Subgroups — #26133646

Order 12032.

526 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688927	C₂	1	2	yes	maximal, complemented
3	#20243417	C₂	1	1	yes	maximal, characteristic
4	#20241479	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#467	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	—	2	4	no	—
7	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
8	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×#13093958	C₂²	1	2	yes	—
10	#15688983	—	2	1	no	—
11	#15688927	—	2	4	no	—
12	#15688886	—	2	1	no	—
13	C₂²×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₄₇	—	2	1	no	—
16	C₄₇×#467	—	2	2	no	—
17	C₄₇×#467	C₄	1	4	yes	complemented
18	C₂×#107×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#103×C₄₇	—	2	1	no	—
20	C₂×C₄⋊Dic₄₇	—	2	2	no	—
21	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
22	C₂×C₄₇⋊C₁₆	—	2	8	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₂×C₄₇⋊C₁₆	—	2	2	no	—
25	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
26	#13093958	—	2	2	no	—
27	#13093958	—	2	2	no	—
28	C₄₇⋊#95	—	4	1	no	—
29	C₂²×Q₈×C₄₇	—	2	1	no	—
30	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂×Q₁₆×C₄₇	—	2	2	no	—
32	C₂×Q₁₆×C₄₇	—	2	1	no	—
33	C₂×Q₁₆×C₄₇	—	2	2	no	—
34	C₂×Q₁₆×C₄₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₄₇	—	2	2	no	—
36	C₂²×C₈×C₄₇	—	2	2	no	—
37	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₇	D₄	1	2	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
41	C₂²⋊Q₈×C₄₇	—	2	4	no	—
42	C₂²⋊Q₈×C₄₇	—	4	2	no	—
43	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
45	C₂×C₂²⋊C₄×C₄₇	—	4	1	no	—
46	#107×C₄₇	—	2	1	no	—
47	#107×C₄₇	C₂×C₄	1	2	yes	—
48	#103×C₄₇	—	2	4	no	—
49	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
50	C₂×Q₈×C₄₇	—	2	2	no	—
51	C₂×Q₈×C₄₇	—	4	1	no	—
52	C₂×Q₈×C₄₇	—	4	2	no	—
53	C₂×Q₈×C₄₇	—	4	2	no	—
54	C₂²×C₄×C₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	—	2	2	no	—
56	C₂²×C₄×C₄₇	—	4	1	no	—
57	C₂²×C₄×C₄₇	—	4	1	no	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₇	—	4	2	no	—
63	Q₁₆×C₄₇	—	2	4	no	—
64	Q₁₆×C₄₇	—	4	1	no	—
65	Q₁₆×C₄₇	—	4	1	no	—
66	C₂×C₈×C₄₇	—	2	2	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₇	#41	1	2	yes	—
75	C₄⋊C₄×C₄₇	—	2	2	no	—
76	C₄⋊C₄×C₄₇	—	4	4	no	—
77	C₄⋊C₄×C₄₇	—	4	1	no	—
78	C₂²⋊C₄×C₄₇	—	4	4	no	—
79	C₂²⋊Dic₄₇	—	8	2	no	—
80	C₄⋊Dic₄₇	—	4	2	no	—
81	C₄⋊Dic₄₇	—	4	2	no	—
82	C₄₇⋊C₁₆	—	2	4	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	2	2	no	—
85	C₂³×C₄₇	#102	1	2	yes	—
86	C₂³×C₄₇	SD₃₂	1	2	yes	—
87	C₂³×C₄₇	Q₃₂	1	2	yes	—
88	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₄₇	—	4	2	no	—
90	Q₈×C₄₇	—	4	2	no	—
91	Q₈×C₄₇	—	8	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	4	no	—
96	C₂×C₄×C₄₇	—	4	2	no	—
97	C₂×C₄×C₄₇	—	4	2	no	—
98	C₂×C₄×C₄₇	—	8	2	no	—
99	C₂×C₄×C₄₇	—	8	2	no	—
100	C₂×C₄×C₄₇	#102	1	2	yes	—
101	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	#121	1	2	yes	—
103	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₄₇	#135	1	2	yes	—
105	C₂×Dic₄₇	—	8	2	no	—
106	C₂×Dic₄₇	—	8	2	no	—
107	C₂×Dic₄₇	—	8	2	no	—
108	C₈×C₄₇	—	2	2	no	—
109	C₈×C₄₇	C₄×D₄	1	2	yes	—
110	C₈×C₄₇	#120	1	1	yes	characteristic
111	C₈×C₄₇	#123	1	1	yes	characteristic
112	#13060	—	47	1	no	maximal, Sylow
113	C₂²×C₄₇	—	2	2	no	—
114	C₂²×C₄₇	—	2	2	no	—
115	C₂²×C₄₇	—	2	4	no	—
116	C₂²×C₄₇	—	2	4	no	—
117	C₂²×C₄₇	#358	1	4	yes	complemented
118	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
119	C₂²×C₄₇	#466	1	2	yes	—
120	C₂²×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
121	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₄₇	#508	1	2	yes	—
123	C₄×C₄₇	—	2	2	no	—
124	C₄×C₄₇	—	8	2	no	—
125	C₄×C₄₇	—	8	2	no	—
126	C₄×C₄₇	#386	1	1	yes	characteristic
127	C₄×C₄₇	#416	1	1	yes	characteristic
128	C₄×C₄₇	#465	1	2	yes	—
129	Dic₄₇	—	16	2	no	—
130	C₂³×C₁₆	—	47	1	no	—
131	C₂×#467	D₄₇	1	1	yes	characteristic
