Subgroups — #26316026

Order 12032.

710 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20251751	C₂	1	1	yes	maximal, characteristic
2	C₄×#13094024	C₂	1	1	yes	maximal, characteristic
3	#20244947	C₂	1	2	yes	maximal
4	#20244635	C₂	1	2	yes	maximal
5	#20244078	C₂	1	4	yes	maximal
6	#20243394	C₂	1	2	yes	maximal
7	#20243394	C₂	1	2	yes	maximal
8	C₄₇×#1767	C₂	1	1	yes	maximal, characteristic
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×#13094016	—	2	1	no	—
11	C₂×C₄×Dic₉₄	C₂²	1	2	yes	—
12	C₂×#13093949	C₂²	1	4	yes	—
13	C₂×#13093949	C₂²	1	4	yes	—
14	#15689029	C₂²	1	2	yes	—
15	#15689021	—	2	1	no	—
16	#15689021	—	2	1	no	—
17	#15689021	—	2	1	no	—
18	#15689021	—	2	1	no	—
19	#15689021	C₂²	1	2	yes	—
20	#15689021	C₂²	1	2	yes	—
21	#15688993	C₂²	1	1	yes	characteristic
22	#15688993	C₂²	1	1	yes	characteristic
23	#15688993	C₂²	1	1	yes	characteristic
24	#15688989	—	2	1	no	—
25	C₄×C₄⋊Dic₄₇	—	2	1	no	—
26	C₄×C₄⋊Dic₄₇	—	2	1	no	—
27	#15688886	C₂²	1	4	yes	—
28	#15688886	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
30	C₂×#97×C₄₇	C₂²	1	2	yes	—
31	C₄³×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#336	C₂²	1	2	yes	—
33	C₂²×Dic₉₄	C₂³	1	2	yes	—
34	#13094024	—	2	4	no	—
35	#13094016	—	4	4	no	—
36	C₄×Dic₉₄	—	2	4	no	—
37	C₄×Dic₉₄	—	2	4	no	—
38	C₂×C₄⋊Dic₄₇	—	2	1	no	—
39	C₂×C₄⋊Dic₄₇	—	2	1	no	—
40	C₂×C₄⋊Dic₄₇	—	2	1	no	—
41	C₂×C₄⋊Dic₄₇	—	2	1	no	—
42	C₂×C₄⋊Dic₄₇	—	2	1	no	—
43	C₂×C₄⋊Dic₄₇	—	2	1	no	—
44	C₂×C₄⋊Dic₄₇	—	2	1	no	—
45	C₂×C₄⋊Dic₄₇	—	2	1	no	—
46	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
49	C₂×Dic₄₇⋊C₄	—	2	2	no	—
50	C₂×Dic₄₇⋊C₄	—	2	2	no	—
51	C₂×Dic₄₇⋊C₄	—	2	2	no	—
52	C₂×C₄×Dic₄₇	—	2	2	no	—
53	#13093949	—	2	8	no	—
54	#13093949	—	4	4	no	—
55	C₄₇⋊#95	—	2	2	no	—
56	C₄₇⋊#95	—	2	2	no	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
60	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₄₇	—	2	4	no	—
68	#97×C₄₇	—	4	2	no	—
69	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	2	2	no	—
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂×Dic₉₄	—	2	2	no	—
81	C₂×Dic₉₄	—	2	2	no	—
82	C₂×Dic₉₄	—	2	2	no	—
83	C₂×Dic₉₄	—	2	2	no	—
84	C₂×Dic₉₄	—	2	2	no	—
85	C₂×Dic₉₄	#41	1	4	yes	—
86	C₂×C₈×C₄₇	—	2	4	no	—
87	C₂×C₈×C₄₇	—	2	4	no	—
88	C₂×C₈×C₄₇	—	2	4	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	C₄²×C₄₇	—	2	2	no	—
94	C₄²×C₄₇	—	2	1	no	—
95	C₄²×C₄₇	—	2	1	no	—
96	C₄²×C₄₇	#41	1	4	yes	—
97	C₄⋊Dic₄₇	—	2	4	no	—
98	C₄⋊Dic₄₇	—	2	4	no	—
99	C₄⋊Dic₄₇	—	2	4	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	—	2	2	no	—
103	C₄⋊Dic₄₇	—	2	2	no	—
104	C₄⋊Dic₄₇	—	4	2	no	—
105	C₄⋊Dic₄₇	—	4	2	no	—
106	C₄⋊Dic₄₇	—	4	2	no	—
107	C₄⋊Dic₄₇	—	4	2	no	—
108	C₄⋊Dic₄₇	—	4	2	no	—
109	C₄⋊Dic₄₇	—	4	2	no	—
110	Dic₄₇⋊C₄	—	4	4	no	—
111	Dic₄₇⋊C₄	—	4	4	no	—
112	Dic₄₇⋊C₄	—	4	4	no	—
113	C₄×Dic₄₇	—	4	4	no	—
114	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₇	—	2	1	no	—
116	C₂×C₄×C₄₇	—	2	1	no	—
117	C₂×C₄×C₄₇	—	2	1	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	2	1	no	—
121	C₂×C₄×C₄₇	—	2	1	no	—
122	C₂×C₄×C₄₇	—	2	1	no	—
123	C₂×C₄×C₄₇	—	2	1	no	—
124	C₂×C₄×C₄₇	—	2	1	no	—
125	C₂×C₄×C₄₇	—	2	1	no	—
126	C₂×C₄×C₄₇	—	2	1	no	—
