Subgroups — #26107112

Order 12032.

538 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689202	C₂	1	1	yes	maximal, characteristic
2	#20243255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13093946	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688852	C₂	1	1	yes	maximal, characteristic
5	#20241228	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	—	2	1	no	—
9	C₂×C₄×D₁₈₈	C₂²	1	1	yes	characteristic
10	#15689202	—	2	2	no	—
11	#15689202	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×#13093946	C₄	1	2	yes	—
15	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
16	C₂×#13093934	—	2	1	no	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
18	#15688886	C₄	1	2	yes	—
19	#15688877	—	2	1	no	—
20	#15688852	C₄	1	4	yes	complemented
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄₇×#322	—	2	2	no	—
25	C₂²×D₁₈₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₇	—	2	1	no	—
27	C₂×#10625324	—	2	4	no	—
28	C₂×#10625324	—	4	2	no	—
29	C₂×#10625324	—	4	2	no	—
30	C₂²×C₄₇⋊C₈	—	2	1	no	—
31	C₂²×C₄₇⋊C₈	—	2	1	no	—
32	C₄×D₁₈₈	—	2	1	no	—
33	C₄×D₁₈₈	—	2	1	no	—
34	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
35	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×(C₂×D₄₇)⋊C₄	—	2	1	no	—
39	C₂×C₄×Dic₄₇	—	2	1	no	—
40	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
41	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
42	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
43	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
44	#13093946	—	2	4	no	—
45	(C₂×D₄₇)⋊C₈	—	2	4	no	—
46	#13093934	—	4	2	no	—
47	C₄₇⋊#95	—	2	1	no	—
48	C₂²×C₈×C₄₇	—	2	1	no	—
49	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₄₇	—	2	1	no	—
54	C₄×C₈×C₄₇	—	2	1	no	—
55	C₄×C₈×C₄₇	—	2	1	no	—
56	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₄₇	D₄	1	2	yes	—
59	C₂³×D₄₇	—	2	1	no	—
60	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
63	C₂²×Dic₄₇	—	2	1	no	—
64	C₂²×Dic₄₇	—	2	2	no	—
65	C₂×D₁₈₈	—	2	2	no	—
66	C₂×D₁₈₈	—	2	2	no	—
67	C₂×D₁₈₈	—	2	1	no	—
68	C₂×D₁₈₈	C₄²	1	2	yes	—
69	C₂×C₄×D₄₇	—	2	2	no	—
70	C₂×C₄×D₄₇	—	2	2	no	—
71	C₂×C₄×D₄₇	—	4	1	no	—
72	C₂×C₄×D₄₇	—	4	1	no	—
73	C₂×C₄×D₄₇	—	4	1	no	—
74	C₂×C₄×D₄₇	—	4	1	no	—
75	C₂×C₈×C₄₇	—	2	2	no	—
76	C₂×C₈×C₄₇	—	2	2	no	—
77	C₂×C₈×C₄₇	—	2	1	no	—
78	C₂×C₈×C₄₇	—	2	1	no	—
79	C₂×C₈×C₄₇	—	2	1	no	—
80	C₂×C₈×C₄₇	—	2	1	no	—
81	C₂×C₈×C₄₇	—	2	2	no	—
82	C₂×C₈×C₄₇	—	4	2	no	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₄₇	M₁₆	1	4	yes	complemented
86	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₄₇	#41	1	1	yes	characteristic
90	(C₂×D₄₇)⋊C₄	—	2	2	no	—
91	(C₂×D₄₇)⋊C₄	—	2	2	no	—
92	C₄⋊Dic₄₇	—	2	1	no	—
93	C₄⋊Dic₄₇	—	2	2	no	—
94	C₄⋊Dic₄₇	—	2	2	no	—
95	C₄⋊Dic₄₇	C₄²	1	2	yes	—
96	C₄×Dic₄₇	—	4	1	no	—
97	C₄×Dic₄₇	—	4	1	no	—
98	C₂×C₄₇⋊C₈	—	2	2	no	—
99	C₂×C₄₇⋊C₈	—	2	2	no	—
100	C₂×C₄₇⋊C₈	—	4	1	no	—
101	C₂×C₄₇⋊C₈	—	4	1	no	—
102	C₂×C₄₇⋊C₈	—	4	1	no	—
103	C₂×C₄₇⋊C₈	—	4	1	no	—
104	#10625324	—	4	8	no	—
105	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
106	C₂²×D₄₇	—	2	2	no	—
107	C₂²×D₄₇	—	4	2	no	—
108	C₂²×D₄₇	—	4	1	no	—
109	C₂²×D₄₇	—	4	1	no	—
110	C₂²×D₄₇	—	4	2	no	—
111	C₂×C₄×C₄₇	—	2	2	no	—
112	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₄₇	#131	1	2	yes	—
