Subgroups — #26200317

Order 12032.

706 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20253475	C₂	1	2	yes	maximal, complemented
2	C₂×#15689202	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242918	C₂	1	4	yes	maximal, complemented
4	#20242908	C₂	1	2	yes	maximal, complemented
5	#20242864	C₂	1	2	yes	maximal, complemented
6	#20242151	C₂	1	1	yes	maximal, characteristic, complemented
7	#20242083	C₂	1	2	yes	maximal, complemented
8	C₄₇×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#10625324	—	2	1	no	—
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689439	—	2	2	no	—
12	#15689410	C₂²	1	4	yes	—
13	#15689315	C₂²	1	4	yes	—
14	#15689256	—	2	2	no	—
15	#15689233	—	2	4	no	—
16	#15689202	—	2	1	no	—
17	#15689202	—	2	1	no	—
18	#15689202	C₂²	1	2	yes	—
19	#15689202	C₂²	1	2	yes	—
20	#15689180	C₂²	1	2	yes	—
21	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
23	#15688876	C₂²	1	2	yes	complemented
24	#15688874	C₂²	1	2	yes	complemented
25	#15688872	C₂²	1	4	yes	complemented
26	#15688867	C₂²	1	4	yes	complemented
27	C₄₇×#532	C₂²	1	2	yes	—
28	C₄₇×#500	C₂²	1	1	yes	characteristic, complemented
29	C₄₇×#493	C₂²	1	1	yes	characteristic, complemented
30	C₄₇×#486	C₂²	1	2	yes	complemented
31	C₄₇×#466	—	2	1	no	—
32	C₄₇×#466	—	2	1	no	—
33	C₄×D₈×C₄₇	—	2	1	no	—
34	C₄×D₈×C₄₇	—	2	1	no	—
35	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
36	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₄₇	—	2	1	no	—
38	C₂×#10625334	—	2	4	no	—
39	C₂×#10625324	—	2	2	no	—
40	C₂×#10625324	—	2	2	no	—
41	C₂×#10625324	—	4	4	no	—
42	C₂²×C₄₇⋊C₈	—	2	1	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	—	2	1	no	—
45	C₄×D₁₈₈	D₄	1	2	yes	—
46	C₄×D₁₈₈	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
49	#13093955	—	2	4	no	—
50	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
51	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
52	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
53	#13093945	—	2	4	no	—
54	#13093944	—	2	4	no	—
55	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
56	(C₂×D₄₇)⋊C₈	—	2	2	no	—
57	(C₂×D₄₇)⋊C₈	—	2	2	no	—
58	#13093935	—	2	2	no	—
59	C₂×#41×C₄₇	—	2	2	no	—
60	C₂×D₈×C₄₇	—	2	1	no	—
61	C₂×D₈×C₄₇	—	2	1	no	—
62	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
65	#127×C₄₇	C₂³	1	2	yes	—
66	#124×C₄₇	—	2	2	no	—
67	#121×C₄₇	—	2	2	no	—
68	#121×C₄₇	—	2	2	no	—
69	C₄×D₄×C₄₇	—	2	2	no	—
70	C₄×D₄×C₄₇	—	2	2	no	—
71	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
72	#107×C₄₇	—	2	1	no	—
73	#107×C₄₇	—	2	1	no	—
74	#102×C₄₇	—	2	2	no	—
75	#102×C₄₇	—	2	2	no	—
76	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
77	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
78	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
