Subgroups — #26131937

Order 12032.

948 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688933	C₂	1	4	yes	maximal, complemented
2	#20244999	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243413	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#135×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	complemented
6	C₂×#13093956	C₂²	1	2	yes	complemented
7	#15688983	—	2	1	no	—
8	#15688933	—	2	8	no	—
9	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
10	C₂²⋊(C₄₇⋊C₁₆)	—	2	1	no	—
11	#15688886	—	2	1	no	—
12	C₂²×#41×C₄₇	—	2	1	no	—
13	C₂×#135×C₄₇	—	2	8	no	—
14	C₂×#135×C₄₇	—	2	1	no	—
15	C₂×#135×C₄₇	—	2	1	no	—
16	C₂×#135×C₄₇	C₄	1	2	yes	complemented
17	C₂×#135×C₄₇	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₄₇	C₄	1	1	yes	characteristic, complemented
19	C₂²×SD₁₆×C₄₇	—	2	1	no	—
20	C₂²×D₈×C₄₇	C₄	1	1	yes	characteristic, complemented
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
24	#13093964	—	2	4	no	—
25	C₂×C₄₇⋊C₁₆	—	2	2	no	—
26	#13093956	—	2	4	no	—
27	C₄₇⋊#95	—	4	1	no	—
28	C₂×#41×C₄₇	—	2	8	no	—
29	C₂×#41×C₄₇	—	2	2	no	—
30	C₂×#41×C₄₇	—	2	2	no	—
31	C₂×#41×C₄₇	—	4	4	no	—
32	C₂×#41×C₄₇	—	4	1	no	—
33	C₂×#41×C₄₇	—	4	1	no	—
34	C₂²×Q₈×C₄₇	—	2	1	no	—
35	C₂²×D₄×C₄₇	—	2	1	no	—
36	C₂²×D₄×C₄₇	—	4	1	no	—
37	C₂³×C₄×C₄₇	—	4	1	no	—
38	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
39	#135×C₄₇	—	2	8	no	—
40	#135×C₄₇	—	2	8	no	—
41	#135×C₄₇	—	2	4	no	—
42	#135×C₄₇	—	2	4	no	—
43	#135×C₄₇	—	2	4	no	—
44	#135×C₄₇	D₄	1	8	yes	—
45	C₂×Q₁₆×C₄₇	—	2	4	no	—
46	C₂×Q₁₆×C₄₇	—	2	1	no	—
47	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
48	C₂×SD₁₆×C₄₇	—	2	8	no	—
49	C₂×SD₁₆×C₄₇	—	2	2	no	—
50	C₂×SD₁₆×C₄₇	—	2	2	no	—
51	C₂×D₈×C₄₇	—	2	4	no	—
52	C₂×D₈×C₄₇	—	2	1	no	—
53	C₂×D₈×C₄₇	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₄₇	—	2	4	no	—
55	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₂⁴×C₄₇	—	4	1	no	—
62	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
63	#41×C₄₇	—	2	8	no	—
64	#41×C₄₇	—	2	8	no	—
65	#41×C₄₇	—	4	8	no	—
66	#41×C₄₇	—	4	4	no	—
67	#41×C₄₇	—	4	8	no	—
68	#41×C₄₇	—	4	4	no	—
69	C₂×Q₈×C₄₇	—	2	4	no	—
70	C₂×Q₈×C₄₇	—	2	2	no	—
71	C₂×Q₈×C₄₇	—	4	2	no	—
72	C₂×Q₈×C₄₇	—	4	1	no	—
73	C₂×D₄×C₄₇	—	2	4	no	—
74	C₂×D₄×C₄₇	—	2	2	no	—
75	C₂×D₄×C₄₇	—	4	2	no	—
76	C₂×D₄×C₄₇	—	4	8	no	—
77	C₂×D₄×C₄₇	—	4	2	no	—
78	C₂×D₄×C₄₇	—	4	2	no	—
79	C₂×D₄×C₄₇	—	4	1	no	—
80	C₂²×C₄×C₄₇	—	2	2	no	—
81	C₂²×C₄×C₄₇	—	2	2	no	—
82	C₂²×C₄×C₄₇	—	4	8	no	—
83	C₂²×C₄×C₄₇	—	4	1	no	—
84	C₂²×C₄×C₄₇	—	4	2	no	—
85	C₂²×C₄×C₄₇	—	4	2	no	—
86	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
91	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×Dic₄₇	—	4	2	no	—
93	Q₁₆×C₄₇	—	2	4	no	—
94	Q₁₆×C₄₇	—	2	2	no	—
