Subgroups — #26371680

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689553	C₂	1	2	yes	maximal, complemented
3	#20253812	C₂	1	2	yes	maximal, complemented
4	#20245650	C₂	1	1	yes	maximal, characteristic, complemented
5	#20245648	C₂	1	2	yes	maximal, complemented
6	#20245646	C₂	1	1	yes	maximal, characteristic, complemented
7	#20245642	C₂	1	2	yes	maximal, complemented
8	#20245639	C₂	1	1	yes	maximal, characteristic, complemented
9	#20245636	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄₇×#508	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13094046	C₂²	1	2	yes	—
13	C₂×#13094044	—	2	2	no	—
14	#15689669	—	2	2	no	—
15	#15689555	—	2	4	no	—
16	#15689553	—	2	8	no	—
17	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
18	C₂×#13093970	C₂²	1	2	yes	—
19	C₂×#13093969	C₂²	1	2	yes	—
20	C₂×#13093968	C₂²	1	2	yes	—
21	C₂×#13093967	C₂²	1	2	yes	—
22	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
23	C₂×#13093965	C₂²	1	2	yes	—
24	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
25	#15689508	C₂²	1	2	yes	—
26	#15689506	C₂²	1	2	yes	—
27	#15689504	—	2	2	no	—
28	#15689501	C₂²	1	2	yes	—
29	#15689498	C₂²	1	2	yes	—
30	#15689494	—	2	2	no	—
31	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
36	C₂×#135×C₄₇	C₂²	1	2	yes	—
37	C₄₇×#508	—	2	4	no	—
38	C₂×Q₃₂×C₄₇	C₂²	1	1	yes	characteristic
39	C₂×SD₃₂×C₄₇	C₂²	1	2	yes	—
40	C₂×D₁₆×C₄₇	C₂²	1	1	yes	characteristic
41	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
42	C₂×#10625350	—	2	2	no	—
43	#13094046	—	2	4	no	—
44	#13094046	—	2	4	no	—
45	#13094044	—	4	8	no	—
46	C₂×Dic₁₈₈	C₂³	1	2	yes	—
47	C₂×#10625337	—	2	2	no	—
48	C₂×#10625325	—	2	2	no	—
49	C₂×#10625335	—	2	2	no	—
50	C₂×#10625336	—	2	2	no	—
51	C₂×D₃₇₆	C₂³	1	2	yes	—
52	C₂×#10625334	—	2	2	no	—
53	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
54	#13094022	—	2	2	no	—
55	#13094020	—	2	2	no	—
56	#13094013	—	2	2	no	—
57	#13094011	—	2	2	no	—
58	#13093996	—	4	4	no	—
59	C₄×C₂²⋊D₄₇	—	4	4	no	—
60	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
61	Dic₃₇₆	—	2	2	no	—
62	#13093970	—	2	4	no	—
63	#13093969	—	2	4	no	—
64	#13093968	—	2	4	no	—
65	#13093967	—	2	4	no	—
66	D₇₅₂	—	2	2	no	—
67	#13093965	—	2	4	no	—
68	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	complemented
69	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₈×Dic₄₇	D₄	1	1	yes	characteristic
72	C₈×Dic₄₇	D₄	1	1	yes	characteristic
73	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
74	C₈×Dic₄₇	C₂³	1	1	yes	characteristic
75	C₂×#41×C₄₇	—	2	2	no	—
76	#135×C₄₇	—	2	4	no	—
77	#135×C₄₇	—	2	4	no	—
78	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₄₇	—	2	2	no	—
80	C₂×D₈×C₄₇	C₂³	1	2	yes	—
81	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
82	Q₃₂×C₄₇	—	2	2	no	—
83	SD₃₂×C₄₇	—	2	4	no	—
84	D₁₆×C₄₇	—	2	2	no	—
85	C₂×C₁₆×C₄₇	D₄	1	4	yes	complemented
