Subgroups — #26117519

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689474	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689404	C₂	1	1	yes	maximal, characteristic
3	#20248557	C₂	1	1	yes	maximal, characteristic
4	#20244311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688871	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688852	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094024	C₂²	1	1	yes	characteristic
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
11	#15689474	—	2	2	no	—
12	#15689474	—	2	2	no	—
13	#15689404	—	2	2	no	—
14	#15689404	—	2	2	no	—
15	C₂×#13093958	—	2	1	no	—
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×#13093949	—	2	2	no	—
18	C₂×#13093947	—	2	2	no	—
19	C₂×#13093946	—	2	1	no	—
20	#15689036	—	2	2	no	—
21	#15689025	—	2	1	no	—
22	#15688989	—	2	1	no	—
23	#15688980	—	2	2	no	—
24	#15688871	—	2	2	no	—
25	#15688852	—	2	2	no	—
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×Dic₉₄	—	2	1	no	—
30	C₂²×Dic₉₄	D₄	1	2	yes	—
31	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₄₇	—	2	1	no	—
33	C₂²×C₄₇⋊C₈	—	2	1	no	—
34	#13094024	—	2	1	no	—
35	#13094024	—	2	1	no	—
36	#13094024	—	2	1	no	—
37	#13094024	—	2	1	no	—
38	#13094024	—	2	1	no	—
39	#13094024	—	2	1	no	—
40	#13094024	—	2	1	no	—
41	#13094024	D₄	1	2	yes	—
42	C₄×D₁₈₈	—	2	1	no	—
43	C₄×D₁₈₈	—	2	1	no	—
44	C₄×D₁₈₈	—	2	1	no	—
45	C₄×D₁₈₈	—	2	1	no	—
46	C₂×C₄⋊Dic₄₇	—	2	1	no	—
47	C₂×C₄⋊Dic₄₇	—	2	1	no	—
48	C₂×C₄⋊Dic₄₇	—	2	1	no	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
51	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
52	C₂×(C₂×D₄₇)⋊C₄	—	2	4	no	—
53	#13093958	—	2	4	no	—
54	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
55	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
56	#13093949	—	2	8	no	—
57	#13093947	—	4	4	no	—
58	#13093946	—	2	4	no	—
59	C₄₇⋊#95	—	2	4	no	—
60	C₂²×C₈×C₄₇	—	2	1	no	—
61	C₂²×C₈×C₄₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
63	C₂×C₄²×C₄₇	—	2	1	no	—
64	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₄₇	—	2	1	no	—
66	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
67	C₄×C₈×C₄₇	—	2	1	no	—
68	C₄×C₈×C₄₇	D₄	1	2	yes	—
69	#95×C₄₇	—	2	1	no	—
70	C₂³×D₄₇	—	2	1	no	—
71	C₂²×C₄×C₄₇	—	2	2	no	—
72	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₄₇	—	2	1	no	—
76	C₂²×Dic₄₇	—	2	2	no	—
77	C₂²×Dic₄₇	—	4	1	no	—
78	C₂×D₁₈₈	—	2	1	no	—
79	C₂×D₁₈₈	—	2	1	no	—
80	C₂×D₁₈₈	—	2	1	no	—
81	C₂×D₁₈₈	—	2	1	no	—
82	C₂×D₁₈₈	—	2	1	no	—
83	C₂×D₁₈₈	Q₁₆	1	2	yes	—
84	C₂×C₄×D₄₇	—	4	1	no	—
85	C₂×C₄×D₄₇	—	4	1	no	—
86	C₂×C₄×D₄₇	—	4	2	no	—
87	C₂×C₄×D₄₇	—	4	2	no	—
88	C₂×Dic₉₄	—	2	2	no	—
89	C₂×Dic₉₄	—	2	2	no	—
90	C₂×Dic₉₄	—	2	2	no	—
91	C₂×Dic₉₄	—	4	4	no	—
92	C₂×Dic₉₄	—	4	2	no	—
93	C₂×Dic₉₄	—	4	2	no	—
94	C₂×Dic₉₄	—	4	1	no	—
95	C₂×C₈×C₄₇	—	2	2	no	—
96	C₂×C₈×C₄₇	—	2	2	no	—