120	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
124	C₂×Dic₄₇	—	2	2	no	—
125	C₂×Dic₄₇	—	4	1	no	—
126	C₂×Dic₄₇	—	4	1	no	—
127	C₂×Dic₄₇	—	4	2	no	—
128	C₂×Dic₄₇	—	4	2	no	—
129	C₂×Dic₄₇	—	4	2	no	—
130	D₁₈₈	—	2	2	no	—
131	D₁₈₈	—	2	2	no	—
132	D₁₈₈	—	4	2	no	—
133	C₄×D₄₇	—	4	2	no	—
134	C₄×D₄₇	—	4	2	no	—
135	C₄×D₄₇	—	4	2	no	—
136	C₄×D₄₇	—	4	2	no	—
137	C₈×C₄₇	—	2	2	no	—
138	C₈×C₄₇	—	2	2	no	—
139	C₈×C₄₇	—	2	4	no	—
140	C₈×C₄₇	—	4	2	no	—
141	C₄₇⋊C₈	—	4	2	no	—
142	C₄₇⋊C₈	—	4	2	no	—
143	#8208	—	47	1	no	maximal, Sylow
144	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄₇	#436	1	2	yes	—
146	C₂²×C₄₇	#440	1	1	yes	characteristic
147	C₂²×C₄₇	#442	1	1	yes	characteristic
148	C₂²×C₄₇	#444	1	2	yes	—
149	C₂×D₄₇	—	4	2	no	—
150	C₂×D₄₇	—	4	2	no	—
151	C₂×D₄₇	—	4	2	no	—
152	C₂×D₄₇	—	8	2	no	—
153	C₂×D₄₇	—	8	2	no	—
154	C₄×C₄₇	—	2	1	no	—
155	C₄×C₄₇	—	2	1	no	—
156	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
157	C₄×C₄₇	#405	1	1	yes	characteristic
158	C₄×C₄₇	#417	1	1	yes	characteristic
159	C₄×C₄₇	#418	1	1	yes	characteristic
160	C₄×C₄₇	#435	1	2	yes	—
161	C₄×C₄₇	#440	1	1	yes	characteristic
162	C₄×C₄₇	#442	1	1	yes	characteristic
163	Dic₄₇	—	4	2	no	—
164	Dic₄₇	—	8	2	no	—
165	C₂×#436	—	47	1	no	—
166	#1770	—	47	1	no	—
167	C₄×#102	—	47	1	no	—
168	C₂×D₄⋊C₈	—	47	1	no	—
169	C₂×#322	D₄₇	1	1	yes	characteristic, complemented
170	#1279	—	47	1	no	—
171	#1277	—	47	1	no	—
172	C₂×C₄₇	#1592	1	2	yes	—
173	C₂×C₄₇	#1594	1	2	yes	—
174	C₂×C₄₇	#1753	1	1	yes	characteristic
175	C₂×C₄₇	#1762	1	1	yes	characteristic
176	C₂×C₄₇	#1765	1	1	yes	characteristic
177	D₄₇	—	8	2	no	—
178	C₂²×M₁₆	—	94	1	no	—
179	C₂×C₄×D₄	—	47	1	no	—
180	#436	—	47	4	no	—
181	#436	—	94	2	no	—
182	C₂×C₄⋊C₈	—	47	1	no	—
183	C₂×C₄⋊C₈	—	47	1	no	—
184	C₂×#102	—	47	2	no	—
185	C₂×C₂²⋊C₈	—	94	1	no	—
186	C₂×#97	—	94	1	no	—
187	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	47	1	no	—
191	#340	—	47	2	no	—
192	#336	—	94	1	no	—
193	D₄⋊C₈	—	47	4	no	—
194	#322	—	2	2	no	—
195	C₄₇	#8208	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	47	1	no	—
197	C₂³×C₄	—	94	1	no	—
198	C₂×M₁₆	—	94	4	no	—
199	C₂×M₁₆	—	188	2	no	—
200	C₂×M₁₆	—	188	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	94	1	no	—
203	C₂²×C₈	—	94	1	no	—
204	C₂²×C₈	C₄×D₄₇	1	2	yes	—
205	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
206	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
207	C₄×D₄	—	47	2	no	—
208	C₄×D₄	—	47	2	no	—
209	C₄×D₄	—	94	1	no	—
210	C₄×D₄	—	94	1	no	—
211	C₂×C₄⋊C₄	—	47	1	no	—
212	C₂×C₄⋊C₄	—	47	2	no	—
213	C₂×C₂²⋊C₄	—	94	1	no	—
214	C₂×C₄²	—	94	1	no	—
215	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
216	C₄⋊C₈	—	47	2	no	—
217	C₄⋊C₈	—	47	2	no	—
218	C₄⋊C₈	—	94	1	no	—
219	C₄⋊C₈	—	94	1	no	—
220	#102	—	94	4	no	—
221	C₂²⋊C₈	—	94	4	no	—
222	#97	—	188	2	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	C₄×D₄₇	1	2	yes	—
227	C₄×C₈	C₄×D₄₇	1	2	yes	—
228	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
229	#95	—	94	1	no	—
230	C₂⁴	—	94	1	no	—
231	C₂×D₄	—	47	2	no	—
232	C₂×D₄	—	94	1	no	—
233	C₂×D₄	—	94	2	no	—
234	C₂×D₄	—	94	2	no	—
235	C₂²×C₄	—	94	1	no	—
236	C₂²×C₄	—	94	2	no	—
237	C₂²×C₄	—	94	2	no	—
238	C₂²×C₄	—	94	2	no	—
239	C₂²×C₄	—	188	1	no	—
240	C₂²×C₄	—	188	1	no	—
241	C₂²×C₄	—	188	1	no	—
242	C₂²×C₄	—	188	1	no	—
