Subgroups — #26118856

Order 12032.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689521	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689414	C₂	1	1	yes	maximal, characteristic
3	C₂×#15689329	C₂	1	1	yes	maximal, characteristic
4	#20244182	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688871	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15688870	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625325	—	2	1	no	—
9	C₂×#13094024	C₂²	1	1	yes	characteristic
10	C₂×#13094024	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#15689521	—	2	2	no	—
13	#15689521	—	2	2	no	—
14	#15689414	—	2	2	no	—
15	#15689414	—	2	2	no	—
16	#15689329	—	2	2	no	—
17	#15689329	—	2	2	no	—
18	C₂×#13093956	—	2	1	no	—
19	C₂×#13093956	—	2	2	no	—
20	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×#13093949	—	2	1	no	—
22	C₂×#13093946	—	2	1	no	—
23	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
24	#15688871	—	2	2	no	—
25	#15688870	—	2	2	no	—
26	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₂²×Dic₉₄	—	2	1	no	—
30	C₂²×Dic₉₄	—	2	1	no	—
31	C₂²×D₁₈₈	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₄₇	—	2	1	no	—
33	C₂×#10625325	—	2	4	no	—
34	C₂×#10625325	—	4	2	no	—
35	C₂×#10625325	—	4	2	no	—
36	C₂²×C₄₇⋊C₈	—	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	—	2	1	no	—
42	#13094024	—	2	1	no	—
43	#13094024	D₄	1	2	yes	—
44	#13094024	D₄	1	2	yes	—
45	C₄×D₁₈₈	—	2	1	no	—
46	C₄×D₁₈₈	—	2	1	no	—
47	C₄×D₁₈₈	—	2	1	no	—
48	C₄×D₁₈₈	—	2	1	no	—
49	C₂×C₄⋊Dic₄₇	—	2	1	no	—
50	C₂×C₄⋊Dic₄₇	—	2	1	no	—
51	C₂×C₄⋊Dic₄₇	—	2	1	no	—
52	C₂×C₄⋊Dic₄₇	—	2	1	no	—
53	C₂×C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
54	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
55	#13093956	—	2	8	no	—
56	#13093956	—	2	4	no	—
57	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
58	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
59	#13093949	—	2	4	no	—
60	#13093946	—	2	4	no	—
61	(C₂×D₄₇)⋊C₈	—	4	4	no	—
62	C₂²×C₈×C₄₇	—	2	1	no	—
63	C₂²×C₈×C₄₇	—	2	1	no	—
64	C₂²×C₈×C₄₇	D₄	1	2	yes	—
65	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₇	—	2	1	no	—
67	C₄⋊C₈×C₄₇	—	2	1	no	—
68	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
69	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
70	C₄×C₈×C₄₇	—	2	1	no	—
71	C₄×C₈×C₄₇	—	2	1	no	—
72	C₂³×D₄₇	—	2	1	no	—
73	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
76	C₂²×Dic₄₇	—	2	1	no	—
77	C₂²×Dic₄₇	—	4	1	no	—
78	C₂²×Dic₄₇	—	4	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	—	2	1	no	—
84	C₂×D₁₈₈	SD₁₆	1	2	yes	—
85	C₂×C₄×D₄₇	—	4	1	no	—
86	C₂×C₄×D₄₇	—	4	1	no	—
87	C₂×C₄×D₄₇	—	4	2	no	—
88	C₂×C₄×D₄₇	—	4	2	no	—
89	C₂×Dic₉₄	—	2	2	no	—
90	C₂×Dic₉₄	—	2	2	no	—
91	C₂×Dic₉₄	—	4	1	no	—
92	C₂×Dic₉₄	—	4	1	no	—
93	C₂×Dic₉₄	—	4	1	no	—
94	C₂×Dic₉₄	—	4	1	no	—
95	C₂×Dic₉₄	—	4	1	no	—
96	C₂×Dic₉₄	—	4	1	no	—
97	C₂×Dic₉₄	—	8	2	no	—
