Subgroups — #26124381

Order 12032.

682 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093960	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688891	C₂	1	2	yes	maximal
3	#20243109	C₂	1	2	yes	maximal
4	C₂×#15688886	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₇×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10625329	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×#13093960	—	2	4	no	—
9	C₂×#13093960	—	2	2	no	—
10	#15688891	—	2	4	no	—
11	#15688886	—	2	2	no	—
12	#15688885	—	2	2	no	—
13	#15688885	—	2	2	no	—
14	C₂²×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#424	—	2	2	no	—
18	C₄₇×#424	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂³×Dic₄₇	—	2	2	no	—
21	C₂×#10625329	—	2	2	no	—
22	C₂×#10625329	—	2	2	no	—
23	C₂×#10625329	—	2	2	no	—
24	C₂×#10625329	—	2	2	no	—
25	C₂×#10625329	—	2	2	no	—
26	C₂×#10625329	D₄	1	4	yes	—
27	C₂²×C₄₇⋊C₈	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄⋊Dic₄₇	Q₈	1	4	yes	—
34	#13093960	—	2	8	no	—
35	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₄₇	—	2	1	no	—
37	C₂×M₁₆×C₄₇	—	2	2	no	—
38	C₂×M₁₆×C₄₇	—	2	2	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₄₇	D₄	1	2	yes	—
45	C₂²×C₈×C₄₇	D₄	1	2	yes	—
46	#117×C₄₇	—	2	1	no	—
47	#117×C₄₇	—	2	1	no	—
48	#117×C₄₇	C₂×C₄	1	2	yes	—
49	#117×C₄₇	Q₈	1	2	yes	—
50	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
53	C₂×C₄²×C₄₇	—	2	1	no	—
54	C₄⋊C₈×C₄₇	—	2	8	no	—
55	C₂⁴×C₄₇	#41	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	—	2	1	no	—
57	C₂²×C₄×C₄₇	—	2	1	no	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	2	yes	—
66	C₂²×C₄×C₄₇	#41	1	2	yes	—
67	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂²×Dic₄₇	—	2	4	no	—
69	C₂²×Dic₄₇	—	4	2	no	—
70	C₂²×Dic₄₇	—	4	2	no	—
71	C₂²×Dic₄₇	—	8	2	no	—
72	C₂²×Dic₄₇	—	8	2	no	—
73	M₁₆×C₄₇	—	2	4	no	—
74	M₁₆×C₄₇	—	4	1	no	—
75	M₁₆×C₄₇	—	4	1	no	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	2	no	—
78	C₂×C₈×C₄₇	—	2	4	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	—	2	2	no	—
81	C₂×C₈×C₄₇	—	4	2	no	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
83	C₄⋊C₄×C₄₇	—	2	1	no	—
84	C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
86	C₄⋊C₄×C₄₇	C₄⋊C₄	1	2	yes	—
87	C₂²⋊C₄×C₄₇	—	2	2	no	—
88	C₂²⋊C₄×C₄₇	—	2	2	no	—
89	C₄²×C₄₇	—	2	2	no	—
90	C₄²×C₄₇	—	2	2	no	—
91	C₄⋊Dic₄₇	—	2	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄⋊Dic₄₇	—	4	2	no	—
95	#10625329	—	2	4	no	—
96	#10625329	—	2	4	no	—
97	#10625329	—	4	2	no	—
98	#10625329	—	4	2	no	—
99	C₂×C₄₇⋊C₈	—	2	4	no	—
100	C₂×C₄₇⋊C₈	—	4	2	no	—
101	C₂×C₄₇⋊C₈	—	4	2	no	—
102	C₂³×C₄₇	—	2	2	no	—
103	C₂³×C₄₇	—	2	1	no	—
104	C₂³×C₄₇	—	2	1	no	—
105	C₂³×C₄₇	#117	1	1	yes	characteristic
106	C₂³×C₄₇	#121	1	1	yes	characteristic
107	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
108	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₄₇	#126	1	2	yes	—
110	C₂×C₄×C₄₇	—	2	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	—	4	2	no	—
113	C₂×C₄×C₄₇	—	4	1	no	—
114	C₂×C₄×C₄₇	—	4	1	no	—
115	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#121	1	2	yes	—
121	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#121	1	2	yes	—
123	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