95	Q₁₆×C₄₇	C₂²⋊C₄	1	4	yes	complemented
96	SD₁₆×C₄₇	—	2	8	no	—
97	SD₁₆×C₄₇	—	2	8	no	—
98	D₈×C₄₇	—	2	4	no	—
99	D₈×C₄₇	—	2	2	no	—
100	D₈×C₄₇	C₂²⋊C₄	1	4	yes	complemented
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	2	2	no	—
103	C₂×C₈×C₄₇	—	2	2	no	—
104	C₂×C₈×C₄₇	—	2	2	no	—
105	C₂×C₈×C₄₇	—	2	2	no	—
106	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
110	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
111	C₂²⋊Dic₄₇	—	8	2	no	—
112	C₄⋊Dic₄₇	—	4	2	no	—
113	C₄⋊Dic₄₇	—	4	2	no	—
114	C₄₇⋊C₁₆	—	4	2	no	—
115	C₂³×C₄₇	—	2	2	no	—
116	C₂³×C₄₇	—	2	2	no	—
117	C₂³×C₄₇	—	4	1	no	—
118	C₂³×C₄₇	—	4	4	no	—
119	C₂³×C₄₇	—	4	2	no	—
120	C₂³×C₄₇	—	8	2	no	—
121	C₂³×C₄₇	—	8	2	no	—
122	C₂³×C₄₇	#102	1	1	yes	characteristic
123	C₂³×C₄₇	#102	1	1	yes	characteristic
124	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
125	Q₈×C₄₇	—	2	4	no	—
126	Q₈×C₄₇	—	4	4	no	—
127	Q₈×C₄₇	—	4	2	no	—
128	D₄×C₄₇	—	2	4	no	—
129	D₄×C₄₇	—	4	4	no	—
130	D₄×C₄₇	—	4	8	no	—
131	D₄×C₄₇	—	4	8	no	—
132	D₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	—	2	2	no	—
134	C₂×C₄×C₄₇	—	2	1	no	—
135	C₂×C₄×C₄₇	—	2	1	no	—
136	C₂×C₄×C₄₇	—	2	1	no	—
137	C₂×C₄×C₄₇	—	2	1	no	—
138	C₂×C₄×C₄₇	—	2	2	no	—
139	C₂×C₄×C₄₇	—	2	2	no	—
140	C₂×C₄×C₄₇	—	4	8	no	—
141	C₂×C₄×C₄₇	—	4	4	no	—
142	C₂×C₄×C₄₇	—	4	2	no	—
143	C₂×C₄×C₄₇	—	4	8	no	—
144	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	#120	1	2	yes	—
148	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×Dic₄₇	—	8	2	no	—
151	C₂×Dic₄₇	—	8	2	no	—
152	C₂×Dic₄₇	—	8	2	no	—
153	C₈×C₄₇	—	2	1	no	—
154	C₈×C₄₇	—	2	1	no	—
155	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
156	C₈×C₄₇	#120	1	2	yes	—
157	#12735	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	—	2	1	no	—
159	C₂²×C₄₇	—	2	1	no	—
160	C₂²×C₄₇	—	2	1	no	—
161	C₂²×C₄₇	—	2	1	no	—
162	C₂²×C₄₇	—	2	1	no	—
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	4	4	no	—
166	C₂²×C₄₇	—	4	4	no	—
167	C₂²×C₄₇	—	8	8	no	—
168	C₂²×C₄₇	—	8	2	no	—
169	C₂²×C₄₇	—	8	2	no	—
170	C₂²×C₄₇	#360	1	2	yes	complemented
171	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
172	C₂²×C₄₇	#447	1	2	yes	—
173	C₄×C₄₇	—	2	1	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	4	4	no	—
176	C₄×C₄₇	#379	1	1	yes	characteristic
177	C₄×C₄₇	#418	1	1	yes	characteristic
178	C₄×C₄₇	#450	1	2	yes	—
179	Dic₄₇	—	16	2	no	—
180	C₂²×#135	D₄₇	1	1	yes	characteristic
181	C₂×#360	—	47	4	no	—
182	#2112	—	47	1	no	—
183	#1807	—	47	1	no	—
184	C₂×C₄₇	—	2	2	no	—
185	C₂×C₄₇	—	2	2	no	—
186	C₂×C₄₇	—	4	1	no	—
187	C₂×C₄₇	—	8	4	no	—
188	C₂×C₄₇	#1786	1	1	yes	characteristic
189	C₂×C₄₇	C₂×#360	1	1	yes	characteristic
190	C₂×C₄₇	#2145	1	1	yes	characteristic
191	C₂²×#41	—	2	1	no	—
192	C₂×#135	—	2	8	no	—
