Subgroups — #26125171

Order 12032.

814 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688885	C₂	1	2	yes	maximal
3	C₂×#15688877	C₂	1	1	yes	maximal, characteristic
4	#20241617	C₂	1	2	yes	maximal
5	C₂×C₄₇×#340	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂×#13093958	—	2	2	no	—
10	C₂×#13093958	—	2	2	no	—
11	C₂×#13093958	C₄	1	4	yes	complemented
12	C₂×C₄₇⋊#95	—	2	4	no	—
13	#15688885	—	2	4	no	—
14	#15688877	—	2	2	no	—
15	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₄₇×#340	—	2	2	no	—
18	C₂³×Dic₄₇	—	2	2	no	—
19	C₂³×Dic₄₇	D₄	1	2	yes	—
20	C₂²×C₄₇⋊C₈	—	2	1	no	—
21	C₂²×C₄₇⋊C₈	—	2	1	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₂²×C₄₇⋊C₈	—	2	2	no	—
24	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₄₇	—	2	1	no	—
32	C₂×C₄×Dic₄₇	—	2	2	no	—
33	C₂×C₄×Dic₄₇	—	2	2	no	—
34	C₂×C₄×Dic₄₇	—	2	1	no	—
35	#13093958	—	2	8	no	—
36	C₄₇⋊#95	—	4	8	no	—
37	C₂³×C₄×C₄₇	—	2	2	no	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	—	2	2	no	—
40	C₂²×C₈×C₄₇	—	2	2	no	—
41	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₄₇	D₄	1	2	yes	—
43	C₂²×C₈×C₄₇	D₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₄₇	Q₈	1	4	yes	—
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	4	2	no	—
53	C₂²×C₄×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	4	4	no	—
55	C₂²×C₄×C₄₇	—	4	4	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₇	#41	1	2	yes	—
62	C₂²×C₄×C₄₇	#41	1	2	yes	—
63	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₇	#41	1	2	yes	—
65	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
66	C₂²×Dic₄₇	—	2	2	no	—
67	C₂²×Dic₄₇	—	2	4	no	—
68	C₂²×Dic₄₇	—	2	2	no	—
69	C₂²×Dic₄₇	—	2	4	no	—
70	C₂²×Dic₄₇	—	2	4	no	—
71	C₂²×Dic₄₇	—	4	2	no	—
72	C₂²×Dic₄₇	—	4	2	no	—
73	C₂²×Dic₄₇	—	4	4	no	—
74	C₂²×Dic₄₇	—	4	4	no	—
75	C₂²×Dic₄₇	Q₁₆	1	4	yes	—
76	C₂×C₈×C₄₇	—	2	4	no	—
77	C₂×C₈×C₄₇	—	2	4	no	—
78	C₂×C₈×C₄₇	—	2	2	no	—
79	C₂×C₈×C₄₇	—	2	2	no	—
80	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₄₇	—	2	4	no	—
82	C₄⋊C₄×C₄₇	—	2	4	no	—
83	C₄⋊C₄×C₄₇	—	4	2	no	—
84	C₄⋊C₄×C₄₇	—	4	2	no	—
85	C₄⋊Dic₄₇	—	2	4	no	—
86	C₄⋊Dic₄₇	—	2	4	no	—
87	C₄⋊Dic₄₇	—	4	2	no	—
88	C₄⋊Dic₄₇	—	4	2	no	—
89	C₄×Dic₄₇	—	4	1	no	—
90	C₄×Dic₄₇	—	4	1	no	—
91	C₄×Dic₄₇	—	4	2	no	—
92	C₂×C₄₇⋊C₈	—	2	2	no	—
93	C₂×C₄₇⋊C₈	—	2	4	no	—
94	C₂×C₄₇⋊C₈	—	4	1	no	—
95	C₂×C₄₇⋊C₈	—	4	1	no	—
96	C₂×C₄₇⋊C₈	—	4	2	no	—
97	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₄₇	#121	1	1	yes	characteristic
99	C₂³×C₄₇	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄₇	#123	1	2	yes	—
101	C₂³×C₄₇	#127	1	1	yes	characteristic
102	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
103	C₂³×C₄₇	#135	1	2	yes	—
104	C₂³×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₄₇	—	4	4	no	—
106	C₂×C₄×C₄₇	—	4	8	no	—
107	C₂×C₄×C₄₇	—	8	4	no	—
108	C₂×C₄×C₄₇	—	8	2	no	—
109	C₂×C₄×C₄₇	—	8	2	no	—
110	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₇	#117	1	2	yes	—
112	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₇	#123	1	2	yes	—
118	C₂×C₄×C₄₇	#123	1	2	yes	—
119	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
120	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
121	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#125	1	2	yes	—
124	C₂×C₄×C₄₇	#125	1	2	yes	—