97	C₂×C₈×C₄₇	—	4	2	no	—
98	C₂×C₈×C₄₇	—	4	1	no	—
99	C₂×C₈×C₄₇	—	4	1	no	—
100	C₄⋊C₄×C₄₇	—	2	2	no	—
101	C₄⋊C₄×C₄₇	—	2	2	no	—
102	C₄²×C₄₇	—	4	1	no	—
103	C₄²×C₄₇	—	4	1	no	—
104	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₇	#41	1	1	yes	characteristic
108	(C₂×D₄₇)⋊C₄	—	4	1	no	—
109	(C₂×D₄₇)⋊C₄	—	4	1	no	—
110	(C₂×D₄₇)⋊C₄	—	8	4	no	—
111	C₄⋊Dic₄₇	—	2	2	no	—
112	C₄⋊Dic₄₇	—	2	2	no	—
113	C₄⋊Dic₄₇	—	2	1	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	4	1	no	—
116	C₄⋊Dic₄₇	—	4	1	no	—
117	C₄⋊Dic₄₇	—	4	1	no	—
118	C₄⋊Dic₄₇	—	4	1	no	—
119	C₄⋊Dic₄₇	—	4	1	no	—
120	C₄⋊Dic₄₇	—	4	1	no	—
121	C₂×C₄₇⋊C₈	—	4	1	no	—
122	C₂×C₄₇⋊C₈	—	4	2	no	—
123	C₂³×C₄₇	#120	1	1	yes	characteristic
124	C₂²×D₄₇	—	2	2	no	—
125	C₂²×D₄₇	—	4	1	no	—
126	C₂²×D₄₇	—	4	2	no	—
127	C₂²×D₄₇	—	4	1	no	—
128	C₂²×D₄₇	—	4	2	no	—
129	C₂×C₄×C₄₇	—	2	2	no	—
130	C₂×C₄×C₄₇	—	4	2	no	—
131	C₂×C₄×C₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	4	2	no	—
133	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
136	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
137	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
138	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
139	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
141	C₂×C₄×C₄₇	#135	1	2	yes	—
142	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	1	no	—
147	C₂×Dic₄₇	—	4	2	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	4	4	no	—
150	C₂×Dic₄₇	—	8	2	no	—
151	D₁₈₈	—	2	2	no	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	—	4	1	no	—
154	D₁₈₈	—	4	1	no	—
155	C₄×D₄₇	—	8	2	no	—
156	C₄×D₄₇	—	8	1	no	—
157	C₄×D₄₇	—	8	1	no	—
158	Dic₉₄	—	4	4	no	—
159	Dic₉₄	—	4	4	no	—
160	Dic₉₄	—	8	1	no	—
161	Dic₉₄	—	8	1	no	—
162	Dic₉₄	—	8	2	no	—
163	C₈×C₄₇	—	4	2	no	—
164	C₈×C₄₇	—	4	2	no	—
165	C₄₇⋊C₈	—	8	1	no	—
166	#10072	—	47	1	no	maximal, Sylow
167	C₂²×C₄₇	#392	1	1	yes	characteristic
168	C₂²×C₄₇	#449	1	2	yes	—
169	C₂²×C₄₇	#451	1	1	yes	characteristic
170	C₂²×C₄₇	#452	1	1	yes	characteristic
171	C₂²×C₄₇	#455	1	2	yes	—
172	C₂×D₄₇	—	4	4	no	—
173	C₂×D₄₇	—	4	2	no	—
174	C₂×D₄₇	—	8	2	no	—
175	C₂×D₄₇	—	8	1	no	—
176	C₂×D₄₇	—	8	1	no	—
177	C₄×C₄₇	—	2	1	no	—
178	C₄×C₄₇	—	2	1	no	—
179	C₄×C₄₇	—	8	2	no	—
180	C₄×C₄₇	#451	1	1	yes	characteristic
181	C₄×C₄₇	#452	1	1	yes	characteristic
182	C₄×C₄₇	#463	1	2	yes	—
183	C₄×C₄₇	#467	1	1	yes	characteristic
184	C₄×C₄₇	#470	1	1	yes	characteristic
185	C₄×C₄₇	#495	1	1	yes	characteristic
186	C₄×C₄₇	#496	1	1	yes	characteristic
187	Dic₄₇	—	8	2	no	—
188	Dic₄₇	—	8	2	no	—
189	Dic₄₇	—	8	1	no	—
190	C₂×#487	—	47	1	no	—
191	C₂×#474	—	47	1	no	—
192	#2563	—	47	1	no	—
193	#1981	—	47	1	no	—
194	C₂×#333	—	47	1	no	—
195	C₂×D₄⋊C₈	—	47	1	no	—
196	#1279	D₄₇	1	1	yes	characteristic, complemented
197	C₂×C₄₇	#1644	1	2	yes	—
198	C₂×C₄₇	#1646	1	2	yes	—
199	C₂×C₄₇	#2001	1	1	yes	characteristic