80	C₂³×D₄₇	—	2	1	no	—
81	#41×C₄₇	—	8	4	no	—
82	C₂×Q₈×C₄₇	—	2	2	no	—
83	C₂×D₄×C₄₇	—	2	2	no	—
84	C₂×D₄×C₄₇	—	2	2	no	—
85	C₂×D₄×C₄₇	—	2	2	no	—
86	C₂×D₄×C₄₇	—	4	2	no	—
87	C₂²×C₄×C₄₇	—	4	2	no	—
88	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
91	C₂²×Dic₄₇	—	2	1	no	—
92	C₂×D₁₈₈	—	2	1	no	—
93	C₂×D₁₈₈	—	4	2	no	—
94	C₂×D₁₈₈	C₂×D₄	1	2	yes	—
95	C₂×C₄×D₄₇	—	2	2	no	—
96	C₂×C₄×D₄₇	—	2	2	no	—
97	C₂×C₄×D₄₇	—	4	2	no	—
98	C₂×C₄×D₄₇	—	4	2	no	—
99	D₈×C₄₇	—	4	2	no	—
100	D₈×C₄₇	—	4	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	4	1	no	—
103	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₄₇	—	2	2	no	—
108	C₄⋊C₄×C₄₇	—	2	2	no	—
109	C₂²⋊C₄×C₄₇	—	4	2	no	—
110	C₂²⋊C₄×C₄₇	—	4	2	no	—
111	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₄₇	#41	1	1	yes	characteristic
114	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
115	#10625334	—	4	8	no	—
116	(C₂×D₄₇)⋊C₄	—	2	2	no	—
117	(C₂×D₄₇)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₄₇	—	2	1	no	—
119	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
120	C₂×C₄₇⋊C₈	—	2	4	no	—
121	C₂×C₄₇⋊C₈	—	2	2	no	—
122	C₂×C₄₇⋊C₈	—	4	2	no	—
123	#10625324	—	4	4	no	—
124	#10625324	—	4	4	no	—
125	C₂³×C₄₇	—	4	2	no	—
126	C₂³×C₄₇	#121	1	1	yes	characteristic
127	C₂²×D₄₇	—	2	2	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	C₂²×D₄₇	—	4	2	no	—
130	C₂²×D₄₇	—	4	2	no	—
131	Q₈×C₄₇	—	8	2	no	—
132	D₄×C₄₇	—	4	2	no	—
133	D₄×C₄₇	—	4	4	no	—
134	D₄×C₄₇	—	4	2	no	—
135	D₄×C₄₇	—	8	2	no	—
136	D₄×C₄₇	—	8	4	no	—
137	C₂×C₄×C₄₇	—	2	2	no	—
138	C₂×C₄×C₄₇	—	2	1	no	—
139	C₂×C₄×C₄₇	—	2	1	no	—
140	C₂×C₄×C₄₇	—	2	2	no	—
141	C₂×C₄×C₄₇	—	4	2	no	—
142	C₂×C₄×C₄₇	—	8	4	no	—
143	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
146	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₄₇	—	2	2	no	—
150	C₂×Dic₄₇	—	4	2	no	—
151	D₁₈₈	—	2	4	no	—
152	D₁₈₈	—	4	2	no	—
153	C₄×D₄₇	—	4	4	no	—
154	C₄×D₄₇	—	4	2	no	—
155	C₄×D₄₇	—	4	2	no	—
156	C₈×C₄₇	—	2	1	no	—
157	C₈×C₄₇	—	2	1	no	—
158	C₈×C₄₇	—	2	1	no	—
159	C₈×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
160	C₄₇⋊C₈	—	4	2	no	—
161	C₄₇⋊C₈	—	4	4	no	—
162	#19283	—	47	1	no	maximal, Sylow
163	C₂²×C₄₇	—	2	1	no	—
164	C₂²×C₄₇	—	2	1	no	—
165	C₂²×C₄₇	—	8	2	no	—
166	C₂²×C₄₇	—	8	2	no	—
167	C₂²×C₄₇	—	8	2	no	—
168	C₂²×C₄₇	#472	1	2	yes	—
169	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
170	C₂×D₄₇	—	4	2	no	—
171	C₂×D₄₇	—	4	2	no	—
172	C₂×D₄₇	—	4	2	no	—
173	C₂×D₄₇	—	8	4	no	—
174	C₄×C₄₇	—	2	1	no	—
175	C₄×C₄₇	—	2	1	no	—
176	C₄×C₄₇	—	2	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	8	2	no	—
179	C₄×C₄₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₄₇	#575	1	2	yes	—
181	C₄×C₄₇	#575	1	1	yes	characteristic
182	Dic₄₇	—	4	2	no	—
183	#3169	D₄₇	1	1	yes	characteristic, complemented
184	#3109	—	47	2	no	—
