Subgroups — #26134033

Order 12032.

332 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245146	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15688906	C₂	1	1	yes	maximal, characteristic
3	#20245001	C₂	1	1	yes	maximal, characteristic
4	#20243225	C₂	1	1	yes	maximal, characteristic
5	#20241476	C₂	1	1	yes	maximal, characteristic, complemented
6	#20241470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13094030	C₂²	1	1	yes	characteristic
9	C₂×#13093964	C₂²	1	1	yes	characteristic
10	C₂×#13093964	C₂²	1	1	yes	characteristic
11	C₂×#13093952	C₄	1	1	yes	characteristic
12	#15689120	C₄	1	1	yes	characteristic
13	#15689078	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₁₆	—	2	2	no	—
15	#15688906	—	2	2	no	—
16	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
17	#15688888	C₄	1	1	yes	characteristic
18	#15688883	C₄	1	1	yes	characteristic
19	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₄₇	—	2	2	no	—
23	C₂×#10625350	C₂×C₄	1	1	yes	characteristic
24	#13094030	—	2	1	no	—
25	#13094030	—	2	1	no	—
26	#13094030	—	2	1	no	—
27	#13094030	—	2	1	no	—
28	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
29	C₂×#10625329	C₂×C₄	1	1	yes	characteristic
30	C₂×#10625324	—	2	1	no	—
31	C₂×C₈×D₄₇	—	2	1	no	—
32	#13093983	C₂×C₄	1	1	yes	characteristic
33	#13093964	—	2	1	no	—
34	#13093964	—	2	1	no	—
35	#13093964	—	2	1	no	—
36	#13093964	—	2	1	no	—
37	C₂×C₄₇⋊C₁₆	—	2	1	no	—
38	C₂×C₄₇⋊C₁₆	—	2	1	no	—
39	#13093952	—	2	2	no	—
40	#13093949	C₈	1	2	yes	—
41	#13093946	C₈	1	2	yes	—
42	#13093934	—	2	1	no	—
43	C₈×Dic₄₇	—	2	1	no	—
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₄₇	—	2	1	no	—
48	C₂×C₁₆×C₄₇	—	2	1	no	—
49	C₂×C₁₆×C₄₇	—	2	1	no	—
50	C₂×C₁₆×C₄₇	—	2	1	no	—
51	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
52	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
53	C₄×C₈×C₄₇	—	2	1	no	—
54	C₄×C₈×C₄₇	—	2	1	no	—
55	C₂²×C₄×C₄₇	—	2	1	no	—
56	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₄₇	—	2	1	no	—
58	#10625350	—	2	1	no	—
59	#10625350	—	2	1	no	—
60	#10625350	—	2	1	no	—
61	#10625350	—	2	1	no	—
62	C₂×D₁₈₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₄₇	—	2	1	no	—
64	C₂×Dic₉₄	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₄₇	—	2	1	no	—
66	C₂×C₈×C₄₇	—	2	1	no	—
67	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
71	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
72	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
73	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
74	C₄²×C₄₇	—	2	1	no	—
75	C₄²×C₄₇	M₁₆	1	2	yes	—
76	C₂²⋊Dic₄₇	—	2	1	no	—
77	C₄⋊Dic₄₇	C₂×C₈	1	2	yes	—
78	C₄×Dic₄₇	—	2	1	no	—
79	#10625329	—	2	1	no	—
80	#10625329	—	2	1	no	—
81	#10625329	—	2	1	no	—
82	#10625329	—	2	1	no	—
83	C₂×C₄₇⋊C₈	—	2	1	no	—
84	C₂×C₄₇⋊C₈	—	2	1	no	—
85	#10625324	—	4	1	no	—
86	#10625324	—	4	1	no	—
87	C₈×D₄₇	—	4	1	no	—
88	C₈×D₄₇	—	4	1	no	—
89	C₁₆×C₄₇	—	2	2	no	—
90	C₁₆×C₄₇	—	2	2	no	—
91	C₄₇⋊C₁₆	—	4	1	no	—
92	C₄₇⋊C₁₆	—	4	1	no	—
93	C₂³×C₄₇	#131	1	1	yes	characteristic
94	C₂²×D₄₇	—	2	1	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#131	1	1	yes	characteristic
103	C₂²⋊D₄₇	—	4	1	no	—
104	C₂²⋊D₄₇	—	4	1	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	—	2	1	no	—
107	D₁₈₈	—	2	1	no	—
108	D₁₈₈	—	2	1	no	—
109	C₄×D₄₇	—	4	1	no	—
110	C₄×D₄₇	—	4	1	no	—
111	Dic₉₄	—	2	1	no	—
112	Dic₉₄	—	2	1	no	—
113	C₈×C₄₇	—	2	2	no	—
114	C₈×C₄₇	#121	1	1	yes	characteristic
115	C₈×C₄₇	#123	1	1	yes	characteristic
116	C₈×C₄₇	#124	1	1	yes	characteristic
117	C₈×C₄₇	#126	1	1	yes	characteristic
118	C₄₇⋊C₈	—	4	1	no	—
119	C₄₇⋊C₈	—	4	1	no	—
120	#13124	—	47	1	no	maximal, Sylow
121	C₂²×C₄₇	—	2	2	no	—
122	C₂²×C₄₇	#433	1	1	yes	characteristic
123	C₂²×C₄₇	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₄₇	#435	1	1	yes	characteristic
125	C₂×D₄₇	—	4	2	no	—
126	C₂×D₄₇	—	4	1	no	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	#388	1	1	yes	characteristic
129	C₄×C₄₇	#431	1	1	yes	characteristic
130	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
131	C₄×C₄₇	#436	1	1	yes	characteristic
132	Dic₄₇	—	4	2	no	—
133	Dic₄₇	—	4	1	no	—
134	#2151	—	47	1	no	—
135	C₂×#350	—	47	1	no	—
136	#2113	—	47	1	no	—
137	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic, complemented
138	#1764	—	47	1	no	—
139	#1379	—	47	1	no	—
140	#1376	—	47	1	no	—
