Subgroups — #26199575

Order 12032.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20254336	C₂	1	1	yes	maximal, characteristic, complemented
2	#20254331	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689471	C₂	1	1	yes	maximal, characteristic
4	#20252318	C₂	1	1	yes	maximal, characteristic, complemented
5	#20242959	C₂	1	4	yes	maximal, complemented
6	#20242835	C₂	1	4	yes	maximal, complemented
7	#20242134	C₂	1	1	yes	maximal, characteristic, complemented
8	#20242057	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#1549	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689680	C₂²	1	1	yes	characteristic
11	C₂×#13094021	C₂²	1	1	yes	characteristic
12	#15689502	—	2	1	no	—
13	#15689496	—	2	1	no	—
14	#15689473	—	2	1	no	—
15	#15689471	—	2	1	no	—
16	#15689471	—	2	1	no	—
17	#15689471	C₂²	1	2	yes	—
18	#15689471	C₂²	1	2	yes	—
19	#15689470	—	2	1	no	—
20	#15689330	C₂²	1	4	yes	complemented
21	#15689324	C₂²	1	4	yes	—
22	#15689279	—	2	2	no	—
23	#15689276	—	2	2	no	—
24	#15689250	—	2	1	no	—
25	#15689250	C₂²	1	2	yes	complemented
26	#15689247	—	2	1	no	—
27	#15689247	C₂²	1	2	yes	complemented
28	#15689176	C₂²	1	1	yes	characteristic
29	#15689151	—	2	1	no	—
30	C₂×#13093944	—	2	1	no	—
31	C₂×#13093944	—	2	1	no	—
32	#15689081	—	2	1	no	—
33	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
34	#15688875	C₂²	1	2	yes	complemented
35	#15688873	C₂²	1	2	yes	complemented
36	#15688866	C₂²	1	4	yes	—
37	C₂×C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
38	C₄₇×#431	C₂²	1	1	yes	characteristic
39	C₄₇×#423	C₂²	1	1	yes	characteristic
40	C₄₇×#332	C₂²	1	4	yes	complemented
41	C₂²×D₁₈₈	—	2	1	no	—
42	C₂²×D₁₈₈	—	2	1	no	—
43	C₂²×D₁₈₈	—	2	1	no	—
44	C₂×#10625337	—	2	4	no	—
45	C₂×#10625325	—	2	4	no	—
46	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
47	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
48	#13094024	D₄	1	1	yes	characteristic
49	#13094024	D₄	1	1	yes	characteristic
50	#13094021	—	2	1	no	—
51	#13094021	—	2	1	no	—
52	#13094021	—	2	1	no	—
53	#13094021	C₂³	1	2	yes	—
54	#13094016	—	2	1	no	—
55	#13094003	—	2	1	no	—
56	#13094003	D₄	1	2	yes	—
57	C₄×Dic₉₄	—	2	1	no	—
58	C₄×Dic₉₄	C₂³	1	2	yes	—
59	#13093983	—	2	1	no	—
60	C₄⋊(C₄₇⋊C₈)	C₂³	1	2	yes	—
61	Dic₄₇⋊C₈	—	2	2	no	—
62	Dic₄₇⋊C₈	—	2	2	no	—
63	#13093948	—	2	4	no	—
64	#13093946	—	2	4	no	—
65	#13093944	—	2	2	no	—
66	#13093944	—	2	2	no	—
67	#13093944	—	2	2	no	—
68	#13093944	—	2	2	no	—
69	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
70	#13093934	—	2	2	no	—
71	C₄×C₄₇⋊C₈	—	2	1	no	—
72	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
76	C₈×Dic₄₇	—	2	2	no	—
77	C₂²×Q₈×C₄₇	—	2	1	no	—
78	C₂×M₁₆×C₄₇	—	2	1	no	—
79	C₄⋊Q₈×C₄₇	—	2	1	no	—
80	C₄⋊Q₈×C₄₇	—	2	1	no	—
81	C₄⋊Q₈×C₄₇	—	2	1	no	—
82	C₄⋊Q₈×C₄₇	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
85	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₄₇	—	2	1	no	—
87	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
88	C₄⋊C₈×C₄₇	C₂³	1	2	yes	—
