Subgroups — #26214420

Order 12032.

970 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689185	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689182	C₂	1	1	yes	maximal, characteristic
3	#20245727	C₂	1	1	yes	maximal, characteristic
4	#20243244	C₂	1	2	yes	maximal, complemented
5	#20243236	C₂	1	2	yes	maximal, complemented
6	#20243153	C₂	1	2	yes	maximal
7	#20243145	C₂	1	2	yes	maximal
8	#20243140	C₂	1	1	yes	maximal, characteristic, complemented
9	#97×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#1746	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13093981	C₂²	1	1	yes	characteristic
13	C₂×#13093981	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
15	#15689185	C₄	1	8	yes	complemented
16	#15689182	C₄	1	8	yes	complemented
17	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
18	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
19	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
20	C₂×(C₂×D₄₇)⋊C₈	C₂²	1	2	yes	—
21	C₂×#13093935	C₂²	1	1	yes	characteristic
22	C₂×#13093934	C₂²	1	2	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
25	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
26	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
27	#15688974	C₄	1	4	yes	—
28	#15688970	C₄	1	4	yes	—
29	#15688969	C₂²	1	1	yes	characteristic
30	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
32	#15688878	C₂²	1	2	yes	—
33	#15688878	C₂²	1	2	yes	—
34	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₄₇×#376	C₂²	1	1	yes	characteristic
37	C₄₇×#336	C₂²	1	2	yes	—
38	C₄₇×#336	C₂²	1	2	yes	—
39	C₂²×C₄×D₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
43	#13093981	—	2	4	no	—
44	#13093981	C₂×C₄	1	4	yes	—
45	#13093981	C₂×C₄	1	4	yes	—
46	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×Dic₄₇⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×(C₂×D₄₇)⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
52	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
56	C₄²×D₄₇	—	2	4	no	—
57	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
58	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
59	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
60	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
61	#13093935	C₂×C₄	1	4	yes	complemented
62	#13093934	—	2	4	no	—
63	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
64	C₈×Dic₄₇	—	2	4	no	—
65	C₈×Dic₄₇	—	2	4	no	—
66	C₈×Dic₄₇	—	2	4	no	—
67	C₄₇⋊#95	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#97×C₄₇	C₂×C₄	1	4	yes	complemented
74	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
75	#95×C₄₇	C₂×C₄	1	4	yes	—
76	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₄×D₄₇	—	2	2	no	—
86	C₂×C₄×D₄₇	—	2	2	no	—
87	C₂×C₄×D₄₇	—	2	4	no	—
88	C₂×C₄×D₄₇	—	2	8	no	—
89	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₇	—	2	2	no	—
92	C₂×C₈×C₄₇	—	2	2	no	—
93	C₂×C₈×C₄₇	—	2	2	no	—
94	C₂×C₈×C₄₇	—	2	2	no	—
95	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
97	C₄²×C₄₇	—	2	2	no	—
98	C₄²×C₄₇	—	2	2	no	—
99	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
101	(C₂×D₄₇)⋊C₄	—	2	4	no	—
102	(C₂×D₄₇)⋊C₄	C₄²	1	8	yes	—
103	Dic₄₇⋊C₄	—	2	4	no	—
104	Dic₄₇⋊C₄	C₄²	1	8	yes	—
105	C₄×Dic₄₇	—	2	4	no	—
106	C₄×Dic₄₇	—	2	8	no	—
107	C₄×Dic₄₇	—	2	8	no	—
108	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
109	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
110	C₄×Dic₄₇	#41	1	4	yes	—
111	C₄×Dic₄₇	#41	1	4	yes	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	—	2	2	no	—
