Subgroups — #26181865

Order 12032.

922 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20252344	C₂	1	2	yes	maximal
2	C₄×#13094030	C₂	1	1	yes	maximal, characteristic
3	#20241670	C₂	1	2	yes	maximal, complemented
4	#20241669	C₂	1	2	yes	maximal, complemented
5	#20241667	C₂	1	4	yes	maximal, complemented
6	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈²×D₄₇	C₂	1	2	yes	maximal, complemented
8	C₄₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13094030	C₄	1	2	yes	—
10	C₄×#10625350	C₂²	1	1	yes	characteristic
11	#15689247	C₄	1	4	yes	—
12	#15689188	C₄	1	4	yes	—
13	(C₂²⋊D₄₇)⋊C₈	C₄	1	8	yes	—
14	#15689174	C₂²	1	4	yes	—
15	#15689078	C₄	1	2	yes	—
16	#15689077	C₄	1	4	yes	—
17	C₄×#10625329	C₂²	1	1	yes	characteristic
18	C₄×#10625324	C₂²	1	2	yes	—
19	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
20	C₄×C₈×D₄₇	C₂²	1	4	yes	—
21	C₄×C₈×D₄₇	C₂²	1	2	yes	—
22	#15688845	C₂²	1	4	yes	—
23	C₈×C₄₇⋊C₈	C₂²	1	2	yes	—
24	C₈×C₄₇⋊C₈	C₂²	1	4	yes	—
25	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
27	C₄₇×#431	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
29	C₄₇×#322	C₂²	1	2	yes	—
30	C₈²×C₄₇	C₂²	1	2	yes	—
31	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
32	#13094030	C₈	1	16	yes	complemented
33	C₂×#10625329	C₂×C₄	1	2	yes	—
34	C₂×#10625324	C₂×C₄	1	4	yes	—
35	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₄₇	C₂×C₄	1	8	yes	—
37	C₂×C₈×D₄₇	C₂×C₄	1	4	yes	—
38	C₄×Dic₉₄	C₂×C₄	1	2	yes	—
39	C₄×D₁₈₈	C₂×C₄	1	2	yes	—
40	C₄×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
41	#13093983	C₂×C₄	1	1	yes	characteristic
42	#13093981	C₂×C₄	1	2	yes	—
43	C₄²×D₄₇	C₂³	1	2	yes	—
44	Dic₄₇⋊C₈	C₂×C₄	1	8	yes	—
45	(C₂×D₄₇)⋊C₈	C₂×C₄	1	8	yes	—
46	#13093935	C₂×C₄	1	2	yes	—
47	#13093934	C₂×C₄	1	4	yes	—
48	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
49	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
50	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
51	C₄×C₄₇⋊C₈	C₂³	1	2	yes	—
52	C₈×Dic₄₇	C₂×C₄	1	8	yes	—
53	C₈×Dic₄₇	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
57	C₂²⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₄₇	C₂³	1	4	yes	—
60	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	C₄²	1	2	yes	—
65	#10625350	C₂×C₈	1	8	yes	—
66	C₂×D₁₈₈	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₄₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×D₄₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₉₄	C₄²	1	1	yes	characteristic
70	C₂×C₈×C₄₇	C₂×C₈	1	8	yes	complemented
71	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
72	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₄₇	C₄²	1	2	yes	—
78	(C₂×D₄₇)⋊C₄	C₄²	1	4	yes	—
79	C₄⋊Dic₄₇	C₄²	1	2	yes	—
80	Dic₄₇⋊C₄	C₄²	1	4	yes	—
81	C₄×Dic₄₇	C₂²×C₄	1	2	yes	—
82	C₄×Dic₄₇	C₂²×C₄	1	4	yes	—
