Subgroups — #26213766

Order 12032.

1022 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×M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic
3	#117×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	#20243138	C₂	1	4	yes	maximal, complemented
5	#20243135	C₂	1	4	yes	maximal, complemented
6	#20243086	C₂	1	1	yes	maximal, characteristic
7	#20243085	C₂	1	1	yes	maximal, characteristic
8	#20243084	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1729	C₂	1	1	yes	maximal, characteristic
10	C₂²×#10625329	C₂²	1	1	yes	characteristic
11	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×#13093983	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	M₁₆×Dic₄₇	C₄	1	8	yes	complemented
15	C₂×#13093934	C₄	1	8	yes	complemented
16	C₂×#13093934	C₂²	1	2	yes	—
17	C₂×#13093934	C₂²	1	2	yes	—
18	C₂×#13093934	C₂²	1	2	yes	—
19	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₄₇	C₄	1	4	yes	—
21	C₂²⋊C₄×Dic₄₇	C₄	1	4	yes	—
22	#15688969	C₂²	1	2	yes	—
23	C₄²×Dic₄₇	C₂²	1	2	yes	—
24	#15688878	C₂²	1	4	yes	—
25	#15688877	C₂²	1	4	yes	—
26	#15688877	C₂²	1	4	yes	—
27	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
28	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
30	C₄₇×#336	C₂²	1	4	yes	—
31	C₂³×Dic₄₇	C₂×C₄	1	2	yes	—
32	C₂×#10625329	—	2	2	no	—
33	C₂×#10625329	—	2	2	no	—
34	C₂×#10625329	C₂×C₄	1	4	yes	—
35	C₂²×C₄₇⋊C₈	—	2	2	no	—
36	C₂²×C₄₇⋊C₈	—	2	2	no	—
37	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
40	#13093983	—	2	2	no	—
41	#13093983	—	2	2	no	—
42	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
45	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₄₇	C₂³	1	1	yes	characteristic
51	#13093934	C₂×C₄	1	8	yes	—
52	#13093934	C₂×C₄	1	8	yes	—
53	#13093934	C₂×C₄	1	8	yes	—
54	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
55	C₄₇⋊#95	C₂×C₄	1	8	yes	—
56	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₄₇	—	2	2	no	—
58	C₂×M₁₆×C₄₇	—	2	2	no	—
59	C₂×M₁₆×C₄₇	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₄₇	—	2	2	no	—
61	C₂²×C₈×C₄₇	—	2	2	no	—
62	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
65	#117×C₄₇	—	2	2	no	—
66	#117×C₄₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
70	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₇	#41	1	2	yes	—
76	C₂²×C₄×C₄₇	#41	1	2	yes	—
77	C₂²×C₄×C₄₇	#41	1	2	yes	—
78	C₂²×C₄×C₄₇	#41	1	2	yes	—
79	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₄₇	—	4	4	no	—
81	C₂²×Dic₄₇	C₄²	1	8	yes	—
82	C₂²×Dic₄₇	C₂²×C₄	1	4	yes	—
83	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₄₇	C₂²×C₄	1	2	yes	—
85	M₁₆×C₄₇	—	2	8	no	—
86	C₂×C₈×C₄₇	—	2	8	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	2	2	no	—
91	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
93	C₄⋊C₄×C₄₇	—	2	4	no	—
94	C₂²⋊C₄×C₄₇	—	2	4	no	—
95	C₄²×C₄₇	—	2	2	no	—
96	C₄²×C₄₇	—	2	2	no	—
97	C₂²⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄×Dic₄₇	—	2	2	no	—
100	C₄×Dic₄₇	—	2	2	no	—
101	C₄×Dic₄₇	—	4	8	no	—
102	C₄×Dic₄₇	C₄²	1	8	yes	—
103	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
104	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
105	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
106	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
107	#10625329	—	2	8	no	—
108	C₂×C₄₇⋊C₈	—	2	8	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	—	2	2	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	—	2	2	no	—
