Subgroups — #26107474

Order 12032.

446 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093934	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688875	C₂	1	1	yes	maximal, characteristic
5	C₂×#15688873	C₂	1	1	yes	maximal, characteristic
6	#20241223	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#13093934	C₄	1	2	yes	—
11	C₂×#13093934	C₄	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₇	—	2	1	no	—
14	C₂×C₈×Dic₄₇	—	2	1	no	—
15	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
16	#15688877	—	2	1	no	—
17	#15688877	—	2	1	no	—
18	#15688875	C₄	1	4	yes	complemented
19	#15688873	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	2	1	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
31	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
33	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
34	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
35	#13093934	—	2	4	no	—
36	#13093934	C₈	1	8	yes	complemented
37	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
39	C₈×Dic₄₇	—	4	2	no	—
40	C₈×Dic₄₇	—	4	2	no	—
41	C₂²×C₈×C₄₇	—	2	1	no	—
42	C₂²×C₈×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₄₇	—	2	1	no	—
46	C₄⋊C₈×C₄₇	—	2	1	no	—
47	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₄₇	—	2	1	no	—
50	C₄×C₈×C₄₇	—	2	1	no	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
55	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	2	2	no	—
59	C₂×C₈×C₄₇	—	2	2	no	—
60	C₂×C₈×C₄₇	—	4	1	no	—
61	C₂×C₈×C₄₇	—	4	1	no	—
62	C₂×C₈×C₄₇	—	4	1	no	—
63	C₂×C₈×C₄₇	—	4	1	no	—
64	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₄₇	#41	1	1	yes	characteristic
68	C₄²×C₄₇	#41	1	1	yes	characteristic
69	C₄×Dic₄₇	—	2	1	no	—
70	C₄×Dic₄₇	—	2	1	no	—
71	C₄×Dic₄₇	—	2	1	no	—
72	C₄×Dic₄₇	—	2	1	no	—
73	C₄×Dic₄₇	—	4	2	no	—
74	C₄×Dic₄₇	C₂×C₈	1	4	yes	—
75	C₄×Dic₄₇	M₁₆	1	4	yes	—
76	C₂×C₄₇⋊C₈	—	2	2	no	—
77	C₂×C₄₇⋊C₈	—	2	2	no	—
78	C₂×C₄₇⋊C₈	—	2	2	no	—
79	C₂×C₄₇⋊C₈	—	4	1	no	—
80	C₂×C₄₇⋊C₈	—	4	1	no	—
81	C₂×C₄₇⋊C₈	—	4	1	no	—
82	C₂×C₄₇⋊C₈	—	4	1	no	—
83	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
84	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
85	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₇	—	2	2	no	—
88	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
99	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	2	no	—
104	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
105	C₂×Dic₄₇	C₄⋊C₈	1	4	yes	—
106	C₈×C₄₇	—	2	4	no	—
107	C₈×C₄₇	—	4	2	no	—
108	C₈×C₄₇	—	4	2	no	—
109	C₄₇⋊C₈	—	2	2	no	—
110	C₄₇⋊C₈	—	4	2	no	—
111	C₄₇⋊C₈	—	4	2	no	—
112	C₄₇⋊C₈	C₄×C₈	1	4	yes	complemented
113	#8275	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
116	C₂²×C₄₇	#428	1	1	yes	characteristic
117	C₂²×C₄₇	#428	1	1	yes	characteristic
118	C₂²×C₄₇	#430	1	2	yes	—
119	C₄×C₄₇	—	2	1	no	—
120	C₄×C₄₇	—	2	1	no	—
121	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
122	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
123	C₄×C₄₇	#431	1	2	yes	—
124	C₄×C₄₇	#440	1	1	yes	characteristic
125	C₄×C₄₇	#440	1	1	yes	characteristic
126	C₄×C₄₇	#441	1	1	yes	characteristic
127	C₄×C₄₇	#442	1	1	yes	characteristic
128	Dic₄₇	—	2	4	no	—
129	Dic₄₇	—	2	4	no	—
130	C₄×C₄⋊C₈	—	47	1	no	—
131	C₄×C₄⋊C₈	—	47	1	no	—
132	C₄×#97	—	47	1	no	—
133	C₂×#335	—	47	1	no	—
134	C₂×#334	—	47	1	no	—
135	#1277	—	47	1	no	—
136	#1277	D₄₇	1	1	yes	characteristic
137	C₂×C₄₇	#1565	1	2	yes	—
138	C₂×C₄₇	#1565	1	2	yes	—
139	C₂×C₄₇	C₄×C₄⋊C₈	1	1	yes	characteristic
140	C₂×C₄₇	#1776	1	1	yes	characteristic
141	C₂×C₄₇	#1778	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	47	1	no	—
143	C₂×C₄⋊C₈	—	47	1	no	—
144	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
146	C₂×#97	—	47	2	no	—
147	C₂×#97	—	47	2	no	—
148	C₂×C₄×C₈	—	47	1	no	—
149	C₂×C₄×C₈	—	94	1	no	—
150	C₂×C₄×C₈	—	94	1	no	—