86	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
87	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
88	#41×C₄₇	—	4	4	no	—
89	#41×C₄₇	—	4	4	no	—
90	C₂×Q₈×C₄₇	—	2	2	no	—
91	C₂×D₄×C₄₇	—	2	2	no	—
92	C₂×D₄×C₄₇	—	4	2	no	—
93	C₂²×C₄×C₄₇	—	4	2	no	—
94	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₂²⋊D₄₇	—	4	2	no	—
96	C₂×C₂²⋊D₄₇	—	4	4	no	—
97	C₂²×Dic₄₇	D₈	1	2	yes	—
98	#10625350	—	4	4	no	—
99	#10625350	—	4	4	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×C₄×D₄₇	—	4	2	no	—
102	C₂×Dic₉₄	—	2	2	no	—
103	Q₁₆×C₄₇	—	2	2	no	—
104	Q₁₆×C₄₇	—	2	2	no	—
105	SD₁₆×C₄₇	—	4	4	no	—
106	D₈×C₄₇	—	2	2	no	—
107	D₈×C₄₇	—	2	2	no	—
108	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂²⋊Dic₄₇	—	4	4	no	—
113	#10625337	—	4	4	no	—
114	#10625336	—	4	4	no	—
115	#10625335	—	4	4	no	—
116	#10625334	—	4	4	no	—
117	(C₂×D₄₇)⋊C₄	—	4	4	no	—
118	Dic₄₇⋊C₄	—	4	4	no	—
119	C₄×Dic₄₇	D₈	1	2	yes	—
120	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₄₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
123	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	Dic₁₈₈	—	2	2	no	—
126	Dic₁₈₈	—	2	2	no	—
127	D₃₇₆	—	2	2	no	—
128	D₃₇₆	—	2	2	no	—
129	#10625325	—	4	4	no	—
130	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
131	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
132	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
133	C₄₇⋊C₁₆	C₂×D₄	1	2	yes	—
134	C₂³×C₄₇	—	4	2	no	—
135	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₄₇	—	4	2	no	—
137	Q₈×C₄₇	—	4	2	no	—
138	Q₈×C₄₇	—	4	2	no	—
139	D₄×C₄₇	—	4	2	no	—
140	D₄×C₄₇	—	4	2	no	—
141	D₄×C₄₇	—	8	4	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	8	4	no	—
144	C₂×C₄×C₄₇	#127	1	2	yes	—
145	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₄₇	—	8	4	no	—
149	C₂²⋊D₄₇	—	8	8	no	—
150	C₂×Dic₄₇	—	2	4	no	—
151	C₂×Dic₄₇	—	4	2	no	—
152	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
153	C₂×Dic₄₇	C₂×D₈	1	2	yes	—
154	D₁₈₈	—	4	2	no	—
155	D₁₈₈	—	4	2	no	—
156	C₄×D₄₇	—	8	4	no	—
157	Dic₉₄	—	4	2	no	—
158	Dic₉₄	—	4	2	no	—
159	C₈×C₄₇	#127	1	2	yes	—
160	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
162	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
163	C₄₇⋊C₈	C₂×D₈	1	2	yes	—
164	#33713	—	47	1	no	maximal, Sylow
165	C₂²×C₄₇	—	2	2	no	—
166	C₂²×C₄₇	—	8	2	no	—
167	C₂²×C₄₇	—	8	4	no	—
168	C₂²×C₄₇	#493	1	2	yes	—
169	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₄₇	—	8	2	no	—
171	C₂×D₄₇	—	8	4	no	—
172	C₄×C₄₇	—	8	2	no	—
173	C₄×C₄₇	#493	1	2	yes	—
174	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
175	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
176	Dic₄₇	—	2	2	no	—
177	Dic₄₇	—	8	2	no	—
178	Dic₄₇	#508	1	4	yes	complemented
179	C₂×#508	—	47	1	no	—
180	C₂×#508	—	47	2	no	—
181	C₂×#508	D₄₇	1	1	yes	characteristic, complemented