200	C₂×C₄₇	#2004	1	1	yes	characteristic
201	C₂×C₄₇	#2012	1	1	yes	characteristic
202	D₄₇	—	8	2	no	—
203	C₂×C₄⋊Q₈	—	47	1	no	—
204	C₂×C₄⋊Q₈	—	47	1	no	—
205	C₂×C₄×D₄	—	47	1	no	—
206	#487	—	94	2	no	—
207	#487	—	94	2	no	—
208	#474	—	94	2	no	—
209	#474	—	94	2	no	—
210	C₂×#107	—	94	1	no	—
211	C₂×C₄⋊C₈	—	47	1	no	—
212	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
213	C₂×#103	—	94	2	no	—
214	C₂×#103	—	94	2	no	—
215	C₂×#102	—	94	1	no	—
216	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
217	#393	—	94	2	no	—
218	#390	—	94	1	no	—
219	#382	—	94	1	no	—
220	#380	—	94	2	no	—
221	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
222	#333	—	94	2	no	—
223	D₄⋊C₈	—	94	2	no	—
224	C₄₇	#10072	1	1	yes	characteristic, complemented, Sylow
225	C₂²×Q₈	—	47	2	no	—
226	C₂²×Q₈	—	94	1	no	—
227	C₂²×D₄	—	47	1	no	—
228	C₂³×C₄	—	94	1	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	94	1	no	—
232	C₄⋊Q₈	—	47	2	no	—
233	C₄⋊Q₈	—	94	1	no	—
234	C₄⋊Q₈	—	94	1	no	—
235	C₄⋊Q₈	—	94	1	no	—
236	C₄⋊Q₈	—	94	1	no	—
237	C₄⋊Q₈	—	94	1	no	—
238	C₄⋊Q₈	—	94	1	no	—
239	C₄⋊Q₈	—	94	1	no	—
240	C₄×D₄	—	94	1	no	—
241	C₄×D₄	—	94	1	no	—
242	C₄×D₄	—	94	1	no	—
243	C₄×D₄	—	94	1	no	—
244	C₂×C₄⋊C₄	—	47	1	no	—
245	C₂×C₄⋊C₄	—	94	1	no	—
246	C₂×C₄⋊C₄	—	94	1	no	—
247	C₂×C₄⋊C₄	—	94	1	no	—
248	C₂×C₄⋊C₄	—	94	1	no	—
249	C₂×C₄⋊C₄	D₁₈₈	1	2	yes	—
250	C₂×C₂²⋊C₄	—	94	4	no	—
251	C₂×C₂²⋊C₄	—	94	1	no	—
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
254	#107	—	94	4	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	—	94	1	no	—
257	C₄⋊C₈	—	94	1	no	—
258	C₄⋊C₈	D₁₈₈	1	2	yes	—
259	#103	—	94	8	no	—
260	#103	—	188	4	no	—
261	#102	—	94	4	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	D₁₈₈	1	2	yes	—
264	#95	—	2	1	no	—
265	#95	—	94	4	no	—
266	C₂⁴	—	94	1	no	—
267	C₂×Q₈	—	94	2	no	—
268	C₂×Q₈	—	94	2	no	—
269	C₂×Q₈	—	94	2	no	—
270	C₂×Q₈	—	188	4	no	—
271	C₂×Q₈	—	188	2	no	—
272	C₂×Q₈	—	188	2	no	—
273	C₂×Q₈	—	188	1	no	—
274	C₂×D₄	—	47	2	no	—
275	C₂×D₄	—	94	1	no	—
276	C₂×D₄	—	94	1	no	—
277	C₂×D₄	—	94	1	no	—
278	C₂×D₄	—	94	1	no	—
279	C₂×D₄	—	94	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	94	1	no	—
282	C₂²×C₄	—	94	2	no	—
283	C₂²×C₄	—	188	1	no	—
284	C₂²×C₄	—	188	1	no	—
285	C₂²×C₄	—	188	1	no	—
286	C₂²×C₄	—	188	2	no	—
287	C₂²×C₄	—	188	2	no	—
288	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	4	1	no	—
296	C₂×C₈	—	188	1	no	—
297	C₂×C₈	—	188	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	94	2	no	—
301	C₄⋊C₄	—	94	2	no	—
302	C₄⋊C₄	—	94	1	no	—
303	C₄⋊C₄	—	94	1	no	—
304	C₄⋊C₄	—	188	1	no	—
305	C₄⋊C₄	—	188	1	no	—
306	C₄⋊C₄	—	188	1	no	—
307	C₄⋊C₄	—	188	1	no	—
308	C₄⋊C₄	—	188	1	no	—
309	C₄⋊C₄	—	188	1	no	—
310	C₂²⋊C₄	—	376	4	no	—
311	C₂²⋊C₄	—	188	1	no	—
312	C₂²⋊C₄	—	188	1	no	—
313	C₄²	—	4	1	no	—
314	C₄²	—	4	1	no	—
