	2	2	no	—
19	C₄₇×#432	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
22	C₂×#97×C₄₇	—	2	1	no	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	8	no	—
25	C₂×C₄⋊Dic₄₇	—	2	4	no	—
26	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄×Dic₄₇	—	2	2	no	—
29	C₂×C₄×Dic₄₇	—	2	4	no	—
30	C₂×C₄×Dic₄₇	—	4	4	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	C₄₇⋊#95	—	2	2	no	—
36	C₄₇⋊#95	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
49	C₄⋊C₈×C₄₇	—	2	4	no	—
50	C₂²⋊C₈×C₄₇	—	2	2	no	—
51	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
52	#97×C₄₇	—	2	4	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	2	no	—
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	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
72	C₂²×Dic₄₇	—	2	8	no	—
73	C₂²×Dic₄₇	—	2	4	no	—
74	C₂²×Dic₄₇	—	2	4	no	—
75	C₂²×Dic₄₇	—	4	4	no	—
76	C₂²×Dic₄₇	—	4	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	4	4	no	—
79	C₄²×C₄₇	C₄²	1	2	yes	—
80	C₄²×C₄₇	C₄²	1	2	yes	—
81	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
84	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
85	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₂²⋊Dic₄₇	—	4	2	no	—
88	C₂²⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	2	16	no	—
90	C₄⋊Dic₄₇	—	4	4	no	—
91	C₄×Dic₄₇	—	4	8	no	—
92	C₄×Dic₄₇	—	4	4	no	—
93	C₂³×C₄₇	—	2	2	no	—
94	C₂³×C₄₇	—	2	2	no	—
95	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂³×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
99	C₂³×C₄₇	#137	1	2	yes	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	2	4	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	—	4	1	no	—
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	#95	1	2	yes	—
118	C₂×C₄×C₄₇	#104	1	4	yes	—
119	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×Dic₄₇	—	2	8	no	—
124	C₂×Dic₄₇	—	4	4	no	—
125	C₂×Dic₄₇	—	4	16	no	—
126	C₂×Dic₄₇	—	4	4	no	—
127	C₂×Dic₄₇	—	4	4	no	—
128	C₂×Dic₄₇	—	4	4	no	—
129	C₂×Dic₄₇	—	4	4	no	—
130	C₈×C₄₇	—	4	4	no	—
131	#8021	—	47	1	no	maximal, Sylow
132	C₂²×C₄₇	—	2	1	no	—
133	C₂²×C₄₇	—	2	1	no	—
134	C₂²×C₄₇	—	2	1	no	—
135	C₂²×C₄₇	—	2	1	no	—
136	C₂²×C₄₇	—	2	4	no	—
137	C₂²×C₄₇	—	2	4	no	—
138	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
139	C₂²×C₄₇	#410	1	2	yes	—
140	C₂²×C₄₇	#417	1	2	yes	—
141	C₂²×C₄₇	#418	1	1	yes	characteristic
142	C₂²×C₄₇	#419	1	1	yes	characteristic
143	C₂²×C₄₇	C₂×#104	1	2	yes	—
144	C₂²×C₄₇	#428	1	2	yes	—
145	C₄×C₄₇	—	2	2	no	—
146	C₄×C₄₇	—	2	2	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	1	no	—
149	C₄×C₄₇	—	2	1	no	—
150	C₄×C₄₇	—	2	1	no	—
151	C₄×C₄₇	—	2	1	no	—
152	C₄×C₄₇	#339	1	4	yes	—
153	Dic₄₇	—	4	8	no	—
154	Dic₄₇	—	8	4	no	—
155	C₂×#432	D₄₇	1	1	yes	characteristic
156	#2283	—	47	1	no	—
157	C₂×C₄×C₄⋊C₄	—	47	1	no	—
158	#1294	—	47	4	no	—
159	C₂×C₄₇	—	2	1	no	—
160	C₂×C₄₇	—	2	1	no	—
161	C₂×C₄₇	—	2	2	no	—
162	C₂×C₄₇	#1509	1	4	yes	—
163	C₂×C₄₇	C₂×#339	1	1	yes	characteristic
164	C₂×C₄₇	#1736	1	1	yes	characteristic
165	C₂×C₄₇	#1740	1	1	yes	characteristic
166	C₂²×C₄⋊C₄	—	94	1	no	—
167	C₂²×C₄²	—	94	1	no	—
168	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
169	#432	—	2	2	no	—
170	#432	Dic₄₇	1	4	yes	complemented
171	C₂×C₄⋊C₈	—	2	1	no	—
172	C₂×C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
173	C₂×#97	—	2	1	no	—
174	#383	—	94	1	no	—
175	#382	—	47	2	no	—
176	#381	—	94	1	no	—
177	#380	—	47	2	no	—
178	C₄×C₄⋊C₄	—	47	4	no	—
179	C₄×C₄⋊C₄	—	47	2	no	—
180	C₄×C₄⋊C₄	—	94	2	no	—
181	C₄×C₂²⋊C₄	—	94	1	no	—
182	#376	—	94	1	no	—
183	C₂×#95	—	47	2	no	—
184	C₄₇	#8021	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	—	94	2	no	—
