Subgroups — #26115065

Order 12032.

582 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
5	C₄×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	2	no	—
10	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
11	D₄⋊(C₄₇⋊C₈)	—	2	4	no	—
12	C₂×#124×C₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#391	C₄	1	1	yes	characteristic
16	C₄₇×#390	C₄	1	1	yes	characteristic
17	C₄₇×#388	—	2	2	no	—
18	C₄₇×#382	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
20	C₄³×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₂²×C₄₇⋊C₈	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
28	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₇	—	2	1	no	—
31	#124×C₄₇	—	2	2	no	—
32	#124×C₄₇	C₈	1	4	yes	complemented
33	C₄×Q₈×C₄₇	—	2	2	no	—
34	C₄×Q₈×C₄₇	—	2	2	no	—
35	C₄×D₄×C₄₇	—	2	2	no	—
36	C₄×D₄×C₄₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
43	C₂×C₄²×C₄₇	—	2	1	no	—
44	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	D₄	1	2	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	#95×C₄₇	—	2	2	no	—
50	#95×C₄₇	—	2	2	no	—
51	C₂⁴×C₄₇	—	2	1	no	—
52	C₂×Q₈×C₄₇	—	2	1	no	—
53	C₂×Q₈×C₄₇	—	2	2	no	—
54	C₂×Q₈×C₄₇	—	2	2	no	—
55	C₂×Q₈×C₄₇	C₂×C₈	1	2	yes	—
56	C₂×D₄×C₄₇	—	2	2	no	—
57	C₂×D₄×C₄₇	—	2	1	no	—
58	C₂×D₄×C₄₇	—	2	2	no	—
59	C₂×D₄×C₄₇	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₄₇	—	2	1	no	—
61	C₂²×C₄×C₄₇	—	2	1	no	—
62	C₂²×C₄×C₄₇	—	2	1	no	—
63	C₂²×C₄×C₄₇	—	4	1	no	—
64	C₂²×C₄×C₄₇	—	4	1	no	—
65	C₂²×C₄×C₄₇	—	4	2	no	—
66	C₂²×C₄×C₄₇	—	4	2	no	—
67	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₄₇	—	2	2	no	—
73	C₄⋊C₄×C₄₇	—	2	2	no	—
74	C₄⋊C₄×C₄₇	—	4	2	no	—
75	C₂²⋊C₄×C₄₇	—	2	8	no	—
76	C₂²⋊C₄×C₄₇	—	4	4	no	—
77	C₂²⋊C₄×C₄₇	—	4	2	no	—
78	C₄²×C₄₇	—	2	2	no	—
79	C₄²×C₄₇	—	2	2	no	—
80	C₄²×C₄₇	—	2	2	no	—
81	C₄²×C₄₇	—	4	2	no	—
82	C₄²×C₄₇	—	4	1	no	—
83	C₄²×C₄₇	—	4	1	no	—
84	C₄²×C₄₇	C₂×C₈	1	2	yes	—
85	C₄²×C₄₇	M₁₆	1	2	yes	—
86	C₄²×C₄₇	C₂×D₄	1	2	yes	—
87	C₄²×C₄₇	#41	1	2	yes	—
88	C₂×C₄₇⋊C₈	—	2	4	no	—
89	C₂×C₄₇⋊C₈	—	4	4	no	—
90	C₂×C₄₇⋊C₈	—	4	2	no	—
91	C₂×C₄₇⋊C₈	—	4	4	no	—
92	C₂³×C₄₇	—	2	2	no	—
93	C₂³×C₄₇	—	4	1	no	—
94	C₂³×C₄₇	—	4	1	no	—
95	C₂³×C₄₇	—	4	2	no	—
96	C₂³×C₄₇	—	4	2	no	—
97	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	Q₈×C₄₇	—	2	4	no	—
99	Q₈×C₄₇	—	4	2	no	—
100	D₄×C₄₇	—	2	4	no	—
101	D₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	2	no	—
104	C₂×C₄×C₄₇	—	2	2	no	—
105	C₂×C₄×C₄₇	—	2	2	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	2	no	—
109	C₂×C₄×C₄₇	—	4	1	no	—
110	C₂×C₄×C₄₇	—	4	1	no	—
111	C₂×C₄×C₄₇	—	4	1	no	—
112	C₂×C₄×C₄₇	—	4	1	no	—
113	C₂×C₄×C₄₇	—	4	2	no	—
114	C₂×C₄×C₄₇	—	4	2	no	—
115	C₂×C₄×C₄₇	—	8	2	no	—
116	C₂×C₄×C₄₇	—	8	1	no	—
117	C₂×C₄×C₄₇	—	8	1	no	—
118	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
119	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₄₇	#121	1	2	yes	—
121	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
122	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
123	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₄₇	#131	1	2	yes	—
126	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
127	C₂×C₄×C₄₇	#137	1	2	yes	—
128	C₄₇⋊C₈	—	4	4	no	—
129	C₄₇⋊C₈	—	8	2	no	—
130	#9640	—	47	1	no	maximal, Sylow
131	C₂²×C₄₇	—	4	2	no	—
132	C₂²×C₄₇	—	4	4	no	—
133	C₂²×C₄₇	—	8	1	no	—
134	C₂²×C₄₇	—	8	1	no	—
135	C₂²×C₄₇	—	8	2	no	—
136	C₂²×C₄₇	#390	1	1	yes	characteristic
137	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₄₇	#408	1	1	yes	characteristic
139	C₂²×C₄₇	#417	1	2	yes	—
140	C₂²×C₄₇	#421	1	2	yes	—
141	C₄×C₄₇	—	2	2	no	—
142	C₄×C₄₇	—	2	4	no	—
143	C₄×C₄₇	—	4	2	no	—
144	C₄×C₄₇	—	4	2	no	—
145	C₄×C₄₇	—	8	1	no	—
146	C₄×C₄₇	—	8	1	no	—
147	C₄×C₄₇	C₈×D₄	1	2	yes	—