125	C₂×C₄×C₄₇	#126	1	2	yes	—
126	C₂×C₄×C₄₇	#126	1	2	yes	—
127	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×Dic₄₇	—	2	4	no	—
129	C₂×Dic₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	4	2	no	—
131	C₂×Dic₄₇	—	4	2	no	—
132	C₂×Dic₄₇	—	4	4	no	—
133	C₂×Dic₄₇	—	4	4	no	—
134	C₂×Dic₄₇	—	4	8	no	—
135	C₂×Dic₄₇	—	8	4	no	—
136	C₂×Dic₄₇	—	8	2	no	—
137	C₂×Dic₄₇	—	8	2	no	—
138	C₈×C₄₇	—	2	4	no	—
139	C₄₇⋊C₈	—	4	2	no	—
140	#11377	—	47	1	no	maximal, Sylow
141	C₂²×C₄₇	#387	1	2	yes	—
142	C₂²×C₄₇	#390	1	1	yes	characteristic
143	C₂²×C₄₇	#397	1	1	yes	characteristic
144	C₂²×C₄₇	#398	1	1	yes	characteristic
145	C₂²×C₄₇	#399	1	2	yes	—
146	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
147	C₂²×C₄₇	#442	1	2	yes	—
148	C₂²×C₄₇	#467	1	2	yes	—
149	C₂²×C₄₇	#469	1	2	yes	—
150	C₂²×C₄₇	#477	1	4	yes	—
151	C₂²×C₄₇	#478	1	4	yes	—
152	C₂²×C₄₇	#482	1	4	yes	—
153	C₂²×C₄₇	#484	1	4	yes	—
154	C₂²×C₄₇	#494	1	1	yes	characteristic
155	C₂²×C₄₇	#495	1	1	yes	characteristic
156	C₂²×C₄₇	#496	1	2	yes	—
157	C₄×C₄₇	—	8	4	no	—
158	C₄×C₄₇	#381	1	1	yes	characteristic
159	C₄×C₄₇	#387	1	2	yes	—
160	C₄×C₄₇	#391	1	1	yes	characteristic
161	C₄×C₄₇	#397	1	1	yes	characteristic
162	C₄×C₄₇	#398	1	1	yes	characteristic
163	C₄×C₄₇	#400	1	2	yes	—
164	Dic₄₇	—	4	4	no	—
165	Dic₄₇	—	8	4	no	—
166	C₂²×#107	—	47	1	no	—
167	C₂×#340	—	47	2	no	—
168	C₂×#340	D₄₇	1	1	yes	characteristic
169	C₂×#336	—	47	1	no	—
170	C₄⋊#95	—	47	2	no	—
171	C₂×C₄₇	#1445	1	1	yes	characteristic
172	C₂×C₄₇	#1921	1	4	yes	—
173	C₂×C₄₇	#1969	1	1	yes	characteristic
174	C₂×C₄₇	#1972	1	1	yes	characteristic
175	C₂×C₄₇	#2076	1	2	yes	—
176	C₂×C₄₇	#2087	1	2	yes	—
177	C₂×C₄₇	#2092	1	2	yes	—
178	C₂×C₄₇	#2095	1	2	yes	—
179	C₂³×C₈	—	47	1	no	—
180	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	—	47	2	no	—
182	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
183	C₂²×C₄²	—	47	1	no	—
184	C₂×#107	—	47	4	no	—
185	C₂×#107	—	94	2	no	—
186	C₂×#107	—	94	2	no	—
187	C₂×#95	—	94	4	no	—
188	#340	—	2	2	no	—
189	#340	—	94	4	no	—
190	#336	—	94	2	no	—
191	C₄₇	#11377	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	—	47	2	no	—
194	C₂³×C₄	—	94	2	no	—
195	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	47	2	no	—
199	C₂²×C₈	—	94	1	no	—
200	C₂²×C₈	—	94	2	no	—
201	C₂²×C₈	—	94	2	no	—
202	C₂²×C₈	—	94	1	no	—
203	C₂²×C₈	C₄×D₄₇	1	2	yes	—
204	C₂²×C₈	D₁₈₈	1	2	yes	—
205	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	47	4	no	—
212	C₂×C₄⋊C₄	—	94	2	no	—
213	C₂×C₄⋊C₄	—	94	2	no	—
214	C₂×C₄⋊C₄	—	94	2	no	—
215	C₂×C₄⋊C₄	—	94	2	no	—
216	C₂×C₄⋊C₄	—	94	2	no	—
217	C₂×C₄⋊C₄	Dic₉₄	1	4	yes	—
218	C₂×C₄²	—	94	1	no	—
219	C₂×C₄²	—	94	2	no	—
220	C₂×C₄²	—	94	2	no	—
221	C₂×C₄²	—	94	1	no	—
222	#107	—	94	8	no	—
223	#95	—	188	8	no	—
224	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	47	4	no	—
231	C₂²×C₄	—	94	2	no	—
232	C₂²×C₄	—	94	4	no	—
233	C₂²×C₄	—	94	2	no	—
234	C₂²×C₄	—	94	4	no	—
235	C₂²×C₄	—	94	4	no	—
236	C₂²×C₄	—	188	2	no	—
237	C₂²×C₄	—	188	2	no	—
238	C₂²×C₄	—	188	4	no	—
239	C₂²×C₄	—	188	4	no	—
240	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
243	C₂²×C₄	#10625350	1	2	yes	—
244	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
245	C₂²×C₄	#10625352	1	2	yes	—
246	C₂²×C₄	#10625352	1	2	yes	—
