Subgroups — #26216493

Order 12032.

1014 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688993	C₂	1	1	yes	maximal, characteristic
3	#20243522	C₂	1	4	yes	maximal, complemented
4	#20243516	C₂	1	4	yes	maximal, complemented
5	#20243108	C₂	1	4	yes	maximal
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#13093983	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₄₇	—	2	2	no	—
9	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	#15688993	—	2	2	no	—
11	#15688993	C₂²	1	2	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688993	C₂²	1	2	yes	—
14	#15688989	C₂²	1	2	yes	—
15	#15688983	—	2	2	no	—
16	C₄×C₄⋊Dic₄₇	—	2	2	no	—
17	#15688886	C₂²	1	16	yes	—
18	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
21	C₂×#97×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
23	C₂³×Dic₄₇	—	2	2	no	—
24	#13093983	—	2	8	no	—
25	C₂×C₄⋊Dic₄₇	—	2	8	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	—	2	2	no	—
29	C₂×C₄⋊Dic₄₇	—	2	4	no	—
30	C₂×C₄⋊Dic₄₇	—	2	4	no	—
31	C₂×C₄⋊Dic₄₇	—	4	8	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
34	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
35	C₂×C₄×Dic₄₇	—	2	2	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₂³×C₄×C₄₇	D₄	1	2	yes	—
38	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₇	—	2	4	no	—
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
42	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₇	D₄	1	2	yes	—
45	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
46	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	#97×C₄₇	C₂×C₄	1	8	yes	complemented
49	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
50	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	—	2	4	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₇	#41	1	4	yes	—
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	2	4	no	—
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂²×Dic₄₇	—	4	8	no	—
65	C₂²×Dic₄₇	—	4	4	no	—
66	M₁₆×C₄₇	—	2	16	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	4	no	—
69	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
70	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
71	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
74	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
75	C₂²⋊Dic₄₇	—	4	4	no	—
76	C₄⋊Dic₄₇	—	2	8	no	—
77	C₄⋊Dic₄₇	—	2	8	no	—
78	C₄⋊Dic₄₇	—	4	8	no	—
79	C₄⋊Dic₄₇	—	4	4	no	—
80	C₄⋊Dic₄₇	—	4	4	no	—
81	C₄⋊Dic₄₇	—	4	8	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₄×Dic₄₇	—	4	4	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	—	2	2	no	—
86	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₄₇	#124	1	2	yes	—
88	C₂³×C₄₇	#124	1	2	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	—	2	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	16	no	—
93	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₄₇	#117	1	8	yes	—
97	C₂×C₄×C₄₇	#124	1	2	yes	—
98	C₂×C₄×C₄₇	#124	1	2	yes	—
99	C₂×C₄×C₄₇	#125	1	8	yes	—
100	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
101	C₂×C₄×C₄₇	#137	1	4	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
105	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
106	C₂×Dic₄₇	—	4	4	no	—
107	C₂×Dic₄₇	—	4	4	no	—
108	C₂×Dic₄₇	—	4	8	no	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	8	8	no	—
112	C₈×C₄₇	—	2	8	no	—
113	#20764	—	47	1	no	maximal, Sylow
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	2	no	—
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	—	2	4	no	—
118	C₂²×C₄₇	#382	1	4	yes	—
119	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
122	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
123	C₂²×C₄₇	C₂×#137	1	2	yes	—
124	C₄×C₄₇	—	2	4	no	—
125	C₄×C₄₇	#382	1	4	yes	—
126	C₄×C₄₇	#428	1	8	yes	—
127	C₄×C₄₇	C₂×#117	1	2	yes	—
128	C₄×C₄₇	C₂×#125	1	2	yes	—
129	Dic₄₇	—	8	4	no	—
130	Dic₄₇	—	8	4	no	—
131	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
132	#2285	—	47	1	no	—
133	C₂×#384	—	47	1	no	—
134	#1836	—	47	4	no	—
135	#1834	—	47	4	no	—
136	#1736	—	47	4	no	—
137	C₂×C₄₇	—	2	4	no	—
138	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
139	C₂×C₄₇	C₂×#428	1	2	yes	—
140	C₂×C₄₇	#3138	1	1	yes	characteristic
141	C₂×C₄₇	#3139	1	1	yes	characteristic
142	C₂×C₄₇	#3141	1	2	yes	—
143	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
