Subgroups — #26185758

Order 12032.

714 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252497	C₂	1	2	yes	maximal, complemented
2	C₄×#13094046	C₂	1	1	yes	maximal, characteristic
3	#20242199	C₂	1	4	yes	maximal, complemented
4	#20242194	C₂	1	2	yes	maximal, complemented
5	#20242182	C₂	1	2	yes	maximal, complemented
6	#20241773	C₂	1	2	yes	maximal, complemented
7	#20241671	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	C₄	1	1	yes	characteristic
10	C₄×#10625350	C₂²	1	2	yes	—
11	#15689247	C₂²	1	4	yes	—
12	C₄×Dic₁₈₈	C₄	1	1	yes	characteristic
13	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
14	C₄×#10625325	C₄	1	2	yes	—
15	C₄×#10625325	C₂²	1	2	yes	—
16	C₄×D₃₇₆	C₄	1	1	yes	characteristic
17	C₄×D₃₇₆	C₂²	1	1	yes	characteristic
18	#15689188	C₂²	1	4	yes	—
19	C₈×C₂²⋊D₄₇	—	2	4	no	—
20	#15689176	C₂²	1	1	yes	characteristic
21	#15689176	C₂²	1	1	yes	characteristic
22	#15689175	—	2	2	no	—
23	#15689161	C₄	1	1	yes	characteristic
24	C₄⋊C₈×D₄₇	—	2	2	no	—
25	#15689120	C₄	1	2	yes	—
26	#15688853	C₂²	1	4	yes	—
27	#15688850	C₂²	1	4	yes	—
28	#15688844	C₂²	1	2	yes	—
29	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	C₄₇×#335	C₂²	1	2	yes	—
33	C₄₇×#335	C₂²	1	2	yes	—
34	C₂×#10625350	C₂×C₄	1	2	yes	—
35	#13094046	C₈	1	8	yes	complemented
36	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#10625325	C₂×C₄	1	2	yes	—
38	C₂×D₃₇₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₄₇	—	2	4	no	—
40	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
41	C₄×Dic₉₄	C₂³	1	2	yes	—
42	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
43	C₄×D₁₈₈	C₂³	1	2	yes	—
44	C₄×C₂²⋊D₄₇	—	2	2	no	—
45	C₄×C₂²⋊D₄₇	—	2	2	no	—
46	#13093983	C₂×C₄	1	2	yes	—
47	#13093981	—	2	2	no	—
48	C₄²×D₄₇	—	2	2	no	—
49	#13093958	C₂×C₄	1	1	yes	characteristic
50	#13093958	C₂×C₄	1	1	yes	characteristic
51	#13093956	C₂×C₄	1	2	yes	—
52	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
53	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
54	C₄⋊(C₄₇⋊C₈)	D₄	1	1	yes	characteristic
55	Dic₄₇⋊C₈	—	2	4	no	—
56	#13093949	C₂×C₄	1	2	yes	—
57	#13093949	C₂×C₄	1	2	yes	—
58	#13093946	C₂×C₄	1	2	yes	—
59	#13093946	C₂×C₄	1	2	yes	—
60	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
61	C₂²⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
62	(C₂×D₄₇)⋊C₈	—	2	4	no	—
63	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
64	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
65	C₈×Dic₄₇	—	2	4	no	—
66	C₂²×C₈×C₄₇	—	2	2	no	—
67	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₄₇	—	2	2	no	—
71	C₄⋊C₈×C₄₇	C₂³	1	4	yes	—
72	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₄₇	—	2	2	no	—
80	#10625350	—	2	4	no	—
