Subgroups — #26122097

Order 12032.

1126 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15688993	C₂	1	2	yes	maximal
2	C₂×#15688886	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₄₇	—	2	4	no	—
5	C₂²×C₄⋊Dic₄₇	C₂²	1	4	yes	—
6	#15688993	—	2	4	no	—
7	#15688993	C₄	1	8	yes	complemented
8	#15688886	—	2	8	no	—
9	C₂³×C₈×C₄₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₄₇	—	2	4	no	—
12	C₂×C₄×C₈×C₄₇	C₄	1	4	yes	complemented
13	C₂³×Dic₄₇	—	2	4	no	—
14	C₂×C₄⋊Dic₄₇	—	2	8	no	—
15	C₂×C₄⋊Dic₄₇	—	2	8	no	—
16	C₂×C₄⋊Dic₄₇	—	2	16	no	—
17	C₂×C₄⋊Dic₄₇	—	2	4	no	—
18	C₂×C₄⋊Dic₄₇	—	4	8	no	—
19	C₂×C₄⋊Dic₄₇	—	4	8	no	—
20	C₂×C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₄₇	D₄	1	2	yes	—
22	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₄₇	—	2	4	no	—
24	C₂²×C₈×C₄₇	—	2	4	no	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₄₇	D₄	1	4	yes	—
27	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
28	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₄₇	D₄	1	4	yes	—
31	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
32	C₄×C₈×C₄₇	—	2	8	no	—
33	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
35	C₂²×C₄×C₄₇	C₄⋊C₄	1	4	yes	—
36	C₂²×C₄×C₄₇	D₈	1	4	yes	—
37	C₂²×C₄×C₄₇	SD₁₆	1	8	yes	—
38	C₂²×C₄×C₄₇	Q₁₆	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂²×C₄×C₄₇	#41	1	4	yes	—
46	C₂²×Dic₄₇	—	2	8	no	—
47	C₂²×Dic₄₇	—	4	4	no	—
48	C₂²×Dic₄₇	—	4	4	no	—
49	C₂²×Dic₄₇	—	4	8	no	—
50	C₂²×Dic₄₇	—	4	8	no	—
51	C₂×C₈×C₄₇	—	2	4	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₂×C₈×C₄₇	—	2	8	no	—
54	C₂×C₈×C₄₇	—	2	8	no	—
55	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	complemented
56	C₄²×C₄₇	C₄²	1	4	yes	—
57	C₄²×C₄₇	C₂²⋊C₄	1	4	yes	—
58	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
59	C₄⋊Dic₄₇	—	2	16	no	—
60	C₄⋊Dic₄₇	—	4	32	no	—
61	C₄⋊Dic₄₇	—	4	8	no	—
62	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
63	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂³×C₄₇	#124	1	2	yes	—
65	C₂³×C₄₇	#127	1	1	yes	characteristic
66	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂³×C₄₇	C₂×D₈	1	2	yes	—
68	C₂³×C₄₇	C₂×SD₁₆	1	4	yes	—
69	C₂³×C₄₇	C₂×Q₁₆	1	2	yes	—
70	C₂×C₄×C₄₇	#95	1	8	yes	—
71	C₂×C₄×C₄₇	#102	1	16	yes	—
72	C₂×C₄×C₄₇	#103	1	16	yes	—
73	C₂×C₄×C₄₇	#106	1	8	yes	—
74	C₂×C₄×C₄₇	#107	1	8	yes	—
75	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₄₇	#117	1	2	yes	—
80	C₂×C₄×C₄₇	C₄×D₄	1	4	yes	—
81	C₂×C₄×C₄₇	C₄×Q₈	1	4	yes	—
82	C₂×C₄×C₄₇	#121	1	4	yes	—
83	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₄×C₄₇	#125	1	2	yes	—
85	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
86	C₂×Dic₄₇	—	4	8	no	—
87	C₂×Dic₄₇	—	4	16	no	—
