Subgroups — #26124195

Order 12032.

842 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245871	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688884	C₂	1	2	yes	maximal
3	C₂×#15688884	C₂	1	2	yes	maximal
4	C₄×C₄₇⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	C₂²	1	1	yes	characteristic
7	C₂×#13093983	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
9	#15689021	C₄	1	2	yes	complemented
10	#15689021	C₄	1	2	yes	complemented
11	#15688996	—	2	2	no	—
12	C₄×Dic₄₇⋊C₄	—	2	2	no	—
13	C₂×C₄₇⋊#95	—	2	2	no	—
14	#15688884	—	2	4	no	—
15	#15688884	—	2	4	no	—
16	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#423	—	2	4	no	—
19	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂³×Dic₄₇	—	2	2	no	—
21	#13093983	—	2	1	no	—
22	#13093983	—	2	1	no	—
23	#13093983	—	2	2	no	—
24	#13093983	—	2	2	no	—
25	#13093983	D₄	1	2	yes	—
26	#13093983	Q₈	1	2	yes	—
27	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₄₇⋊C₄	—	2	2	no	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
32	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄×Dic₄₇	—	2	1	no	—
34	C₂×C₄×Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₂×C₄×Dic₄₇	—	2	2	no	—
37	C₂×C₄×Dic₄₇	—	2	2	no	—
38	C₂×C₄×Dic₄₇	—	2	1	no	—
39	C₂×C₄×Dic₄₇	—	2	2	no	—
40	C₂×C₄×Dic₄₇	Q₈	1	4	yes	—
41	C₄₇⋊#95	—	2	8	no	—
42	C₄₇⋊#95	—	2	2	no	—
43	C₄₇⋊#95	—	2	2	no	—
44	C₂³×C₄×C₄₇	D₄	1	2	yes	—
45	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₄₇	—	2	2	no	—
47	C₂×M₁₆×C₄₇	—	2	2	no	—
48	C₂×M₁₆×C₄₇	—	2	2	no	—
49	C₂×M₁₆×C₄₇	—	2	2	no	—
50	C₂×M₁₆×C₄₇	—	2	2	no	—
51	C₂×M₁₆×C₄₇	D₄	1	4	yes	—
52	C₂²×C₈×C₄₇	—	2	2	no	—
53	C₂×C₄²×C₄₇	—	2	2	no	—
54	C₂×C₄²×C₄₇	—	2	2	no	—
55	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₄₇	D₄	1	2	yes	—
58	C₄⋊C₈×C₄₇	—	2	8	no	—
59	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	—	2	4	no	—
61	C₂²×C₄×C₄₇	—	2	4	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×C₄×C₄₇	#41	1	2	yes	—
75	C₂²×Dic₄₇	—	2	8	no	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	2	4	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	4	4	no	—
80	C₂²×Dic₄₇	—	4	2	no	—
81	C₂²×Dic₄₇	—	4	2	no	—
82	M₁₆×C₄₇	—	2	4	no	—
83	M₁₆×C₄₇	—	2	4	no	—
84	M₁₆×C₄₇	—	4	2	no	—
85	M₁₆×C₄₇	—	4	2	no	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	4	2	no	—
88	C₂×C₈×C₄₇	—	4	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄²×C₄₇	—	2	2	no	—
92	C₄²×C₄₇	—	2	2	no	—
93	C₂²⋊Dic₄₇	—	2	2	no	—
94	C₂²⋊Dic₄₇	—	2	2	no	—
95	C₂²⋊Dic₄₇	—	4	2	no	—
96	C₄⋊Dic₄₇	—	2	2	no	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
99	Dic₄₇⋊C₄	—	4	4	no	—
100	Dic₄₇⋊C₄	—	4	4	no	—
101	C₄×Dic₄₇	—	2	2	no	—
102	C₄×Dic₄₇	—	2	2	no	—
103	C₄×Dic₄₇	—	2	4	no	—
104	C₄×Dic₄₇	—	2	4	no	—
105	C₄×Dic₄₇	—	4	1	no	—
106	C₄×Dic₄₇	—	4	2	no	—
107	C₄×Dic₄₇	—	4	1	no	—
108	C₄×Dic₄₇	—	4	2	no	—
109	C₂³×C₄₇	—	2	2	no	—
110	C₂³×C₄₇	—	2	2	no	—
111	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂³×C₄₇	#121	1	2	yes	—
113	C₂³×C₄₇	#121	1	1	yes	characteristic
114	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	—	2	2	no	—
117	C₂×C₄×C₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	2	no	—
119	C₂×C₄×C₄₇	—	2	2	no	—
120	C₂×C₄×C₄₇	—	2	2	no	—
121	C₂×C₄×C₄₇	—	4	8	no	—
122	C₂×C₄×C₄₇	#104	1	4	yes	—
123	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
127	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
128	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#121	1	2	yes	—
