Subgroups — #26216876

Order 12032.

930 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245795	C₂	1	2	yes	maximal
2	#20243548	C₂	1	1	yes	maximal, characteristic, complemented
3	#20243545	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243516	C₂	1	2	yes	maximal, complemented
5	#20243421	C₂	1	4	yes	maximal
6	#20243285	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093983	—	2	2	no	—
9	C₂×#13093958	C₂²	1	4	yes	—
10	C₂×#13093956	C₂²	1	4	yes	—
11	#15688993	C₂²	1	2	yes	—
12	#15688993	C₂²	1	2	yes	—
13	#15688989	—	2	2	no	—
14	#15688983	—	2	2	no	—
15	#15688983	C₂²	1	4	yes	—
16	C₄×C₄⋊Dic₄₇	—	2	2	no	—
17	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
18	#15688886	C₂²	1	8	yes	—
19	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
22	C₂×#97×C₄₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
25	#13093983	—	4	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	2	no	—
30	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	4	no	—
33	C₂×C₄×Dic₄₇	—	2	4	no	—
34	#13093958	—	2	8	no	—
35	#13093956	—	2	8	no	—
36	C₄₇⋊#95	—	2	4	no	—
37	C₄₇⋊#95	—	2	4	no	—
38	C₂³×C₄×C₄₇	D₄	1	2	yes	—
39	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₄₇	—	2	4	no	—
41	C₂×M₁₆×C₄₇	—	2	4	no	—
42	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₇	D₄	1	2	yes	—
47	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
50	#97×C₄₇	C₂×C₄	1	8	yes	—
51	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₇	—	2	4	no	—
55	C₂²×C₄×C₄₇	—	2	4	no	—
56	C₂²×C₄×C₄₇	—	2	2	no	—
57	C₂²×C₄×C₄₇	—	2	2	no	—
58	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₄₇	—	2	4	no	—
66	C₂²×Dic₄₇	—	2	4	no	—
67	M₁₆×C₄₇	—	2	16	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₄₇	#41	1	4	yes	—
73	C₂×C₈×C₄₇	#41	1	4	yes	—
74	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
75	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
78	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
79	C₂²⋊Dic₄₇	—	4	8	no	—
80	C₄⋊Dic₄₇	—	2	8	no	—
81	C₄⋊Dic₄₇	—	2	8	no	—
82	C₄⋊Dic₄₇	—	4	4	no	—
83	C₄⋊Dic₄₇	—	4	4	no	—
84	C₄⋊Dic₄₇	—	4	4	no	—
85	C₄⋊Dic₄₇	—	4	4	no	—
86	C₄×Dic₄₇	—	4	8	no	—
87	C₂³×C₄₇	—	2	1	no	—
88	C₂³×C₄₇	—	2	1	no	—
89	C₂³×C₄₇	—	2	2	no	—
90	C₂³×C₄₇	—	2	2	no	—
91	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₇	—	2	1	no	—
94	C₂×C₄×C₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	4	no	—
98	C₂×C₄×C₄₇	—	2	4	no	—
99	C₂×C₄×C₄₇	—	2	16	no	—
100	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
104	C₂×C₄×C₄₇	#137	1	4	yes	—
105	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
108	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
109	C₂×Dic₄₇	—	4	4	no	—
110	C₂×Dic₄₇	—	4	4	no	—
111	C₂×Dic₄₇	—	4	4	no	—
112	C₂×Dic₄₇	—	4	4	no	—
113	C₂×Dic₄₇	—	4	4	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₈×C₄₇	—	2	4	no	—
116	C₈×C₄₇	#117	1	8	yes	complemented
117	#20824	—	47	1	no	maximal, Sylow
118	C₂²×C₄₇	—	2	4	no	—
119	C₂²×C₄₇	—	2	2	no	—
120	C₂²×C₄₇	—	4	2	no	—
121	C₂²×C₄₇	—	4	1	no	—
122	C₂²×C₄₇	—	4	1	no	—
123	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₄₇	#532	1	1	yes	characteristic
125	C₂²×C₄₇	#532	1	1	yes	characteristic
126	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
127	C₂²×C₄₇	C₂×#137	1	2	yes	—
128	C₄×C₄₇	—	2	4	no	—
129	C₄×C₄₇	—	2	2	no	—
130	C₄×C₄₇	#428	1	8	yes	—
131	C₄×C₄₇	C₂×#117	1	2	yes	—
132	C₄×C₄₇	#533	1	2	yes	—
133	Dic₄₇	—	8	4	no	—
134	Dic₄₇	—	8	4	no	—
135	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
136	#2285	—	47	2	no	—
137	#1845	—	47	1	no	—
138	#1844	—	47	1	no	—
139	#1834	—	47	2	no	—
140	#1809	—	47	4	no	—
141	#1776	—	47	4	no	—
142	C₂×C₄₇	—	4	2	no	—
143	C₂×C₄₇	#2285	1	1	yes	characteristic
144	C₂×C₄₇	C₂×#428	1	2	yes	—
145	C₂×C₄₇	#3182	1	2	yes	—
146	C₂×C₄₇	#3184	1	1	yes	characteristic
147	C₂×C₄₇	#3185	1	1	yes	characteristic
148	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