113	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
115	C₂³×C₄₇	—	2	4	no	—
116	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
117	C₂³×C₄₇	#117	1	2	yes	—
118	C₂³×C₄₇	#117	1	2	yes	—
119	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₄₇	—	2	4	no	—
121	C₂×C₄×C₄₇	—	2	8	no	—
122	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₄₇	#117	1	4	yes	—
124	C₂×C₄×C₄₇	#117	1	4	yes	—
125	C₂×C₄×C₄₇	#117	1	2	yes	—
126	C₂×C₄×C₄₇	#117	1	4	yes	—
127	C₂×C₄×C₄₇	#117	1	4	yes	—
128	C₂×C₄×C₄₇	#117	1	2	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	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	4	no	—
136	C₂×Dic₄₇	—	2	4	no	—
137	C₂×Dic₄₇	—	2	4	no	—
138	C₂×Dic₄₇	—	2	8	no	—
139	C₂×Dic₄₇	—	4	8	no	—
140	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
141	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
142	C₈×C₄₇	—	2	4	no	—
143	C₈×C₄₇	—	2	4	no	—
144	C₄₇⋊C₈	—	2	4	no	—
145	C₄₇⋊C₈	—	2	4	no	—
146	#20367	—	47	1	no	maximal, Sylow
147	C₂²×C₄₇	—	2	4	no	—
148	C₂²×C₄₇	—	2	4	no	—
149	C₂²×C₄₇	—	2	4	no	—
150	C₂²×C₄₇	#376	1	4	yes	—
151	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
153	C₂²×C₄₇	C₂×#117	1	1	yes	characteristic
154	C₂²×C₄₇	#568	1	4	yes	—
155	C₄×C₄₇	—	4	4	no	—
156	C₄×C₄₇	#376	1	4	yes	—
157	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
158	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
159	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
160	C₄×C₄₇	C₂×#117	1	1	yes	characteristic
161	Dic₄₇	—	2	4	no	—
162	Dic₄₇	—	2	4	no	—
163	Dic₄₇	—	4	4	no	—
164	C₂×C₄×M₁₆	—	47	1	no	—
165	C₂²×#97	—	47	1	no	—
166	C₄×#117	—	47	1	no	—
167	#1746	—	47	4	no	—
168	#1745	—	47	4	no	—
169	#1729	—	47	1	no	—
170	#1729	—	47	1	no	—
171	#1729	—	47	1	no	—
172	#1729	D₄₇	1	1	yes	characteristic
173	C₂×C₄₇	—	2	4	no	—
174	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
175	C₂×C₄₇	#2873	1	2	yes	—
176	C₂×C₄₇	#2923	1	2	yes	—
177	C₂×C₄₇	#2923	1	2	yes	—
178	C₂²×M₁₆	—	47	1	no	—
179	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
180	C₂³×C₈	—	47	1	no	—
181	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
182	C₂×#117	—	47	1	no	—
183	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
184	C₂²×C₄²	—	47	1	no	—
185	C₄×M₁₆	—	47	8	no	—
186	C₂×#97	—	47	8	no	—
187	C₂×#97	—	47	2	no	—
188	C₂×#97	—	47	2	no	—
189	C₂×#97	—	47	2	no	—
190	C₂×C₄×C₈	—	47	2	no	—
191	C₄×C₄⋊C₄	—	47	4	no	—
192	C₄×C₂²⋊C₄	—	47	4	no	—
193	#376	—	47	2	no	—
194	C₄³	—	47	2	no	—
195	#336	—	47	4	no	—
196	#336	—	47	4	no	—
197	#336	—	47	4	no	—
198	#336	C₂×D₄₇	1	4	yes	—
199	C₄₇	#20367	1	1	yes	characteristic, complemented, Sylow
200	C₂³×C₄	—	47	2	no	—
201	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	47	4	no	—
205	C₂×M₁₆	—	94	2	no	—
206	C₂×M₁₆	—	94	2	no	—
207	C₂×M₁₆	C₄×D₄₇	1	4	yes	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	47	4	no	—
211	C₂²×C₈	—	47	2	no	—
212	C₂²×C₈	—	47	2	no	—
213	C₂²×C₈	—	94	2	no	—
214	C₂²×C₈	—	94	2	no	—
215	C₂²×C₈	C₄×D₄₇	1	4	yes	—
216	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
217	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
218	#117	—	2	2	no	—
219	#117	—	2	2	no	—
220	#117	—	94	2	no	—
221	#117	—	94	2	no	—
222	C₂×C₄⋊C₄	—	47	2	no	—
223	C₂×C₄⋊C₄	C₄×D₄₇	1	2	yes	—
224	C₂×C₂²⋊C₄	—	47	2	no	—
225	C₂×C₂²⋊C₄	C₄×D₄₇	1	2	yes	—
226	C₂×C₄²	—	47	2	no	—
227	C₂×C₄²	—	47	4	no	—
228	C₂×C₄²	—	47	4	no	—
229	C₂×C₄²	—	47	1	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	—	47	2	no	—