247	C₂²×C₄	#10625354	1	1	yes	characteristic
248	C₂²×C₄	#10625354	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	94	2	no	—
255	C₂×C₈	—	94	4	no	—
256	C₂×C₈	—	188	1	no	—
257	C₂×C₈	—	188	1	no	—
258	C₂×C₈	—	188	2	no	—
259	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	94	4	no	—
265	C₄⋊C₄	—	94	4	no	—
266	C₄⋊C₄	—	188	2	no	—
267	C₄⋊C₄	—	188	2	no	—
268	C₄²	—	188	1	no	—
269	C₄²	—	188	1	no	—
270	C₄²	—	188	2	no	—
271	C₂³	#13093986	1	1	yes	characteristic
272	C₂³	#13093999	1	1	yes	characteristic
273	C₂³	#13094001	1	2	yes	—
274	C₂³	#13094006	1	2	yes	—
275	C₂³	#13094020	1	1	yes	characteristic
276	C₂³	Q₁₆×D₄₇	1	2	yes	—
277	C₂³	#13094043	1	2	yes	—
278	C₂³	#13094051	1	4	yes	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	—	8	2	no	—
283	C₂×C₄	—	8	4	no	—
284	C₂×C₄	—	94	4	no	—
285	C₂×C₄	—	94	8	no	—
286	C₂×C₄	—	376	2	no	—
287	C₂×C₄	—	376	2	no	—
288	C₂×C₄	—	376	4	no	—
289	C₂×C₄	—	188	8	no	—
290	C₂×C₄	—	188	4	no	—
291	C₂×C₄	—	188	2	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	4	no	—
294	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
295	C₂×C₄	#13093980	1	2	yes	—
296	C₂×C₄	#13093986	1	1	yes	characteristic
297	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
298	C₂×C₄	#13093999	1	1	yes	characteristic
299	C₂×C₄	#13094001	1	2	yes	—
300	C₂×C₄	#13094003	1	2	yes	—
301	C₂×C₄	#13094005	1	2	yes	—
302	C₂×C₄	#13094006	1	2	yes	—
303	C₂×C₄	#13094008	1	1	yes	characteristic
304	C₂×C₄	#13094008	1	1	yes	characteristic
305	C₂×C₄	#13094012	1	1	yes	characteristic
306	C₂×C₄	#13094013	1	1	yes	characteristic
307	C₂×C₄	#13094015	1	2	yes	—
308	C₂×C₄	#13094015	1	2	yes	—
309	C₂×C₄	#13094019	1	2	yes	—
310	C₂×C₄	#13094019	1	2	yes	—
311	C₂×C₄	#13094024	1	1	yes	characteristic
312	C₈	—	2	4	no	—
313	C₈	—	188	2	no	—
314	C₂²	#15689006	1	2	yes	—
315	C₂²	#15689024	1	1	yes	characteristic
316	C₂²	#15689050	1	1	yes	characteristic
317	C₂²	#15689055	1	1	yes	characteristic
318	C₂²	#15689059	1	2	yes	—
319	C₂²	#15689217	1	2	yes	—
320	C₂²	#15689234	1	2	yes	—
321	C₂²	#15689365	1	2	yes	—
322	C₂²	#15689377	1	2	yes	—
323	C₂²	#15689416	1	4	yes	—
324	C₂²	#15689421	1	4	yes	—
325	C₂²	#15689445	1	4	yes	—
326	C₂²	#15689457	1	4	yes	—
327	C₂²	#15689501	1	1	yes	characteristic
328	C₂²	#15689506	1	1	yes	characteristic
329	C₂²	#15689509	1	2	yes	—
330	C₄	—	8	4	no	—
331	C₄	—	376	4	no	—
332	C₄	—	188	4	no	—
333	C₄	#15688984	1	1	yes	characteristic
334	C₄	#15689006	1	2	yes	—
335	C₄	#15689029	1	1	yes	characteristic
336	C₄	#15689050	1	1	yes	characteristic
337	C₄	#15689055	1	1	yes	characteristic
338	C₄	#15689062	1	2	yes	—
339	C₂	#20241657	1	1	yes	characteristic
340	C₂	#20243979	1	4	yes	—
341	C₂	#20244247	1	1	yes	characteristic
342	C₂	#20244262	1	1	yes	characteristic
343	C₂	#20244818	1	2	yes	—
344	C₂	#20244883	1	2	yes	—
345	C₂	#20244907	1	2	yes	—
346	C₂	#20244924	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#340	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#340	D₄₇
hypocenter	C₄₇	#11377
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
11. C₂×#13093958	C₄	semidirect
191. C₄₇	#11377	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13093958	2
C₂×#15688885	2
C₂×#15688877	2
#20241617	2
C₂×C₄₇×#340	2
#11377	47