89	C₂²⋊C₈×C₄₇	—	2	1	no	—
90	C₄×C₈×C₄₇	—	2	1	no	—
91	C₂³×D₄₇	—	4	1	no	—
92	C₂×Q₈×C₄₇	—	2	2	no	—
93	C₂×Q₈×C₄₇	—	4	1	no	—
94	C₂×Q₈×C₄₇	—	8	2	no	—
95	C₂²×C₄×C₄₇	—	4	1	no	—
96	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
99	C₂×D₁₈₈	—	2	2	no	—
100	C₂×D₁₈₈	—	2	2	no	—
101	C₂×D₁₈₈	—	2	2	no	—
102	C₂×D₁₈₈	—	4	2	no	—
103	C₂×D₁₈₈	—	4	2	no	—
104	C₂×D₁₈₈	—	4	1	no	—
105	C₂×D₁₈₈	—	4	1	no	—
106	C₂×D₁₈₈	—	4	1	no	—
107	C₂×D₁₈₈	—	8	2	no	—
108	C₂×D₁₈₈	—	8	2	no	—
109	C₂×Dic₉₄	—	2	1	no	—
110	C₂×Dic₉₄	C₂×D₄	1	2	yes	—
111	M₁₆×C₄₇	—	4	2	no	—
112	C₂×C₈×C₄₇	—	2	2	no	—
113	C₂×C₈×C₄₇	—	2	2	no	—
114	C₄⋊C₄×C₄₇	—	2	2	no	—
115	C₄⋊C₄×C₄₇	—	2	2	no	—
116	C₄⋊C₄×C₄₇	—	4	1	no	—
117	C₄⋊C₄×C₄₇	—	4	1	no	—
118	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₄₇	#41	1	1	yes	characteristic
121	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
122	C₂²⋊Dic₄₇	—	2	2	no	—
123	#10625337	—	8	4	no	—
124	C₄⋊Dic₄₇	—	2	1	no	—
125	C₄⋊Dic₄₇	—	2	1	no	—
126	C₄⋊Dic₄₇	—	2	1	no	—
127	C₄⋊Dic₄₇	C₂×D₄	1	2	yes	—
128	Dic₄₇⋊C₄	—	2	2	no	—
129	Dic₄₇⋊C₄	—	2	2	no	—
130	C₄×Dic₄₇	—	2	2	no	—
131	C₂×C₄₇⋊C₈	—	2	2	no	—
132	C₂×C₄₇⋊C₈	—	2	2	no	—
133	C₂×C₄₇⋊C₈	—	2	2	no	—
134	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄₇⋊C₈	C₂×D₄	1	1	yes	characteristic
138	#10625325	—	8	4	no	—
139	C₂³×C₄₇	#121	1	1	yes	characteristic
140	C₂²×D₄₇	—	4	2	no	—
141	C₂²×D₄₇	—	8	2	no	—
142	C₂²×D₄₇	—	8	2	no	—
143	C₂²×D₄₇	—	8	2	no	—
144	Q₈×C₄₇	—	4	4	no	—
145	Q₈×C₄₇	—	8	2	no	—
146	C₂×C₄×C₄₇	—	2	2	no	—
147	C₂×C₄×C₄₇	—	2	2	no	—
148	C₂×C₄×C₄₇	—	4	2	no	—
149	C₂×C₄×C₄₇	—	8	2	no	—
150	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
151	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
152	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
153	C₂×C₄×C₄₇	#135	1	2	yes	—
154	C₂×C₄×C₄₇	#137	1	2	yes	—
155	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
158	C₂×Dic₄₇	—	2	2	no	—
159	C₂×Dic₄₇	—	2	2	no	—
160	D₁₈₈	—	4	2	no	—
161	D₁₈₈	—	4	2	no	—
162	D₁₈₈	—	4	4	no	—
163	D₁₈₈	—	8	2	no	—
164	D₁₈₈	—	8	2	no	—
165	D₁₈₈	—	8	2	no	—
166	D₁₈₈	—	8	2	no	—
167	Dic₉₄	—	2	4	no	—
168	Dic₉₄	—	4	2	no	—
169	C₈×C₄₇	—	4	2	no	—
170	C₈×C₄₇	—	4	2	no	—
171	C₄₇⋊C₈	—	2	1	no	—
172	C₄₇⋊C₈	—	2	1	no	—
173	C₄₇⋊C₈	—	2	1	no	—
174	C₄₇⋊C₈	—	2	1	no	—
175	C₄₇⋊C₈	—	4	2	no	—
176	#19214	—	47	1	no	maximal, Sylow
177	C₂²×C₄₇	—	2	1	no	—
178	C₂²×C₄₇	—	2	1	no	—
179	C₂²×C₄₇	#464	1	2	yes	—
180	C₂²×C₄₇	C₂×#121	1	1	yes	characteristic
181	C₂×D₄₇	—	8	2	no	—
182	C₂×D₄₇	—	8	2	no	—
183	C₂×D₄₇	—	8	2	no	—
184	C₂×D₄₇	—	16	4	no	—
185	C₄×C₄₇	—	2	1	no	—
186	C₄×C₄₇	—	2	1	no	—
187	C₄×C₄₇	—	8	2	no	—
188	C₄×C₄₇	#464	1	2	yes	—
189	C₄×C₄₇	#471	1	2	yes	—
190	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
191	C₄×C₄₇	C₂×#137	1	1	yes	characteristic
192	C₄×C₄₇	#576	1	2	yes	—
193	Dic₄₇	—	4	2	no	—
194	Dic₄₇	—	4	2	no	—
195	#3209	—	47	1	no	—
196	#3208	—	47	1	no	—
197	C₂×#486	—	47	1	no	—
198	#2960	—	47	1	no	—
199	#1702	—	47	4	no	—
200	#1682	—	47	4	no	—
201	#1564	—	47	1	no	—
202	#1549	D₄₇	1	1	yes	characteristic, complemented