81	#10625350	C₂×C₈	1	8	yes	complemented
82	C₂×D₁₈₈	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₄₇	—	2	2	no	—
84	C₂×C₄×D₄₇	—	2	2	no	—
85	C₂×C₄×D₄₇	—	2	2	no	—
86	C₂×Dic₉₄	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	4	no	—
89	C₂×C₈×C₄₇	—	8	2	no	—
90	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	complemented
91	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₄₇	#41	1	1	yes	characteristic
98	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₄₇	—	2	2	no	—
100	(C₂×D₄₇)⋊C₄	—	2	2	no	—
101	(C₂×D₄₇)⋊C₄	—	2	2	no	—
102	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₄₇	C₂²×C₄	1	2	yes	—
104	Dic₄₇⋊C₄	—	2	2	no	—
105	Dic₄₇⋊C₄	—	2	2	no	—
106	C₄×Dic₄₇	—	2	2	no	—
107	C₄×Dic₄₇	—	2	2	no	—
108	C₄×Dic₄₇	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	2	2	no	—
110	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
111	C₂×C₄₇⋊C₈	C₂×D₄	1	2	yes	—
112	Dic₁₈₈	C₂×C₈	1	4	yes	complemented
113	D₃₇₆	C₂×C₈	1	4	yes	complemented
114	#10625325	C₂×C₈	1	8	yes	complemented
115	C₈×D₄₇	—	8	4	no	—
116	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₇	—	2	2	no	—
118	C₂×C₄×C₄₇	—	2	1	no	—
119	C₂×C₄×C₄₇	—	2	1	no	—
120	C₂×C₄×C₄₇	—	4	1	no	—
121	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₄₇	#131	1	2	yes	—
126	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₄₇	—	2	8	no	—
130	C₂×Dic₄₇	—	2	2	no	—
131	C₂×Dic₄₇	—	2	2	no	—
132	C₂×Dic₄₇	—	2	2	no	—
133	D₁₈₈	—	2	2	no	—
134	D₁₈₈	C₂²×C₈	1	4	yes	—
135	C₄×D₄₇	—	2	8	no	—
136	C₄×D₄₇	—	4	2	no	—
137	C₄×D₄₇	—	4	2	no	—
138	C₄×D₄₇	—	8	2	no	—
139	Dic₉₄	—	2	2	no	—
140	Dic₉₄	C₂²×C₈	1	4	yes	—
141	C₈×C₄₇	—	2	1	no	—
142	C₈×C₄₇	—	2	1	no	—
143	C₈×C₄₇	—	4	4	no	—
144	C₈×C₄₇	C₂²×C₈	1	2	yes	—
145	C₈×C₄₇	C₂²×C₈	1	2	yes	—
146	C₄₇⋊C₈	—	2	2	no	—
147	C₄₇⋊C₈	—	2	2	no	—
148	C₄₇⋊C₈	—	4	2	no	—
149	#17857	—	47	1	no	maximal, Sylow
150	C₂²×C₄₇	—	2	1	no	—
151	C₂²×C₄₇	—	2	1	no	—
152	C₂²×C₄₇	C₈×D₄	1	2	yes	—
153	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
154	C₂×D₄₇	—	2	4	no	—
155	C₂×D₄₇	—	4	2	no	—
156	C₂×D₄₇	—	4	2	no	—
157	C₄×C₄₇	—	2	1	no	—
158	C₄×C₄₇	—	2	1	no	—
159	C₄×C₄₇	—	2	1	no	—
160	C₄×C₄₇	C₈×D₄	1	2	yes	—
161	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
162	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
163	C₄×C₄₇	#576	1	1	yes	characteristic
164	C₄×C₄₇	#578	1	1	yes	characteristic
165	Dic₄₇	—	2	4	no	—
166	Dic₄₇	—	4	2	no	—
167	Dic₄₇	—	4	2	no	—
168	#2988	—	47	2	no	—
169	C₄×#135	—	47	1	no	—
170	SD₁₆⋊C₈	—	47	4	no	—
171	C₈×Q₁₆	—	47	2	no	—
172	C₈×D₈	—	47	2	no	—
173	C₂×#335	D₄₇	1	1	yes	characteristic, complemented
174	#1477	—	47	2	no	—
175	#1450	—	47	1	no	—
176	C₂×C₄₇	—	2	2	no	—