315	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
316	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
317	C₄²	D₄×D₄₇	1	1	yes	characteristic
318	C₄²	#10625354	1	1	yes	characteristic
319	C₂³	—	94	2	no	—
320	C₂³	—	188	2	no	—
321	C₂³	—	188	1	no	—
322	C₂³	—	188	1	no	—
323	C₂³	—	188	2	no	—
324	C₂³	#13093994	1	1	yes	characteristic
325	Q₈	—	376	2	no	—
326	Q₈	—	376	1	no	—
327	Q₈	—	376	1	no	—
328	Q₈	—	188	4	no	—
329	Q₈	—	188	4	no	—
330	D₄	—	94	2	no	—
331	D₄	—	94	2	no	—
332	D₄	—	188	1	no	—
333	D₄	—	188	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	376	2	no	—
339	C₂×C₄	—	376	1	no	—
340	C₂×C₄	—	376	1	no	—
341	C₂×C₄	—	376	2	no	—
342	C₂×C₄	—	188	2	no	—
343	C₂×C₄	—	188	1	no	—
344	C₂×C₄	—	188	2	no	—
345	C₂×C₄	—	188	2	no	—
346	C₂×C₄	—	188	4	no	—
347	C₂×C₄	#13093994	1	1	yes	characteristic
348	C₂×C₄	#13093994	1	1	yes	characteristic
349	C₂×C₄	#13093999	1	1	yes	characteristic
350	C₂×C₄	#13093999	1	1	yes	characteristic
351	C₂×C₄	#13093999	1	1	yes	characteristic
352	C₂×C₄	#13094021	1	1	yes	characteristic
353	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
354	C₂×C₄	#13094043	1	1	yes	characteristic
355	C₂×C₄	#13094046	1	2	yes	—
356	C₂×C₄	#13094052	1	2	yes	—
357	C₂×C₄	#13094053	1	1	yes	characteristic
358	C₂×C₄	#13094053	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	376	1	no	—
362	C₂²	—	376	2	no	—
363	C₂²	—	376	1	no	—
364	C₂²	—	376	1	no	—
365	C₂²	—	188	4	no	—
366	C₂²	—	188	2	no	—
367	C₂²	#15689033	1	1	yes	characteristic
368	C₂²	#15689270	1	2	yes	—
369	C₂²	#15689280	1	1	yes	characteristic
370	C₂²	#15689286	1	1	yes	characteristic
371	C₂²	#15689302	1	2	yes	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	8	2	no	—
375	C₄	—	376	2	no	—
376	C₄	—	376	1	no	—
377	C₄	—	376	2	no	—
378	C₄	#15689280	1	1	yes	characteristic
379	C₄	#15689286	1	1	yes	characteristic
380	C₄	#15689339	1	2	yes	—
381	C₄	#15689365	1	1	yes	characteristic
382	C₄	#15689382	1	1	yes	characteristic
383	C₄	#15689507	1	1	yes	characteristic
384	C₄	#15689512	1	1	yes	characteristic
385	C₂	—	376	2	no	—
386	C₂	#20242604	1	2	yes	—
387	C₂	#20242618	1	2	yes	—
388	C₂	#20244399	1	1	yes	characteristic
389	C₂	#20244420	1	1	yes	characteristic
390	C₂	#20244468	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093994
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1279	C₂
socle	C₂³×C₄₇	#120
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#1279	D₄₇
hypocenter	C₄₇	#10072
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
1. C₂×#15689474	C₂	semidirect
4. #20244311	C₂	semidirect
5. C₂×#15688871	C₂	semidirect
7. C₄₇×#1279	C₂	semidirect
196. #1279	D₄₇	semidirect
224. C₄₇	#10072	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689474	2
C₂×#15689404	2
#20248557	2
#20244311	2
C₂×#15688871	2
C₂×#15688852	2
C₄₇×#1279	2
#10072	47