151	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
152	C₄³	—	47	1	no	—
153	#336	—	94	1	no	—
154	#336	—	94	1	no	—
155	#335	—	47	4	no	—
156	#334	—	47	4	no	—
157	C₄₇	#8275	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	C₂²×C₈	—	47	1	no	—
161	C₂²×C₈	—	47	1	no	—
162	C₂²×C₈	—	94	1	no	—
163	C₂²×C₈	—	94	1	no	—
164	C₂²×C₈	C₄×D₄₇	1	2	yes	—
165	C₂×C₄²	—	47	2	no	—
166	C₂×C₄²	—	47	1	no	—
167	C₂×C₄²	—	47	1	no	—
168	C₂×C₄²	—	47	2	no	—
169	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
170	C₄⋊C₈	—	2	1	no	—
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	47	2	no	—
173	C₄⋊C₈	—	47	2	no	—
174	C₄⋊C₈	—	94	1	no	—
175	C₄⋊C₈	—	94	1	no	—
176	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
177	C₄⋊C₈	C₄×D₄₇	1	2	yes	—
178	#97	—	47	8	no	—
179	#97	—	94	4	no	—
180	C₄×C₈	—	2	1	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	—	47	2	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	—	188	2	no	—
185	C₄×C₈	—	188	2	no	—
186	C₂²×C₄	—	47	2	no	—
187	C₂²×C₄	—	47	2	no	—
188	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
189	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	47	2	no	—
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	47	2	no	—
202	C₂×C₈	—	94	2	no	—
203	C₂×C₈	—	94	2	no	—
204	C₂×C₈	—	94	2	no	—
205	C₂×C₈	—	188	1	no	—
206	C₂×C₈	—	188	1	no	—
207	C₂×C₈	—	188	1	no	—
208	C₂×C₈	—	188	1	no	—
209	C₂×C₈	C₈×D₄₇	1	4	yes	—
210	C₄²	—	47	4	no	—
211	C₄²	—	47	4	no	—
212	C₄²	—	94	1	no	—
213	C₄²	—	94	1	no	—
214	C₄²	—	94	1	no	—
215	C₄²	—	94	1	no	—
216	C₄²	—	188	2	no	—
217	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
219	C₄²	#10625352	1	1	yes	characteristic
220	C₄²	#10625352	1	1	yes	characteristic
221	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	47	4	no	—
224	C₂×C₄	—	47	4	no	—
225	C₂×C₄	—	94	2	no	—
226	C₂×C₄	—	94	2	no	—
227	C₂×C₄	—	94	2	no	—
228	C₂×C₄	—	94	2	no	—
229	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
230	C₂×C₄	#13093980	1	1	yes	characteristic
231	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
232	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
233	C₂×C₄	#13093989	1	1	yes	characteristic
234	C₂×C₄	#13093989	1	1	yes	characteristic
235	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
236	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
237	C₂×C₄	#13094048	1	1	yes	characteristic
238	C₂×C₄	#13094051	1	1	yes	characteristic
239	C₂×C₄	#13094053	1	1	yes	characteristic
240	C₂×C₄	#13094056	1	1	yes	characteristic
241	C₈	—	2	4	no	—
242	C₈	—	4	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	47	4	no	—
245	C₈	—	94	2	no	—
246	C₈	—	188	2	no	—
247	C₈	—	188	2	no	—
248	C₂²	#15688974	1	1	yes	characteristic
249	C₂²	C₄⋊C₈×D₄₇	1	2	yes	—
250	C₂²	#15689162	1	1	yes	characteristic
251	C₂²	#15689164	1	1	yes	characteristic
252	C₂²	#15689170	1	2	yes	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	94	4	no	—
256	C₄	—	94	4	no	—
257	C₄	C₄×C₈×D₄₇	1	1	yes	characteristic
258	C₄	#15689075	1	1	yes	characteristic
259	C₄	#15689173	1	2	yes	—
260	C₄	#15689220	1	1	yes	characteristic
261	C₄	#15689226	1	1	yes	characteristic
262	C₄	#15689227	1	1	yes	characteristic
263	C₄	#15689232	1	1	yes	characteristic
264	C₂	#20242137	1	2	yes	—
265	C₂	#20242139	1	2	yes	—
266	C₂	#20243236	1	1	yes	characteristic
267	C₂	#20243284	1	1	yes	characteristic
268	C₂	#20243293	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1277	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1277	D₄₇
hypocenter	C₄₇	#8275
LCS	G ▷ 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
18. #15688875	C₄	semidirect
19. #15688873	C₄	semidirect
36. #13093934	C₈	semidirect
112. C₄₇⋊C₈	C₄×C₈	semidirect
157. C₄₇	#8275	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₄₇	2
C₄×#13093934	2
C₂×#15688875	2
C₂×#15688873	2
#20241223	2
C₄₇×#1277	2
#8275	47