232	C₂×C₄²	—	47	8	no	—
233	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
234	#97	—	47	8	no	—
235	#97	—	47	8	no	—
236	#97	—	47	8	no	—
237	C₄×C₈	—	47	8	no	—
238	#95	—	47	8	no	—
239	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
240	C₂²×C₄	—	47	2	no	—
241	C₂²×C₄	—	47	2	no	—
242	C₂²×C₄	—	47	8	no	—
243	C₂²×C₄	—	47	4	no	—
244	C₂²×C₄	—	188	4	no	—
245	C₂²×C₄	C₂×C₄×D₄₇	1	4	yes	—
246	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
248	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
249	C₂²×C₄	#10625352	1	2	yes	—
250	C₂²×C₄	#10625352	1	2	yes	—
251	C₂²×C₄	#10625352	1	2	yes	—
252	C₂²×C₄	#10625354	1	2	yes	—
253	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
254	M₁₆	—	2	8	no	—
255	M₁₆	—	94	8	no	—
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	47	4	no	—
262	C₂×C₈	—	47	4	no	—
263	C₂×C₈	—	94	8	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	—	94	2	no	—
266	C₂×C₈	—	94	2	no	—
267	C₂×C₈	—	94	2	no	—
268	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
269	C₂×C₈	C₂×C₄×D₄₇	1	4	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	94	4	no	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	94	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	47	8	no	—
277	C₄²	—	47	2	no	—
278	C₄²	—	47	2	no	—
279	C₄²	—	47	2	no	—
280	C₄²	—	47	2	no	—
281	C₄²	—	94	2	no	—
282	C₄²	—	94	2	no	—
283	C₄²	—	188	8	no	—
284	C₂³	—	2	4	no	—
285	C₂³	C₄²×D₄₇	1	2	yes	—
286	C₂³	#13093980	1	2	yes	—
287	C₂³	#13093982	1	2	yes	—
288	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	47	4	no	—
292	C₂×C₄	—	47	4	no	—
293	C₂×C₄	—	94	4	no	—
294	C₂×C₄	—	94	4	no	—
295	C₂×C₄	—	94	4	no	—
296	C₂×C₄	—	94	8	no	—
297	C₂×C₄	—	188	8	no	—
298	C₂×C₄	C₄²×D₄₇	1	2	yes	—
299	C₂×C₄	#13093980	1	2	yes	—
300	C₂×C₄	#13093980	1	4	yes	—
301	C₂×C₄	#13093980	1	4	yes	—
302	C₂×C₄	#13093982	1	2	yes	—
303	C₂×C₄	#13093982	1	4	yes	—
304	C₂×C₄	#13093982	1	4	yes	—
305	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
307	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
308	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
309	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
310	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	—	94	4	no	—
314	C₈	—	94	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	#15688968	1	4	yes	—
319	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
320	C₂²	C₂×#13093980	1	1	yes	characteristic
321	C₂²	C₂×#13093982	1	1	yes	characteristic
322	C₂²	#15689888	1	4	yes	—
323	C₄	—	4	4	no	—
324	C₄	—	94	4	no	—
325	C₄	—	94	4	no	—
326	C₄	—	188	4	no	—
327	C₄	#15688968	1	4	yes	—
328	C₄	C₂×#13093980	1	1	yes	characteristic
329	C₄	C₂×#13093980	1	1	yes	characteristic
330	C₄	C₂×#13093982	1	1	yes	characteristic
331	C₄	C₂×#13093982	1	1	yes	characteristic
332	C₂	—	2	4	no	—
333	C₂	C₂×#15688968	1	1	yes	characteristic
334	C₂	#20251804	1	2	yes	—
335	C₂	#20252035	1	2	yes	—
336	C₂	#20252038	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Fitting	C₄₇×#1729	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1729	D₄₇
hypocenter	C₄₇	#20367
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
4. #20243138	C₂	semidirect
5. #20243135	C₂	semidirect
14. M₁₆×Dic₄₇	C₄	semidirect
15. C₂×#13093934	C₄	semidirect
199. C₄₇	#20367	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₄₇	2
C₂²×#13093934	2
#117×Dic₄₇	2
#20243138	2
#20243135	2
#20243086	2
#20243085	2
#20243084	2
C₄₇×#1729	2
#20367	47