Subgroups — #26235161

Order 12032.

730 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	C₂×#15689253	C₂	1	1	yes	maximal, characteristic
2	#20246346	C₂	1	2	yes	maximal
3	#20244159	C₂	1	2	yes	maximal
4	#20244156	C₂	1	2	yes	maximal
5	#20244145	C₂	1	2	yes	maximal
6	#20243555	C₂	1	2	yes	maximal
7	#20243552	C₂	1	1	yes	maximal, characteristic
8	#20243287	C₂	1	2	yes	maximal
9	C₄×#107×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	#15689253	C₄	1	8	yes	complemented
12	C₂×#13093958	C₂²	1	1	yes	characteristic
13	C₂×#13093958	C₂²	1	1	yes	characteristic
14	C₂×#13093957	C₂²	1	2	yes	—
15	C₂×#13093956	C₂²	1	2	yes	—
16	C₂×#13093955	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₇⋊C₈	C₂²	1	2	yes	—
19	C₂×#13093934	C₂²	1	2	yes	—
20	#15689027	—	2	2	no	—
21	#15689007	—	2	2	no	—
22	#15689006	—	2	2	no	—
23	#15688992	C₂²	1	2	yes	—
24	#15688989	C₂²	1	2	yes	—
25	#15688988	C₂²	1	2	yes	—
26	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
27	C₄⋊C₄×Dic₄₇	C₂²	1	2	yes	—
28	C₄×Dic₄₇⋊C₄	—	2	2	no	—
29	#15688885	C₂²	1	2	yes	—
30	#15688885	C₂²	1	2	yes	—
31	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
32	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
35	C₄₇×#340	C₂²	1	2	yes	—
36	C₂²×Dic₉₄	D₄	1	1	yes	characteristic
37	C₂²×C₄₇⋊C₈	C₂³	1	2	yes	—
38	C₄×Dic₉₄	C₂×C₄	1	4	yes	—
39	C₄×Dic₉₄	Q₈	1	4	yes	—
40	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇⋊C₄	—	2	2	no	—
44	C₂×Dic₄₇⋊C₄	—	2	2	no	—
45	C₂×Dic₄₇⋊C₄	D₄	1	2	yes	—
46	C₂×C₄×Dic₄₇	—	2	2	no	—
47	C₂×C₄×Dic₄₇	—	2	2	no	—
48	C₂×C₄×Dic₄₇	C₂³	1	2	yes	—
49	#13093958	—	4	1	no	—
50	#13093958	—	4	1	no	—
51	#13093957	—	4	2	no	—
52	#13093956	—	2	4	no	—
53	#13093955	—	4	2	no	—
54	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
55	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
56	#13093934	C₂×C₄	1	8	yes	—
57	C₄₇⋊#95	—	2	2	no	—
58	C₄₇⋊#95	—	2	2	no	—
59	C₄₇⋊#95	—	2	2	no	—
60	C₄₇⋊#95	—	2	2	no	—
61	C₄₇⋊#95	—	2	2	no	—
62	C₄₇⋊#95	—	2	2	no	—
63	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₇	—	2	2	no	—
68	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
69	#107×C₄₇	—	2	2	no	—
70	#107×C₄₇	Q₈	1	4	yes	complemented
71	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
72	#95×C₄₇	—	2	2	no	—
73	C₂²×C₄×C₄₇	—	2	2	no	—
74	C₂²×C₄×C₄₇	—	2	2	no	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
78	C₂²×Dic₄₇	—	2	2	no	—
79	C₂²×Dic₄₇	—	2	2	no	—
80	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
81	C₂²×Dic₄₇	C₂×D₄	1	2	yes	—
82	C₂×Dic₉₄	—	4	1	no	—
83	C₂×Dic₉₄	—	4	1	no	—
84	C₂×Dic₉₄	C₄⋊C₄	1	4	yes	—
85	C₂×C₈×C₄₇	—	2	1	no	—
86	C₂×C₈×C₄₇	—	2	1	no	—
87	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₄₇	#41	1	2	yes	—
91	C₄⋊C₄×C₄₇	—	2	2	no	—
92	C₄⋊C₄×C₄₇	—	2	2	no	—
93	C₄⋊C₄×C₄₇	—	2	2	no	—
94	C₄⋊C₄×C₄₇	C₂×Q₈	1	4	yes	—
95	C₄²×C₄₇	—	8	2	no	—
96	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
98	C₄⋊Dic₄₇	—	2	2	no	—
99	C₄⋊Dic₄₇	—	2	2	no	—
100	C₄⋊Dic₄₇	—	2	2	no	—
101	C₄⋊Dic₄₇	—	2	2	no	—
102	C₄⋊Dic₄₇	C₄⋊C₄	1	4	yes	—
103	Dic₄₇⋊C₄	—	4	4	no	—
104	Dic₄₇⋊C₄	—	8	2	no	—
105	Dic₄₇⋊C₄	C₄⋊C₄	1	8	yes	—
106	C₄×Dic₄₇	—	4	4	no	—
107	C₄×Dic₄₇	—	8	2	no	—
108	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
109	C₄×Dic₄₇	C₂×Q₈	1	4	yes	—
110	C₂×C₄₇⋊C₈	—	2	2	no	—
111	C₂×C₄₇⋊C₈	—	2	2	no	—
112	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