141	C₂×C₄₇	—	2	1	no	—
142	C₂×C₄₇	—	2	1	no	—
143	C₂×C₄₇	#1923	1	1	yes	characteristic
144	C₂×C₄₇	#2170	1	2	yes	—
145	D₄₇	—	8	1	no	—
146	C₂×#131	—	47	1	no	—
147	C₂×M₃₂	—	47	1	no	—
148	C₂×M₃₂	—	47	1	no	—
149	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
151	C₂×#104	—	47	1	no	—
152	#416	—	47	1	no	—
153	#405	—	47	1	no	—
154	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
155	C₄⋊C₁₆	—	94	2	no	—
156	#350	—	94	2	no	—
157	C₂²⋊C₁₆	—	94	2	no	—
158	C₄×C₁₆	—	2	2	no	—
159	#341	—	47	1	no	—
160	#339	—	47	1	no	—
161	C₄₇	#13124	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	47	1	no	—
163	#131	—	94	1	no	—
164	#131	—	94	1	no	—
165	#131	—	94	1	no	—
166	#131	—	94	1	no	—
167	C₂×M₁₆	—	47	1	no	—
168	C₂×M₁₆	—	47	1	no	—
169	C₂×M₁₆	—	94	1	no	—
170	C₂²×C₈	—	94	1	no	—
171	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
173	#117	—	47	1	no	—
174	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
175	M₃₂	—	94	1	no	—
176	M₃₂	—	94	1	no	—
177	M₃₂	—	94	1	no	—
178	M₃₂	—	94	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	94	1	no	—
184	C₂×C₁₆	—	94	1	no	—
185	C₂×C₁₆	D₁₈₈	1	2	yes	—
186	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
187	#104	—	94	2	no	—
188	#103	—	47	2	no	—
189	#102	—	47	2	no	—
190	#97	—	94	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	94	1	no	—
194	#41	—	94	1	no	—
195	#41	—	94	1	no	—
196	#41	—	94	1	no	—
197	#41	—	94	1	no	—
198	C₂×Q₈	—	47	1	no	—
199	C₂×D₄	—	47	1	no	—
200	C₂×D₄	—	94	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	94	1	no	—
203	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
204	M₁₆	—	94	1	no	—
205	M₁₆	—	94	1	no	—
206	M₁₆	—	94	1	no	—
207	M₁₆	—	94	1	no	—
208	M₁₆	—	188	1	no	—
209	M₁₆	—	188	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	94	1	no	—
213	C₂×C₈	—	94	1	no	—
214	C₂×C₈	—	188	1	no	—
215	C₂×C₈	—	188	1	no	—
216	C₂×C₈	C₈×D₄₇	1	2	yes	—
217	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
218	C₂×C₈	#10625350	1	1	yes	characteristic
219	C₂×C₈	#10625350	1	1	yes	characteristic
220	C₂×C₈	#10625350	1	1	yes	characteristic
221	C₂×C₈	#10625350	1	1	yes	characteristic
222	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
223	C₄⋊C₄	—	47	2	no	—
224	C₂²⋊C₄	—	94	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	94	1	no	—
227	C₄²	#10625324	1	2	yes	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	188	1	no	—
231	C₁₆	—	188	1	no	—
232	C₂³	—	94	1	no	—
233	C₂³	#13094030	1	1	yes	characteristic
234	Q₈	—	94	1	no	—
235	Q₈	—	94	1	no	—
236	D₄	—	94	1	no	—
237	D₄	—	94	1	no	—
238	D₄	—	188	1	no	—
239	D₄	—	188	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	94	2	no	—
243	C₂×C₄	—	94	1	no	—
244	C₂×C₄	—	188	1	no	—
245	C₂×C₄	—	188	1	no	—
246	C₂×C₄	#13093981	1	1	yes	characteristic
247	C₂×C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
248	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
250	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
251	C₂×C₄	#13094030	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	188	1	no	—
254	C₈	—	188	1	no	—
255	C₈	#13093998	1	1	yes	characteristic
256	C₈	#13094007	1	1	yes	characteristic
257	C₈	#13094012	1	1	yes	characteristic
258	C₈	#13094018	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	188	1	no	—
261	C₂²	—	188	2	no	—
262	C₂²	#15689182	1	1	yes	characteristic
263	C₂²	C₈×D₁₈₈	1	1	yes	characteristic
264	C₂²	#15689194	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	188	1	no	—
267	C₄	—	188	2	no	—
268	C₄	#15689014	1	1	yes	characteristic
269	C₄	#15689174	1	1	yes	characteristic
270	C₄	C₈×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₄	#15689202	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	376	1	no	—
275	C₂	#20243994	1	1	yes	characteristic
276	C₂	#20245226	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₈×D₄
prime radical	C₂×#10625350	C₂×C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#13124
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #20245146	C₂	semidirect
5. #20241476	C₂	semidirect
6. #20241470	C₂	semidirect
7. C₂×C₄×C₁₆×C₄₇	C₂	semidirect
137. C₂×C₄×C₁₆	D₄₇	semidirect
161. C₄₇	#13124	semidirect

Maximal subgroups

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

subgroup	index
#20245146	2
C₂×#15688906	2
#20245001	2
#20243225	2
#20241476	2
#20241470	2
C₂×C₄×C₁₆×C₄₇	2
#13124	47