130	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₄₇	#123	1	2	yes	—
134	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#125	1	2	yes	—
137	C₂×C₄×C₄₇	#125	1	2	yes	—
138	C₂×C₄×C₄₇	#126	1	2	yes	—
139	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
141	C₂×Dic₄₇	—	2	8	no	—
142	C₂×Dic₄₇	—	2	4	no	—
143	C₂×Dic₄₇	—	4	8	no	—
144	C₂×Dic₄₇	—	4	8	no	—
145	C₂×Dic₄₇	—	4	4	no	—
146	C₂×Dic₄₇	—	4	4	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₈×C₄₇	—	4	4	no	—
153	#11209	—	47	1	no	maximal, Sylow
154	C₂²×C₄₇	—	2	2	no	—
155	C₂²×C₄₇	—	2	2	no	—
156	C₂²×C₄₇	—	2	4	no	—
157	C₂²×C₄₇	—	2	2	no	—
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	#380	1	1	yes	characteristic
160	C₂²×C₄₇	#386	1	1	yes	characteristic
161	C₂²×C₄₇	#391	1	1	yes	characteristic
162	C₂²×C₄₇	#396	1	2	yes	—
163	C₂²×C₄₇	#398	1	2	yes	—
164	C₂²×C₄₇	C₂×#104	1	2	yes	—
165	C₂²×C₄₇	#421	1	2	yes	—
166	C₄×C₄₇	—	2	4	no	—
167	C₄×C₄₇	—	4	2	no	—
168	C₄×C₄₇	#387	1	2	yes	—
169	C₄×C₄₇	#389	1	2	yes	—
170	C₄×C₄₇	#394	1	1	yes	characteristic
171	C₄×C₄₇	#395	1	1	yes	characteristic
172	C₄×C₄₇	#398	1	1	yes	characteristic
173	C₄×C₄₇	#400	1	1	yes	characteristic
174	Dic₄₇	—	4	8	no	—
175	Dic₄₇	—	4	4	no	—
176	Dic₄₇	—	8	2	no	—
177	C₂×#423	D₄₇	1	1	yes	characteristic
178	#2296	—	47	1	no	—
179	C₂×#339	—	47	2	no	—
180	C₂×#339	—	47	2	no	—
181	C₄×#95	—	47	1	no	—
182	C₂×C₄₇	—	2	2	no	—
183	C₂×C₄₇	—	2	2	no	—
184	C₂×C₄₇	#1441	1	1	yes	characteristic
185	C₂×C₄₇	#1806	1	2	yes	—
186	C₂×C₄₇	#1871	1	2	yes	—
187	C₂×C₄₇	#1994	1	1	yes	characteristic
188	C₂×C₄₇	#1995	1	1	yes	characteristic
189	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
190	C₂×#117	—	47	1	no	—
191	C₂×#117	—	47	1	no	—
192	C₂²×C₄²	—	47	2	no	—
193	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
194	#423	—	2	4	no	—
195	C₂×C₄⋊C₈	—	2	2	no	—
196	#389	—	47	2	no	—
197	#389	—	47	2	no	—
198	#385	—	94	2	no	—
199	C₄×C₄⋊C₄	—	94	2	no	—
200	C₂×#95	—	94	2	no	—
201	#339	—	94	4	no	—
202	#339	—	94	4	no	—
203	C₄₇	#11209	1	1	yes	characteristic, complemented, Sylow
204	C₂³×C₄	—	94	2	no	—
205	C₂³×C₄	D₁₈₈	1	2	yes	—
206	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	C₂²⋊D₄₇	1	4	yes	—
213	C₂²×C₈	—	2	2	no	—
214	#117	—	47	2	no	—
215	#117	—	47	2	no	—
216	#117	—	94	1	no	—
217	#117	—	94	1	no	—
218	#117	—	94	2	no	—
219	#117	—	94	2	no	—
220	C₂×C₄⋊C₄	—	47	2	no	—
221	C₂×C₄⋊C₄	—	47	2	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	—	94	2	no	—
224	C₂×C₂²⋊C₄	—	94	1	no	—
225	C₂×C₂²⋊C₄	—	94	1	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	—	2	2	no	—
228	C₂×C₄²	—	47	4	no	—
229	C₂×C₄²	—	94	1	no	—
230	C₂×C₄²	—	94	2	no	—
231	C₂×C₄²	—	94	2	no	—
232	C₂×C₄²	—	94	2	no	—
233	C₂×C₄²	—	94	2	no	—
234	C₂×C₄²	—	94	2	no	—
235	C₂×C₄²	—	94	1	no	—
236	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
237	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
239	C₄⋊C₈	—	2	8	no	—
240	#95	—	94	8	no	—
241	#95	—	94	2	no	—
242	#95	—	94	2	no	—
243	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	94	8	no	—
248	C₂²×C₄	—	94	2	no	—
249	C₂²×C₄	—	94	4	no	—
250	C₂²×C₄	—	94	2	no	—
251	C₂²×C₄	—	188	4	no	—
252	C₂²×C₄	—	188	2	no	—
253	C₂²×C₄	—	188	2	no	—
254	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
255	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
256	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
257	C₂²×C₄	#10625350	1	2	yes	—
