Subgroups — #26107498

Order 12032.

394 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243243	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243133	C₂	1	1	yes	maximal, characteristic
4	C₂×#15688874	C₂	1	1	yes	maximal, characteristic
5	#20241222	C₂	1	1	yes	maximal, characteristic
6	#20241222	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₇×#335	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	—	2	1	no	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₄₇	—	2	1	no	—
13	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15688877	—	2	1	no	—
15	#15688877	—	2	1	no	—
16	#15688877	C₄	1	2	yes	—
17	#15688877	C₄	1	2	yes	—
18	#15688874	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄₇×#335	C₄	1	4	yes	complemented
23	C₂²×C₄₇⋊C₈	—	2	1	no	—
24	C₂²×C₄₇⋊C₈	—	2	1	no	—
25	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄×Dic₄₇	Q₈	1	2	yes	—
30	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
31	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
33	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	2	yes	—
34	#13093934	—	4	2	no	—
35	C₄×C₄₇⋊C₈	—	2	1	no	—
36	C₄×C₄₇⋊C₈	—	2	1	no	—
37	C₈×Dic₄₇	—	4	2	no	—
38	C₂²×C₈×C₄₇	—	2	1	no	—
39	C₂²×C₈×C₄₇	—	2	1	no	—
40	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	—	2	1	no	—
44	C₄⋊C₈×C₄₇	—	2	1	no	—
45	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
46	C₄⋊C₈×C₄₇	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
53	C₂²×Dic₄₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₄₇	—	2	2	no	—
55	C₂×C₈×C₄₇	—	2	2	no	—
56	C₂×C₈×C₄₇	—	2	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₂×C₈×C₄₇	—	4	1	no	—
59	C₂×C₈×C₄₇	—	4	1	no	—
60	C₂×C₈×C₄₇	—	4	1	no	—
61	C₂×C₈×C₄₇	—	4	1	no	—
62	C₂×C₈×C₄₇	C₄²	1	2	yes	—
63	C₂×C₈×C₄₇	C₄²	1	2	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₄₇	#41	1	1	yes	characteristic
67	C₄²×C₄₇	#41	1	1	yes	characteristic
68	C₄×Dic₄₇	—	2	2	no	—
69	C₄×Dic₄₇	—	2	2	no	—
70	C₄×Dic₄₇	—	2	1	no	—
71	C₄×Dic₄₇	—	2	1	no	—
72	C₄×Dic₄₇	—	2	1	no	—
73	C₄×Dic₄₇	—	2	1	no	—
74	C₄×Dic₄₇	—	2	4	no	—
75	C₂×C₄₇⋊C₈	—	2	2	no	—
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	—	2	2	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	—	4	1	no	—
84	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	—	2	2	no	—
86	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
88	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₇	#131	1	2	yes	—
93	C₂×C₄×C₄₇	#131	1	2	yes	—
94	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
97	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
98	C₂×Dic₄₇	—	2	2	no	—
99	C₂×Dic₄₇	—	2	2	no	—
100	C₂×Dic₄₇	—	2	2	no	—
101	C₂×Dic₄₇	—	2	2	no	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	#108	1	4	yes	—
104	C₈×C₄₇	—	2	2	no	—
105	C₈×C₄₇	—	2	2	no	—
106	C₈×C₄₇	—	4	2	no	—
107	C₈×C₄₇	—	4	2	no	—
108	C₄₇⋊C₈	—	4	2	no	—
109	C₄₇⋊C₈	—	4	2	no	—
110	C₄₇⋊C₈	—	4	2	no	—
111	#8279	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄₇	#424	1	2	yes	—
114	C₂²×C₄₇	#428	1	1	yes	characteristic
115	C₂²×C₄₇	#428	1	1	yes	characteristic
116	C₂²×C₄₇	C₂×#108	1	2	yes	—
117	C₄×C₄₇	—	2	1	no	—
118	C₄×C₄₇	—	2	1	no	—
119	C₄×C₄₇	#405	1	1	yes	characteristic
120	C₄×C₄₇	#405	1	1	yes	characteristic
121	C₄×C₄₇	#433	1	2	yes	—
122	C₄×C₄₇	#440	1	1	yes	characteristic
123	C₄×C₄₇	#440	1	1	yes	characteristic
124	C₄×C₄₇	#441	1	1	yes	characteristic
125	C₄×C₄₇	#442	1	1	yes	characteristic
126	Dic₄₇	—	2	4	no	—
127	Dic₄₇	—	4	2	no	—
128	#1767	—	47	1	no	—
129	C₄×C₄⋊C₈	—	47	1	no	—
130	#1744	—	47	1	no	—
131	C₂×#335	D₄₇	1	1	yes	characteristic
132	C₂×#334	—	47	1	no	—
133	#1277	—	47	1	no	—
134	#1277	—	47	1	no	—
135	C₂×C₄₇	#1573	1	2	yes	—
136	C₂×C₄₇	#1574	1	2	yes	—
137	C₂×C₄₇	#1768	1	1	yes	characteristic
138	C₂×C₄₇	#1776	1	1	yes	characteristic
139	C₂×C₄₇	C₄×#108	1	1	yes	characteristic
140	C₂×C₄⋊C₈	—	47	1	no	—
141	C₂×C₄⋊C₈	—	47	1	no	—
142	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
143	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×#97	—	94	1	no	—
145	C₂×C₄×C₈	—	47	1	no	—