185	C₂×#436	—	47	1	no	—
186	#1697	—	47	4	no	—
187	#1695	—	47	2	no	—
188	#1687	—	47	2	no	—
189	#1568	—	47	1	no	—
190	#1551	—	47	2	no	—
191	C₂×C₄₇	—	2	2	no	—
192	C₂×C₄₇	—	16	2	no	—
193	C₂×C₄₇	#3036	1	1	yes	characteristic
194	C₂×C₄₇	#3054	1	1	yes	characteristic
195	C₂×C₄₇	C₂×#472	1	1	yes	characteristic
196	D₄₇	—	8	2	no	—
197	C₂²×M₁₆	—	94	1	no	—
198	#532	C₂×D₄₇	1	2	yes	—
199	C₂×C₄×D₄	—	47	1	no	—
200	#500	C₂×D₄₇	1	1	yes	characteristic, complemented
201	#493	C₂×D₄₇	1	1	yes	characteristic, complemented
202	#486	C₂×D₄₇	1	2	yes	complemented
203	#481	—	94	2	no	—
204	#476	—	47	4	no	—
205	#466	—	2	1	no	—
206	#466	—	2	1	no	—
207	#459	—	47	4	no	—
208	#447	—	94	2	no	—
209	#442	—	94	4	no	—
210	C₄×D₈	—	2	1	no	—
211	C₄×D₈	—	2	1	no	—
212	#436	—	47	2	no	—
213	#436	—	47	2	no	—
214	#436	—	94	1	no	—
215	#436	—	94	1	no	—
216	#432	—	47	2	no	—
217	C₂×C₄⋊C₈	—	47	1	no	—
218	C₂×C₂²⋊C₈	—	94	1	no	—
219	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
220	#335	—	47	2	no	—
221	#334	—	47	2	no	—
222	#333	—	47	4	no	—
223	#331	—	47	4	no	—
224	C₄₇	#19283	1	1	yes	characteristic, complemented, Sylow
225	C₂×#41	—	2	2	no	—
226	C₂²×D₄	—	47	1	no	—
227	C₂³×C₄	—	94	1	no	—
228	C₂×D₈	—	2	1	no	—
229	C₂×D₈	—	2	1	no	—
230	C₂×D₈	—	94	4	no	—
231	C₂×M₁₆	—	94	2	no	—
232	C₂×M₁₆	—	94	2	no	—
233	C₂×M₁₆	—	188	4	no	—
234	C₂²×C₈	—	94	1	no	—
235	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
236	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
237	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
238	#127	C₂²×D₄₇	1	2	yes	—
239	#124	—	2	2	no	—
240	#121	—	2	2	no	—
241	#121	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	47	2	no	—
245	C₄×D₄	—	47	2	no	—
246	C₄×D₄	—	94	1	no	—
247	C₄×D₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	47	1	no	—
249	C₂×C₂²⋊C₄	—	94	1	no	—
250	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
251	#107	—	2	1	no	—
252	#107	—	2	1	no	—
253	#106	—	94	4	no	—
254	C₄⋊C₈	—	47	2	no	—
255	C₄⋊C₈	—	47	4	no	—
256	C₄⋊C₈	—	94	1	no	—
257	#102	—	2	2	no	—
258	#102	—	2	2	no	—
259	#102	—	94	4	no	—
260	#102	—	94	4	no	—
261	C₂²⋊C₈	—	94	2	no	—
262	C₂²⋊C₈	—	94	2	no	—
263	C₂²⋊C₈	—	94	2	no	—
264	#97	—	94	2	no	—
265	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
266	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
269	C₂⁴	—	94	1	no	—
270	#41	—	8	4	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	47	2	no	—
277	C₂×D₄	—	94	1	no	—
278	C₂×D₄	—	188	2	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	94	1	no	—
281	C₂²×C₄	—	94	2	no	—
282	C₂²×C₄	—	94	2	no	—
283	C₂²×C₄	—	188	2	no	—
284	C₂²×C₄	—	188	2	no	—
285	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
286	C₂²×C₄	#10625352	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
288	D₈	—	4	2	no	—
