Subgroups — #26106880

Order 12032.

718 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
4	#20241229	C₂	1	2	yes	maximal, complemented
5	C₄²×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	2	no	—
11	D₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×D₄×C₄₇	C₄	1	2	yes	—
13	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₇	—	2	1	no	—
15	C₄×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊C₄×C₄₇	—	2	2	no	—
18	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	complemented
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	4	2	no	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
27	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	—	2	2	no	—
29	C₂³×C₄×C₄₇	—	2	1	no	—
30	C₄×D₄×C₄₇	—	2	4	no	—
31	C₄×D₄×C₄₇	—	2	2	no	—
32	C₄×D₄×C₄₇	C₈	1	8	yes	complemented
33	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
38	C₂×C₄²×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	—	4	4	no	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	#95×C₄₇	—	2	2	no	—
47	#95×C₄₇	—	2	1	no	—
48	C₂⁴×C₄₇	—	2	1	no	—
49	C₂×D₄×C₄₇	—	2	1	no	—
50	C₂×D₄×C₄₇	—	2	2	no	—
51	C₂×D₄×C₄₇	C₄²	1	2	yes	—
52	C₂×D₄×C₄₇	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₄₇	—	2	1	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	8	no	—
56	C₂²×C₄×C₄₇	—	2	4	no	—
57	C₂²×C₄×C₄₇	—	4	4	no	—
58	C₂²×C₄×C₄₇	—	4	4	no	—
59	C₂²×C₄×C₄₇	—	4	2	no	—
60	C₂²×C₄×C₄₇	—	4	1	no	—
61	C₂²×C₄×C₄₇	—	4	1	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₄₇	—	2	1	no	—
68	C₄⋊C₄×C₄₇	—	2	2	no	—
69	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
70	C₄⋊C₄×C₄₇	C₂×C₈	1	4	yes	—
71	C₂²⋊C₄×C₄₇	—	2	8	no	—
72	C₂²⋊C₄×C₄₇	—	2	4	no	—
73	C₄²×C₄₇	—	2	2	no	—
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	—	2	1	no	—
76	C₄²×C₄₇	—	4	2	no	—
77	C₄²×C₄₇	—	4	4	no	—
78	C₄²×C₄₇	—	4	2	no	—
79	C₄²×C₄₇	C₂×C₈	1	4	yes	—
80	C₄²×C₄₇	M₁₆	1	4	yes	—
81	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₄₇	#41	1	2	yes	—
83	C₂×C₄₇⋊C₈	—	2	4	no	—
84	C₂×C₄₇⋊C₈	—	2	4	no	—
85	C₂×C₄₇⋊C₈	—	4	4	no	—
86	C₂×C₄₇⋊C₈	—	4	2	no	—
87	C₂×C₄₇⋊C₈	—	4	2	no	—
88	C₂³×C₄₇	—	2	2	no	—
89	C₂³×C₄₇	—	2	4	no	—
90	C₂³×C₄₇	—	4	1	no	—
91	C₂³×C₄₇	—	4	1	no	—
92	C₂³×C₄₇	—	4	2	no	—
93	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	D₄×C₄₇	—	2	2	no	—
95	D₄×C₄₇	—	2	4	no	—
96	D₄×C₄₇	C₄×C₈	1	4	yes	complemented
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	2	2	no	—
99	C₂×C₄×C₄₇	—	2	2	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	2	2	no	—
102	C₂×C₄×C₄₇	—	2	2	no	—
103	C₂×C₄×C₄₇	—	2	4	no	—
104	C₂×C₄×C₄₇	—	4	16	no	—
105	C₂×C₄×C₄₇	—	4	1	no	—
106	C₂×C₄×C₄₇	—	4	2	no	—
107	C₂×C₄×C₄₇	—	4	1	no	—
108	C₂×C₄×C₄₇	—	4	2	no	—
109	C₂×C₄×C₄₇	—	4	4	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
112	C₂×C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
115	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
121	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
122	C₄₇⋊C₈	—	4	4	no	—
123	C₄₇⋊C₈	—	4	4	no	—
124	#8170	—	47	1	no	maximal, Sylow
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	4	4	no	—
127	C₂²×C₄₇	—	4	4	no	—
128	C₂²×C₄₇	—	4	2	no	—
129	C₂²×C₄₇	—	4	2	no	—
130	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
132	C₂²×C₄₇	#417	1	1	yes	characteristic
133	C₂²×C₄₇	#418	1	1	yes	characteristic
134	C₂²×C₄₇	#421	1	2	yes	—
135	C₄×C₄₇	—	2	4	no	—
136	C₄×C₄₇	—	2	4	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	4	4	no	—
139	C₄×C₄₇	—	4	2	no	—
140	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
142	C₄×C₄₇	#431	1	2	yes	—
143	C₄×C₄₇	#440	1	2	yes	—
144	C₄×C₄₇	#442	1	2	yes	—
145	C₄²×D₄	D₄₇	1	1	yes	characteristic
146	C₄×C₄⋊C₈	—	47	1	no	—
147	#1744	—	47	1	no	—
148	C₂×D₄⋊C₈	—	47	2	no	—
149	#1279	—	47	2	no	—
150	C₂×C₄₇	—	4	4	no	—
151	C₂×C₄₇	#1477	1	2	yes	—
152	C₂×C₄₇	#1478	1	2	yes	—
153	C₂×C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₄₇	#1759	1	1	yes	characteristic
155	C₂×C₄₇	#1762	1	1	yes	characteristic
156	C₂×C₄×D₄	Dic₄₇	1	2	yes	—