114	C₂×C₄₇⋊C₈	—	2	2	no	—
115	C₂×C₄₇⋊C₈	—	2	2	no	—
116	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
117	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
118	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
119	C₂²×D₄₇	—	2	1	no	—
120	C₂²×D₄₇	—	2	1	no	—
121	C₂²×D₄₇	—	2	4	no	—
122	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	—	2	4	no	—
124	C₂×C₄×C₄₇	—	2	2	no	—
125	C₂×C₄×C₄₇	—	2	2	no	—
126	C₂×C₄×C₄₇	—	2	4	no	—
127	C₂×C₄×C₄₇	—	2	4	no	—
128	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
129	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
132	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
133	C₂×Dic₄₇	—	2	1	no	—
134	C₂×Dic₄₇	—	2	1	no	—
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
138	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
139	C₂×Dic₄₇	#117	1	4	yes	—
140	C₂×Dic₄₇	#117	1	4	yes	—
141	C₂×Dic₄₇	#131	1	8	yes	—
142	C₄×D₄₇	—	2	4	no	—
143	C₄×D₄₇	—	2	4	no	—
144	C₄×D₄₇	—	4	8	no	—
145	C₈×C₄₇	—	2	4	no	—
146	C₈×C₄₇	—	2	4	no	—
147	C₄₇⋊C₈	—	2	4	no	—
148	C₄₇⋊C₈	—	2	4	no	—
149	C₄×#433	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
151	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#131	1	4	yes	—
154	C₂×D₄₇	—	2	4	no	—
155	C₂×D₄₇	—	2	2	no	—
156	C₂×D₄₇	—	4	2	no	—
157	C₂×D₄₇	—	4	2	no	—
158	C₄×C₄₇	—	2	4	no	—
159	C₄×C₄₇	—	4	2	no	—
160	C₄×C₄₇	—	4	2	no	—
161	C₄×C₄₇	C₄×#41	1	2	yes	—
162	C₄×C₄₇	C₄×#41	1	2	yes	—
163	Dic₄₇	—	2	4	no	—
164	Dic₄₇	—	2	4	no	—
165	Dic₄₇	—	2	2	no	—
166	Dic₄₇	#433	1	8	yes	complemented
167	C₂×#433	—	47	1	no	—
168	C₂×#433	—	47	1	no	—
169	C₄×#117	—	47	1	no	—
170	#1768	—	47	2	no	—
171	C₄×C₄⋊C₈	—	47	2	no	—
172	#1750	—	47	2	no	—
173	C₄×C₂²⋊C₈	—	47	2	no	—
174	#1746	—	47	1	no	—
175	#1746	D₄₇	1	1	yes	characteristic, complemented
176	C₄×#97	—	47	1	no	—
177	C₄²×C₈	—	47	1	no	—
178	C₂×C₄₇	C₄×#117	1	1	yes	characteristic
179	C₂×C₄₇	C₄×#131	1	2	yes	—
180	C₂×C₄₇	C₂×#433	1	2	yes	—
181	C₂×C₄₇	#2922	1	2	yes	—
182	D₄₇	—	4	2	no	—
183	C₂×#117	—	47	1	no	—
184	C₂×#117	—	47	1	no	—
185	C₂²×C₄²	—	47	1	no	—
186	#433	—	47	8	no	—
187	#433	—	47	8	no	—
188	C₂×C₄⋊C₈	—	47	2	no	—
189	C₂×C₄⋊C₈	—	47	2	no	—
190	C₂×C₂²⋊C₈	—	47	2	no	—
191	C₂×C₂²⋊C₈	—	47	2	no	—
192	C₂×#97	—	47	1	no	—
193	C₂×#97	—	47	2	no	—
194	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
195	C₂×C₄×C₈	—	47	1	no	—
196	C₂×C₄×C₈	—	47	2	no	—
197	C₂×C₄×C₈	—	47	2	no	—
198	C₂×C₄×C₈	—	47	2	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
200	C₄×C₄⋊C₄	—	47	4	no	—
201	C₄×C₂²⋊C₄	—	47	4	no	—
202	#376	—	47	1	no	—
203	#376	C₂×D₄₇	1	1	yes	characteristic
204	C₄³	—	47	1	no	—
205	C₄³	—	47	1	no	—
206	#336	—	47	2	no	—
207	#336	—	47	2	no	—
208	#336	C₂×D₄₇	1	2	yes	—
209	#336	C₂×D₄₇	1	2	yes	—
210	C₄₇	C₄×#433	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	47	1	no	—
212	C₂³×C₄	—	47	2	no	—
213	C₂²×C₈	—	47	2	no	—
214	C₂²×C₈	—	47	2	no	—
215	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
216	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
217	#117	—	47	4	no	—
218	#117	—	47	4	no	—
219	#117	—	94	4	no	—
220	C₂×C₄⋊C₄	—	47	2	no	—
221	C₂×C₄⋊C₄	—	47	2	no	—
222	C₂×C₂²⋊C₄	—	47	2	no	—
223	C₂×C₂²⋊C₄	—	47	2	no	—
224	C₂×C₄²	—	47	1	no	—
225	C₂×C₄²	—	47	1	no	—
226	C₂×C₄²	—	47	1	no	—
227	C₂×C₄²	—	47	4	no	—
228	C₂×C₄²	—	47	4	no	—
229	C₂×C₄²	—	47	2	no	—
230	C₂×C₄²	—	94	4	no	—
231	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
234	C₄⋊C₈	—	47	8	no	—
235	C₄⋊C₈	—	47	8	no	—
236	C₂²⋊C₈	—	47	8	no	—
237	C₂²⋊C₈	—	47	8	no	—
238	#97	—	47	4	no	—
239	#97	—	94	4	no	—
