Subgroups — #26105748

Order 12032.

662 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689179	C₂	1	1	yes	maximal, characteristic
2	C₂×#15689145	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#10625324	C₂	1	1	yes	maximal, characteristic
4	#20241223	C₂	1	1	yes	maximal, characteristic, complemented
5	#20241222	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241222	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10625324	—	2	1	no	—
9	C₂²×#10625324	C₄	1	2	yes	—
10	C₂×C₄²×D₄₇	C₂²	1	1	yes	characteristic
11	#15689179	—	2	2	no	—
12	#15689179	C₄	1	4	yes	—
13	#15689145	—	2	2	no	—
14	#15689145	C₄	1	4	yes	—
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇⋊C₈	—	2	1	no	—
18	C₂×(C₂×D₄₇)⋊C₈	—	2	1	no	—
19	C₄×#10625324	—	2	2	no	—
20	C₄×#10625324	—	2	2	no	—
21	C₂×#13093934	—	2	1	no	—
22	C₂×#13093934	C₄	1	2	yes	—
23	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₄₇	Q₈	1	1	yes	characteristic
30	C₂×#10625324	—	2	4	no	—
31	C₂×#10625324	—	2	4	no	—
32	C₂×#10625324	—	2	4	no	—
33	C₂×#10625324	—	2	4	no	—
34	C₂×#10625324	—	4	2	no	—
35	C₂×#10625324	—	4	2	no	—
36	C₂²×C₄₇⋊C₈	—	2	1	no	—
37	C₂²×C₄₇⋊C₈	—	2	1	no	—
38	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₇	Q₈	1	1	yes	characteristic
42	C₄²×D₄₇	C₂×C₄	1	2	yes	—
43	C₄²×D₄₇	C₂×C₄	1	2	yes	—
44	C₄²×D₄₇	D₄	1	2	yes	—
45	C₄²×D₄₇	D₄	1	2	yes	—
46	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
49	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
50	Dic₄₇⋊C₈	—	2	4	no	—
51	(C₂×D₄₇)⋊C₈	—	2	4	no	—
52	#13093934	—	2	4	no	—
53	#13093934	—	2	4	no	—
54	C₄×C₄₇⋊C₈	—	2	1	no	—
55	C₄×C₄₇⋊C₈	—	2	1	no	—
56	C₂²×C₈×C₄₇	—	2	1	no	—
57	C₂²×C₈×C₄₇	—	2	1	no	—
58	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₄₇	—	2	1	no	—
61	C₄⋊C₈×C₄₇	—	2	1	no	—
62	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
63	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₄₇	—	2	1	no	—
65	C₄×C₈×C₄₇	—	2	1	no	—
66	C₂³×D₄₇	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
70	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₄₇	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₄₇	—	2	2	no	—
74	C₂×C₄×D₄₇	—	2	2	no	—
75	C₂×C₄×D₄₇	—	2	1	no	—
76	C₂×C₄×D₄₇	—	2	1	no	—
77	C₂×C₄×D₄₇	—	2	1	no	—
78	C₂×C₄×D₄₇	—	2	1	no	—
79	C₂×C₄×D₄₇	—	4	2	no	—
80	C₂×C₄×D₄₇	C₄²	1	2	yes	—
81	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
85	C₂×C₄×D₄₇	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₄₇	—	2	2	no	—
87	C₂×C₈×C₄₇	—	2	2	no	—
88	C₂×C₈×C₄₇	—	2	2	no	—
89	C₂×C₈×C₄₇	—	2	2	no	—
90	C₂×C₈×C₄₇	—	4	1	no	—
91	C₂×C₈×C₄₇	—	4	1	no	—
92	C₂×C₈×C₄₇	—	4	1	no	—
93	C₂×C₈×C₄₇	—	4	1	no	—
94	C₂×C₈×C₄₇	M₁₆	1	4	yes	complemented