177	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
178	C₂×C₄₇	#2951	1	1	yes	characteristic
179	C₂×C₄₇	#2957	1	1	yes	characteristic
180	D₄₇	—	4	4	no	—
181	C₂×#135	—	47	1	no	—
182	C₄×#41	—	47	2	no	—
183	C₈×Q₈	—	47	4	no	—
184	C₄×Q₁₆	—	47	1	no	—
185	C₄×Q₁₆	—	47	1	no	—
186	C₄×SD₁₆	—	47	2	no	—
187	C₄×SD₁₆	—	47	2	no	—
188	C₄×D₈	—	47	1	no	—
189	C₄×D₈	—	47	1	no	—
190	C₈×D₄	—	47	4	no	—
191	C₈×D₄	—	94	4	no	—
192	#431	—	47	1	no	—
193	#431	—	47	1	no	—
194	#431	—	94	2	no	—
195	#427	—	47	1	no	—
196	C₂×C₄⋊C₈	—	94	2	no	—
197	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
198	#416	—	47	2	no	—
199	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
200	#335	C₂×D₄₇	1	2	yes	—
201	#335	C₂×D₄₇	1	2	yes	—
202	Q₈⋊C₈	—	47	4	no	—
203	D₄⋊C₈	—	47	4	no	—
204	#322	—	47	2	no	—
205	C₈²	—	47	2	no	—
206	C₄₇	#17857	1	1	yes	characteristic, complemented, Sylow
207	C₂×#41	—	47	2	no	—
208	#135	—	47	8	no	—
209	C₂×Q₁₆	—	47	1	no	—
210	C₂×SD₁₆	—	47	2	no	—
211	C₂×D₈	—	47	1	no	—
212	C₂²×C₈	—	2	2	no	—
213	C₂²×C₈	—	94	4	no	—
214	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
215	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
216	C₄×Q₈	—	47	2	no	—
217	C₄×Q₈	—	47	2	no	—
218	C₄×D₄	—	47	2	no	—
219	C₄×D₄	—	47	2	no	—
220	C₄×D₄	—	94	2	no	—
221	C₄×D₄	—	94	2	no	—
222	#117	—	47	2	no	—
223	#117	—	94	2	no	—
224	C₂×C₄²	—	94	2	no	—
225	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
226	#107	—	47	1	no	—
227	#107	—	47	1	no	—
228	#106	—	47	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	47	1	no	—
231	C₄⋊C₈	—	47	1	no	—
232	C₄⋊C₈	—	94	4	no	—
233	C₄⋊C₈	—	94	1	no	—
234	C₄⋊C₈	C₂²×D₄₇	1	4	yes	—
235	#103	—	47	2	no	—
236	#103	—	47	2	no	—
237	#102	—	47	2	no	—
238	#102	—	47	2	no	—
239	C₂²⋊C₈	—	47	2	no	—
240	C₂²⋊C₈	—	94	4	no	—
241	C₂²⋊C₈	—	94	1	no	—
242	C₄×C₈	—	47	2	no	—
243	C₄×C₈	—	47	2	no	—
244	C₄×C₈	—	94	4	no	—
245	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
246	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
249	#41	—	47	8	no	—
250	#41	—	94	4	no	—
251	C₂×Q₈	—	47	2	no	—
252	C₂×D₄	—	47	2	no	—
253	C₂×D₄	—	94	2	no	—
254	C₂²×C₄	—	94	2	no	—
255	C₂²×C₄	—	94	2	no	—
256	C₂²×C₄	—	94	2	no	—
257	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
259	C₂²×C₄	#10625354	1	1	yes	characteristic
260	Q₁₆	—	47	4	no	—
261	SD₁₆	—	47	8	no	—
262	D₈	—	47	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	8	2	no	—
266	C₂×C₈	—	47	2	no	—
267	C₂×C₈	—	47	2	no	—
268	C₂×C₈	—	94	2	no	—
269	C₂×C₈	—	376	4	no	—
270	C₂×C₈	C₈×D₄₇	1	4	yes	complemented
271	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
272	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
275	C₄⋊C₄	—	47	2	no	—
276	C₄⋊C₄	—	47	2	no	—
277	C₄⋊C₄	—	94	2	no	—