127	C₂×C₄×C₄₇	#124	1	2	yes	—
128	C₂×C₄×C₄₇	#124	1	2	yes	—
129	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₄₇	#137	1	2	yes	—
131	C₂×C₄×C₄₇	#137	1	2	yes	—
132	C₂×C₄×C₄₇	#137	1	2	yes	—
133	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
134	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
135	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
137	C₂×Dic₄₇	—	2	4	no	—
138	C₂×Dic₄₇	—	4	2	no	—
139	C₂×Dic₄₇	—	4	2	no	—
140	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	—	4	2	no	—
143	C₂×Dic₄₇	—	4	2	no	—
144	C₂×Dic₄₇	—	4	2	no	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	Dic₉₄	—	2	4	no	—
150	Dic₉₄	—	2	4	no	—
151	Dic₉₄	—	4	2	no	—
152	Dic₉₄	—	4	2	no	—
153	C₈×C₄₇	—	4	4	no	—
154	#29132	—	47	1	no	maximal, Sylow
155	C₂²×C₄₇	#525	1	1	yes	characteristic
156	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
157	C₂²×C₄₇	#537	1	1	yes	characteristic
158	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
160	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
161	C₂²×C₄₇	C₂×#137	1	1	yes	characteristic
162	C₄×C₄₇	—	2	2	no	—
163	C₄×C₄₇	—	2	2	no	—
164	C₄×C₄₇	—	2	2	no	—
165	C₄×C₄₇	—	4	1	no	—
166	C₄×C₄₇	—	4	1	no	—
167	C₄×C₄₇	#491	1	4	yes	—
168	C₄×C₄₇	#552	1	2	yes	—
169	C₄×C₄₇	#553	1	2	yes	—
170	Dic₄₇	—	4	4	no	—
171	Dic₄₇	—	8	2	no	—
172	Dic₄₇	—	8	2	no	—
173	Dic₄₇	—	8	2	no	—
174	#2859	—	47	1	no	—
175	C₄×C₄⋊Q₈	—	47	1	no	—
176	#2099	—	47	2	no	—
177	#2043	—	47	2	no	—
178	#1935	—	47	4	no	—
179	#1803	—	47	2	no	—
180	#1803	—	47	2	no	—
181	#1767	D₄₇	1	1	yes	characteristic
182	C₂×C₄₇	#2562	1	1	yes	characteristic
183	C₂×C₄₇	#3112	1	1	yes	characteristic
184	C₂×C₄₇	#3113	1	1	yes	characteristic
185	C₂×C₄₇	C₂×#491	1	1	yes	characteristic
186	C₂×C₄₇	#3141	1	1	yes	characteristic
187	C₂×C₄₇	#3254	1	1	yes	characteristic
188	C₂×C₄₇	#3255	1	1	yes	characteristic
189	C₂×C₄⋊Q₈	—	47	1	no	—
190	C₂×#125	—	94	1	no	—
191	C₂×C₄×Q₈	—	47	2	no	—
192	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
193	C₂×#103	—	47	4	no	—
194	C₂×#103	—	47	4	no	—
195	C₂×#97	C₂×D₄₇	1	2	yes	—
196	#391	—	47	2	no	—
197	#389	—	47	2	no	—
198	#389	—	47	2	no	—
199	#389	—	94	1	no	—
200	#389	—	94	1	no	—
201	#389	—	94	1	no	—
202	#389	—	94	1	no	—
203	#384	—	47	1	no	—
204	#384	—	47	1	no	—
205	#384	—	47	1	no	—
206	#382	—	94	1	no	—
207	C₄×C₄⋊C₄	—	94	1	no	—
208	C₄×C₄⋊C₄	—	94	1	no	—
209	C₄³	C₂×D₄₇	1	1	yes	characteristic
210	#340	—	47	4	no	—
211	#340	—	47	4	no	—
212	#336	C₂×D₄₇	1	2	yes	—
213	C₄₇	#29132	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	47	2	no	—
215	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
216	C₄⋊Q₈	—	94	4	no	—
217	#125	—	188	4	no	—
218	C₄×Q₈	—	94	4	no	—
219	C₄×Q₈	—	94	4	no	—
220	C₂×C₄⋊C₄	—	47	2	no	—
221	C₂×C₄⋊C₄	—	47	2	no	—
222	C₂×C₄⋊C₄	—	47	2	no	—
223	C₂×C₄⋊C₄	—	94	1	no	—
224	C₂×C₄⋊C₄	—	94	1	no	—
225	C₂×C₄⋊C₄	—	94	2	no	—
226	C₂×C₄⋊C₄	—	94	2	no	—
227	C₂×C₄⋊C₄	—	94	2	no	—
228	C₂×C₄⋊C₄	—	94	1	no	—
229	C₂×C₄⋊C₄	—	94	1	no	—
230	C₂×C₄⋊C₄	—	94	1	no	—
231	C₂×C₄⋊C₄	—	94	1	no	—
232	C₂×C₄⋊C₄	—	94	1	no	—
233	C₂×C₄⋊C₄	—	94	1	no	—
234	C₂×C₄²	—	94	2	no	—
235	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