98	C₂×C₈×C₄₇	—	2	2	no	—
99	C₂×C₈×C₄₇	—	2	2	no	—
100	C₂×C₈×C₄₇	—	2	2	no	—
101	C₂×C₈×C₄₇	—	2	2	no	—
102	C₂×C₈×C₄₇	—	4	1	no	—
103	C₂×C₈×C₄₇	—	4	1	no	—
104	C₂×C₈×C₄₇	—	4	1	no	—
105	C₂×C₈×C₄₇	—	4	1	no	—
106	C₂×C₈×C₄₇	SD₁₆	1	4	yes	complemented
107	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
111	(C₂×D₄₇)⋊C₄	—	4	1	no	—
112	(C₂×D₄₇)⋊C₄	—	4	1	no	—
113	C₄⋊Dic₄₇	—	2	2	no	—
114	C₄⋊Dic₄₇	—	2	1	no	—
115	C₄⋊Dic₄₇	—	2	2	no	—
116	C₄⋊Dic₄₇	—	2	2	no	—
117	C₄⋊Dic₄₇	—	2	1	no	—
118	C₄⋊Dic₄₇	—	2	2	no	—
119	C₄⋊Dic₄₇	—	4	1	no	—
120	C₄⋊Dic₄₇	—	4	1	no	—
121	C₄⋊Dic₄₇	—	4	1	no	—
122	C₄⋊Dic₄₇	—	4	1	no	—
123	C₂×C₄₇⋊C₈	—	4	1	no	—
124	C₂×C₄₇⋊C₈	—	4	2	no	—
125	#10625325	—	4	8	no	—
126	C₂³×C₄₇	#121	1	1	yes	characteristic
127	C₂²×D₄₇	—	2	2	no	—
128	C₂²×D₄₇	—	4	1	no	—
129	C₂²×D₄₇	—	4	2	no	—
130	C₂²×D₄₇	—	4	1	no	—
131	C₂²×D₄₇	—	4	2	no	—
132	C₂×C₄×C₄₇	—	2	2	no	—
133	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
135	C₂×C₄×C₄₇	#120	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×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
141	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
142	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
144	C₂×C₄×C₄₇	#137	1	2	yes	—
145	C₂×Dic₄₇	—	4	2	no	—
146	C₂×Dic₄₇	—	4	2	no	—
147	C₂×Dic₄₇	—	4	1	no	—
148	C₂×Dic₄₇	—	4	2	no	—
149	C₂×Dic₄₇	—	8	1	no	—
150	C₂×Dic₄₇	—	8	1	no	—
151	C₂×Dic₄₇	—	8	2	no	—
152	D₁₈₈	—	2	2	no	—
153	D₁₈₈	—	2	2	no	—
154	D₁₈₈	—	4	1	no	—
155	D₁₈₈	—	4	1	no	—
156	C₄×D₄₇	—	8	2	no	—
157	C₄×D₄₇	—	8	1	no	—
158	C₄×D₄₇	—	8	1	no	—
159	Dic₉₄	—	4	2	no	—
160	Dic₉₄	—	4	2	no	—
161	Dic₉₄	—	8	2	no	—
162	Dic₉₄	—	8	1	no	—
163	Dic₉₄	—	8	1	no	—
164	Dic₉₄	—	8	2	no	—
165	C₈×C₄₇	—	2	4	no	—
166	C₈×C₄₇	—	4	2	no	—
167	C₈×C₄₇	—	4	2	no	—
168	C₄₇⋊C₈	—	8	1	no	—
169	#10271	—	47	1	no	maximal, Sylow
170	C₂²×C₄₇	#392	1	1	yes	characteristic
171	C₂²×C₄₇	#461	1	2	yes	—
172	C₂²×C₄₇	#465	1	1	yes	characteristic
173	C₂²×C₄₇	#468	1	1	yes	characteristic
174	C₂²×C₄₇	#471	1	2	yes	—
175	C₂×D₄₇	—	4	4	no	—
176	C₂×D₄₇	—	4	2	no	—
177	C₂×D₄₇	—	8	2	no	—
178	C₂×D₄₇	—	8	1	no	—
179	C₂×D₄₇	—	8	1	no	—
180	C₄×C₄₇	—	2	1	no	—
181	C₄×C₄₇	—	2	1	no	—
182	C₄×C₄₇	#448	1	1	yes	characteristic
183	C₄×C₄₇	#448	1	2	yes	—
184	C₄×C₄₇	#449	1	1	yes	characteristic
185	C₄×C₄₇	#450	1	1	yes	characteristic
186	C₄×C₄₇	#452	1	1	yes	characteristic
187	C₄×C₄₇	#492	1	1	yes	characteristic
188	C₄×C₄₇	#497	1	1	yes	characteristic
189	Dic₄₇	—	8	2	no	—
190	Dic₄₇	—	8	1	no	—
191	Dic₄₇	—	8	2	no	—
192	C₂×#498	—	47	1	no	—
193	C₂×#476	—	47	1	no	—
194	C₂×#461	—	47	1	no	—
195	#1954	—	47	1	no	—
196	C₂×#333	—	47	1	no	—
197	C₂×#332	—	47	1	no	—
198	#1277	D₄₇	1	1	yes	characteristic, complemented
199	C₂×C₄₇	#1671	1	2	yes	—
200	C₂×C₄₇	#1691	1	2	yes	—
201	C₂×C₄₇	#2000	1	1	yes	characteristic
202	C₂×C₄₇	#2013	1	1	yes	characteristic
203	C₂×C₄₇	#2019	1	1	yes	characteristic
204	D₄₇	—	8	2	no	—
205	C₂²×SD₁₆	—	94	1	no	—
206	C₂×C₄⋊Q₈	—	47	1	no	—