278	C₄⋊C₄	—	94	2	no	—
279	C₂²⋊C₄	—	94	2	no	—
280	C₂²⋊C₄	—	94	2	no	—
281	C₂²⋊C₄	—	94	2	no	—
282	C₄²	—	94	2	no	—
283	C₄²	—	94	2	no	—
284	C₄²	—	94	2	no	—
285	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
286	C₄²	D₄×D₄₇	1	1	yes	characteristic
287	C₄²	#10625354	1	1	yes	characteristic
288	C₄²	C₂³×D₄₇	1	1	yes	characteristic
289	C₂³	—	94	2	no	—
290	C₂³	#13093986	1	1	yes	characteristic
291	Q₈	—	47	4	no	—
292	Q₈	—	94	2	no	—
293	D₄	—	47	4	no	—
294	D₄	—	94	8	no	—
295	D₄	—	94	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	94	2	no	—
300	C₂×C₄	—	94	8	no	—
301	C₂×C₄	—	94	2	no	—
302	C₂×C₄	—	94	2	no	—
303	C₂×C₄	—	376	2	no	—
304	C₂×C₄	—	188	2	no	—
305	C₂×C₄	—	188	2	no	—
306	C₂×C₄	#13093986	1	1	yes	characteristic
307	C₂×C₄	#13093986	1	1	yes	characteristic
308	C₂×C₄	#13093986	1	1	yes	characteristic
309	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
310	C₂×C₄	#13094032	1	2	yes	—
311	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
313	C₂×C₄	C₂×#10625354	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	4	no	—
317	C₈	—	94	2	no	—
318	C₈	—	94	2	no	—
319	C₈	—	188	2	no	—
320	C₈	C₂×C₈×D₄₇	1	2	yes	—
321	C₈	C₂×C₈×D₄₇	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	94	4	no	—
325	C₂²	—	188	2	no	—
326	C₂²	—	188	2	no	—
327	C₂²	#15689188	1	2	yes	—
328	C₂²	C₂×#13093986	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	94	4	no	—
333	C₄	—	188	2	no	—
334	C₄	—	188	2	no	—
335	C₄	#15689188	1	2	yes	—
336	C₄	C₂²×C₈×D₄₇	1	1	yes	characteristic
337	C₄	C₂×#13094032	1	1	yes	characteristic
338	C₄	#15689924	1	1	yes	characteristic
339	C₄	#15689936	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	188	4	no	—
342	C₂	C₂×#15689188	1	1	yes	characteristic
343	C₂	#20252262	1	1	yes	characteristic
344	C₂	#20252304	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₇
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle	C₂²×C₄₇	C₂×C₄×D₄
prime radical	C₂×#13094046	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#17857
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
1. #20252497	C₂	semidirect
3. #20242199	C₂	semidirect
4. #20242194	C₂	semidirect
5. #20242182	C₂	semidirect
6. #20241773	C₂	semidirect
7. #20241671	C₂	semidirect
8. C₂×C₄₇×#335	C₂	semidirect
35. #13094046	C₈	semidirect
81. #10625350	C₂×C₈	semidirect
90. C₂×C₈×C₄₇	C₂×C₈	semidirect
112. Dic₁₈₈	C₂×C₈	semidirect
113. D₃₇₆	C₂×C₈	semidirect
114. #10625325	C₂×C₈	semidirect
173. C₂×#335	D₄₇	semidirect
206. C₄₇	#17857	semidirect
270. C₂×C₈	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252497	2
C₄×#13094046	2
#20242199	2
#20242194	2
#20242182	2
#20241773	2
#20241671	2
C₂×C₄₇×#335	2
#17857	47