Subgroups — #26116464

Order 12032.

592 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251740	C₂	1	1	yes	maximal, characteristic
2	#20243368	C₂	1	1	yes	maximal, characteristic
3	#20241267	C₂	1	4	yes	maximal
4	C₄₇×#1434	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094024	C₄	1	1	yes	characteristic, complemented
6	C₂×#13094003	—	2	1	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15689029	—	2	1	no	—
10	#15689021	—	2	1	no	—
11	#15689021	C₄	1	2	yes	complemented
12	#15688993	C₄	1	1	yes	characteristic, complemented
13	#15688983	—	2	1	no	—
14	#15688983	C₂²	1	2	yes	—
15	#15688878	—	2	1	no	—
16	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#95×C₄₇	—	2	1	no	—
18	C₂×#95×C₄₇	—	2	1	no	—
19	C₂×#95×C₄₇	C₂²	1	2	yes	—
20	C₂²×Dic₉₄	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	#13094024	—	4	2	no	—
23	#13094003	—	4	2	no	—
24	#13094003	—	4	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	1	no	—
26	C₂×C₄⋊Dic₄₇	—	2	1	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
32	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
33	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
34	C₄₇⋊#95	—	2	2	no	—
35	C₄₇⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₇	—	2	2	no	—
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	C₂×C₄²×C₄₇	—	4	2	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	4	2	no	—
47	#95×C₄₇	—	4	2	no	—
48	#95×C₄₇	Q₈	1	4	yes	complemented
49	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	—	2	2	no	—
51	C₂²×C₄×C₄₇	—	2	2	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	4	1	no	—
54	C₂²×C₄×C₄₇	—	4	1	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	—	4	2	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	2	no	—
61	C₂²×C₄×C₄₇	—	4	2	no	—
62	C₂²×C₄×C₄₇	—	4	2	no	—
63	C₂²×C₄×C₄₇	—	4	2	no	—
64	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	D₈	1	2	yes	—
69	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₄₇	#41	1	2	yes	—
74	C₂²×Dic₄₇	—	2	2	no	—
75	C₂²×Dic₄₇	—	2	2	no	—
76	C₂×Dic₉₄	—	4	2	no	—
77	C₂×Dic₉₄	—	4	2	no	—
78	C₂×Dic₉₄	—	4	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	4	1	no	—
82	C₄²×C₄₇	—	4	1	no	—
83	C₂²⋊Dic₄₇	—	4	2	no	—
84	C₂²⋊Dic₄₇	—	4	2	no	—
85	C₄⋊Dic₄₇	—	4	2	no	—
86	C₄⋊Dic₄₇	—	4	2	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄⋊Dic₄₇	—	4	2	no	—
90	Dic₄₇⋊C₄	—	4	4	no	—
91	C₂×C₄₇⋊C₈	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	4	2	no	—
93	C₂×C₄₇⋊C₈	—	4	2	no	—
94	C₂³×C₄₇	—	2	1	no	—
95	C₂³×C₄₇	—	2	1	no	—
96	C₂³×C₄₇	—	2	1	no	—
97	C₂³×C₄₇	—	2	1	no	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
101	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
102	C₂³×C₄₇	#137	1	2	yes	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	—	2	1	no	—
106	C₂×C₄×C₄₇	—	2	1	no	—
107	C₂×C₄×C₄₇	—	2	1	no	—
108	C₂×C₄×C₄₇	—	2	1	no	—
109	C₂×C₄×C₄₇	—	2	1	no	—
110	C₂×C₄×C₄₇	—	2	1	no	—
111	C₂×C₄×C₄₇	—	2	1	no	—
112	C₂×C₄×C₄₇	—	2	1	no	—
113	C₂×C₄×C₄₇	—	2	1	no	—
114	C₂×C₄×C₄₇	—	2	1	no	—