193	C₂×#135	—	2	1	no	—
194	C₂×#135	—	2	1	no	—
195	C₂×#135	Dic₄₇	1	2	yes	complemented
196	C₂×#135	C₂×D₄₇	1	2	yes	—
197	C₂²×Q₁₆	Dic₄₇	1	1	yes	characteristic, complemented
198	C₂²×SD₁₆	—	2	1	no	—
199	C₂²×D₈	Dic₄₇	1	1	yes	characteristic, complemented
200	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
201	C₂×M₃₂	—	47	2	no	—
202	C₂×#106	—	47	2	no	—
203	#380	—	94	1	no	—
204	#360	—	94	8	no	—
205	C₂²⋊C₁₆	—	94	1	no	—
206	C₂²⋊C₁₆	—	94	1	no	—
207	#340	—	94	1	no	—
208	C₄₇	#12735	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	2	8	no	—
210	C₂×#41	—	2	2	no	—
211	C₂×#41	—	2	2	no	—
212	C₂×#41	—	4	4	no	—
213	C₂×#41	—	4	1	no	—
214	C₂×#41	—	4	1	no	—
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×D₄	—	2	1	no	—
217	C₂²×D₄	—	4	1	no	—
218	C₂³×C₄	—	4	1	no	—
219	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
220	#135	—	2	8	no	—
221	#135	—	2	8	no	—
222	#135	—	2	4	no	—
223	#135	—	2	4	no	—
224	#135	—	2	4	no	—
225	#135	C₂²⋊D₄₇	1	8	yes	—
226	C₂×Q₁₆	—	2	4	no	—
227	C₂×Q₁₆	—	2	1	no	—
228	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
229	C₂×SD₁₆	—	2	8	no	—
230	C₂×SD₁₆	—	2	2	no	—
231	C₂×SD₁₆	—	2	2	no	—
232	C₂×D₈	—	2	4	no	—
233	C₂×D₈	—	2	1	no	—
234	C₂×D₈	C₂×Dic₄₇	1	2	yes	—
235	C₂²×C₈	—	2	4	no	—
236	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
237	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
238	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
239	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
240	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
241	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
242	C₂×C₄⋊C₄	—	94	2	no	—
243	C₂×C₂²⋊C₄	—	188	1	no	—
244	M₃₂	—	94	4	no	—
245	C₂×C₁₆	—	94	2	no	—
246	#106	—	94	4	no	—
247	#95	—	188	1	no	—
248	C₂⁴	—	4	1	no	—
249	C₂⁴	#10625334	1	1	yes	characteristic
250	#41	—	2	8	no	—
251	#41	—	2	8	no	—
252	#41	—	4	8	no	—
253	#41	—	4	8	no	—
254	#41	—	4	4	no	—
255	#41	—	4	4	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	8	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	4	8	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	188	2	no	—
274	C₂²×C₄	#10625334	1	1	yes	characteristic
275	C₂²×C₄	#10625335	1	1	yes	characteristic
276	C₂²×C₄	#10625335	1	1	yes	characteristic
277	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
278	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
280	Q₁₆	—	2	4	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	C₂²⋊Dic₄₇	1	4	yes	complemented
283	SD₁₆	—	2	8	no	—
284	SD₁₆	—	2	8	no	—
285	D₈	—	2	4	no	—
286	D₈	—	2	2	no	—
287	D₈	C₂²⋊Dic₄₇	1	4	yes	complemented
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
294	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
