Subgroups — #26371720

Order 12032.

834 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689564	C₂	1	2	yes	maximal, complemented
2	C₂×#15689559	C₂	1	2	yes	maximal, complemented
3	#20253816	C₂	1	1	yes	maximal, characteristic, complemented
4	#20245660	C₂	1	2	yes	maximal, complemented
5	#20245655	C₂	1	2	yes	maximal, complemented
6	#20245647	C₂	1	4	yes	maximal, complemented
7	C₄×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094046	—	2	1	no	—
10	C₂×#13094046	C₂²	1	2	yes	—
11	#15689673	—	2	1	no	—
12	#15689564	—	2	8	no	—
13	#15689559	—	2	8	no	—
14	C₂×#13093970	C₂²	1	4	yes	complemented
15	C₂×#13093968	C₂²	1	4	yes	—
16	C₂×#13093967	C₂²	1	4	yes	complemented
17	C₂×#13093965	C₂²	1	4	yes	—
18	C₂×#13093964	C₂²	1	2	yes	—
19	#15689507	—	2	1	no	—
20	#15689507	C₂²	1	2	yes	complemented
21	#15689503	C₂²	1	1	yes	characteristic, complemented
22	#15689500	C₂²	1	1	yes	characteristic, complemented
23	#15689496	—	2	1	no	—
24	#15688899	C₂²	1	2	yes	complemented
25	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
26	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×SD₁₆×C₄₇	—	2	1	no	—
28	C₂²×D₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂×#124×C₄₇	—	2	1	no	—
30	C₄₇×#495	—	2	1	no	—
31	C₄₇×#495	—	2	1	no	—
32	C₄₇×#495	C₂²	1	2	yes	—
33	C₄₇×#495	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×#10625350	—	2	2	no	—
36	#13094046	—	2	4	no	—
37	#13094046	—	2	4	no	—
38	#13094046	—	4	4	no	—
39	C₂×Dic₁₈₈	—	2	1	no	—
40	C₂×Dic₁₈₈	C₂³	1	2	yes	—
41	C₂×#10625325	—	2	2	no	—
42	C₂×#10625325	—	2	2	no	—
43	C₂×D₃₇₆	—	2	1	no	—
44	C₂×D₃₇₆	C₂³	1	2	yes	—
45	#13094024	—	2	1	no	—
46	#13094021	—	2	1	no	—
47	#13094012	—	2	2	no	—
48	#13093998	—	4	2	no	—
49	C₄×D₁₈₈	—	4	2	no	—
50	#13093970	—	2	8	no	—
51	#13093968	—	2	8	no	—
52	#13093967	—	2	8	no	—
53	#13093965	—	2	8	no	—
54	#13093964	D₄	1	8	yes	complemented
55	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	—
56	C₂²×Q₈×C₄₇	—	2	1	no	—
57	C₂²×D₄×C₄₇	—	2	1	no	—
58	C₂×Q₁₆×C₄₇	—	2	4	no	—
59	C₂×Q₁₆×C₄₇	—	2	1	no	—
60	C₂×Q₁₆×C₄₇	C₂³	1	2	yes	—
61	C₂×SD₁₆×C₄₇	—	2	2	no	—
62	C₂×SD₁₆×C₄₇	—	2	2	no	—
63	C₂×SD₁₆×C₄₇	—	4	4	no	—
64	C₂×D₈×C₄₇	—	2	4	no	—
65	C₂×D₈×C₄₇	—	2	1	no	—
66	C₂×D₈×C₄₇	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
69	#124×C₄₇	—	2	2	no	—
70	#124×C₄₇	—	2	2	no	—
71	#124×C₄₇	—	4	2	no	—
72	C₂×C₂²⋊C₄×C₄₇	—	4	2	no	—
73	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
74	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
75	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
76	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
78	C₂⁴×C₄₇	—	4	1	no	—
79	C₂×Q₈×C₄₇	—	2	2	no	—