83	#10625329	C₂×C₈	1	8	yes	complemented
84	C₂×C₄₇⋊C₈	—	2	8	no	—
85	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
86	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄₇⋊C₈	C₂²×C₄	1	4	yes	—
88	#10625324	C₂×C₈	1	16	yes	complemented
89	C₈×D₄₇	—	2	16	no	—
90	C₈×D₄₇	C₂×C₈	1	16	yes	complemented
91	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
92	C₂²×D₄₇	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂³×C₄	1	2	yes	—
99	C₂²⋊D₄₇	C₄×C₈	1	8	yes	—
100	C₂×Dic₄₇	C₂×C₄²	1	4	yes	—
101	C₂×Dic₄₇	C₂×C₄²	1	2	yes	—
102	D₁₈₈	C₄×C₈	1	4	yes	—
103	C₄×D₄₇	—	2	8	no	—
104	C₄×D₄₇	C₂²×C₈	1	8	yes	—
105	Dic₉₄	C₄×C₈	1	4	yes	—
106	C₈×C₄₇	—	2	8	no	—
107	C₈×C₄₇	C₂²×C₈	1	4	yes	—
108	C₈×C₄₇	C₂²×C₈	1	4	yes	—
109	C₄₇⋊C₈	—	2	4	no	—
110	C₄₇⋊C₈	C₂²×C₈	1	8	yes	—
111	C₄₇⋊C₈	#131	1	8	yes	complemented
112	C₈×#131	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	C₂×C₄×C₈	1	2	yes	—
115	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
116	C₂×D₄₇	—	2	4	no	—
117	C₂×D₄₇	C₂×C₄×C₈	1	4	yes	—
118	C₄×C₄₇	—	2	2	no	—
119	C₄×C₄₇	C₂×C₄×C₈	1	2	yes	—
120	C₄×C₄₇	C₂³×C₈	1	2	yes	—
121	C₄×C₄₇	C₂×#131	1	4	yes	—
122	Dic₄₇	—	2	4	no	—
123	Dic₄₇	C₂×C₄×C₈	1	4	yes	—
124	C₈×#41	—	47	2	no	—
125	C₄×#131	—	47	1	no	—
126	#1450	—	47	2	no	—
127	#1450	D₄₇	1	1	yes	characteristic, complemented
128	C₈×M₁₆	—	47	2	no	—
129	C₈×M₁₆	—	47	4	no	—
130	C₂×C₈²	—	47	1	no	—
131	C₂×C₈²	—	47	2	no	—
132	C₂×C₄₇	—	2	2	no	—
133	C₂×C₄₇	C₂²×C₄×C₈	1	1	yes	characteristic
134	C₂×C₄₇	C₄×#131	1	2	yes	—
135	D₄₇	—	2	4	no	—
136	C₂×#131	—	47	2	no	—
137	C₄×#41	—	47	1	no	—
138	C₈×Q₈	—	47	4	no	—
139	C₈×D₄	—	47	4	no	—
140	C₈×D₄	—	47	8	no	—
141	#431	—	47	4	no	—
142	#431	C₂×D₄₇	1	2	yes	—
143	#405	—	47	2	no	—
144	#405	—	47	4	no	—
145	C₄×M₁₆	—	47	1	no	—
146	C₄×M₁₆	—	47	2	no	—
147	C₂×C₄×C₈	—	47	2	no	—
148	C₂×C₄×C₈	—	47	4	no	—
149	C₂×C₄×C₈	—	47	2	no	—
150	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
151	#322	—	47	4	no	—
152	#322	C₂×D₄₇	1	2	yes	—
153	C₈²	—	47	4	no	—
154	C₈²	—	47	2	no	—
155	C₈²	—	47	1	no	—
156	C₈²	—	47	1	no	—
157	C₈²	C₂×D₄₇	1	2	yes	—
158	C₄₇	C₈×#131	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	47	1	no	—
160	#131	—	47	16	no	—
161	C₂×M₁₆	—	47	2	no	—
162	C₂×M₁₆	—	47	4	no	—
163	C₂²×C₈	—	47	4	no	—
164	C₂²×C₈	—	47	4	no	—
165	C₂²×C₈	—	47	8	no	—
166	C₂²×C₈	C₄×D₄₇	1	2	yes	—
167	C₄×Q₈	—	47	2	no	—
168	C₄×D₄	—	47	4	no	—
169	C₄×D₄	—	47	2	no	—
170	#117	—	47	1	no	—
171	#117	—	47	2	no	—
172	C₂×C₄²	—	47	2	no	—
173	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	C₄⋊C₈	—	47	8	no	—
175	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
176	C₂²⋊C₈	—	47	8	no	—
177	C₂²⋊C₈	C₄×D₄₇	1	4	yes	—
178	#97	—	47	4	no	—
179	#97	—	47	2	no	—
180	C₄×C₈	—	47	4	no	—
181	C₄×C₈	—	47	8	no	—
182	C₄×C₈	—	47	4	no	—
183	C₄×C₈	—	47	2	no	—
184	C₄×C₈	—	47	2	no	—
185	C₄×C₈	—	47	2	no	—