127	C₂×C₄×C₄₇	#123	1	2	yes	—
128	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
129	C₂×C₄×C₄₇	#125	1	2	yes	—
130	C₂×C₄×C₄₇	#125	1	2	yes	—
131	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	8	4	no	—
135	C₂×Dic₄₇	—	8	2	no	—
136	C₂×Dic₄₇	—	8	2	no	—
137	C₂×Dic₄₇	—	8	4	no	—
138	C₈×C₄₇	—	2	4	no	—
139	C₈×C₄₇	—	4	2	no	—
140	C₄₇⋊C₈	—	4	4	no	—
141	#11242	—	47	1	no	maximal, Sylow
142	C₂²×C₄₇	—	2	2	no	—
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	1	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	#380	1	1	yes	characteristic
151	C₂²×C₄₇	#387	1	2	yes	—
152	C₂²×C₄₇	#395	1	1	yes	characteristic
153	C₂²×C₄₇	#396	1	2	yes	—
154	C₂²×C₄₇	#400	1	1	yes	characteristic
155	C₂²×C₄₇	#471	1	2	yes	—
156	C₂²×C₄₇	#472	1	1	yes	characteristic
157	C₂²×C₄₇	#473	1	1	yes	characteristic
158	C₄×C₄₇	—	4	2	no	—
159	C₄×C₄₇	—	4	2	no	—
160	C₄×C₄₇	#386	1	1	yes	characteristic
161	C₄×C₄₇	#389	1	2	yes	—
162	C₄×C₄₇	#391	1	1	yes	characteristic
163	C₄×C₄₇	#394	1	1	yes	characteristic
164	C₄×C₄₇	#398	1	1	yes	characteristic
165	C₄×C₄₇	#398	1	2	yes	—
166	Dic₄₇	—	8	4	no	—
167	C₂²×#108	—	47	1	no	—
168	C₂×#424	D₄₇	1	1	yes	characteristic
169	C₂×#343	—	47	2	no	—
170	#1736	—	47	2	no	—
171	C₂×#340	—	47	1	no	—
172	C₂×C₄₇	—	2	2	no	—
173	C₂×C₄₇	—	2	1	no	—
174	C₂×C₄₇	—	2	1	no	—
175	C₂×C₄₇	#1441	1	1	yes	characteristic
176	C₂×C₄₇	#1815	1	1	yes	characteristic
177	C₂×C₄₇	#1816	1	1	yes	characteristic
178	C₂×C₄₇	#2063	1	2	yes	—
179	C₂×C₄₇	#2064	1	2	yes	—
180	C₂²×M₁₆	—	47	2	no	—
181	C₂²×M₁₆	C₂×D₄₇	1	1	yes	characteristic
182	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
183	C₂×#117	C₂×D₄₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₄	—	47	2	no	—
185	C₂×#108	—	94	4	no	—
186	C₂×#108	—	94	2	no	—
187	#424	—	2	2	no	—
188	#424	Dic₄₇	1	4	yes	complemented
189	C₂×C₄⋊C₈	—	2	2	no	—
190	#343	—	94	4	no	—
191	#340	—	94	2	no	—
192	#340	—	94	2	no	—
193	#340	—	94	2	no	—
194	C₄₇	#11242	1	1	yes	characteristic, complemented, Sylow
195	C₂³×C₄	—	94	2	no	—
196	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	47	4	no	—
201	C₂×M₁₆	—	94	2	no	—
202	C₂×M₁₆	—	94	2	no	—
203	C₂×M₁₆	—	94	2	no	—
204	C₂×M₁₆	—	94	2	no	—
205	C₂×M₁₆	—	94	2	no	—
206	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	94	2	no	—
211	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
212	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
213	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
214	#117	—	2	1	no	—
215	#117	—	2	1	no	—
216	#117	Dic₉₄	1	2	yes	—
217	#117	C₂×Dic₄₇	1	2	yes	—
218	C₂×C₄⋊C₄	—	47	4	no	—
219	C₂×C₄⋊C₄	—	94	2	no	—
220	C₂×C₄⋊C₄	—	94	2	no	—
221	C₂×C₄⋊C₄	—	94	2	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	—	94	2	no	—
224	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
225	C₂×C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	2	1	no	—
228	#108	—	94	8	no	—
229	C₄⋊C₈	—	2	8	no	—
230	C₂⁴	#10625350	1	1	yes	characteristic
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	94	4	no	—
234	C₂²×C₄	—	376	2	no	—
235	C₂²×C₄	—	376	2	no	—
236	C₂²×C₄	—	188	2	no	—
237	C₂²×C₄	—	188	2	no	—
238	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
240	C₂²×C₄	#10625350	1	1	yes	characteristic