203	#1545	—	47	1	no	—
204	C₂×C₄₇	—	2	2	no	—
205	C₂×C₄₇	C₂×#464	1	1	yes	characteristic
206	C₂×C₄₇	#3058	1	1	yes	characteristic
207	C₂×C₄₇	C₂×#471	1	1	yes	characteristic
208	D₄₇	—	16	2	no	—
209	#533	—	47	1	no	—
210	C₂×C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
211	C₂×#127	—	47	1	no	—
212	#494	—	94	1	no	—
213	#492	—	94	1	no	—
214	#487	—	94	1	no	—
215	#486	—	47	2	no	—
216	#486	—	47	2	no	—
217	#486	—	94	1	no	—
218	#486	—	94	1	no	—
219	#486	—	94	1	no	—
220	#462	—	47	4	no	—
221	#460	—	47	4	no	—
222	#451	—	94	2	no	—
223	#450	—	94	2	no	—
224	#446	—	47	2	no	—
225	#446	—	94	1	no	—
226	C₈×Q₈	—	47	2	no	—
227	C₈×Q₈	—	94	1	no	—
228	#431	—	47	1	no	—
229	#431	C₂×D₄₇	1	1	yes	characteristic
230	#424	—	94	1	no	—
231	#423	C₂×D₄₇	1	1	yes	characteristic
232	C₂×#102	—	94	1	no	—
233	C₂×#102	—	94	1	no	—
234	#405	—	94	1	no	—
235	C₂×C₄×C₈	—	47	1	no	—
236	#335	—	47	2	no	—
237	#334	—	47	2	no	—
238	#332	C₂×D₄₇	1	4	yes	complemented
239	#331	—	47	4	no	—
240	C₄₇	#19214	1	1	yes	characteristic, complemented, Sylow
241	C₂²×Q₈	—	2	1	no	—
242	C₂²×D₄	—	94	1	no	—
243	C₂²×D₄	—	94	1	no	—
244	C₂²×D₄	—	94	1	no	—
245	C₂×Q₁₆	—	94	4	no	—
246	C₂×SD₁₆	—	94	4	no	—
247	C₂×M₁₆	—	2	1	no	—
248	C₂²×C₈	—	47	1	no	—
249	C₂²×C₈	—	47	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	47	1	no	—
254	C₄⋊Q₈	—	47	1	no	—
255	C₄⋊Q₈	C₂²×D₄₇	1	2	yes	—
256	#127	—	47	2	no	—
257	#127	—	94	1	no	—
258	#127	—	94	1	no	—
259	#127	—	94	1	no	—
260	#125	—	94	1	no	—
261	C₂²⋊Q₈	—	47	2	no	—
262	C₂²⋊Q₈	—	94	1	no	—
263	C₄×Q₈	—	47	2	no	—
264	C₄×Q₈	—	94	1	no	—
265	#117	—	94	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	47	2	no	—
271	C₄⋊C₈	—	94	2	no	—
272	C₄⋊C₈	—	94	2	no	—
273	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
274	C₄⋊C₈	C₂²×D₄₇	1	2	yes	—
275	#103	—	94	4	no	—
276	#102	—	94	2	no	—
277	#102	—	94	2	no	—
278	#102	—	94	2	no	—
279	#102	—	94	2	no	—
280	#102	—	94	4	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	94	1	no	—
283	#97	—	94	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	47	1	no	—
286	C₄×C₈	—	47	1	no	—
287	C₄×C₈	—	47	1	no	—
288	C₄×C₈	—	47	1	no	—
289	C₄×C₈	—	94	2	no	—
290	C₄×C₈	—	94	1	no	—
291	C₂⁴	—	188	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	8	2	no	—
295	C₂×Q₈	—	47	2	no	—
296	C₂×Q₈	—	94	1	no	—
297	C₂×D₄	—	94	2	no	—
298	C₂×D₄	—	94	2	no	—
299	C₂×D₄	—	94	2	no	—
300	C₂×D₄	—	376	2	no	—
301	C₂×D₄	—	376	2	no	—
302	C₂×D₄	—	188	1	no	—
303	C₂×D₄	—	188	1	no	—
304	C₂×D₄	—	188	1	no	—
305	C₂×D₄	—	188	2	no	—
306	C₂×D₄	—	188	2	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	#10625350	1	1	yes	characteristic
309	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
310	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
311	Q₁₆	—	376	4	no	—
312	SD₁₆	—	376	4	no	—
313	M₁₆	—	4	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	47	1	no	—
317	C₂×C₈	—	47	1	no	—
318	C₂×C₈	—	47	1	no	—
319	C₂×C₈	—	47	1	no	—
320	C₂×C₈	—	94	2	no	—
321	C₂×C₈	—	94	2	no	—