113	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	—	2	1	no	—
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	4	no	—
119	C₂×C₄×C₄₇	—	4	2	no	—
120	C₂×C₄×C₄₇	—	4	2	no	—
121	C₂×C₄×C₄₇	—	4	2	no	—
122	C₂×C₄×C₄₇	—	4	2	no	—
123	C₂×C₄×C₄₇	—	4	2	no	—
124	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×Dic₄₇	—	2	2	no	—
133	C₂×Dic₄₇	—	2	2	no	—
134	C₂×Dic₄₇	—	2	2	no	—
135	C₂×Dic₄₇	—	2	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	C₂×Dic₄₇	—	4	2	no	—
141	C₂×Dic₄₇	—	4	2	no	—
142	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
144	Dic₉₄	—	4	4	no	—
145	C₈×C₄₇	—	2	2	no	—
146	C₈×C₄₇	C₄×Q₈	1	4	yes	—
147	C₄₇⋊C₈	—	2	4	no	—
148	#22731	—	47	1	no	maximal, Sylow
149	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₄₇	D₄×Q₈	1	1	yes	characteristic
151	C₂²×C₄₇	#550	1	1	yes	characteristic
152	C₂²×C₄₇	#575	1	1	yes	characteristic
153	C₂²×C₄₇	#575	1	1	yes	characteristic
154	C₂²×C₄₇	#576	1	1	yes	characteristic
155	C₂²×C₄₇	#578	1	1	yes	characteristic
156	C₄×C₄₇	—	2	2	no	—
157	C₄×C₄₇	—	2	2	no	—
158	C₄×C₄₇	—	4	4	no	—
159	C₄×C₄₇	—	8	2	no	—
160	C₄×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₄×C₄₇	#520	1	1	yes	characteristic
162	C₄×C₄₇	#557	1	1	yes	characteristic
163	C₄×C₄₇	Q₈²	1	1	yes	characteristic
164	Dic₄₇	—	2	4	no	—
165	Dic₄₇	—	2	4	no	—
166	Dic₄₇	—	8	2	no	—
167	Dic₄₇	—	8	2	no	—
168	C₂×#446	—	47	1	no	—
169	#2355	—	47	2	no	—
170	#1949	—	47	2	no	—
171	#1949	—	47	2	no	—
172	#1947	—	47	2	no	—
173	#1847	—	47	2	no	—
174	#1846	—	47	1	no	—
175	#1776	—	47	2	no	—
176	C₄×#107	D₄₇	1	1	yes	characteristic
177	C₂×C₄₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄₇	#2911	1	1	yes	characteristic
179	C₂×C₄₇	#2913	1	1	yes	characteristic
180	C₂×C₄₇	#3389	1	1	yes	characteristic
181	C₂×C₄₇	#3390	1	1	yes	characteristic
182	C₂×C₄₇	#3402	1	1	yes	characteristic
183	C₂×C₄₇	#3403	1	1	yes	characteristic
184	C₂×C₄×Q₈	—	47	1	no	—
185	#446	—	47	8	no	—
186	C₂×#107	—	47	1	no	—
187	C₂×#107	—	47	1	no	—
188	C₂×#107	—	47	2	no	—
189	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
190	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
191	C₂×#106	—	47	2	no	—
192	C₂×#106	—	47	2	no	—
193	C₂×C₄⋊C₈	—	47	2	no	—
194	C₂×C₄⋊C₈	—	47	1	no	—
195	C₂×#97	—	47	2	no	—
196	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
197	#391	—	94	2	no	—
198	#387	—	94	2	no	—
199	#387	—	94	2	no	—
200	#384	—	47	2	no	—
201	#382	—	47	2	no	—
202	#382	—	47	2	no	—
203	C₄×C₄⋊C₄	—	47	2	no	—
204	C₄×C₄⋊C₄	—	47	2	no	—
205	C₄×C₄⋊C₄	—	94	2	no	—
206	C₄×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
207	#340	—	47	2	no	—
208	#340	—	47	2	no	—
209	#340	C₂×D₄₇	1	2	yes	—
210	C₄₇	#22731	1	1	yes	characteristic, complemented, Sylow
211	C₂²×Q₈	—	47	1	no	—
212	C₂²×C₈	—	47	2	no	—
213	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
215	C₄×Q₈	—	47	4	no	—
216	C₄×Q₈	—	47	4	no	—
217	C₂×C₄⋊C₄	—	47	2	no	—
218	C₂×C₄⋊C₄	—	47	1	no	—
219	C₂×C₄⋊C₄	—	47	2	no	—
220	C₂×C₄⋊C₄	—	47	2	no	—
221	C₂×C₄⋊C₄	—	94	2	no	—
222	C₂×C₄⋊C₄	—	94	2	no	—
223	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
224	C₂×C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
225	C₂×C₄²	—	2	2	no	—
226	C₂×C₄²	—	47	2	no	—
227	C₂×C₄²	—	94	2	no	—
228	C₂×C₄²	—	94	2	no	—
229	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
230	#107	—	2	2	no	—
231	#107	—	188	2	no	—
232	#107	—	188	1	no	—