186	C₄×C₈	C₄×D₄₇	1	2	yes	—
187	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₄₇	1	4	yes	—
189	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
190	#41	—	47	8	no	—
191	C₂×Q₈	—	47	1	no	—
192	C₂×D₄	—	47	2	no	—
193	C₂×D₄	—	47	1	no	—
194	C₂²×C₄	—	47	2	no	—
195	C₂²×C₄	—	47	4	no	—
196	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
198	M₁₆	—	47	8	no	—
199	M₁₆	—	47	16	no	—
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	47	4	no	—
202	C₂×C₈	—	47	16	no	—
203	C₂×C₈	—	47	4	no	—
204	C₂×C₈	—	94	16	no	—
205	C₂×C₈	—	94	8	no	—
206	C₂×C₈	C₈×D₄₇	1	8	yes	complemented
207	C₂×C₈	C₂×C₄×D₄₇	1	8	yes	—
208	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
210	C₄⋊C₄	—	47	4	no	—
211	C₄⋊C₄	—	47	2	no	—
212	C₂²⋊C₄	—	47	2	no	—
213	C₂²⋊C₄	—	47	4	no	—
214	C₄²	—	47	2	no	—
215	C₄²	—	47	4	no	—
216	C₄²	C₂×C₄×D₄₇	1	2	yes	—
217	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
218	C₄²	C₂³×D₄₇	1	1	yes	characteristic
219	C₂³	—	47	2	no	—
220	C₂³	C₄²×D₄₇	1	1	yes	characteristic
221	Q₈	—	47	4	no	—
222	D₄	—	47	8	no	—
223	D₄	—	47	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	47	2	no	—
226	C₂×C₄	—	47	8	no	—
227	C₂×C₄	—	47	4	no	—
228	C₂×C₄	—	94	8	no	—
229	C₂×C₄	C₄²×D₄₇	1	2	yes	—
230	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₄₇	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₄₇	1	2	yes	—
234	C₈	—	2	8	no	—
235	C₈	—	47	8	no	—
236	C₈	—	47	8	no	—
237	C₈	—	94	4	no	—
238	C₈	C₂×C₈×D₄₇	1	4	yes	—
239	C₈	C₂×C₈×D₄₇	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	47	4	no	—
242	C₂²	—	94	4	no	—
243	C₂²	C₄×C₈×D₄₇	1	2	yes	—
244	C₂²	C₂×C₄²×D₄₇	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	47	4	no	—
247	C₄	—	94	4	no	—
248	C₄	C₄×C₈×D₄₇	1	2	yes	—
249	C₄	C₂²×C₈×D₄₇	1	2	yes	—
250	C₄	#131×D₄₇	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	94	4	no	—
253	C₂	C₂×C₄×C₈×D₄₇	1	1	yes	characteristic
254	C₂	#20251810	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₄₇
commutator	C₂×C₄₇	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₄₇×#1450	C₂
socle	C₂²×C₄₇	C₂²×C₄²
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	#1450	D₄₇
hypocenter	C₄₇	C₈×#131
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20241670	C₂	semidirect
4. #20241669	C₂	semidirect
5. #20241667	C₂	semidirect
6. C₂×C₈×C₄₇⋊C₈	C₂	semidirect
7. C₈²×D₄₇	C₂	semidirect
8. C₄₇×#1450	C₂	semidirect
32. #13094030	C₈	semidirect
70. C₂×C₈×C₄₇	C₂×C₈	semidirect
83. #10625329	C₂×C₈	semidirect
88. #10625324	C₂×C₈	semidirect
90. C₈×D₄₇	C₂×C₈	semidirect
111. C₄₇⋊C₈	#131	semidirect
127. #1450	D₄₇	semidirect
158. C₄₇	C₈×#131	semidirect
206. C₂×C₈	C₈×D₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20252344	2
C₄×#13094030	2
#20241670	2
#20241669	2
#20241667	2
C₂×C₈×C₄₇⋊C₈	2
C₈²×D₄₇	2
C₄₇×#1450	2
C₈×#131	47