182	#3147	—	47	2	no	—
183	#2249	—	47	1	no	—
184	#2249	—	47	2	no	—
185	#2249	—	47	1	no	—
186	#2248	—	47	2	no	—
187	#2247	—	47	1	no	—
188	#2246	—	47	1	no	—
189	C₂×C₄×C₁₆	—	47	1	no	—
190	C₂×C₄₇	—	2	2	no	—
191	C₂×C₄₇	—	16	2	no	—
192	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
193	C₂×C₄₇	C₂×#508	1	2	yes	—
194	D₄₇	—	16	2	no	—
195	C₂×#135	—	47	2	no	—
196	C₂×#135	—	94	2	no	—
197	C₂×#135	C₂×D₄₇	1	2	yes	—
198	#532	—	94	2	no	—
199	#508	—	2	4	no	—
200	#508	—	94	8	no	—
201	#508	—	94	4	no	—
202	C₂×Q₃₂	—	47	2	no	—
203	C₂×Q₃₂	—	47	1	no	—
204	C₂×Q₃₂	C₂×D₄₇	1	1	yes	characteristic
205	C₂×SD₃₂	—	47	2	no	—
206	C₂×SD₃₂	—	47	2	no	—
207	C₂×SD₃₂	—	47	2	no	—
208	C₂×SD₃₂	C₂×D₄₇	1	2	yes	—
209	C₂×D₁₆	—	47	2	no	—
210	C₂×D₁₆	—	47	1	no	—
211	C₂×D₁₆	C₂×D₄₇	1	1	yes	characteristic
212	C₂²×C₁₆	—	47	1	no	—
213	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
214	#495	—	47	2	no	—
215	#495	—	47	2	no	—
216	#495	—	94	2	no	—
217	#494	—	47	2	no	—
218	#493	—	47	2	no	—
219	#492	—	94	2	no	—
220	C₂×C₄×C₈	—	47	1	no	—
221	C₄×C₁₆	—	47	1	no	—
222	C₄×C₁₆	—	47	1	no	—
223	C₄×C₁₆	—	47	1	no	—
224	C₄×C₁₆	—	47	1	no	—
225	C₄₇	#33713	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	2	2	no	—
227	C₂×#41	—	94	2	no	—
228	#135	—	2	4	no	—
229	#135	—	2	4	no	—
230	#135	—	94	4	no	—
231	#135	—	94	4	no	—
232	#135	—	188	8	no	—
233	C₂×Q₁₆	—	47	2	no	—
234	C₂×Q₁₆	—	94	2	no	—
235	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	94	2	no	—
238	C₂×SD₁₆	—	94	2	no	—
239	C₂×SD₁₆	—	94	2	no	—
240	C₂×D₈	—	47	2	no	—
241	C₂×D₈	—	94	2	no	—
242	C₂×D₈	C₂²×D₄₇	1	2	yes	—
243	C₂²×C₈	—	47	1	no	—
244	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
245	C₄⋊Q₈	—	94	2	no	—
246	#127	—	94	2	no	—
247	#124	—	94	2	no	—
248	#124	—	94	2	no	—
249	#121	—	188	4	no	—
250	C₄×D₄	—	188	4	no	—
251	C₂×C₄²	—	47	1	no	—
252	Q₃₂	—	2	2	no	—
253	Q₃₂	—	94	4	no	—
254	Q₃₂	—	94	2	no	—
255	SD₃₂	—	2	4	no	—
256	SD₃₂	—	94	4	no	—
257	SD₃₂	—	94	4	no	—
258	SD₃₂	—	94	4	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	94	4	no	—
261	D₁₆	—	94	2	no	—
262	C₂×C₁₆	—	47	4	no	—
263	C₂×C₁₆	—	47	1	no	—
264	C₂×C₁₆	—	47	1	no	—
265	C₂×C₁₆	C₂²⋊D₄₇	1	4	yes	complemented
266	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
267	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
268	C₄×C₈	—	47	1	no	—
269	C₄×C₈	—	47	1	no	—
270	C₄×C₈	—	47	1	no	—
271	C₄×C₈	—	47	1	no	—
272	#41	—	4	4	no	—
273	#41	—	4	4	no	—
274	#41	—	188	4	no	—
275	#41	—	188	4	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	94	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	94	2	no	—
281	C₂×D₄	—	188	4	no	—
282	C₂×D₄	—	188	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	47	2	no	—
285	C₂²×C₄	—	188	2	no	—
286	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	94	2	no	—
290	Q₁₆	—	94	2	no	—
291	Q₁₆	—	188	4	no	—
292	SD₁₆	—	4	4	no	—