322	C₂×C₈	—	94	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	47	2	no	—
328	C₄⋊C₄	—	94	2	no	—
329	C₄⋊C₄	—	94	1	no	—
330	C₄⋊C₄	—	94	1	no	—
331	C₄⋊C₄	—	94	1	no	—
332	C₄⋊C₄	—	94	2	no	—
333	C₂²⋊C₄	—	94	2	no	—
334	C₄²	—	94	2	no	—
335	C₄²	#10625350	1	1	yes	characteristic
336	C₄²	D₄×D₄₇	1	1	yes	characteristic
337	C₄²	D₄×D₄₇	1	1	yes	characteristic
338	C₄²	C₂³×D₄₇	1	1	yes	characteristic
339	C₂³	—	376	2	no	—
340	C₂³	—	376	2	no	—
341	C₂³	—	376	2	no	—
342	C₂³	—	188	2	no	—
343	C₂³	#13093997	1	1	yes	characteristic
344	Q₈	—	4	4	no	—
345	Q₈	—	8	2	no	—
346	Q₈	—	94	4	no	—
347	Q₈	—	188	2	no	—
348	D₄	—	376	2	no	—
349	D₄	—	376	2	no	—
350	D₄	—	376	2	no	—
351	D₄	—	376	2	no	—
352	D₄	—	188	2	no	—
353	D₄	—	188	2	no	—
354	D₄	—	188	4	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	94	2	no	—
360	C₂×C₄	—	94	2	no	—
361	C₂×C₄	#13093997	1	1	yes	characteristic
362	C₂×C₄	#13093997	1	1	yes	characteristic
363	C₂×C₄	#13093997	1	1	yes	characteristic
364	C₂×C₄	#13094045	1	2	yes	—
365	C₂×C₄	#13094056	1	2	yes	—
366	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
367	C₂×C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
368	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
369	C₈	—	4	2	no	—
370	C₈	—	4	2	no	—
371	C₈	—	94	1	no	—
372	C₈	—	94	1	no	—
373	C₈	—	94	1	no	—
374	C₈	—	94	1	no	—
375	C₈	—	188	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	376	2	no	—
379	C₂²	—	376	2	no	—
380	C₂²	—	376	2	no	—
381	C₂²	—	752	4	no	—
382	C₂²	#15689348	1	2	yes	—
383	C₂²	C₂×#13093997	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	188	2	no	—
388	C₄	—	188	2	no	—
389	C₄	#15689348	1	2	yes	—
390	C₄	#15689389	1	2	yes	—
391	C₄	C₂×#13094045	1	1	yes	characteristic
392	C₄	C₂×#13094056	1	1	yes	characteristic
393	C₄	#15689927	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	752	2	no	—
396	C₂	C₂×#15689348	1	1	yes	characteristic
397	C₂	#20253035	1	1	yes	characteristic
398	C₂	C₂×#15689389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13093997
commutator, cosocle	C₄²×C₄₇	C₂⁴
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1549	C₂
socle	C₂²×C₄₇	C₂×#121
prime radical	C₂×#13094021	C₂²
hypercenter	#1549	D₄₇
hypocenter	C₄₇	#19214
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

Complements

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

N	H ≅ G/N	type
1. #20254336	C₂	semidirect
4. #20252318	C₂	semidirect
5. #20242959	C₂	semidirect
6. #20242835	C₂	semidirect
7. #20242134	C₂	semidirect
9. C₄₇×#1549	C₂	semidirect
20. #15689330	C₂²	semidirect
25. #15689250	C₂²	semidirect
27. #15689247	C₂²	semidirect
34. #15688875	C₂²	semidirect
35. #15688873	C₂²	semidirect
40. C₄₇×#332	C₂²	semidirect
202. #1549	D₄₇	semidirect
238. #332	C₂×D₄₇	semidirect
240. C₄₇	#19214	semidirect

Maximal subgroups

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

subgroup	index
#20254336	2
#20254331	2
C₂×#15689471	2
#20252318	2
#20242959	2
#20242835	2
#20242134	2
#20242057	2
C₄₇×#1549	2
#19214	47