95	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₄₇	C₄²	1	2	yes	—
100	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
101	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
102	C₄×Dic₄₇	C₂²⋊C₄	1	2	yes	—
103	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
105	C₂×C₄₇⋊C₈	—	2	2	no	—
106	C₂×C₄₇⋊C₈	—	2	2	no	—
107	C₂×C₄₇⋊C₈	—	2	2	no	—
108	C₂×C₄₇⋊C₈	—	2	2	no	—
109	C₂×C₄₇⋊C₈	—	4	1	no	—
110	C₂×C₄₇⋊C₈	—	4	1	no	—
111	C₂×C₄₇⋊C₈	—	4	1	no	—
112	C₂×C₄₇⋊C₈	—	4	1	no	—
113	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	complemented
114	#10625324	—	2	16	no	—
115	#10625324	—	4	8	no	—
116	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×D₄₇	—	2	2	no	—
118	C₂²×D₄₇	—	2	1	no	—
119	C₂²×D₄₇	—	2	1	no	—
120	C₂²×D₄₇	—	2	2	no	—
121	C₂²×D₄₇	#95	1	2	yes	—
122	C₂×C₄×C₄₇	—	2	2	no	—
123	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
134	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
135	C₂×Dic₄₇	—	2	2	no	—
136	C₂×Dic₄₇	—	2	1	no	—
137	C₂×Dic₄₇	—	2	1	no	—
138	C₂×Dic₄₇	—	2	2	no	—
139	C₂×Dic₄₇	#95	1	2	yes	—
140	C₂×Dic₄₇	#95	1	2	yes	—
141	C₂×Dic₄₇	#95	1	2	yes	—
142	C₄×D₄₇	—	2	2	no	—
143	C₄×D₄₇	—	2	2	no	—
144	C₄×D₄₇	—	2	4	no	—
145	C₄×D₄₇	—	2	2	no	—
146	C₄×D₄₇	—	2	2	no	—
147	C₄×D₄₇	—	2	2	no	—
148	C₄×D₄₇	—	2	2	no	—
149	C₄×D₄₇	—	4	2	no	—
150	C₄×D₄₇	—	4	2	no	—
151	C₈×C₄₇	—	2	4	no	—
152	C₈×C₄₇	—	4	2	no	—
153	C₈×C₄₇	—	4	2	no	—
154	C₄₇⋊C₈	—	2	4	no	—
155	C₄₇⋊C₈	—	4	2	no	—
156	C₄₇⋊C₈	—	4	2	no	—
157	#7876	—	47	1	no	maximal, Sylow
158	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
159	C₂²×C₄₇	#423	1	2	yes	—
160	C₂²×C₄₇	#428	1	1	yes	characteristic
161	C₂²×C₄₇	#428	1	1	yes	characteristic
162	C₂²×C₄₇	#430	1	2	yes	—
163	C₂×D₄₇	—	2	2	no	—
164	C₂×D₄₇	—	2	2	no	—
165	C₂×D₄₇	—	2	2	no	—
166	C₂×D₄₇	—	4	2	no	—
167	C₂×D₄₇	—	4	2	no	—
168	C₄×C₄₇	—	2	1	no	—
169	C₄×C₄₇	—	2	1	no	—
170	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
171	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
172	C₄×C₄₇	#407	1	2	yes	—
173	C₄×C₄₇	#417	1	1	yes	characteristic
174	C₄×C₄₇	#417	1	1	yes	characteristic
175	C₄×C₄₇	#418	1	1	yes	characteristic
176	C₄×C₄₇	#419	1	1	yes	characteristic
177	Dic₄₇	—	2	2	no	—
178	Dic₄₇	—	2	2	no	—
179	Dic₄₇	—	2	2	no	—
180	C₂×#432	—	47	1	no	—
181	C₂×#423	—	47	1	no	—
182	C₂×C₄×M₁₆	—	47	1	no	—
183	#1277	—	47	1	no	—
184	#1277	—	47	1	no	—
185	#1277	—	47	1	no	—
186	#1277	D₄₇	1	1	yes	characteristic, complemented
187	C₂×C₄₇	#1569	1	2	yes	—
188	C₂×C₄₇	#1569	1	2	yes	—
189	C₂×C₄₇	#1728	1	1	yes	characteristic
190	C₂×C₄₇	#1736	1	1	yes	characteristic
191	C₂×C₄₇	#1738	1	1	yes	characteristic
192	D₄₇	—	4	2	no	—
193	C₂²×M₁₆	—	47	2	no	—
194	C₂²×M₁₆	—	94	1	no	—