293	SD₁₆	—	188	4	no	—
294	SD₁₆	—	188	4	no	—
295	SD₁₆	—	188	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	94	2	no	—
299	D₈	—	94	2	no	—
300	D₈	—	188	4	no	—
301	C₂×C₈	—	47	4	no	—
302	C₂×C₈	—	47	1	no	—
303	C₂×C₈	—	47	1	no	—
304	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
305	C₂×C₈	D₄×D₄₇	1	2	yes	—
306	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
307	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
308	C₄⋊C₄	—	188	4	no	—
309	C₂²⋊C₄	—	188	4	no	—
310	C₂²⋊C₄	—	188	4	no	—
311	C₄²	—	47	2	no	—
312	C₄²	—	47	1	no	—
313	C₄²	—	47	1	no	—
314	C₁₆	—	47	2	no	—
315	C₁₆	—	47	2	no	—
316	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
317	C₁₆	C₂×C₂²⋊D₄₇	1	2	yes	—
318	C₂³	—	4	2	no	—
319	C₂³	—	188	2	no	—
320	C₂³	D₈×D₄₇	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	188	2	no	—
324	Q₈	—	188	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	8	4	no	—
328	D₄	—	376	8	no	—
329	D₄	—	376	4	no	—
330	D₄	—	188	2	no	—
331	D₄	—	188	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	47	2	no	—
335	C₂×C₄	—	47	2	no	—
336	C₂×C₄	—	94	4	no	—
337	C₂×C₄	—	376	4	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	#13094020	1	2	yes	—
340	C₂×C₄	D₈×D₄₇	1	2	yes	—
341	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
343	C₈	—	47	2	no	—
344	C₈	—	47	2	no	—
345	C₈	#13094020	1	2	yes	—
346	C₈	C₂×D₄×D₄₇	1	1	yes	characteristic
347	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
348	C₂²	—	2	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	4	no	—
351	C₂²	—	376	2	no	—
352	C₂²	—	376	4	no	—
353	C₂²	#15689498	1	2	yes	—
354	C₂²	C₂×D₈×D₄₇	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	47	4	no	—
357	C₄	—	94	2	no	—
358	C₄	—	376	2	no	—
359	C₄	#15689498	1	2	yes	—
360	C₄	C₂×#13094020	1	1	yes	characteristic
361	C₄	C₂×D₈×D₄₇	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	16	2	no	—
364	C₂	—	752	2	no	—
365	C₂	C₂×#15689498	1	1	yes	characteristic
366	C₂	#508×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#508	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#508	D₄₇
hypocenter	C₄₇	#33713
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#15689555	C₂	semidirect
2. C₂×#15689553	C₂	semidirect
3. #20253812	C₂	semidirect
4. #20245650	C₂	semidirect
5. #20245648	C₂	semidirect
6. #20245646	C₂	semidirect
7. #20245642	C₂	semidirect
8. #20245639	C₂	semidirect
9. #20245636	C₂	semidirect
10. C₂×C₁₆×Dic₄₇	C₂	semidirect
11. C₂×C₄₇×#508	C₂	semidirect
68. C₂×C₄₇⋊C₁₆	D₄	semidirect
85. C₂×C₁₆×C₄₇	D₄	semidirect
178. Dic₄₇	#508	semidirect
181. C₂×#508	D₄₇	semidirect
225. C₄₇	#33713	semidirect
265. C₂×C₁₆	C₂²⋊D₄₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689555	2
C₂×#15689553	2
#20253812	2
#20245650	2
#20245648	2
#20245646	2
#20245642	2
#20245639	2
#20245636	2
C₂×C₁₆×Dic₄₇	2
C₂×C₄₇×#508	2
#33713	47