207	C₂×C₄⋊Q₈	—	47	1	no	—
208	C₂×C₄×D₄	—	47	1	no	—
209	#498	—	94	2	no	—
210	#498	—	94	2	no	—
211	#476	—	94	2	no	—
212	#476	—	94	2	no	—
213	#461	—	94	2	no	—
214	#461	—	94	2	no	—
215	C₂×#106	—	94	1	no	—
216	C₂×#106	—	94	2	no	—
217	C₂×C₄⋊C₈	—	47	1	no	—
218	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
220	C₂×#103	—	94	1	no	—
221	C₂×#102	—	94	1	no	—
222	C₂×C₂²⋊C₈	—	94	2	no	—
223	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
224	#333	—	94	2	no	—
225	#332	—	94	2	no	—
226	C₄₇	#10271	1	1	yes	characteristic, complemented, Sylow
227	C₂²×Q₈	—	94	1	no	—
228	C₂²×Q₈	—	94	1	no	—
229	C₂²×D₄	—	47	1	no	—
230	C₂³×C₄	—	94	1	no	—
231	C₂×SD₁₆	—	94	4	no	—
232	C₂×SD₁₆	—	188	2	no	—
233	C₂×SD₁₆	—	188	2	no	—
234	C₂²×C₈	—	2	1	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	94	1	no	—
237	C₂²×C₈	D₁₈₈	1	2	yes	—
238	C₄⋊Q₈	—	47	2	no	—
239	C₄⋊Q₈	—	47	2	no	—
240	C₄⋊Q₈	—	94	1	no	—
241	C₄⋊Q₈	—	94	1	no	—
242	C₄⋊Q₈	—	94	1	no	—
243	C₄⋊Q₈	—	94	1	no	—
244	C₄⋊Q₈	—	94	1	no	—
245	C₄⋊Q₈	—	94	1	no	—
246	C₄×D₄	—	94	1	no	—
247	C₄×D₄	—	94	1	no	—
248	C₄×D₄	—	94	1	no	—
249	C₄×D₄	—	94	1	no	—
250	C₂×C₄⋊C₄	—	47	1	no	—
251	C₂×C₄⋊C₄	—	94	1	no	—
252	C₂×C₄⋊C₄	—	94	1	no	—
253	C₂×C₄⋊C₄	—	94	1	no	—
254	C₂×C₄⋊C₄	—	94	1	no	—
255	C₂×C₂²⋊C₄	—	94	1	no	—
256	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
257	#106	—	94	4	no	—
258	#106	—	94	8	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	94	1	no	—
262	C₄⋊C₈	—	94	1	no	—
263	C₄⋊C₈	D₁₈₈	1	2	yes	—
264	C₄⋊C₈	D₁₈₈	1	2	yes	—
265	#103	—	94	4	no	—
266	#102	—	94	4	no	—
267	C₂²⋊C₈	—	188	4	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	2	1	no	—
270	C₂⁴	—	94	1	no	—
271	C₂×Q₈	—	94	2	no	—
272	C₂×Q₈	—	94	2	no	—
273	C₂×Q₈	—	376	2	no	—
274	C₂×Q₈	—	188	1	no	—
275	C₂×Q₈	—	188	1	no	—
276	C₂×Q₈	—	188	1	no	—
277	C₂×Q₈	—	188	1	no	—
278	C₂×Q₈	—	188	1	no	—
279	C₂×Q₈	—	188	1	no	—
280	C₂×D₄	—	47	2	no	—
281	C₂×D₄	—	94	1	no	—
282	C₂×D₄	—	94	1	no	—
283	C₂×D₄	—	94	1	no	—
284	C₂×D₄	—	94	1	no	—
285	C₂×D₄	—	94	1	no	—
286	C₂²×C₄	—	94	1	no	—
287	C₂²×C₄	—	188	1	no	—
288	C₂²×C₄	—	188	1	no	—
289	C₂²×C₄	—	188	1	no	—
290	C₂²×C₄	—	188	1	no	—
291	C₂²×C₄	—	188	2	no	—
292	C₂²×C₄	—	188	2	no	—
293	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
294	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
295	C₂²×C₄	#10625354	1	1	yes	characteristic
296	SD₁₆	—	188	8	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	1	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	—	188	1	no	—
306	C₂×C₈	—	188	2	no	—
307	C₂×C₈	#10625325	1	4	yes	complemented
308	C₄⋊C₄	—	94	2	no	—
309	C₄⋊C₄	—	94	1	no	—
310	C₄⋊C₄	—	94	1	no	—
311	C₄⋊C₄	—	94	2	no	—
312	C₄⋊C₄	—	94	2	no	—
313	C₄⋊C₄	—	94	2	no	—
314	C₄⋊C₄	—	188	1	no	—
315	C₄⋊C₄	—	188	1	no	—
316	C₄⋊C₄	—	188	1	no	—
317	C₄⋊C₄	—	188	1	no	—
318	C₂²⋊C₄	—	188	1	no	—
319	C₂²⋊C₄	—	188	1	no	—
320	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