80	C₂×Q₈×C₄₇	—	4	1	no	—
81	C₂×Q₈×C₄₇	—	4	2	no	—
82	C₂×Q₈×C₄₇	—	4	2	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	4	2	no	—
85	C₂×D₄×C₄₇	—	4	2	no	—
86	C₂×D₄×C₄₇	—	4	1	no	—
87	C₂²×C₄×C₄₇	—	4	1	no	—
88	C₂²×C₄×C₄₇	D₈	1	2	yes	—
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₂²⋊D₄₇	—	4	2	no	—
91	#10625350	—	4	4	no	—
92	#10625350	—	4	4	no	—
93	C₂×D₁₈₈	—	2	2	no	—
94	C₂×D₁₈₈	—	4	2	no	—
95	C₂×C₄×D₄₇	—	4	2	no	—
96	C₂×Dic₉₄	—	2	2	no	—
97	Q₁₆×C₄₇	—	2	4	no	—
98	Q₁₆×C₄₇	—	4	2	no	—
99	SD₁₆×C₄₇	—	4	4	no	—
100	SD₁₆×C₄₇	—	4	4	no	—
101	D₈×C₄₇	—	2	4	no	—
102	D₈×C₄₇	—	4	2	no	—
103	C₂×C₈×C₄₇	D₈	1	4	yes	—
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₂²⋊C₄×C₄₇	—	4	4	no	—
110	C₂²⋊C₄×C₄₇	—	4	4	no	—
111	C₄²×C₄₇	D₈	1	2	yes	—
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
114	(C₂×D₄₇)⋊C₄	—	4	4	no	—
115	C₄⋊Dic₄₇	—	4	2	no	—
116	Dic₁₈₈	—	2	2	no	—
117	Dic₁₈₈	—	2	2	no	—
118	Dic₁₈₈	—	4	2	no	—
119	D₃₇₆	—	2	2	no	—
120	D₃₇₆	—	2	2	no	—
121	D₃₇₆	—	4	2	no	—
122	#10625325	—	4	4	no	—
123	#10625325	—	4	4	no	—
124	C₄₇⋊C₁₆	C₂×D₄	1	8	yes	—
125	C₂³×C₄₇	—	4	2	no	—
126	C₂³×C₄₇	—	8	4	no	—
127	C₂³×C₄₇	—	8	2	no	—
128	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
129	C₂²×D₄₇	—	4	2	no	—
130	Q₈×C₄₇	—	4	2	no	—
131	Q₈×C₄₇	—	4	2	no	—
132	Q₈×C₄₇	—	8	2	no	—
133	D₄×C₄₇	—	4	2	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	8	2	no	—
136	C₂×C₄×C₄₇	—	4	2	no	—
137	C₂×C₄×C₄₇	—	4	2	no	—
138	C₂×C₄×C₄₇	—	8	2	no	—
139	C₂×C₄×C₄₇	#127	1	2	yes	—
140	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₄₇	—	8	4	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	D₁₈₈	—	4	2	no	—
148	D₁₈₈	—	4	2	no	—
149	D₁₈₈	—	8	4	no	—
150	C₄×D₄₇	—	8	4	no	—
151	Dic₉₄	—	4	2	no	—
152	Dic₉₄	—	4	2	no	—
153	C₈×C₄₇	#127	1	2	yes	—
154	C₈×C₄₇	C₂×D₈	1	2	yes	—
155	C₈×C₄₇	C₂×D₈	1	2	yes	—
156	C₈×C₄₇	C₂²×D₄	1	2	yes	—
157	#33719	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	2	no	—
159	C₂²×C₄₇	—	8	2	no	—
160	C₂²×C₄₇	—	8	4	no	—
161	C₂²×C₄₇	—	16	4	no	—
162	C₂²×C₄₇	#493	1	2	yes	—
163	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
164	C₂×D₄₇	—	8	2	no	—
165	C₂×D₄₇	—	8	4	no	—
166	C₄×C₄₇	—	2	2	no	—
167	C₄×C₄₇	—	2	2	no	—
168	C₄×C₄₇	—	8	2	no	—
169	C₄×C₄₇	#493	1	2	yes	—
170	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
171	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
172	Dic₄₇	—	8	2	no	—
173	C₂×#510	—	47	2	no	—
174	C₂×#509	—	47	2	no	—
175	#3147	—	47	1	no	—
176	C₂×#495	D₄₇	1	1	yes	characteristic, complemented
177	#2251	—	47	2	no	—
178	#2250	—	47	2	no	—
179	#2249	—	47	4	no	—
180	C₄×M₃₂	—	47	1	no	—
181	C₂×C₄₇	—	2	2	no	—
182	C₂×C₄₇	—	16	2	no	—
183	C₂×C₄₇	C₂×#493	1	1	yes	characteristic