149	C₂×#117	—	94	2	no	—
150	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
151	C₂×#107	—	47	4	no	—
152	C₂×#106	—	47	4	no	—
153	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
154	C₂×#97	C₂×D₄₇	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
157	#384	—	47	2	no	—
158	#384	—	47	2	no	—
159	#382	—	94	2	no	—
160	#380	—	47	4	no	—
161	#380	—	94	2	no	—
162	C₄×C₄⋊C₄	—	47	4	no	—
163	C₄×C₄⋊C₄	—	94	2	no	—
164	#340	—	47	8	no	—
165	C₄₇	#20824	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	D₁₈₈	1	2	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
171	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
172	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
173	#117	—	188	8	no	—
174	C₂×C₄⋊C₄	—	47	4	no	—
175	C₂×C₄⋊C₄	—	47	4	no	—
176	C₂×C₄⋊C₄	—	94	2	no	—
177	C₂×C₄⋊C₄	—	94	2	no	—
178	C₂×C₄⋊C₄	—	94	2	no	—
179	C₂×C₄⋊C₄	—	94	2	no	—
180	C₂×C₂²⋊C₄	—	94	4	no	—
181	C₂×C₄²	—	94	4	no	—
182	C₂×C₄²	Dic₉₄	1	4	yes	—
183	C₂×C₄²	D₁₈₈	1	2	yes	—
184	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
185	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
187	#107	—	94	8	no	—
188	#106	—	94	8	no	—
189	#97	C₂×Dic₄₇	1	8	yes	—
190	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
191	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
192	#95	—	94	4	no	—
193	#95	—	94	4	no	—
194	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	94	4	no	—
200	C₂²×C₄	—	94	4	no	—
201	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
202	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
203	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
205	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	#10625350	1	4	yes	—
212	C₂×C₈	#10625350	1	4	yes	—
213	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
214	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
215	C₄⋊C₄	—	94	8	no	—
216	C₄⋊C₄	—	94	8	no	—
217	C₄⋊C₄	—	188	4	no	—
218	C₄⋊C₄	—	188	4	no	—
219	C₄⋊C₄	—	188	4	no	—
220	C₄⋊C₄	—	188	4	no	—
221	C₂²⋊C₄	—	188	8	no	—
222	C₄²	—	188	8	no	—
223	C₄²	C₄⋊Dic₄₇	1	8	yes	—
224	C₄²	C₂×Dic₉₄	1	2	yes	—
225	C₄²	C₂×Dic₉₄	1	2	yes	—
226	C₄²	C₂²×Dic₄₇	1	2	yes	—
227	C₄²	C₂²×Dic₄₇	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
233	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	16	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	4	no	—
245	C₂×C₄	—	188	4	no	—
246	C₂×C₄	—	188	4	no	—
247	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
250	C₂×C₄	#13094052	1	4	yes	—
251	C₂×C₄	#13094057	1	4	yes	—
252	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625350	1	2	yes	—
255	C₂×C₄	C₂×#10625350	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	#13093983	1	8	yes	complemented
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
264	C₂²	#15689673	1	1	yes	characteristic
265	C₂²	#15689673	1	1	yes	characteristic
266	C₂²	C₂×#13094052	1	2	yes	—
267	C₂²	C₂×#13094057	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	376	4	no	—
271	C₄	—	376	4	no	—
272	C₄	#15689165	1	8	yes	—
273	C₄	C₂×#13093983	1	2	yes	—
274	C₄	#15689680	1	2	yes	—
275	C₂	—	4	2	no	—
276	C₂	#20245795	1	1	yes	characteristic
277	C₂	C₂×#15689165	1	2	yes	—
278	C₂	#20254106	1	2	yes	—
279	C₂	#20254126	1	1	yes	characteristic
280	C₂	#20254133	1	1	yes	characteristic

	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	#20824
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
2. #20243548	C₂	semidirect
3. #20243545	C₂	semidirect
4. #20243516	C₂	semidirect
6. #20243285	C₂	semidirect
21. C₄×M₁₆×C₄₇	C₄	semidirect
116. C₈×C₄₇	#117	semidirect
153. C₄×M₁₆	Dic₄₇	semidirect
165. C₄₇	#20824	semidirect
257. C₈	#13093983	semidirect

Maximal subgroups

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

subgroup	index
#20245795	2
#20243548	2
#20243545	2
#20243516	2
#20243421	2
#20243285	2
C₂×C₄×M₁₆×C₄₇	2
#20824	47