289	D₈	—	4	2	no	—
290	D₈	—	188	8	no	—
291	M₁₆	—	188	4	no	—
292	M₁₆	—	188	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	94	4	no	—
296	C₂×C₈	—	94	2	no	—
297	C₂×C₈	—	188	2	no	—
298	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	47	2	no	—
305	C₄⋊C₄	—	94	1	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	94	2	no	—
309	C₂²⋊C₄	—	94	2	no	—
310	C₄²	D₄×D₄₇	1	1	yes	characteristic
311	C₄²	#10625352	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
313	C₄²	C₂³×D₄₇	1	1	yes	characteristic
314	C₂³	—	4	2	no	—
315	C₂³	—	94	2	no	—
316	C₂³	—	188	2	no	—
317	C₂³	—	188	2	no	—
318	C₂³	—	188	2	no	—
319	C₂³	#13093995	1	1	yes	characteristic
320	Q₈	—	8	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	4	no	—
326	D₄	—	94	4	no	—
327	D₄	—	188	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	94	2	no	—
335	C₂×C₄	—	188	4	no	—
336	C₂×C₄	—	188	2	no	—
337	C₂×C₄	—	188	2	no	—
338	C₂×C₄	—	188	2	no	—
339	C₂×C₄	#13093995	1	1	yes	characteristic
340	C₂×C₄	#13093995	1	1	yes	characteristic
341	C₂×C₄	#13093995	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
344	C₂×C₄	C₂×#10625352	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	188	2	no	—
349	C₈	—	188	4	no	—
350	C₈	#13094049	1	2	yes	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	376	4	no	—
357	C₂²	—	188	2	no	—
358	C₂²	—	188	2	no	—
359	C₂²	—	188	2	no	—
360	C₂²	#15689390	1	2	yes	—
361	C₂²	C₂×#13093995	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	188	2	no	—
368	C₄	C₂×#13094049	1	1	yes	characteristic
369	C₄	#15689920	1	2	yes	—
370	C₄	#15689921	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	376	2	no	—
374	C₂	#20252848	1	1	yes	characteristic
375	C₂	#20252993	1	1	yes	characteristic
376	C₂	C₂×#15689390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093995
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#3169	C₂
socle	C₂²×C₄₇	C₂×#121
hypercenter	#3169	D₄₇
hypocenter	C₄₇	#19283
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20253475	C₂	semidirect
2. C₂×#15689202	C₂	semidirect
3. #20242918	C₂	semidirect
4. #20242908	C₂	semidirect
5. #20242864	C₂	semidirect
6. #20242151	C₂	semidirect
7. #20242083	C₂	semidirect
8. C₄₇×#3169	C₂	semidirect
23. #15688876	C₂²	semidirect
24. #15688874	C₂²	semidirect
25. #15688872	C₂²	semidirect
26. #15688867	C₂²	semidirect
28. C₄₇×#500	C₂²	semidirect
29. C₄₇×#493	C₂²	semidirect
30. C₄₇×#486	C₂²	semidirect
183. #3169	D₄₇	semidirect
200. #500	C₂×D₄₇	semidirect
201. #493	C₂×D₄₇	semidirect
202. #486	C₂×D₄₇	semidirect
224. C₄₇	#19283	semidirect

Maximal subgroups

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

subgroup	index
#20253475	2
C₂×#15689202	2
#20242918	2
#20242908	2
#20242864	2
#20242151	2
#20242083	2
C₄₇×#3169	2
#19283	47