148	C₄×C₄₇	#436	1	2	yes	—
149	C₄×C₄₇	#489	1	2	yes	—
150	C₄×C₄₇	#496	1	1	yes	characteristic
151	C₄×C₄₇	#497	1	1	yes	characteristic
152	C₄×#124	D₄₇	1	1	yes	characteristic
153	#1842	—	47	1	no	—
154	#1744	—	47	1	no	—
155	C₂×Q₈⋊C₈	—	47	2	no	—
156	C₂×D₄⋊C₈	—	47	2	no	—
157	C₂×C₄₇	—	8	2	no	—
158	C₂×C₄₇	#1477	1	2	yes	—
159	C₂×C₄₇	#1495	1	2	yes	—
160	C₂×C₄₇	#1954	1	1	yes	characteristic
161	C₂×C₄₇	#1959	1	1	yes	characteristic
162	C₂×C₄₇	#1963	1	1	yes	characteristic
163	C₂×#124	Dic₄₇	1	1	yes	characteristic
164	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
165	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	47	2	no	—
167	C₂×C₄⋊C₈	—	94	2	no	—
168	C₂×C₄×C₈	—	47	2	no	—
169	#391	Dic₄₇	1	1	yes	characteristic
170	#390	Dic₄₇	1	1	yes	characteristic
171	#388	—	2	2	no	—
172	#382	Dic₄₇	1	1	yes	characteristic
173	C₄×C₂²⋊C₄	—	2	2	no	—
174	C₄³	C₂×D₄₇	1	1	yes	characteristic
175	#336	—	94	2	no	—
176	Q₈⋊C₈	—	94	4	no	—
177	D₄⋊C₈	—	94	4	no	—
178	C₄₇	#9640	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
180	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂²×C₈	—	94	2	no	—
183	C₂²×C₈	—	94	2	no	—
184	#124	—	2	2	no	—
185	#124	C₄₇⋊C₈	1	4	yes	complemented
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
201	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
202	C₄⋊C₈	—	94	2	no	—
203	C₄⋊C₈	—	94	2	no	—
204	C₄⋊C₈	—	188	4	no	—
205	C₄×C₈	—	47	4	no	—
206	C₄×C₈	—	94	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	C₂⁴	—	2	1	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	C₂×C₄₇⋊C₈	1	2	yes	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₂×C₄₇⋊C₈	1	2	yes	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
226	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
230	C₂×C₈	—	94	4	no	—
231	C₂×C₈	—	188	2	no	—
232	C₂×C₈	—	188	4	no	—
233	C₂×C₈	—	188	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	2	8	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
247	C₄²	#10625329	1	2	yes	—
248	C₄²	D₄×D₄₇	1	2	yes	—
249	C₄²	#10625352	1	2	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
277	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
278	C₂×C₄	#13093995	1	2	yes	—
279	C₂×C₄	#13094011	1	1	yes	characteristic
280	C₂×C₄	#13094020	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
282	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
283	C₂×C₄	#13094031	1	2	yes	—
284	C₂×C₄	#13094049	1	2	yes	—
285	C₂×C₄	#13094054	1	2	yes	—
286	C₈	—	376	2	no	—
287	C₈	—	188	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	#15689026	1	1	yes	characteristic
294	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
295	C₂²	#15689090	1	1	yes	characteristic
296	C₂²	#15689124	1	2	yes	—
297	C₂²	#15689140	1	2	yes	—
298	C₄	—	2	4	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	1	no	—
303	C₄	—	8	1	no	—
304	C₄	D₄×C₄₇⋊C₈	1	2	yes	—
305	C₄	#15689201	1	2	yes	—
306	C₄	#15689482	1	2	yes	—
307	C₄	#15689513	1	1	yes	characteristic
308	C₄	#15689518	1	1	yes	characteristic
309	C₂	—	8	2	no	—
310	C₂	#20241775	1	2	yes	—
311	C₂	#20241853	1	2	yes	—
312	C₂	#20244183	1	1	yes	characteristic
313	C₂	#20244205	1	1	yes	characteristic
314	C₂	#20244221	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄×#124×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄×#124	D₄₇
hypocenter	C₄₇	#9640
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #20243540	C₂	semidirect
2. #20243134	C₂	semidirect
3. C₂×Q₈⋊(C₄₇⋊C₈)	C₂	semidirect
32. #124×C₄₇	C₈	semidirect
178. C₄₇	#9640	semidirect
185. #124	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20243540	2
#20243134	2
C₂×Q₈⋊(C₄₇⋊C₈)	2
C₂×D₄⋊(C₄₇⋊C₈)	2
C₄×#124×C₄₇	2
#9640	47