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	4	2	no	—
117	C₂×C₄×C₄₇	—	4	2	no	—
118	C₂×C₄×C₄₇	—	4	2	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	4	no	—
121	C₂×C₄×C₄₇	—	4	4	no	—
122	C₂×C₄×C₄₇	—	4	4	no	—
123	C₂×C₄×C₄₇	—	4	4	no	—
124	C₂×C₄×C₄₇	—	8	2	no	—
125	C₂×C₄×C₄₇	—	8	2	no	—
126	C₂×C₄×C₄₇	—	8	2	no	—
127	C₂×C₄×C₄₇	—	8	2	no	—
128	C₂×C₄×C₄₇	#99	1	2	yes	—
129	C₂×C₄×C₄₇	#102	1	2	yes	—
130	C₂×C₄×C₄₇	#102	1	2	yes	—
131	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
133	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
134	C₂×Dic₄₇	—	4	2	no	—
135	C₂×Dic₄₇	—	4	2	no	—
136	C₂×Dic₄₇	—	4	2	no	—
137	C₂×Dic₄₇	—	4	2	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	Dic₉₄	—	8	4	no	—
141	C₄₇⋊C₈	—	8	2	no	—
142	#9912	—	47	1	no	maximal, Sylow
143	C₂²×C₄₇	—	2	2	no	—
144	C₂²×C₄₇	—	2	2	no	—
145	C₂²×C₄₇	—	2	2	no	—
146	C₂²×C₄₇	—	2	2	no	—
147	C₂²×C₄₇	—	2	1	no	—
148	C₂²×C₄₇	—	2	1	no	—
149	C₂²×C₄₇	—	2	1	no	—
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	#391	1	1	yes	characteristic
152	C₂²×C₄₇	C₂×#99	1	1	yes	characteristic
153	C₂²×C₄₇	#410	1	1	yes	characteristic
154	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
155	C₂²×C₄₇	#419	1	1	yes	characteristic
156	C₂²×C₄₇	#421	1	1	yes	characteristic
157	C₂²×C₄₇	#421	1	1	yes	characteristic
158	C₂²×C₄₇	#474	1	2	yes	—
159	C₂²×C₄₇	#475	1	2	yes	—
160	C₄×C₄₇	—	2	1	no	—
161	C₄×C₄₇	—	2	2	no	—
162	C₄×C₄₇	—	2	1	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	2	2	no	—
166	C₄×C₄₇	—	4	1	no	—
167	C₄×C₄₇	—	8	4	no	—
168	Dic₄₇	—	8	2	no	—
169	Dic₄₇	—	8	2	no	—
170	#2857	—	47	1	no	—
171	#1797	—	47	1	no	—
172	#1434	D₄₇	1	1	yes	characteristic
173	#1294	—	47	4	no	—
174	C₂×C₄₇	—	2	2	no	—
175	C₂×C₄₇	—	2	2	no	—
176	C₂×C₄₇	#1511	1	1	yes	characteristic
177	C₂×C₄₇	#1515	1	1	yes	characteristic
178	C₂×C₄₇	#1517	1	1	yes	characteristic
179	C₂×C₄₇	#1519	1	1	yes	characteristic
180	C₂×C₄₇	#1988	1	1	yes	characteristic
181	C₂×C₄₇	#1995	1	1	yes	characteristic
182	C₂×C₄₇	#1996	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	—	47	1	no	—
184	C₂×C₂²⋊Q₈	—	94	1	no	—
185	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
186	C₂×C₄⋊C₈	—	94	1	no	—
187	C₂×C₂²⋊C₈	—	47	2	no	—
188	#391	—	94	1	no	—
189	#389	—	47	2	no	—
190	#389	—	94	1	no	—
191	#384	—	47	1	no	—
192	#380	—	47	2	no	—
193	#380	—	94	1	no	—
194	C₂×#95	—	2	1	no	—
195	C₂×#95	—	2	1	no	—
196	C₂×#95	C₂×D₄₇	1	2	yes	—
197	#336	—	94	1	no	—
198	C₄₇	#9912	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	94	1	no	—
200	C₂³×C₄	—	2	2	no	—
201	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
202	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
204	C₂²×C₈	—	94	2	no	—
205	C₄⋊Q₈	—	188	2	no	—
206	C₂²⋊Q₈	—	188	2	no	—
207	C₂²⋊Q₈	—	188	2	no	—
208	C₂×C₄⋊C₄	—	47	2	no	—
209	C₂×C₄⋊C₄	—	47	2	no	—
210	C₂×C₄⋊C₄	—	94	2	no	—
211	C₂×C₄⋊C₄	—	94	1	no	—
212	C₂×C₄⋊C₄	—	94	1	no	—
213	C₂×C₄⋊C₄	—	94	1	no	—
214	C₂×C₂²⋊C₄	—	94	2	no	—
215	C₂×C₄²	—	4	2	no	—
216	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