243	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
245	C₂²×C₄	#10625350	1	1	yes	characteristic
246	M₁₆	—	188	8	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	94	2	no	—
256	C₂×C₈	—	94	2	no	—
257	C₂×C₈	—	188	1	no	—
258	C₂×C₈	—	188	1	no	—
259	C₂×C₈	—	188	1	no	—
260	C₂×C₈	—	188	1	no	—
261	C₂×C₈	#10625324	1	4	yes	complemented
262	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
263	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
264	C₄⋊C₄	—	47	2	no	—
265	C₄⋊C₄	—	94	1	no	—
266	C₄⋊C₄	—	94	2	no	—
267	C₄⋊C₄	—	94	2	no	—
268	C₂²⋊C₄	—	94	2	no	—
269	C₂²⋊C₄	—	94	2	no	—
270	C₄²	—	188	1	no	—
271	C₄²	—	188	1	no	—
272	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
274	C₄²	#10625350	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
276	C₂³	—	94	2	no	—
277	C₂³	—	188	1	no	—
278	C₂³	—	188	1	no	—
279	C₂³	—	188	2	no	—
280	C₂³	—	188	2	no	—
281	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
282	D₄	—	94	2	no	—
283	D₄	—	94	2	no	—
284	D₄	—	188	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	94	2	no	—
287	C₂×C₄	—	188	2	no	—
288	C₂×C₄	—	188	2	no	—
289	C₂×C₄	—	188	2	no	—
290	C₂×C₄	—	188	1	no	—
291	C₂×C₄	—	188	1	no	—
292	C₂×C₄	—	188	2	no	—
293	C₂×C₄	—	188	2	no	—
294	C₂×C₄	—	188	2	no	—
295	C₂×C₄	—	188	2	no	—
296	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	#13093981	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
301	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
302	C₂×C₄	C₂×#10625324	1	2	yes	—
303	C₂×C₄	#13094030	1	2	yes	—
304	C₂×C₄	#13094052	1	1	yes	characteristic
305	C₂×C₄	#13094052	1	1	yes	characteristic
306	C₂×C₄	#13094052	1	1	yes	characteristic
307	C₂×C₄	#13094057	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	4	no	—
311	C₈	—	4	2	no	—
312	C₈	—	188	2	no	—
313	C₈	—	188	2	no	—
314	C₂²	—	376	2	no	—
315	C₂²	—	376	2	no	—
316	C₂²	—	188	2	no	—
317	C₂²	—	188	2	no	—
318	C₂²	—	188	2	no	—
319	C₂²	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
320	C₂²	#15689202	1	2	yes	—
321	C₂²	#15689225	1	1	yes	characteristic
322	C₂²	#15689236	1	1	yes	characteristic
323	C₂²	#15689246	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	376	2	no	—
327	C₄	—	188	2	no	—
328	C₄	C₄×#10625324	1	1	yes	characteristic
329	C₄	#15689077	1	1	yes	characteristic
330	C₄	#15689125	1	1	yes	characteristic
331	C₄	#15689129	1	1	yes	characteristic
332	C₄	#15689194	1	2	yes	—
333	C₄	#15689225	1	1	yes	characteristic
334	C₄	#15689236	1	1	yes	characteristic
335	C₂	—	376	2	no	—
336	C₂	#20242292	1	2	yes	—
337	C₂	#20242303	1	2	yes	—
338	C₂	#20243171	1	1	yes	characteristic
339	C₂	#20243214	1	1	yes	characteristic
340	C₂	#20243232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₁₈₈	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#8208
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #20243255	C₂	semidirect
5. #20241228	C₂	semidirect
6. #20241223	C₂	semidirect
7. C₂×C₄₇×#322	C₂	semidirect
11. #15689202	C₄	semidirect
20. #15688852	C₄	semidirect
85. C₂×C₈×C₄₇	M₁₆	semidirect
169. C₂×#322	D₄₇	semidirect
195. C₄₇	#8208	semidirect
261. C₂×C₈	#10625324	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15689202	2
#20243255	2
C₄×#13093946	2
C₂×#15688852	2
#20241228	2
#20241223	2
C₂×C₄₇×#322	2
#8208	47