88	C₂×Dic₄₇	—	8	8	no	—
89	C₂×Dic₄₇	—	8	4	no	—
90	C₂×Dic₄₇	—	8	4	no	—
91	C₈×C₄₇	—	2	8	no	—
92	#10841	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	C₂×#95	1	1	yes	characteristic
94	C₂²×C₄₇	#382	1	1	yes	characteristic
95	C₂²×C₄₇	#384	1	1	yes	characteristic
96	C₂²×C₄₇	#390	1	2	yes	—
97	C₂²×C₄₇	#391	1	2	yes	—
98	C₂²×C₄₇	C₂×#102	1	4	yes	—
99	C₂²×C₄₇	C₂×#103	1	4	yes	—
100	C₂²×C₄₇	C₂×#106	1	2	yes	—
101	C₂²×C₄₇	C₂×#107	1	2	yes	—
102	C₂²×C₄₇	#486	1	4	yes	—
103	C₂²×C₄₇	#487	1	4	yes	—
104	C₂²×C₄₇	#492	1	2	yes	—
105	C₂²×C₄₇	#493	1	1	yes	characteristic
106	C₂²×C₄₇	#494	1	1	yes	characteristic
107	C₂²×C₄₇	#498	1	2	yes	—
108	C₂²×C₄₇	#500	1	2	yes	—
109	C₄×C₄₇	#340	1	16	yes	complemented
110	C₄×C₄₇	C₄×C₄⋊C₄	1	2	yes	—
111	C₄×C₄₇	#380	1	2	yes	—
112	C₄×C₄₇	#389	1	4	yes	—
113	Dic₄₇	—	8	8	no	—
114	C₂²×C₄×C₈	D₄₇	1	1	yes	characteristic
115	C₂×#384	—	47	2	no	—
116	C₂×#340	—	47	4	no	—
117	C₂×C₄₇	C₄⋊#95	1	1	yes	characteristic
118	C₂×C₄₇	C₂×#340	1	2	yes	—
119	C₂×C₄₇	#1834	1	2	yes	—
120	C₂×C₄₇	#1844	1	1	yes	characteristic
121	C₂×C₄₇	#1845	1	1	yes	characteristic
122	C₂×C₄₇	#1956	1	2	yes	—
123	C₂×C₄₇	#1957	1	2	yes	—
124	C₂×C₄₇	#1990	1	4	yes	—
125	C₂³×C₈	C₂×D₄₇	1	2	yes	—
126	C₂²×C₄⋊C₄	—	47	4	no	—
127	C₂²×C₄⋊C₄	—	94	4	no	—
128	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	4	no	—
130	C₂×C₄×C₈	Dic₄₇	1	4	yes	complemented
131	#384	—	47	8	no	—
132	#384	—	94	4	no	—
133	#340	—	94	8	no	—
134	C₄₇	#10841	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₄	—	94	4	no	—
136	C₂³×C₄	D₁₈₈	1	2	yes	—
137	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	4	no	—
140	C₂²×C₈	Dic₉₄	1	4	yes	—
141	C₂²×C₈	D₁₈₈	1	4	yes	—
142	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
143	C₂×C₄⋊C₄	—	47	8	no	—
144	C₂×C₄⋊C₄	—	94	8	no	—
145	C₂×C₄⋊C₄	—	94	8	no	—
146	C₂×C₄⋊C₄	—	94	16	no	—
147	C₂×C₄⋊C₄	—	94	4	no	—
148	C₂×C₄⋊C₄	—	188	8	no	—
149	C₂×C₄⋊C₄	—	188	8	no	—
150	C₂×C₄²	Dic₉₄	1	2	yes	—
151	C₂×C₄²	C₄×D₄₇	1	4	yes	—
152	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
153	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
154	C₄×C₈	—	2	8	no	—
155	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
156	C₂²×C₄	—	94	8	no	—
157	C₂²×C₄	—	188	4	no	—
158	C₂²×C₄	—	188	4	no	—
159	C₂²×C₄	—	188	8	no	—
160	C₂²×C₄	—	188	8	no	—
161	C₂²×C₄	#10625325	1	8	yes	—
162	C₂²×C₄	D₃₇₆	1	4	yes	—
163	C₂²×C₄	Dic₁₈₈	1	4	yes	—
164	C₂²×C₄	C₄⋊Dic₄₇	1	4	yes	—
165	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	8	yes	—
166	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
169	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
170	C₂²×C₄	#10625350	1	4	yes	—
171	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	complemented