184	C₂×C₄₇	#3414	1	2	yes	—
185	D₄₇	—	16	2	no	—
186	C₂×#135	—	47	2	no	—
187	C₂×#135	—	94	1	no	—
188	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
189	C₂²×SD₁₆	—	2	1	no	—
190	C₂²×D₈	C₂×D₄₇	1	1	yes	characteristic
191	#532	—	94	1	no	—
192	C₂×#124	—	2	1	no	—
193	#510	—	94	8	no	—
194	#509	—	94	8	no	—
195	C₂×Q₃₂	—	47	4	no	—
196	C₂×SD₃₂	—	47	4	no	—
197	C₂×SD₃₂	—	47	4	no	—
198	C₂×D₁₆	—	47	4	no	—
199	C₂×M₃₂	—	47	2	no	—
200	#495	—	2	1	no	—
201	#495	—	2	1	no	—
202	#495	—	47	2	no	—
203	#495	—	94	1	no	—
204	#495	C₂×D₄₇	1	2	yes	—
205	#495	C₂×D₄₇	1	2	yes	—
206	#494	—	47	1	no	—
207	#493	—	47	1	no	—
208	#492	—	94	1	no	—
209	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
210	#346	—	47	2	no	—
211	C₄×C₁₆	—	47	2	no	—
212	C₄₇	#33719	1	1	yes	characteristic, complemented, Sylow
213	C₂×#41	—	94	2	no	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×D₄	—	2	1	no	—
216	#135	—	94	4	no	—
217	#135	—	94	4	no	—
218	#135	—	188	4	no	—
219	C₂×Q₁₆	—	2	4	no	—
220	C₂×Q₁₆	—	2	1	no	—
221	C₂×Q₁₆	—	47	2	no	—
222	C₂×Q₁₆	—	94	1	no	—
223	C₂×Q₁₆	C₂²×D₄₇	1	2	yes	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	4	4	no	—
227	C₂×SD₁₆	—	94	2	no	—
228	C₂×SD₁₆	—	94	2	no	—
229	C₂×D₈	—	2	4	no	—
230	C₂×D₈	—	2	1	no	—
231	C₂×D₈	—	47	2	no	—
232	C₂×D₈	—	94	1	no	—
233	C₂×D₈	C₂²×D₄₇	1	2	yes	—
234	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
236	C₄⋊Q₈	—	94	1	no	—
237	#127	—	94	1	no	—
238	#124	—	2	2	no	—
239	#124	—	2	2	no	—
240	#124	—	4	2	no	—
241	#124	—	94	2	no	—
242	#121	—	188	2	no	—
243	C₄×D₄	—	188	2	no	—
244	C₂×C₂²⋊C₄	—	4	2	no	—
245	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
246	Q₃₂	—	94	8	no	—
247	SD₃₂	—	94	8	no	—
248	SD₃₂	—	94	8	no	—
249	D₁₆	—	94	8	no	—
250	M₃₂	—	47	8	no	—
251	C₂×C₁₆	—	47	4	no	—
252	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
253	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
255	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
256	C₂⁴	—	4	1	no	—
257	#41	—	188	4	no	—
258	#41	—	188	4	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×Q₈	—	4	2	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	94	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	94	2	no	—
269	C₂×D₄	—	188	2	no	—
270	C₂×D₄	—	188	2	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	188	2	no	—
273	C₂²×C₄	#10625334	1	2	yes	—
274	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
275	Q₁₆	—	2	4	no	—
276	Q₁₆	—	4	2	no	—
277	Q₁₆	—	94	2	no	—
278	Q₁₆	—	94	2	no	—
279	Q₁₆	—	188	2	no	—
280	SD₁₆	—	4	4	no	—
281	SD₁₆	—	4	4	no	—
282	SD₁₆	—	188	4	no	—
283	SD₁₆	—	188	4	no	—
284	D₈	—	2	4	no	—
285	D₈	—	4	2	no	—
286	D₈	—	94	2	no	—