258	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	#10625352	1	1	yes	characteristic
260	C₂²×C₄	#10625352	1	2	yes	—
261	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
262	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
263	C₂²×C₄	#10625354	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
265	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
266	M₁₆	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	M₁₆	—	4	2	no	—
269	M₁₆	—	4	2	no	—
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₄⋊C₄	—	47	4	no	—
274	C₄⋊C₄	—	94	2	no	—
275	C₄⋊C₄	—	94	4	no	—
276	C₄⋊C₄	—	188	4	no	—
277	C₄⋊C₄	—	188	4	no	—
278	C₂²⋊C₄	—	94	2	no	—
279	C₂²⋊C₄	—	94	2	no	—
280	C₂²⋊C₄	—	188	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	94	2	no	—
286	C₄²	—	94	2	no	—
287	C₄²	—	94	4	no	—
288	C₄²	—	94	4	no	—
289	C₄²	—	188	1	no	—
290	C₄²	—	188	2	no	—
291	C₄²	—	188	1	no	—
292	C₄²	—	188	2	no	—
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
296	C₂³	#13093998	1	2	yes	—
297	C₂³	#13093999	1	1	yes	characteristic
298	C₂³	#13094004	1	1	yes	characteristic
299	C₂³	#13094004	1	2	yes	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	94	8	no	—
307	C₂×C₄	—	94	4	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	4	no	—
312	C₂×C₄	—	188	8	no	—
313	C₂×C₄	—	188	8	no	—
314	C₂×C₄	—	188	4	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	#13093950	1	4	yes	—
318	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
319	C₂×C₄	#13093982	1	1	yes	characteristic
320	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
321	C₂×C₄	#13093989	1	2	yes	—
322	C₂×C₄	#13093993	1	1	yes	characteristic
323	C₂×C₄	#13093995	1	1	yes	characteristic
324	C₂×C₄	#13093996	1	2	yes	—
325	C₂×C₄	#13094000	1	2	yes	—
326	C₂×C₄	#13094000	1	1	yes	characteristic
327	C₂×C₄	#13094002	1	2	yes	—
328	C₂×C₄	#13094005	1	2	yes	—
329	C₂×C₄	#13094007	1	1	yes	characteristic
330	C₂×C₄	#13094011	1	1	yes	characteristic
331	C₂×C₄	#13094014	1	2	yes	—
332	C₂×C₄	#13094014	1	2	yes	—
333	C₂×C₄	#13094019	1	2	yes	—
334	C₂×C₄	#13094022	1	2	yes	—
335	C₂×C₄	#13094022	1	1	yes	characteristic
336	C₈	—	4	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	4	no	—
342	C₂²	#15688980	1	1	yes	characteristic
343	C₂²	#15689002	1	1	yes	characteristic
344	C₂²	#15689028	1	1	yes	characteristic
345	C₂²	#15689045	1	2	yes	—
346	C₂²	#15689053	1	2	yes	—
347	C₂²	C₂×#13093950	1	2	yes	—
348	C₂²	#15689139	1	2	yes	—
349	C₄	—	2	4	no	—
350	C₄	—	4	2	no	—
351	C₄	—	376	2	no	—
352	C₄	—	188	8	no	—
353	C₄	—	188	4	no	—
354	C₄	#15689004	1	2	yes	—
355	C₄	#15689018	1	2	yes	—
356	C₄	#15689037	1	1	yes	characteristic
357	C₄	#15689042	1	1	yes	characteristic
358	C₄	#15689054	1	1	yes	characteristic
359	C₄	#15689061	1	1	yes	characteristic
360	C₂	—	2	2	no	—
361	C₂	—	2	2	no	—
362	C₂	#20241638	1	1	yes	characteristic
363	C₂	#20243405	1	2	yes	—
364	C₂	#20243690	1	2	yes	—
365	C₂	#20244364	1	1	yes	characteristic
366	C₂	#20244368	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#423	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#423	D₄₇
hypocenter	C₄₇	#11209
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
9. #15689021	C₄	semidirect
10. #15689021	C₄	semidirect
203. C₄₇	#11209	semidirect

Maximal subgroups

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

subgroup	index
#20245871	2
C₂×#15688884	2
C₄×C₄₇⋊#95	2
C₂×C₄₇×#423	2
#11209	47