321	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
322	C₄²	D₄×D₄₇	1	1	yes	characteristic
323	C₄²	D₄×D₄₇	1	1	yes	characteristic
324	C₂³	—	94	2	no	—
325	C₂³	—	188	1	no	—
326	C₂³	—	188	2	no	—
327	C₂³	—	188	1	no	—
328	C₂³	—	188	2	no	—
329	C₂³	#13093999	1	1	yes	characteristic
330	Q₈	—	376	2	no	—
331	Q₈	—	376	2	no	—
332	Q₈	—	376	1	no	—
333	Q₈	—	376	1	no	—
334	Q₈	—	188	2	no	—
335	Q₈	—	188	2	no	—
336	D₄	—	94	2	no	—
337	D₄	—	94	2	no	—
338	D₄	—	188	1	no	—
339	D₄	—	188	1	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	376	1	no	—
342	C₂×C₄	—	376	1	no	—
343	C₂×C₄	—	376	2	no	—
344	C₂×C₄	—	376	1	no	—
345	C₂×C₄	—	376	1	no	—
346	C₂×C₄	—	376	2	no	—
347	C₂×C₄	—	188	1	no	—
348	C₂×C₄	—	188	2	no	—
349	C₂×C₄	—	188	2	no	—
350	C₂×C₄	—	188	2	no	—
351	C₂×C₄	#13093994	1	1	yes	characteristic
352	C₂×C₄	#13093994	1	1	yes	characteristic
353	C₂×C₄	#13093994	1	1	yes	characteristic
354	C₂×C₄	#13093999	1	1	yes	characteristic
355	C₂×C₄	#13093999	1	1	yes	characteristic
356	C₂×C₄	#13094021	1	1	yes	characteristic
357	C₂×C₄	SD₁₆×D₄₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#10625325	1	2	yes	—
359	C₂×C₄	#13094045	1	1	yes	characteristic
360	C₂×C₄	#13094048	1	1	yes	characteristic
361	C₂×C₄	#13094056	1	1	yes	characteristic
362	C₂×C₄	#13094057	1	2	yes	—
363	C₈	—	2	4	no	—
364	C₈	—	4	2	no	—
365	C₈	—	4	2	no	—
366	C₈	—	376	1	no	—
367	C₂²	—	376	1	no	—
368	C₂²	—	376	2	no	—
369	C₂²	—	376	1	no	—
370	C₂²	—	188	4	no	—
371	C₂²	—	188	2	no	—
372	C₂²	#15689033	1	1	yes	characteristic
373	C₂²	#15689329	1	2	yes	—
374	C₂²	#15689353	1	1	yes	characteristic
375	C₂²	#15689370	1	1	yes	characteristic
376	C₂²	#15689387	1	2	yes	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	376	2	no	—
380	C₄	—	376	2	no	—
381	C₄	—	376	1	no	—
382	C₄	#15689263	1	1	yes	characteristic
383	C₄	#15689264	1	2	yes	—
384	C₄	#15689269	1	1	yes	characteristic
385	C₄	#15689275	1	1	yes	characteristic
386	C₄	#15689288	1	1	yes	characteristic
387	C₄	#15689496	1	1	yes	characteristic
388	C₄	#15689517	1	1	yes	characteristic
389	C₂	—	376	2	no	—
390	C₂	#20242764	1	2	yes	—
391	C₂	#20242888	1	2	yes	—
392	C₂	#20244394	1	1	yes	characteristic
393	C₂	#20244473	1	1	yes	characteristic
394	C₂	#20244505	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13093999
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle	C₂³×C₄₇	#121
prime radical	C₂²×D₁₈₈	D₄
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#10271
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
1. C₂×#15689521	C₂	semidirect
5. C₂×#15688871	C₂	semidirect
6. C₂×#15688870	C₂	semidirect
7. C₄₇×#1277	C₂	semidirect
106. C₂×C₈×C₄₇	SD₁₆	semidirect
198. #1277	D₄₇	semidirect
226. C₄₇	#10271	semidirect
307. C₂×C₈	#10625325	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689521	2
C₂×#15689414	2
C₂×#15689329	2
#20244182	2
C₂×#15688871	2
C₂×#15688870	2
C₄₇×#1277	2
#10271	47