236	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
237	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
238	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
242	C₄⋊C₈	—	2	4	no	—
243	#103	—	94	8	no	—
244	#103	—	188	4	no	—
245	#97	—	4	2	no	—
246	#95	—	94	2	no	—
247	#95	—	94	2	no	—
248	#95	—	94	2	no	—
249	#95	—	94	2	no	—
250	C₂×Q₈	—	47	4	no	—
251	C₂×Q₈	—	94	2	no	—
252	C₂×Q₈	—	94	2	no	—
253	C₂×Q₈	—	94	2	no	—
254	C₂×Q₈	—	94	2	no	—
255	C₂×Q₈	—	94	2	no	—
256	C₂²×C₄	—	94	2	no	—
257	C₂²×C₄	—	94	2	no	—
258	C₂²×C₄	—	94	2	no	—
259	C₂²×C₄	—	94	2	no	—
260	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	4	no	—
270	C₄⋊C₄	—	94	4	no	—
271	C₄⋊C₄	—	94	4	no	—
272	C₄⋊C₄	—	94	4	no	—
273	C₄⋊C₄	—	94	2	no	—
274	C₄⋊C₄	—	94	2	no	—
275	C₄⋊C₄	—	94	2	no	—
276	C₄⋊C₄	—	94	2	no	—
277	C₄⋊C₄	—	188	4	no	—
278	C₄⋊C₄	—	188	2	no	—
279	C₄⋊C₄	—	188	2	no	—
280	C₄⋊C₄	—	188	4	no	—
281	C₄⋊C₄	—	188	4	no	—
282	C₄⋊C₄	—	188	2	no	—
283	C₄⋊C₄	—	188	2	no	—
284	C₄⋊C₄	—	188	2	no	—
285	C₄⋊C₄	—	188	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	188	4	no	—
294	C₄²	#10625350	1	4	yes	—
295	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
296	Q₈	—	94	4	no	—
297	Q₈	—	94	4	no	—
298	Q₈	—	188	2	no	—
299	Q₈	—	188	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	94	4	no	—
313	C₂×C₄	—	188	2	no	—
314	C₂×C₄	—	188	2	no	—
315	C₂×C₄	—	188	2	no	—
316	C₂×C₄	—	188	2	no	—
317	C₂×C₄	—	188	2	no	—
318	C₂×C₄	—	188	2	no	—
319	C₂×C₄	—	188	2	no	—
320	C₂×C₄	—	188	2	no	—
321	C₂×C₄	—	188	2	no	—
322	C₂×C₄	—	188	2	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	#13094012	1	2	yes	—
325	C₂×C₄	#13094012	1	2	yes	—
326	C₂×C₄	#13094052	1	2	yes	—
327	C₂×C₄	#13094057	1	2	yes	—
328	C₂×C₄	#13094057	1	2	yes	—
329	C₂×C₄	#13094057	1	2	yes	—
330	C₂×C₄	#41×D₄₇	1	2	yes	—
331	C₂×C₄	C₂×#10625350	1	2	yes	—
332	C₂×C₄	#13094069	1	1	yes	characteristic
333	C₂×C₄	#13094072	1	1	yes	characteristic
334	C₈	—	4	4	no	—
335	C₂²	#15689635	1	1	yes	characteristic
336	C₂²	C₂×#13094012	1	1	yes	characteristic
337	C₂²	#15689704	1	1	yes	characteristic
338	C₂²	C₂×#13094052	1	1	yes	characteristic
339	C₂²	C₂×#13094057	1	1	yes	characteristic
340	C₂²	C₂×#13094057	1	1	yes	characteristic
341	C₂²	C₂×#13094057	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	376	2	no	—
348	C₄	—	376	2	no	—
349	C₄	—	376	2	no	—
350	C₄	—	188	4	no	—
351	C₄	#15689492	1	4	yes	—
352	C₄	#15689814	1	2	yes	—
353	C₄	#15689823	1	2	yes	—
354	C₂	#20248549	1	1	yes	characteristic
355	C₂	#20253511	1	1	yes	characteristic
356	C₂	#20253519	1	1	yes	characteristic
357	C₂	C₂×#15689492	1	1	yes	characteristic
358	C₂	#20253782	1	1	yes	characteristic
359	C₂	#20254776	1	1	yes	characteristic
360	C₂	#20254782	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1767	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1767	D₄₇
hypocenter	C₄₇	#29132
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
213. C₄₇	#29132	semidirect

Maximal subgroups

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

subgroup	index
#20251751	2
C₄×#13094024	2
#20244947	2
#20244635	2
#20244078	2
#20243394	2
C₄₇×#1767	2
#29132	47