178	C₄⋊C₄	—	94	16	no	—
179	C₄⋊C₄	—	188	32	no	—
180	C₄⋊C₄	—	188	8	no	—
181	C₄²	C₄×Dic₄₇	1	4	yes	—
182	C₄²	Dic₄₇⋊C₄	1	8	yes	—
183	C₄²	C₂²⋊Dic₄₇	1	4	yes	—
184	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	2	yes	—
185	C₂³	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
186	C₂³	#13094012	1	2	yes	—
187	C₂³	#13094021	1	1	yes	characteristic
188	C₂³	#13094024	1	1	yes	characteristic
189	C₂³	C₂×D₃₇₆	1	2	yes	—
190	C₂³	C₂×#10625325	1	4	yes	—
191	C₂³	C₂×Dic₁₈₈	1	2	yes	—
192	C₂×C₄	—	376	4	no	—
193	C₂×C₄	—	376	4	no	—
194	C₂×C₄	—	376	8	no	—
195	C₂×C₄	—	188	8	no	—
196	C₂×C₄	—	188	16	no	—
197	C₂×C₄	C₄₇⋊#95	1	8	yes	—
198	C₂×C₄	#13093946	1	16	yes	—
199	C₂×C₄	#13093949	1	16	yes	—
200	C₂×C₄	#13093956	1	8	yes	—
201	C₂×C₄	#13093958	1	8	yes	—
202	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
204	C₂×C₄	C₂×Dic₄₇⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
206	C₂×C₄	#13093983	1	2	yes	—
207	C₂×C₄	C₄×D₁₈₈	1	4	yes	—
208	C₂×C₄	C₄×Dic₉₄	1	4	yes	—
209	C₂×C₄	#13093998	1	4	yes	—
210	C₂×C₄	#13094003	1	4	yes	—
211	C₂×C₄	#13094016	1	2	yes	—
212	C₂×C₄	#13094024	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₂²	C₂×C₄₇⋊#95	1	1	yes	characteristic
215	C₂²	#15688989	1	1	yes	characteristic
216	C₂²	#15688993	1	1	yes	characteristic
217	C₂²	#15689025	1	2	yes	—
218	C₂²	#15689029	1	2	yes	—
219	C₂²	C₂×#13093946	1	4	yes	—
220	C₂²	C₂×#13093949	1	4	yes	—
221	C₂²	C₂×#13093956	1	2	yes	—
222	C₂²	C₂×#13093958	1	2	yes	—
223	C₂²	#15689470	1	4	yes	—
224	C₂²	#15689474	1	4	yes	—
225	C₂²	#15689496	1	2	yes	—
226	C₂²	#15689500	1	1	yes	characteristic
227	C₂²	#15689503	1	1	yes	characteristic
228	C₂²	#15689521	1	2	yes	—
229	C₂²	#15689528	1	2	yes	—
230	C₄	—	376	8	no	—
231	C₄	#15688886	1	16	yes	complemented
232	C₄	C₄×C₄⋊Dic₄₇	1	2	yes	—
233	C₄	#15688983	1	2	yes	—
234	C₄	#15689021	1	4	yes	—
235	C₂	#20241618	1	1	yes	characteristic
236	C₂	C₂×#15688886	1	2	yes	—
237	C₂	#20243516	1	2	yes	—
238	C₂	#20243545	1	1	yes	characteristic
239	C₂	#20243548	1	1	yes	characteristic
240	C₂	#20244190	1	2	yes	—
241	C₂	#20244194	1	2	yes	—
242	C₂	#20244349	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×C₄×C₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄×C₈	D₄₇
hypocenter	C₄₇	#10841
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #15688993	C₄	semidirect
12. C₂×C₄×C₈×C₄₇	C₄	semidirect
55. C₂×C₈×C₄₇	C₄⋊C₄	semidirect
109. C₄×C₄₇	#340	semidirect
130. C₂×C₄×C₈	Dic₄₇	semidirect
134. C₄₇	#10841	semidirect
177. C₂×C₈	C₄⋊Dic₄₇	semidirect
231. C₄	#15688886	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15688993	2
C₂×#15688886	2
C₂²×C₄×C₈×C₄₇	2
#10841	47