240	#97	C₄×D₄₇	1	4	yes	complemented
241	C₄×C₈	—	47	4	no	—
242	C₄×C₈	—	94	4	no	—
243	C₄×C₈	—	94	4	no	—
244	C₄×C₈	—	94	4	no	—
245	C₄×C₈	C₄×D₄₇	1	4	yes	complemented
246	#95	—	47	4	no	—
247	#95	C₄×D₄₇	1	4	yes	—
248	C₂⁴	—	47	1	no	—
249	C₂²×C₄	—	47	2	no	—
250	C₂²×C₄	—	47	2	no	—
251	C₂²×C₄	—	47	2	no	—
252	C₂²×C₄	—	47	1	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	94	4	no	—
256	C₂²×C₄	—	94	8	no	—
257	C₂²×C₄	—	94	2	no	—
258	C₂²×C₄	—	94	2	no	—
259	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
260	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
261	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
262	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	47	4	no	—
268	C₂×C₈	—	47	4	no	—
269	C₂×C₈	—	94	2	no	—
270	C₂×C₈	—	94	2	no	—
271	C₂×C₈	—	94	2	no	—
272	C₂×C₈	—	94	2	no	—
273	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
274	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
275	C₄⋊C₄	—	47	8	no	—
276	C₄⋊C₄	—	94	4	no	—
277	C₂²⋊C₄	—	47	8	no	—
278	C₂²⋊C₄	—	94	4	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	47	2	no	—
282	C₄²	—	47	2	no	—
283	C₄²	—	47	4	no	—
284	C₄²	—	47	4	no	—
285	C₄²	—	94	8	no	—
286	C₄²	—	94	8	no	—
287	C₄²	—	94	4	no	—
288	C₄²	C₂×C₄×D₄₇	1	2	yes	—
289	C₄²	C₂×C₄×D₄₇	1	2	yes	—
290	C₂³	—	47	2	no	—
291	C₂³	—	94	1	no	—
292	C₂³	—	94	1	no	—
293	C₂³	—	94	4	no	—
294	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	47	4	no	—
301	C₂×C₄	—	47	8	no	—
302	C₂×C₄	—	47	4	no	—
303	C₂×C₄	—	47	2	no	—
304	C₂×C₄	—	47	4	no	—
305	C₂×C₄	—	94	1	no	—
306	C₂×C₄	—	94	1	no	—
307	C₂×C₄	—	94	4	no	—
308	C₂×C₄	—	94	4	no	—
309	C₂×C₄	—	94	4	no	—
310	C₂×C₄	—	94	4	no	—
311	C₂×C₄	—	188	8	no	—
312	C₂×C₄	C₄²×D₄₇	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	#41×D₄₇	1	2	yes	—
316	C₂×C₄	#41×D₄₇	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	94	4	no	—
320	C₈	—	94	4	no	—
321	C₂²	—	94	4	no	—
322	C₂²	—	94	2	no	—
323	C₂²	—	188	2	no	—
324	C₂²	—	188	2	no	—
325	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
326	C₂²	#117×D₄₇	1	1	yes	characteristic
327	C₂²	#117×D₄₇	1	1	yes	characteristic
328	C₂²	#131×D₄₇	1	4	yes	—
329	C₄	—	2	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	47	8	no	—
333	C₄	—	94	4	no	—
334	C₄	—	94	4	no	—
335	C₄	—	94	2	no	—
336	C₄	#15689591	1	2	yes	—
337	C₄	#15689591	1	2	yes	—
338	C₂	—	188	2	no	—
339	C₂	#20245727	1	1	yes	characteristic
340	C₂	#20251810	1	2	yes	—
341	C₂	#433×D₄₇	1	2	yes	—
342	C₂	#20252029	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1746	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1746	D₄₇
hypocenter	C₄₇	C₄×#433
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1746

Complements

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

N	H ≅ G/N	type
4. #20243244	C₂	semidirect
5. #20243236	C₂	semidirect
8. #20243140	C₂	semidirect
9. #97×Dic₄₇	C₂	semidirect
10. C₄×C₈×Dic₄₇	C₂	semidirect
11. C₄₇×#1746	C₂	semidirect
15. #15689185	C₄	semidirect
16. #15689182	C₄	semidirect
57. Dic₄₇⋊C₈	C₂×C₄	semidirect
58. Dic₄₇⋊C₈	C₂×C₄	semidirect
61. #13093935	C₂×C₄	semidirect
63. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
73. #97×C₄₇	C₂×C₄	semidirect
74. C₄×C₈×C₄₇	C₂×C₄	semidirect
166. Dic₄₇	#433	semidirect
175. #1746	D₄₇	semidirect
210. C₄₇	C₄×#433	semidirect
240. #97	C₄×D₄₇	semidirect
245. C₄×C₈	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×#15689185	2
C₂×#15689182	2
#20245727	2
#20243244	2
#20243236	2
#20243153	2
#20243145	2
#20243140	2
#97×Dic₄₇	2
C₄×C₈×Dic₄₇	2
C₄₇×#1746	2
C₄×#433	47