186	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
187	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
188	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
189	C₂²×C₈	—	2	2	no	—
190	C₂×C₄⋊C₄	—	47	4	no	—
191	C₂×C₄⋊C₄	—	94	8	no	—
192	C₂×C₄⋊C₄	—	94	4	no	—
193	C₂×C₂²⋊C₄	—	94	2	no	—
194	C₂×C₄²	—	94	2	no	—
195	C₂×C₄²	—	94	4	no	—
196	C₂×C₄²	—	188	4	no	—
197	C₂×C₄²	Dic₉₄	1	2	yes	—
198	C₂×C₄²	C₄×D₄₇	1	2	yes	—
199	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
201	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
202	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
204	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
205	C₄⋊C₈	—	2	4	no	—
206	C₂²⋊C₈	—	2	2	no	—
207	C₂²⋊C₈	C₂×Dic₄₇	1	4	yes	—
208	#97	—	2	4	no	—
209	#95	—	47	4	no	—
210	#95	—	94	2	no	—
211	#95	—	94	2	no	—
212	#95	—	94	2	no	—
213	#95	—	94	2	no	—
214	#95	—	94	2	no	—
215	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	94	8	no	—
221	C₂²×C₄	—	94	4	no	—
222	C₂²×C₄	—	94	4	no	—
223	C₂²×C₄	—	188	4	no	—
224	C₂²×C₄	—	188	4	no	—
225	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
226	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
227	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
228	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
229	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	4	no	—
241	C₄⋊C₄	—	94	16	no	—
242	C₄⋊C₄	—	188	4	no	—
243	C₂²⋊C₄	—	188	2	no	—
244	C₂²⋊C₄	—	188	2	no	—
245	C₄²	—	188	8	no	—
246	C₄²	—	188	4	no	—
247	C₄²	C₄×Dic₄₇	1	2	yes	—
248	C₄²	C₄×Dic₄₇	1	2	yes	—
249	C₄²	Dic₄₇⋊C₄	1	2	yes	—
250	C₄²	Dic₄₇⋊C₄	1	2	yes	—
251	C₄²	Dic₄₇⋊C₄	1	2	yes	—
252	C₄²	Dic₄₇⋊C₄	1	2	yes	—
253	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
254	C₄²	C₂²⋊Dic₄₇	1	2	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
258	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
259	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
260	C₂³	#13094052	1	2	yes	—
261	C₂³	#13094057	1	2	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	94	8	no	—
277	C₂×C₄	—	188	4	no	—
278	C₂×C₄	—	188	4	no	—
279	C₂×C₄	—	188	4	no	—
280	C₂×C₄	—	188	4	no	—
281	C₂×C₄	—	188	16	no	—
282	C₂×C₄	—	188	4	no	—
283	C₂×C₄	C₄₇⋊#95	1	2	yes	—
284	C₂×C₄	C₄₇⋊#95	1	2	yes	—
285	C₂×C₄	C₄₇⋊#95	1	2	yes	—
286	C₂×C₄	C₄₇⋊#95	1	2	yes	—
287	C₂×C₄	#13093950	1	4	yes	—
288	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
290	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
292	C₈	—	4	4	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
300	C₂²	#15689096	1	2	yes	—
301	C₂²	#15689125	1	2	yes	—
302	C₂²	#15689129	1	1	yes	characteristic
303	C₂²	#15689133	1	1	yes	characteristic
304	C₂²	C₂×#13093950	1	2	yes	—
305	C₂²	#15689165	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	376	4	no	—
314	C₄	—	188	8	no	—
315	C₄	#15688884	1	4	yes	—
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	#20241914	1	4	yes	—
320	C₂	C₂×#15688884	1	1	yes	characteristic
321	C₂	#20243108	1	1	yes	characteristic
322	C₂	#20243120	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#432	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#432	D₄₇
hypocenter	C₄₇	#8021
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
8. #15688989	C₄	semidirect
12. C₄×C₄⋊Dic₄₇	C₄	semidirect
13. C₄×C₄⋊Dic₄₇	C₄	semidirect
19. C₄₇×#432	C₄	semidirect
170. #432	Dic₄₇	semidirect
184. C₄₇	#8021	semidirect

Maximal subgroups

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

subgroup	index
#20245786	2
C₂×C₄×C₄⋊Dic₄₇	2
#20241268	2
C₂×C₄₇×#432	2
#8021	47