287	D₈	—	94	2	no	—
288	D₈	—	188	2	no	—
289	C₂×C₈	#10625334	1	4	yes	—
290	C₂×C₈	D₄×D₄₇	1	4	yes	—
291	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
294	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
295	C₄⋊C₄	—	188	2	no	—
296	C₂²⋊C₄	—	4	4	no	—
297	C₂²⋊C₄	—	4	4	no	—
298	C₂²⋊C₄	—	188	4	no	—
299	C₄²	#10625334	1	2	yes	—
300	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
302	C₁₆	—	47	8	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	8	2	no	—
306	C₂³	—	188	2	no	—
307	C₂³	C₂×#10625334	1	1	yes	characteristic
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	8	2	no	—
311	Q₈	—	188	2	no	—
312	Q₈	—	188	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	2	no	—
316	D₄	—	376	4	no	—
317	D₄	—	376	4	no	—
318	D₄	—	188	2	no	—
319	D₄	—	188	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	376	4	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	#13094020	1	2	yes	—
326	C₂×C₄	C₂×#10625334	1	2	yes	—
327	C₂×C₄	C₂×#10625334	1	2	yes	—
328	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
329	C₂×C₄	C₂×#10625334	1	2	yes	—
330	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
331	C₈	#13094020	1	2	yes	—
332	C₈	C₂×#10625334	1	2	yes	—
333	C₈	C₂×#10625334	1	2	yes	—
334	C₈	C₂×D₄×D₄₇	1	2	yes	—
335	C₂²	—	2	2	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	16	4	no	—
339	C₂²	—	376	4	no	—
340	C₂²	—	376	2	no	—
341	C₂²	#15689499	1	2	yes	—
342	C₂²	C₂²×#10625334	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	376	2	no	—
347	C₄	#15689499	1	2	yes	—
348	C₄	C₂×#13094020	1	1	yes	characteristic
349	C₄	C₂²×#10625334	1	1	yes	characteristic
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	—	752	2	no	—
353	C₂	C₂×#15689499	1	1	yes	characteristic
354	C₂	#20256522	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₂×C₄₇×#495	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	C₂×#495	D₄₇
hypocenter	C₄₇	#33719
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#15689564	C₂	semidirect
2. C₂×#15689559	C₂	semidirect
3. #20253816	C₂	semidirect
4. #20245660	C₂	semidirect
5. #20245655	C₂	semidirect
6. #20245647	C₂	semidirect
7. C₄×#13093964	C₂	semidirect
8. C₂×C₄₇×#495	C₂	semidirect
14. C₂×#13093970	C₂²	semidirect
16. C₂×#13093967	C₂²	semidirect
20. #15689507	C₂²	semidirect
21. #15689503	C₂²	semidirect
22. #15689500	C₂²	semidirect
24. #15688899	C₂²	semidirect
25. C₄×C₄₇⋊C₁₆	C₂²	semidirect
54. #13093964	D₄	semidirect
176. C₂×#495	D₄₇	semidirect
212. C₄₇	#33719	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689564	2
C₂×#15689559	2
#20253816	2
#20245660	2
#20245655	2
#20245647	2
C₄×#13093964	2
C₂×C₄₇×#495	2
#33719	47