Subgroups — #26109073

Order 12032.

516 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
2	#20243540	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688871	C₂	1	2	yes	maximal
4	#20241272	C₂	1	2	yes	maximal
5	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₉₄	C₄	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#15689029	C₄	1	2	yes	—
12	#15689021	—	2	1	no	—
13	#15689021	—	2	1	no	—
14	#15688993	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊Dic₄₇	—	2	1	no	—
16	C₄×C₄⋊Dic₄₇	—	2	1	no	—
17	#15688871	—	2	4	no	—
18	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
19	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
21	C₄³×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×Dic₉₄	C₂×C₄	1	2	yes	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	#13094024	—	2	1	no	—
25	#13094024	—	2	1	no	—
26	#13094024	D₄	1	2	yes	—
27	#13094024	Q₈	1	2	yes	—
28	C₄×Dic₉₄	—	4	4	no	—
29	C₂×C₄⋊Dic₄₇	—	2	1	no	—
30	C₂×C₄⋊Dic₄₇	—	2	1	no	—
31	C₂×C₄⋊Dic₄₇	—	2	1	no	—
32	C₂×C₄⋊Dic₄₇	—	2	1	no	—
33	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
34	C₂×Dic₄₇⋊C₄	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
39	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
40	C₄₇⋊#95	—	2	2	no	—
41	C₄₇⋊#95	—	2	2	no	—
42	C₂²×C₈×C₄₇	—	2	2	no	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₄₇	—	2	2	no	—
51	C₄⋊C₈×C₄₇	—	2	2	no	—
52	C₄⋊C₈×C₄₇	—	4	2	no	—
53	C₄⋊C₈×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
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×Dic₄₇	—	2	2	no	—
62	C₂²×Dic₄₇	—	2	2	no	—
63	C₂×Dic₉₄	—	2	2	no	—
64	C₂×Dic₉₄	—	4	2	no	—
65	C₂×Dic₉₄	—	4	2	no	—
66	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
67	C₂×C₈×C₄₇	—	4	2	no	—
68	C₂×C₈×C₄₇	—	4	2	no	—
69	C₂×C₈×C₄₇	—	4	2	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	2	1	no	—
72	C₄²×C₄₇	—	2	1	no	—
73	C₄²×C₄₇	—	2	1	no	—
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	—	4	1	no	—
76	C₄²×C₄₇	SD₁₆	1	2	yes	—
77	C₄²×C₄₇	SD₁₆	1	2	yes	—
78	C₄²×C₄₇	Q₁₆	1	2	yes	—
79	C₄²×C₄₇	Q₁₆	1	2	yes	—
80	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₄₇	#41	1	1	yes	characteristic
83	C₄²×C₄₇	#41	1	1	yes	characteristic
84	C₄⋊Dic₄₇	—	2	2	no	—
85	C₄⋊Dic₄₇	—	2	2	no	—
86	C₄⋊Dic₄₇	—	2	2	no	—
87	C₄⋊Dic₄₇	—	2	2	no	—
88	C₄⋊Dic₄₇	—	4	1	no	—
89	C₄⋊Dic₄₇	—	4	1	no	—
90	C₄⋊Dic₄₇	—	4	1	no	—
91	C₄⋊Dic₄₇	—	4	1	no	—
92	C₄⋊Dic₄₇	—	4	2	no	—
93	Dic₄₇⋊C₄	—	4	4	no	—
94	C₄×Dic₄₇	—	4	4	no	—
95	C₂×C₄₇⋊C₈	—	4	2	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂×C₄₇⋊C₈	—	4	2	no	—
98	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	—	2	1	no	—
100	C₂×C₄×C₄₇	—	2	1	no	—
101	C₂×C₄×C₄₇	—	2	1	no	—
102	C₂×C₄×C₄₇	—	2	1	no	—
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	#100	1	2	yes	—
109	C₂×C₄×C₄₇	#103	1	2	yes	—
110	C₂×C₄×C₄₇	#103	1	2	yes	—
111	C₂×C₄×C₄₇	#103	1	2	yes	—
112	C₂×C₄×C₄₇	#103	1	2	yes	—
113	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
116	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
123	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