219	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
220	C₄⋊C₈	—	188	2	no	—
221	C₂²⋊C₈	—	94	4	no	—
222	#95	—	2	2	no	—
223	#95	—	4	2	no	—
224	#95	—	4	2	no	—
225	#95	—	94	2	no	—
226	#95	—	94	2	no	—
227	#95	Dic₉₄	1	4	yes	complemented
228	C₂⁴	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂×Q₈	—	188	2	no	—
230	C₂×Q₈	—	188	2	no	—
231	C₂×Q₈	—	188	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	94	2	no	—
247	C₂²×C₄	—	94	2	no	—
248	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	#10625334	1	2	yes	—
253	C₂²×C₄	#10625336	1	2	yes	—
254	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
255	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
257	C₂²×C₄	#10625350	1	2	yes	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	—	188	2	no	—
260	C₂×C₈	—	188	2	no	—
261	C₄⋊C₄	—	188	2	no	—
262	C₄⋊C₄	—	188	2	no	—
263	C₄⋊C₄	—	188	2	no	—
264	C₄⋊C₄	—	188	2	no	—
265	C₄⋊C₄	—	188	4	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₂²⋊C₄	—	188	2	no	—
268	C₂²⋊C₄	—	188	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
278	C₂³	#13094003	1	2	yes	—
279	C₂³	C₂×#10625334	1	1	yes	characteristic
280	C₂³	C₂×#10625336	1	1	yes	characteristic
281	C₂³	#13094054	1	2	yes	—
282	Q₈	—	376	4	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	188	2	no	—
309	C₂×C₄	—	188	2	no	—
310	C₂×C₄	—	188	2	no	—
311	C₂×C₄	—	188	2	no	—
312	C₂×C₄	—	188	2	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	#13093938	1	2	yes	—
315	C₂×C₄	#13093944	1	2	yes	—
316	C₂×C₄	#13093944	1	2	yes	—
317	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
318	C₂×C₄	#13094012	1	1	yes	characteristic
319	C₂×C₄	#13094024	1	1	yes	characteristic
320	C₈	—	376	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	#15689029	1	1	yes	characteristic
330	C₂²	C₂×#13093938	1	1	yes	characteristic
331	C₂²	#15689096	1	1	yes	characteristic
332	C₂²	C₂×#13093944	1	1	yes	characteristic
333	C₂²	#15689131	1	1	yes	characteristic
334	C₂²	#15689141	1	1	yes	characteristic
335	C₂²	#15689141	1	1	yes	characteristic
336	C₂²	#15689403	1	2	yes	—
337	C₂²	#15689408	1	2	yes	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	8	4	no	—
346	C₄	—	376	2	no	—
347	C₄	—	376	2	no	—
348	C₂	—	2	2	no	—
349	C₂	—	2	2	no	—
350	C₂	#20241920	1	1	yes	characteristic
351	C₂	#20241938	1	1	yes	characteristic
352	C₂	#20241947	1	1	yes	characteristic
353	C₂	#20241955	1	1	yes	characteristic
354	C₂	#20244340	1	1	yes	characteristic
355	C₂	#20244369	1	1	yes	characteristic
356	C₂	#20244373	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#1434	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#1434	D₄₇
hypocenter	C₄₇	#9912
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
5. C₂×#13094024	C₄	semidirect
11. #15689021	C₄	semidirect
12. #15688993	C₄	semidirect
48. #95×C₄₇	Q₈	semidirect
198. C₄₇	#9912	semidirect
227. #95	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#20251740	2
#20243368	2
#20241267	2
C₄₇×#1434	2
#9912	47