144	C₂×#117	—	47	2	no	—
145	C₂²×C₄⋊C₄	—	47	2	no	—
146	C₂²×C₄⋊C₄	—	94	2	no	—
147	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
148	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
149	C₂×#97	C₂×D₄₇	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
151	#384	—	47	2	no	—
152	#384	—	47	2	no	—
153	#384	—	47	2	no	—
154	#384	—	94	2	no	—
155	#382	—	47	2	no	—
156	#380	—	94	2	no	—
157	C₄×C₄⋊C₄	—	94	2	no	—
158	#340	—	47	16	no	—
159	C₄₇	#20764	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	—	94	2	no	—
161	C₂³×C₄	D₁₈₈	1	2	yes	—
162	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
166	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
167	#117	—	94	8	no	—
168	C₂×C₄⋊C₄	—	47	4	no	—
169	C₂×C₄⋊C₄	—	47	4	no	—
170	C₂×C₄⋊C₄	—	94	8	no	—
171	C₂×C₄⋊C₄	—	94	2	no	—
172	C₂×C₄⋊C₄	—	94	4	no	—
173	C₂×C₄⋊C₄	—	94	4	no	—
174	C₂×C₄⋊C₄	—	94	2	no	—
175	C₂×C₄⋊C₄	—	94	2	no	—
176	C₂×C₄⋊C₄	—	188	8	no	—
177	C₂×C₂²⋊C₄	—	94	2	no	—
178	C₂×C₄²	—	94	2	no	—
179	C₂×C₄²	Dic₉₄	1	4	yes	—
180	C₂×C₄²	D₁₈₈	1	2	yes	—
181	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
182	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
184	#97	C₂×Dic₄₇	1	8	yes	complemented
185	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
186	#95	—	94	4	no	—
187	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	94	4	no	—
191	C₂²×C₄	—	94	4	no	—
192	C₂²×C₄	—	188	8	no	—
193	C₂²×C₄	—	188	4	no	—
194	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
195	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
196	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
197	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
199	C₂²×C₄	#10625350	1	4	yes	—
200	C₂²×C₄	#10625350	1	4	yes	—
201	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
203	M₁₆	—	2	16	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
207	C₄⋊C₄	—	94	8	no	—
208	C₄⋊C₄	—	94	8	no	—
209	C₄⋊C₄	—	188	8	no	—
210	C₄⋊C₄	—	188	4	no	—
211	C₄⋊C₄	—	188	4	no	—
212	C₄⋊C₄	—	188	8	no	—
213	C₄⋊C₄	—	188	4	no	—
214	C₂²⋊C₄	—	188	4	no	—
215	C₄²	—	188	4	no	—
216	C₄²	C₄⋊Dic₄₇	1	8	yes	—
217	C₄²	C₂×Dic₉₄	1	2	yes	—
218	C₄²	C₂×Dic₉₄	1	2	yes	—
219	C₄²	C₂²×Dic₄₇	1	2	yes	—
220	C₄²	C₂²×Dic₄₇	1	2	yes	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
224	C₂³	#13094012	1	2	yes	—
225	C₂³	#13094012	1	2	yes	—
226	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	376	8	no	—
231	C₂×C₄	—	188	4	no	—
232	C₂×C₄	—	188	4	no	—
233	C₂×C₄	—	188	8	no	—
234	C₂×C₄	—	188	4	no	—
235	C₂×C₄	—	188	4	no	—
236	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
237	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
239	C₂×C₄	#13093983	1	8	yes	—
240	C₂×C₄	#13094012	1	2	yes	—
241	C₂×C₄	#13094012	1	2	yes	—
242	C₂×C₄	#13094016	1	8	yes	—
243	C₂×C₄	#13094052	1	4	yes	—
244	C₂×C₄	#13094057	1	4	yes	—
245	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
246	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
247	C₂×C₄	C₂×#10625350	1	2	yes	—
248	C₂×C₄	C₂×#10625350	1	2	yes	—
249	C₈	—	2	8	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	#15688989	1	4	yes	—
255	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
256	C₂²	C₂×#13094012	1	1	yes	characteristic
257	C₂²	C₂×#13094012	1	1	yes	characteristic
258	C₂²	C₂×#13094052	1	2	yes	—
259	C₂²	C₂×#13094057	1	2	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	376	4	no	—
262	C₄	—	376	4	no	—
263	C₄	#15688989	1	4	yes	—
264	C₄	#15689165	1	8	yes	—
265	C₄	C₂×#13093983	1	2	yes	—
266	C₄	C₂×#13094016	1	2	yes	—
267	C₂	—	2	4	no	—
268	C₂	C₂×#15688989	1	1	yes	characteristic
269	C₂	C₂×#15689165	1	2	yes	—
270	C₂	#20253763	1	1	yes	characteristic
271	C₂	#20253769	1	1	yes	characteristic
272	C₂	#20253782	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×D₄
prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20764
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20243522	C₂	semidirect
4. #20243516	C₂	semidirect
20. C₄×M₁₆×C₄₇	C₄	semidirect
48. #97×C₄₇	C₂×C₄	semidirect
49. C₄×C₈×C₄₇	C₂×C₄	semidirect
148. C₄×M₁₆	Dic₄₇	semidirect
159. C₄₇	#20764	semidirect
184. #97	C₂×Dic₄₇	semidirect
185. C₄×C₈	C₂×Dic₄₇	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
C₂×#15688993	2
#20243522	2
#20243516	2
#20243108	2
C₂×C₄×M₁₆×C₄₇	2
#20764	47