126	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
127	C₂×Dic₄₇	—	2	4	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	2	no	—
133	Dic₉₄	—	4	4	no	—
134	Dic₉₄	—	4	4	no	—
135	C₈×C₄₇	—	8	2	no	—
136	C₄₇⋊C₈	—	8	2	no	—
137	#8587	—	47	1	no	maximal, Sylow
138	C₂²×C₄₇	#380	1	1	yes	characteristic
139	C₂²×C₄₇	C₂×#100	1	1	yes	characteristic
140	C₂²×C₄₇	#413	1	1	yes	characteristic
141	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
142	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
143	C₂²×C₄₇	#417	1	1	yes	characteristic
144	C₂²×C₄₇	#417	1	1	yes	characteristic
145	C₄×C₄₇	—	2	1	no	—
146	C₄×C₄₇	—	2	1	no	—
147	C₄×C₄₇	—	2	2	no	—
148	C₄×C₄₇	—	2	2	no	—
149	C₄×C₄₇	—	4	1	no	—
150	C₄×C₄₇	#333	1	4	yes	complemented
151	C₄×C₄₇	#460	1	1	yes	characteristic
152	C₄×C₄₇	#460	1	1	yes	characteristic
153	C₄×C₄₇	#462	1	1	yes	characteristic
154	C₄×C₄₇	#462	1	1	yes	characteristic
155	C₄×C₄₇	#475	1	1	yes	characteristic
156	C₄×C₄₇	#475	1	1	yes	characteristic
157	C₄×C₄₇	#477	1	1	yes	characteristic
158	C₄×C₄₇	#477	1	1	yes	characteristic
159	Dic₄₇	—	4	4	no	—
160	Dic₄₇	—	8	2	no	—
161	C₄×C₄⋊Q₈	—	47	1	no	—
162	#1842	—	47	1	no	—
163	#1842	D₄₇	1	1	yes	characteristic
164	C₂×#333	—	47	2	no	—
165	#1296	—	47	2	no	—
166	C₂×C₄₇	C₂×#333	1	1	yes	characteristic
167	C₂×C₄₇	#1550	1	1	yes	characteristic
168	C₂×C₄₇	#1550	1	1	yes	characteristic
169	C₂×C₄₇	#1560	1	1	yes	characteristic
170	C₂×C₄₇	#1800	1	1	yes	characteristic
171	C₂×C₄₇	#1803	1	1	yes	characteristic
172	C₂×C₄₇	#1803	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	—	47	1	no	—
174	C₂×C₄×Q₈	—	47	2	no	—
175	C₂×C₄⋊C₈	—	2	1	no	—
176	C₂×C₄⋊C₈	—	2	1	no	—
177	C₂×C₄⋊C₈	—	47	2	no	—
178	C₂×C₄⋊C₈	—	94	1	no	—
179	C₂×C₄⋊C₈	—	94	1	no	—
180	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
181	#391	—	47	2	no	—
182	#389	—	94	1	no	—
183	#389	—	94	1	no	—
184	#384	—	47	1	no	—
185	C₄×C₄⋊C₄	—	94	1	no	—
186	C₄×C₄⋊C₄	—	94	1	no	—
187	C₄³	C₂×D₄₇	1	1	yes	characteristic
188	#333	—	94	4	no	—
189	C₄₇	#8587	1	1	yes	characteristic, complemented, Sylow
190	C₂²×Q₈	—	47	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	94	2	no	—
193	C₄⋊Q₈	—	47	2	no	—
194	C₄⋊Q₈	—	47	2	no	—
195	C₄⋊Q₈	—	94	1	no	—
196	C₄⋊Q₈	—	94	1	no	—
197	C₄×Q₈	—	188	4	no	—
198	C₂×C₄⋊C₄	—	47	2	no	—
199	C₂×C₄⋊C₄	—	94	1	no	—
200	C₂×C₄⋊C₄	—	94	1	no	—
201	C₂×C₄⋊C₄	—	94	2	no	—
202	C₂×C₄⋊C₄	—	94	1	no	—
203	C₂×C₄⋊C₄	—	94	1	no	—
204	C₂×C₄²	—	94	2	no	—
205	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
207	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
208	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
209	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
211	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	4	2	no	—
215	C₄⋊C₈	—	4	2	no	—
216	C₄⋊C₈	—	94	2	no	—
217	C₄⋊C₈	—	94	2	no	—
218	C₄⋊C₈	—	188	2	no	—
219	C₄⋊C₈	—	188	2	no	—
220	#95	—	94	2	no	—
221	#95	—	94	2	no	—
222	C₂×Q₈	—	47	4	no	—
223	C₂×Q₈	—	94	2	no	—
224	C₂×Q₈	—	188	2	no	—
225	C₂×Q₈	—	188	2	no	—
226	C₂²×C₄	—	94	2	no	—
227	C₂²×C₄	—	94	2	no	—
228	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	188	2	no	—
239	C₂×C₈	—	188	2	no	—