233	#107	—	188	1	no	—
234	#107	Dic₉₄	1	4	yes	complemented
235	#106	—	94	4	no	—
236	#106	—	188	2	no	—
237	C₄⋊C₈	—	47	8	no	—
238	C₄⋊C₈	—	47	4	no	—
239	#97	—	47	8	no	—
240	C₄×C₈	C₄×D₄₇	1	4	yes	—
241	#95	—	2	2	no	—
242	#95	—	94	2	no	—
243	#95	—	94	2	no	—
244	#95	—	94	2	no	—
245	#95	—	94	2	no	—
246	#95	—	94	2	no	—
247	#95	—	94	2	no	—
248	C₂×Q₈	—	47	4	no	—
249	C₂×Q₈	—	188	1	no	—
250	C₂×Q₈	—	188	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	47	2	no	—
254	C₂²×C₄	—	47	2	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	—	94	2	no	—
257	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	47	4	no	—
263	C₂×C₈	—	94	2	no	—
264	C₂×C₈	—	94	2	no	—
265	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
267	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
268	C₂×C₈	#10625350	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	47	8	no	—
273	C₄⋊C₄	—	47	4	no	—
274	C₄⋊C₄	—	94	2	no	—
275	C₄⋊C₄	—	94	2	no	—
276	C₄⋊C₄	—	94	2	no	—
277	C₄⋊C₄	—	94	2	no	—
278	C₄⋊C₄	—	376	2	no	—
279	C₄⋊C₄	—	188	4	no	—
280	C₄⋊C₄	C₂×Dic₉₄	1	4	yes	—
281	C₄²	—	8	2	no	—
282	C₄²	—	47	4	no	—
283	C₄²	—	47	4	no	—
284	C₄²	—	376	2	no	—
285	C₄²	—	188	4	no	—
286	C₄²	C₂×C₄×D₄₇	1	2	yes	—
287	C₄²	Q₈×D₄₇	1	2	yes	—
288	C₂³	C₂×D₄×D₄₇	1	1	yes	characteristic
289	Q₈	—	188	4	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	4	no	—
295	C₂×C₄	—	4	2	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	—	47	4	no	—
301	C₂×C₄	—	47	4	no	—
302	C₂×C₄	—	94	2	no	—
303	C₂×C₄	—	94	2	no	—
304	C₂×C₄	—	94	2	no	—
305	C₂×C₄	—	94	2	no	—
306	C₂×C₄	—	188	2	no	—
307	C₂×C₄	—	188	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	C₄⋊C₄×D₄₇	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
314	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
317	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
318	C₂×C₄	#13094070	1	1	yes	characteristic
319	C₂×C₄	#13094071	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	94	4	no	—
322	C₈	C₄×Dic₉₄	1	4	yes	—
323	C₂²	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
324	C₂²	D₄×Dic₉₄	1	1	yes	characteristic
325	C₂²	#15689794	1	1	yes	characteristic
326	C₂²	#15689920	1	1	yes	characteristic
327	C₂²	#15689920	1	1	yes	characteristic
328	C₂²	#15689924	1	1	yes	characteristic
329	C₂²	#15689936	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	4	no	—
333	C₄	—	8	2	no	—
334	C₄	—	94	4	no	—
335	C₄	—	94	4	no	—
336	C₄	—	376	2	no	—
337	C₄	—	376	2	no	—
338	C₄	C₂×C₄×Dic₉₄	1	1	yes	characteristic
339	C₄	#15689603	1	1	yes	characteristic
340	C₄	#15689851	1	1	yes	characteristic
341	C₄	Q₈×Dic₉₄	1	1	yes	characteristic
342	C₂	C₄⋊C₄×Dic₉₄	1	1	yes	characteristic
343	C₂	#20251976	1	1	yes	characteristic
344	C₂	#20251984	1	1	yes	characteristic
345	C₂	#20256344	1	1	yes	characteristic
346	C₂	#20256353	1	1	yes	characteristic
347	C₂	#20256465	1	1	yes	characteristic
348	C₂	#20256474	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄×#107×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₄×#107	D₄₇
hypocenter	C₄₇	#22731
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #15689253	C₄	semidirect
70. #107×C₄₇	Q₈	semidirect
210. C₄₇	#22731	semidirect
234. #107	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689253	2
#20246346	2
#20244159	2
#20244156	2
#20244145	2
#20243555	2
#20243552	2
#20243287	2
C₄×#107×C₄₇	2
#22731	47