157	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
158	C₂²×C₄²	—	2	1	no	—
159	C₂×C₄⋊C₈	—	47	2	no	—
160	C₂×C₄×C₈	—	47	2	no	—
161	C₂×C₄×C₈	—	94	1	no	—
162	C₄×C₄⋊C₄	Dic₄₇	1	2	yes	—
163	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
164	C₄×C₂²⋊C₄	—	2	1	no	—
165	C₄×C₂²⋊C₄	—	2	2	no	—
166	C₄³	C₂×D₄₇	1	1	yes	characteristic
167	#336	—	94	2	no	—
168	#336	—	94	1	no	—
169	D₄⋊C₈	—	47	8	no	—
170	C₄₇	#8170	1	1	yes	characteristic, complemented, Sylow
171	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
172	C₂³×C₄	—	2	2	no	—
173	C₂³×C₄	—	2	1	no	—
174	C₂²×C₈	—	94	2	no	—
175	C₂²×C₈	—	94	2	no	—
176	C₄×D₄	—	2	4	no	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	C₄₇⋊C₈	1	8	yes	complemented
179	C₄×D₄	C₄×D₄₇	1	4	yes	—
180	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	1	no	—
186	C₂×C₄²	—	4	4	no	—
187	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
188	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
191	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
192	C₄⋊C₈	—	47	4	no	—
193	C₄⋊C₈	—	94	2	no	—
194	C₄×C₈	—	47	4	no	—
195	C₄×C₈	—	94	2	no	—
196	C₄×C₈	—	188	2	no	—
197	#95	—	2	2	no	—
198	#95	—	2	1	no	—
199	C₂⁴	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₂×C₄₇⋊C₈	1	4	yes	—
203	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
218	C₂×C₈	—	94	4	no	—
219	C₂×C₈	—	94	4	no	—
220	C₂×C₈	—	188	4	no	—
221	C₂×C₈	—	188	2	no	—
222	C₂×C₈	—	188	2	no	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₂×C₄₇⋊C₈	1	4	yes	—
226	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
227	C₂²⋊C₄	—	2	8	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	4	2	no	—
233	C₄²	—	4	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	C₂×C₄₇⋊C₈	1	4	yes	—
236	C₄²	#10625329	1	4	yes	—
237	C₄²	C₂×C₄×D₄₇	1	2	yes	—
238	C₄²	#10625350	1	2	yes	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	4	1	no	—
243	C₂³	—	4	2	no	—
244	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
245	D₄	—	2	2	no	—
246	D₄	—	2	4	no	—
247	D₄	C₄×C₄₇⋊C₈	1	4	yes	complemented
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	16	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
263	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
265	C₂×C₄	#13093983	1	1	yes	characteristic
266	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
267	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
268	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
269	C₂×C₄	C₂×#10625329	1	2	yes	—
270	C₂×C₄	#13094049	1	1	yes	characteristic
271	C₂×C₄	#13094050	1	2	yes	—
272	C₂×C₄	#13094054	1	1	yes	characteristic
273	C₈	—	188	4	no	—
274	C₈	—	188	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
281	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
282	C₂²	#15689124	1	1	yes	characteristic
283	C₂²	#15689128	1	1	yes	characteristic
284	C₂²	#15689140	1	2	yes	—
285	C₄	—	2	4	no	—
286	C₄	—	2	4	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
291	C₄	C₄×#10625329	1	1	yes	characteristic
292	C₄	#15689176	1	2	yes	—
293	C₄	#15689223	1	2	yes	—
294	C₄	#15689233	1	2	yes	—
295	C₂	—	4	4	no	—
296	C₂	#20241775	1	2	yes	—
297	C₂	#20241779	1	2	yes	—
298	C₂	C₄×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
299	C₂	#20243198	1	1	yes	characteristic
300	C₂	#20243213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄²×D₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₄²×D₄	D₄₇
hypocenter	C₄₇	#8170
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×D₄

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. #20243134	C₂	semidirect
4. #20241229	C₂	semidirect
11. D₄⋊(C₄₇⋊C₈)	C₄	semidirect
23. C₄⋊(C₄₇⋊C₈)	C₂×C₄	semidirect
26. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
32. C₄×D₄×C₄₇	C₈	semidirect
96. D₄×C₄₇	C₄×C₈	semidirect
170. C₄₇	#8170	semidirect
178. C₄×D₄	C₄₇⋊C₈	semidirect
247. D₄	C₄×C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243134	2
C₂×D₄⋊(C₄₇⋊C₈)	2
#20241229	2
C₄²×D₄×C₄₇	2
#8170	47