241	C₂²×C₄	D₄×D₄₇	1	2	yes	—
242	C₂²×C₄	#10625352	1	2	yes	—
243	C₂²×C₄	Q₈×D₄₇	1	2	yes	—
244	C₂²×C₄	#10625354	1	2	yes	—
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	M₁₆	—	2	4	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	94	4	no	—
252	M₁₆	—	94	4	no	—
253	M₁₆	—	188	2	no	—
254	M₁₆	—	188	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	94	4	no	—
262	C₂×C₈	—	188	2	no	—
263	C₂×C₈	—	188	2	no	—
264	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	94	4	no	—
268	C₄⋊C₄	—	94	4	no	—
269	C₄⋊C₄	—	188	2	no	—
270	C₄⋊C₄	—	188	2	no	—
271	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
272	C₄⋊C₄	C₄⋊Dic₄₇	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	#13093983	1	1	yes	characteristic
281	C₂³	#13093998	1	1	yes	characteristic
282	C₂³	#13094002	1	2	yes	—
283	C₂³	#13094003	1	1	yes	characteristic
284	C₂³	#13094019	1	2	yes	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	376	4	no	—
291	C₂×C₄	—	376	2	no	—
292	C₂×C₄	—	376	2	no	—
293	C₂×C₄	—	376	4	no	—
294	C₂×C₄	—	188	4	no	—
295	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
297	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
298	C₂×C₄	Q₈×Dic₄₇	1	2	yes	—
299	C₂×C₄	#13093992	1	1	yes	characteristic
300	C₂×C₄	#13093996	1	2	yes	—
301	C₂×C₄	#13093998	1	1	yes	characteristic
302	C₂×C₄	#13093999	1	2	yes	—
303	C₂×C₄	#13094000	1	2	yes	—
304	C₂×C₄	#13094003	1	1	yes	characteristic
305	C₂×C₄	#13094004	1	2	yes	—
306	C₂×C₄	#13094005	1	1	yes	characteristic
307	C₂×C₄	#13094009	1	2	yes	—
308	C₂×C₄	#13094013	1	1	yes	characteristic
309	C₂×C₄	#13094014	1	2	yes	—
310	C₂×C₄	#13094015	1	2	yes	—
311	C₂×C₄	#13094022	1	1	yes	characteristic
312	C₂×C₄	#13094023	1	2	yes	—
313	C₈	—	2	4	no	—
314	C₈	—	4	2	no	—
315	C₈	—	188	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	#15688983	1	1	yes	characteristic
325	C₂²	#15689009	1	2	yes	—
326	C₂²	#15689043	1	1	yes	characteristic
327	C₂²	#15689045	1	2	yes	—
328	C₂²	#15689062	1	1	yes	characteristic
329	C₂²	#15689386	1	2	yes	—
330	C₂²	#15689393	1	1	yes	characteristic
331	C₂²	#15689398	1	1	yes	characteristic
332	C₄	—	4	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	376	4	no	—
335	C₄	#15689003	1	1	yes	characteristic
336	C₄	#15689022	1	2	yes	—
337	C₄	#15689030	1	1	yes	characteristic
338	C₄	#15689040	1	1	yes	characteristic
339	C₄	#15689053	1	1	yes	characteristic
340	C₄	#15689056	1	2	yes	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	#20241641	1	1	yes	characteristic
345	C₂	#20243447	1	1	yes	characteristic
346	C₂	#20243452	1	1	yes	characteristic
347	C₂	#20244747	1	2	yes	—
348	C₂	#20244754	1	2	yes	—

	subgroup	G/H
center	C₂³	#13093998
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#424	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂⁴×C₄₇	#41
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#424	D₄₇
hypocenter	C₄₇	#11242
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#424

Complements

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

N	H ≅ G/N	type
18. C₄₇×#424	C₄	semidirect
188. #424	Dic₄₇	semidirect
194. C₄₇	#11242	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093960	2
C₂×#15688891	2
#20243109	2
C₂×#15688886	2
C₂×C₄₇×#424	2
#11242	47