132	C₂×#358	—	47	2	no	—
133	#1808	—	47	1	no	—
134	#1380	—	47	2	no	—
135	C₂×C₄₇	—	2	4	no	—
136	C₂×C₄₇	#1893	1	1	yes	characteristic
137	C₂×C₄₇	C₂×#358	1	1	yes	characteristic
138	C₂×C₄₇	#2138	1	1	yes	characteristic
139	C₂×C₄₇	#2216	1	2	yes	—
140	C₂×C₄₇	#2217	1	2	yes	—
141	C₂²×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
142	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
143	C₂×C₂²⋊Q₈	—	2	1	no	—
144	C₂²×C₁₆	—	47	2	no	—
145	C₂²×C₁₆	—	94	4	no	—
146	C₂²×C₁₆	—	94	2	no	—
147	#467	—	2	2	no	—
148	#467	Dic₄₇	1	4	yes	complemented
149	C₂×#107	—	47	2	no	—
150	C₂×#107	C₂×D₄₇	1	1	yes	characteristic
151	C₂×#103	—	2	1	no	—
152	#380	—	94	1	no	—
153	#358	—	94	4	no	—
154	#340	—	94	1	no	—
155	C₄₇	#13060	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	2	1	no	—
157	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂×Dic₄₇	1	2	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
165	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
167	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
168	C₂²⋊Q₈	—	2	4	no	—
169	C₂²⋊Q₈	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	4	1	no	—
172	C₂×C₄⋊C₄	—	94	2	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	188	1	no	—
175	C₂×C₁₆	—	47	4	no	—
176	C₂×C₁₆	—	94	8	no	—
177	C₂×C₁₆	—	94	2	no	—
178	C₂×C₁₆	—	94	2	no	—
179	#107	—	2	1	no	—
180	#107	—	94	2	no	—
181	#107	—	94	2	no	—
182	#107	C₂×Dic₄₇	1	2	yes	—
183	#103	—	2	4	no	—
184	#95	—	188	1	no	—
185	C₂⁴	#10625334	1	1	yes	characteristic
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	188	2	no	—
195	C₂²×C₄	#10625334	1	1	yes	characteristic
196	C₂²×C₄	#10625335	1	2	yes	—
197	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	4	1	no	—
201	Q₁₆	—	4	1	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
207	C₂×C₈	D₄×D₄₇	1	2	yes	—
208	C₂×C₈	#10625352	1	2	yes	—
209	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₄⋊C₄	—	188	2	no	—
215	C₄⋊C₄	—	188	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	376	2	no	—
218	C₁₆	—	94	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	#13093945	1	2	yes	—
222	C₂³	#13093967	1	2	yes	—
223	C₂³	#13093970	1	2	yes	—
224	C₂³	C₂×#10625334	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	8	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	376	2	no	—
237	C₂×C₄	—	376	2	no	—
238	C₂×C₄	—	376	2	no	—
239	C₂×C₄	#13093945	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
241	C₂×C₄	#13093996	1	2	yes	—
242	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
243	C₂×C₄	#13094044	1	2	yes	—
244	C₈	—	2	2	no	—
245	C₈	D₄×Dic₄₇	1	2	yes	—
246	C₈	#13093992	1	1	yes	characteristic
247	C₈	#13094005	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#15688927	1	4	yes	complemented
253	C₂²	C₂×#13093945	1	1	yes	characteristic
254	C₂²	#15689357	1	2	yes	—
255	C₂²	C₂×#13093967	1	1	yes	characteristic
256	C₂²	C₂×#13093970	1	1	yes	characteristic
257	C₂²	#15689553	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	—	752	2	no	—
262	C₄	#15689003	1	1	yes	characteristic
263	C₄	#15689119	1	1	yes	characteristic
264	C₄	#15689350	1	2	yes	—
265	C₂	—	2	4	no	—
266	C₂	#20243817	1	1	yes	characteristic
267	C₂	C₂×#15688927	1	1	yes	characteristic
268	C₂	#20245095	1	1	yes	characteristic
269	C₂	#20245477	1	2	yes	—
270	C₂	#20245486	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#467	C₂
socle	C₂³×C₄₇	C₂×D₈
prime radical	C₂⁴×C₄₇	D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#467	D₄₇
hypocenter	C₄₇	#13060
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. C₂×#15688927	C₂	semidirect
4. #20241479	C₂	semidirect
17. C₄₇×#467	C₄	semidirect
117. C₂²×C₄₇	#358	semidirect
148. #467	Dic₄₇	semidirect
155. C₄₇	#13060	semidirect
252. C₂²	#15688927	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄₇⋊C₁₆	2
C₂×#15688927	2
#20243417	2
#20241479	2
C₂×C₄₇×#467	2
#13060	47