195	C₂²×C₄²	—	47	1	no	—
196	#432	—	47	4	no	—
197	#432	—	94	2	no	—
198	#423	—	47	4	no	—
199	#423	—	94	2	no	—
200	C₂×C₄⋊C₈	—	47	1	no	—
201	C₂×C₄⋊C₈	—	47	1	no	—
202	C₂×C₄⋊C₈	—	94	1	no	—
203	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
205	C₂×C₂²⋊C₈	—	94	1	no	—
206	C₄×M₁₆	—	94	2	no	—
207	C₄×M₁₆	—	94	2	no	—
208	C₂×#97	—	47	2	no	—
209	C₂×#97	—	94	1	no	—
210	C₂×C₄×C₈	—	47	1	no	—
211	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
212	C₄₇	#7876	1	1	yes	characteristic, complemented, Sylow
213	C₂³×C₄	—	47	1	no	—
214	C₂³×C₄	—	47	1	no	—
215	C₂³×C₄	—	47	1	no	—
216	C₂×M₁₆	—	94	4	no	—
217	C₂×M₁₆	—	94	4	no	—
218	C₂×M₁₆	—	94	4	no	—
219	C₂×M₁₆	—	94	4	no	—
220	C₂×M₁₆	—	188	2	no	—
221	C₂×M₁₆	—	188	2	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	47	2	no	—
225	C₂²×C₈	—	94	1	no	—
226	C₂²×C₈	—	94	1	no	—
227	C₂²×C₈	C₄×D₄₇	1	2	yes	—
228	C₂×C₄²	—	47	1	no	—
229	C₂×C₄²	—	47	1	no	—
230	C₂×C₄²	—	47	1	no	—
231	C₂×C₄²	—	47	2	no	—
232	C₂×C₄²	—	47	2	no	—
233	C₂×C₄²	—	47	2	no	—
234	C₂×C₄²	—	47	2	no	—
235	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	47	2	no	—
239	C₄⋊C₈	—	47	2	no	—
240	C₄⋊C₈	—	94	4	no	—
241	C₄⋊C₈	—	94	1	no	—
242	C₄⋊C₈	—	94	1	no	—
243	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
244	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
245	C₂²⋊C₈	—	94	4	no	—
246	#97	—	94	4	no	—
247	#97	—	94	4	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	2	1	no	—
250	C₄×C₈	—	94	1	no	—
251	C₄×C₈	—	94	1	no	—
252	C₂⁴	—	47	1	no	—
253	C₂²×C₄	—	47	1	no	—
254	C₂²×C₄	—	47	1	no	—
255	C₂²×C₄	—	47	1	no	—
256	C₂²×C₄	—	47	2	no	—
257	C₂²×C₄	—	47	2	no	—
258	C₂²×C₄	—	47	2	no	—
259	C₂²×C₄	—	47	2	no	—
260	C₂²×C₄	—	47	2	no	—
261	C₂²×C₄	—	47	2	no	—
262	C₂²×C₄	—	94	2	no	—
263	C₂²×C₄	—	94	2	no	—
264	C₂²×C₄	—	94	1	no	—
265	C₂²×C₄	—	94	1	no	—
266	C₂²×C₄	—	94	1	no	—
267	C₂²×C₄	—	94	1	no	—
268	C₂²×C₄	—	188	2	no	—
269	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
270	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
271	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
272	M₁₆	—	94	16	no	—
273	M₁₆	—	188	8	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	47	4	no	—
283	C₂×C₈	—	94	2	no	—
284	C₂×C₈	—	94	2	no	—
285	C₂×C₈	—	94	2	no	—
286	C₂×C₈	—	94	2	no	—
287	C₂×C₈	—	188	1	no	—
288	C₂×C₈	—	188	1	no	—
289	C₂×C₈	—	188	1	no	—
290	C₂×C₈	—	188	1	no	—
291	C₂×C₈	#10625324	1	4	yes	complemented
292	C₄²	—	47	2	no	—
293	C₄²	—	47	2	no	—
294	C₄²	—	47	2	no	—
295	C₄²	—	47	2	no	—
296	C₄²	—	47	2	no	—
297	C₄²	—	47	2	no	—
298	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
300	C₄²	D₄×D₄₇	1	1	yes	characteristic
301	C₄²	D₄×D₄₇	1	1	yes	characteristic