240	C₂×C₈	—	188	2	no	—
241	C₄⋊C₄	—	94	2	no	—
242	C₄⋊C₄	—	94	2	no	—
243	C₄⋊C₄	—	94	2	no	—
244	C₄⋊C₄	—	94	2	no	—
245	C₄⋊C₄	—	188	2	no	—
246	C₄⋊C₄	—	188	4	no	—
247	C₄⋊C₄	—	188	1	no	—
248	C₄⋊C₄	—	188	1	no	—
249	C₄⋊C₄	—	188	1	no	—
250	C₄⋊C₄	—	188	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	188	4	no	—
258	C₄²	#10625325	1	2	yes	—
259	C₄²	Dic₁₈₈	1	2	yes	—
260	C₄²	#10625335	1	2	yes	—
261	C₄²	#10625337	1	2	yes	—
262	C₄²	D₄×D₄₇	1	1	yes	characteristic
263	C₄²	#10625352	1	1	yes	characteristic
264	C₄²	Q₈×D₄₇	1	1	yes	characteristic
265	C₄²	#10625354	1	1	yes	characteristic
266	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
267	Q₈	—	188	4	no	—
268	Q₈	—	188	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	94	4	no	—
279	C₂×C₄	—	188	2	no	—
280	C₂×C₄	—	188	2	no	—
281	C₂×C₄	—	188	2	no	—
282	C₂×C₄	—	188	2	no	—
283	C₂×C₄	—	188	2	no	—
284	C₂×C₄	#13093940	1	2	yes	—
285	C₂×C₄	#13093947	1	2	yes	—
286	C₂×C₄	#13093947	1	2	yes	—
287	C₂×C₄	#13093949	1	2	yes	—
288	C₂×C₄	#13093949	1	2	yes	—
289	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
290	C₂×C₄	#13093982	1	1	yes	characteristic
291	C₂×C₄	#13093995	1	1	yes	characteristic
292	C₂×C₄	#13093999	1	1	yes	characteristic
293	C₂×C₄	#13094000	1	1	yes	characteristic
294	C₂×C₄	#13094004	1	1	yes	characteristic
295	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
296	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
297	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
298	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
299	C₂×C₄	#13094050	1	1	yes	characteristic
300	C₂×C₄	#13094052	1	1	yes	characteristic
301	C₂×C₄	#13094055	1	1	yes	characteristic
302	C₂×C₄	#13094057	1	1	yes	characteristic
303	C₈	—	8	2	no	—
304	C₈	—	376	2	no	—
305	C₂²	#15688980	1	1	yes	characteristic
306	C₂²	C₂×#13093940	1	1	yes	characteristic
307	C₂²	#15689105	1	1	yes	characteristic
308	C₂²	C₂×#13093947	1	1	yes	characteristic
309	C₂²	C₂×#13093949	1	1	yes	characteristic
310	C₂²	#15689122	1	1	yes	characteristic
311	C₂²	#15689125	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	376	2	no	—
318	C₄	—	188	4	no	—
319	C₄	#15688871	1	4	yes	complemented
320	C₄	#15689320	1	1	yes	characteristic
321	C₄	#15689324	1	1	yes	characteristic
322	C₄	#15689330	1	1	yes	characteristic
323	C₄	#15689334	1	1	yes	characteristic
324	C₄	#15689405	1	1	yes	characteristic
325	C₄	#15689409	1	1	yes	characteristic
326	C₄	#15689415	1	1	yes	characteristic
327	C₄	#15689419	1	1	yes	characteristic
328	C₂	C₂×#15688871	1	1	yes	characteristic
329	C₂	#20242076	1	1	yes	characteristic
330	C₂	#20242079	1	1	yes	characteristic
331	C₂	#20242121	1	1	yes	characteristic
332	C₂	#20243377	1	1	yes	characteristic
333	C₂	#20243390	1	1	yes	characteristic
334	C₂	#20243394	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8587
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
150. C₄×C₄₇	#333	semidirect
189. C₄₇	#8587	semidirect
319. C₄	#15688871	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13094024	2
#20243540	2
C₂×#15688871	2
#20241272	2
C₄₇×#1842	2
#8587	47