146	C₂×C₄×C₈	—	94	1	no	—
147	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
148	C₄³	—	47	1	no	—
149	#336	—	47	2	no	—
150	#336	—	47	2	no	—
151	#336	—	94	1	no	—
152	#336	—	94	1	no	—
153	#335	Dic₄₇	1	4	yes	complemented
154	#334	—	47	4	no	—
155	C₄₇	#8279	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	2	1	no	—
158	C₂²×C₈	—	47	2	no	—
159	C₂²×C₈	—	94	1	no	—
160	C₂²×C₈	—	94	1	no	—
161	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
162	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
163	C₂×C₄²	—	47	2	no	—
164	C₂×C₄²	—	47	1	no	—
165	C₂×C₄²	—	47	1	no	—
166	C₂×C₄²	—	47	2	no	—
167	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
168	C₄⋊C₈	—	2	1	no	—
169	C₄⋊C₈	—	2	1	no	—
170	C₄⋊C₈	—	47	2	no	—
171	C₄⋊C₈	—	47	2	no	—
172	C₄⋊C₈	—	94	1	no	—
173	C₄⋊C₈	—	94	1	no	—
174	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
175	C₄⋊C₈	C₂×Dic₄₇	1	2	yes	—
176	#97	—	188	2	no	—
177	C₄×C₈	—	94	1	no	—
178	C₄×C₈	—	94	1	no	—
179	C₄×C₈	—	188	2	no	—
180	C₄×C₈	C₄×D₄₇	1	2	yes	—
181	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
182	C₂²×C₄	—	47	2	no	—
183	C₂²×C₄	—	47	2	no	—
184	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
185	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
186	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	94	2	no	—
196	C₂×C₈	—	94	2	no	—
197	C₂×C₈	—	94	2	no	—
198	C₂×C₈	—	94	2	no	—
199	C₂×C₈	—	94	2	no	—
200	C₂×C₈	—	188	1	no	—
201	C₂×C₈	—	188	1	no	—
202	C₂×C₈	—	188	1	no	—
203	C₂×C₈	—	188	1	no	—
204	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
205	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
206	C₄²	—	94	4	no	—
207	C₄²	—	94	2	no	—
208	C₄²	—	94	2	no	—
209	C₄²	—	94	1	no	—
210	C₄²	—	94	1	no	—
211	C₄²	—	94	1	no	—
212	C₄²	—	94	1	no	—
213	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
214	C₄²	#10625352	1	1	yes	characteristic
215	C₄²	#10625354	1	1	yes	characteristic
216	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
217	C₂³	C₄⋊C₄×D₄₇	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	47	4	no	—
220	C₂×C₄	—	94	2	no	—
221	C₂×C₄	—	94	2	no	—
222	C₂×C₄	—	94	2	no	—
223	C₂×C₄	—	94	2	no	—
224	C₂×C₄	—	94	2	no	—
225	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
226	C₂×C₄	#13093982	1	1	yes	characteristic
227	C₂×C₄	#13093986	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
229	C₂×C₄	#13093989	1	1	yes	characteristic
230	C₂×C₄	Q₈×Dic₄₇	1	1	yes	characteristic
231	C₂×C₄	#13094030	1	2	yes	—
232	C₂×C₄	#13094032	1	2	yes	—
233	C₂×C₄	#13094048	1	1	yes	characteristic
234	C₂×C₄	#13094048	1	1	yes	characteristic
235	C₂×C₄	#13094056	1	1	yes	characteristic
236	C₂×C₄	#13094056	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	4	2	no	—
240	C₈	—	4	2	no	—
241	C₈	—	188	2	no	—
242	C₈	—	188	2	no	—
243	C₈	—	188	2	no	—
244	C₂²	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
245	C₂²	#15689148	1	2	yes	—
246	C₂²	#15689164	1	1	yes	characteristic
247	C₂²	#15689164	1	1	yes	characteristic
248	C₂²	#108×D₄₇	1	2	yes	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	94	4	no	—
252	C₄	—	188	2	no	—
253	C₄	#15689078	1	1	yes	characteristic
254	C₄	#15689081	1	1	yes	characteristic
255	C₄	#15689183	1	2	yes	—
256	C₄	#15689222	1	1	yes	characteristic
257	C₄	#15689222	1	1	yes	characteristic
258	C₄	#15689230	1	1	yes	characteristic
259	C₄	#15689235	1	1	yes	characteristic
260	C₂	#20242172	1	2	yes	—
261	C₂	#20242178	1	2	yes	—
262	C₂	#20243247	1	1	yes	characteristic
263	C₂	#20243287	1	1	yes	characteristic
264	C₂	#108×Dic₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#335	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#335	D₄₇
hypocenter	C₄₇	#8279
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
18. #15688874	C₄	semidirect
22. C₄₇×#335	C₄	semidirect
153. #335	Dic₄₇	semidirect
155. C₄₇	#8279	semidirect

Maximal subgroups

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

subgroup	index
#20243243	2
C₄⋊C₈×Dic₄₇	2
#20243133	2
C₂×#15688874	2
#20241222	2
C₂×C₄₇×#335	2
#8279	47