302	C₂³	—	47	2	no	—
303	C₂³	—	94	2	no	—
304	C₂³	—	94	1	no	—
305	C₂³	—	94	1	no	—
306	C₂³	—	94	2	no	—
307	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	47	2	no	—
310	C₂×C₄	—	47	2	no	—
311	C₂×C₄	—	47	2	no	—
312	C₂×C₄	—	94	2	no	—
313	C₂×C₄	—	94	1	no	—
314	C₂×C₄	—	94	1	no	—
315	C₂×C₄	—	94	2	no	—
316	C₂×C₄	—	94	2	no	—
317	C₂×C₄	—	94	2	no	—
318	C₂×C₄	—	94	4	no	—
319	C₂×C₄	—	94	2	no	—
320	C₂×C₄	—	94	2	no	—
321	C₂×C₄	—	94	2	no	—
322	C₂×C₄	—	94	2	no	—
323	C₂×C₄	—	188	2	no	—
324	C₂×C₄	—	188	2	no	—
325	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
326	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
327	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
328	C₂×C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
329	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
330	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
331	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
332	C₂×C₄	C₂×#10625324	1	2	yes	—
333	C₂×C₄	#13094048	1	1	yes	characteristic
334	C₂×C₄	#13094051	1	1	yes	characteristic
335	C₂×C₄	#13094053	1	1	yes	characteristic
336	C₂×C₄	#13094056	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	94	4	no	—
341	C₈	—	188	2	no	—
342	C₈	—	188	2	no	—
343	C₂²	—	94	2	no	—
344	C₂²	—	94	2	no	—
345	C₂²	—	94	2	no	—
346	C₂²	—	188	2	no	—
347	C₂²	—	188	2	no	—
348	C₂²	#95×D₄₇	1	1	yes	characteristic
349	C₂²	#15689145	1	2	yes	—
350	C₂²	#15689162	1	1	yes	characteristic
351	C₂²	#15689164	1	1	yes	characteristic
352	C₂²	#15689170	1	2	yes	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	94	2	no	—
356	C₄	—	94	2	no	—
357	C₄	—	94	2	no	—
358	C₄	C₄×#10625324	1	1	yes	characteristic
359	C₄	#15689075	1	1	yes	characteristic
360	C₄	#15689085	1	2	yes	—
361	C₄	#15689121	1	1	yes	characteristic
362	C₄	#15689123	1	1	yes	characteristic
363	C₄	#15689126	1	1	yes	characteristic
364	C₄	#15689130	1	1	yes	characteristic
365	C₂	—	188	2	no	—
366	C₂	#20242152	1	2	yes	—
367	C₂	#20242154	1	2	yes	—
368	C₂	#20243080	1	1	yes	characteristic
369	C₂	#20243106	1	1	yes	characteristic
370	C₂	#20243113	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle	C₂³×C₄₇	C₂×C₄⋊C₄
prime radical	C₂³×D₄₇	C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#7876
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
4. #20241223	C₂	semidirect
5. #20241222	C₂	semidirect
6. #20241222	C₂	semidirect
7. C₄₇×#1277	C₂	semidirect
94. C₂×C₈×C₄₇	M₁₆	semidirect
113. C₂×C₄₇⋊C₈	M₁₆	semidirect
186. #1277	D₄₇	semidirect
212. C₄₇	#7876	semidirect
291. 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×#15689179	2
C₂×#15689145	2
C₂×C₄×#10625324	2
#20241223	2
#20241222	2
C₄₇×#1277	2
#7876	47