297	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
298	C₄⋊C₄	—	188	2	no	—
299	C₄⋊C₄	—	188	2	no	—
300	C₂²⋊C₄	—	376	2	no	—
301	C₁₆	—	188	2	no	—
302	C₂³	—	2	2	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	4	1	no	—
305	C₂³	—	4	4	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	#13093945	1	1	yes	characteristic
310	C₂³	#13093945	1	1	yes	characteristic
311	C₂³	C₂×#10625334	1	1	yes	characteristic
312	Q₈	—	2	4	no	—
313	Q₈	—	4	4	no	—
314	Q₈	—	4	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	8	no	—
318	D₄	—	4	8	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	4	8	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	8	no	—
331	C₂×C₄	—	376	2	no	—
332	C₂×C₄	—	376	2	no	—
333	C₂×C₄	—	376	2	no	—
334	C₂×C₄	#13093945	1	1	yes	characteristic
335	C₂×C₄	#13093945	1	1	yes	characteristic
336	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
337	C₂×C₄	#13093993	1	2	yes	—
338	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
339	C₂×C₄	#13094050	1	2	yes	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	C₂×C₂²⋊Dic₄₇	1	2	yes	—
343	C₈	#13093993	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	4	no	—
352	C₂²	—	4	4	no	—
353	C₂²	—	8	8	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	#15688933	1	2	yes	complemented
357	C₂²	C₂×#13093945	1	1	yes	characteristic
358	C₂²	#15689257	1	2	yes	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	4	4	no	—
362	C₄	—	752	2	no	—
363	C₄	#15688979	1	1	yes	characteristic
364	C₄	#15689128	1	1	yes	characteristic
365	C₄	#15689274	1	2	yes	—
366	C₂	—	2	2	no	—
367	C₂	—	2	2	no	—
368	C₂	—	4	1	no	—
369	C₂	—	8	4	no	—
370	C₂	#20243319	1	1	yes	characteristic
371	C₂	C₂×#15688933	1	1	yes	characteristic
372	C₂	#20245124	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093945
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting, prime radical	C₂²×#135×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#102
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#135	D₄₇
hypocenter	C₄₇	#12735
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#15688933	C₂	semidirect
2. #20244999	C₂	semidirect
3. #20243413	C₂	semidirect
5. C₂×#13093964	C₂²	semidirect
6. C₂×#13093956	C₂²	semidirect
16. C₂×#135×C₄₇	C₄	semidirect
18. C₂²×Q₁₆×C₄₇	C₄	semidirect
20. C₂²×D₈×C₄₇	C₄	semidirect
95. Q₁₆×C₄₇	C₂²⋊C₄	semidirect
100. D₈×C₄₇	C₂²⋊C₄	semidirect
170. C₂²×C₄₇	#360	semidirect
195. C₂×#135	Dic₄₇	semidirect
197. C₂²×Q₁₆	Dic₄₇	semidirect
199. C₂²×D₈	Dic₄₇	semidirect
208. C₄₇	#12735	semidirect
282. Q₁₆	C₂²⋊Dic₄₇	semidirect
287. D₈	C₂²⋊Dic₄₇	semidirect
356. C₂²	#15688933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688933	2
#20244999	2
#20243413	2
C₂²×#135×C₄₇	2
#12735	47