Subgroups — #26216499

Order 12032.

926 conjugacy classes of subgroups.

The subgroup lattice has 248 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×#15688989	C₂	1	1	yes	maximal, characteristic
3	#20243522	C₂	1	4	yes	maximal, complemented
4	#20243519	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	C₂²	1	2	yes	—
9	#15688993	C₂²	1	2	yes	—
10	#15688989	—	2	2	no	—
11	#15688989	C₂²	1	2	yes	—
12	#15688989	C₂²	1	2	yes	—
13	#15688989	C₂²	1	2	yes	—
14	#15688983	—	2	2	no	—
15	C₄×C₄⋊Dic₄₇	—	2	2	no	—
16	C₂×C₄₇⋊#95	—	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	4	no	—
27	C₂×C₄⋊Dic₄₇	—	2	2	no	—
28	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
29	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
30	C₂×C₂²⋊Dic₄₇	—	2	2	no	—
31	C₂×C₄×Dic₄₇	—	2	2	no	—
32	C₄₇⋊#95	—	2	4	no	—
33	C₄₇⋊#95	—	2	4	no	—
34	C₄₇⋊#95	—	2	4	no	—
35	C₂³×C₄×C₄₇	D₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	4	no	—
38	C₂×M₁₆×C₄₇	—	2	4	no	—
39	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	Q₈	1	4	yes	—
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
46	#97×C₄₇	C₂×C₄	1	8	yes	complemented
47	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
48	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₇	—	4	4	no	—
50	C₂²×C₄×C₄₇	C₄⋊C₄	1	8	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₄₇	#41	1	4	yes	—
57	C₂²×C₄×C₄₇	#41	1	4	yes	—
58	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₄₇	—	2	4	no	—
60	C₂²×Dic₄₇	—	2	4	no	—
61	C₂²×Dic₄₇	—	4	8	no	—
62	C₂²×Dic₄₇	—	4	4	no	—
63	M₁₆×C₄₇	—	2	16	no	—
64	C₂×C₈×C₄₇	—	2	4	no	—
65	C₂×C₈×C₄₇	—	2	4	no	—
66	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
67	C₄²×C₄₇	C₄⋊C₄	1	8	yes	—
68	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
70	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
71	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
72	C₂²⋊Dic₄₇	—	4	4	no	—
73	C₄⋊Dic₄₇	—	2	8	no	—
74	C₄⋊Dic₄₇	—	2	8	no	—
75	C₄⋊Dic₄₇	—	4	8	no	—
76	C₄⋊Dic₄₇	—	4	4	no	—
77	C₄×Dic₄₇	—	4	4	no	—
78	C₂³×C₄₇	—	2	4	no	—
79	C₂³×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₄₇	#124	1	2	yes	—
81	C₂³×C₄₇	#124	1	2	yes	—
82	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₇	—	2	4	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	—	4	8	no	—
86	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₄₇	#117	1	8	yes	—
90	C₂×C₄×C₄₇	#124	1	2	yes	—
91	C₂×C₄×C₄₇	#124	1	2	yes	—
92	C₂×C₄×C₄₇	#125	1	8	yes	—
93	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
96	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
97	C₂×Dic₄₇	—	4	4	no	—
98	C₂×Dic₄₇	—	4	4	no	—
99	C₂×Dic₄₇	—	4	8	no	—
100	C₂×Dic₄₇	—	4	8	no	—
101	C₂×Dic₄₇	—	8	8	no	—
102	C₈×C₄₇	—	2	8	no	—
103	#20765	—	47	1	no	maximal, Sylow
104	C₂²×C₄₇	—	2	4	no	—
105	C₂²×C₄₇	—	2	4	no	—
106	C₂²×C₄₇	—	2	4	no	—
107	C₂²×C₄₇	#382	1	4	yes	—
108	C₂²×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₇	C₂×#124	1	1	yes	characteristic
111	C₂²×C₄₇	#575	1	4	yes	—
112	C₄×C₄₇	—	2	4	no	—
113	C₄×C₄₇	—	2	4	no	—
114	C₄×C₄₇	#382	1	4	yes	—
115	C₄×C₄₇	C₂×#117	1	2	yes	—
116	C₄×C₄₇	C₂×#125	1	2	yes	—
117	Dic₄₇	—	8	4	no	—
118	Dic₄₇	—	8	4	no	—
119	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
120	#2285	—	47	1	no	—
121	C₂×#382	—	47	1	no	—
122	#1836	—	47	4	no	—
123	#1835	—	47	4	no	—
124	#1736	—	47	4	no	—
125	C₂×C₄₇	—	2	4	no	—
126	C₂×C₄₇	C₂×#382	1	1	yes	characteristic
127	C₂×C₄₇	#2911	1	2	yes	—
128	C₂×C₄₇	#3140	1	2	yes	—
129	C₂×C₄₇	#3142	1	2	yes	—
130	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
131	C₂×#117	—	47	2	no	—
132	C₂²×C₄⋊C₄	—	47	2	no	—
133	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
134	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
135	C₂×#97	C₂×D₄₇	1	2	yes	—
136	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
137	#384	—	47	2	no	—
138	#382	—	47	2	no	—
139	#382	—	47	2	no	—
140	#382	—	47	2	no	—
141	#382	—	94	2	no	—
142	#380	—	94	2	no	—
143	C₄×C₄⋊C₄	—	94	2	no	—
144	C₂×#95	—	94	2	no	—
145	#340	—	47	16	no	—
146	C₄₇	#20765	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	94	2	no	—
148	C₂³×C₄	D₁₈₈	1	2	yes	—
149	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
153	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
154	#117	—	94	8	no	—
155	C₂×C₄⋊C₄	—	47	4	no	—
156	C₂×C₄⋊C₄	—	47	4	no	—
157	C₂×C₄⋊C₄	—	94	8	no	—
158	C₂×C₄⋊C₄	—	94	2	no	—
159	C₂×C₄⋊C₄	—	94	4	no	—
160	C₂×C₂²⋊C₄	—	94	2	no	—
161	C₂×C₄²	—	94	2	no	—
162	C₂×C₄²	Dic₉₄	1	4	yes	—
163	C₂×C₄²	D₁₈₈	1	2	yes	—
164	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
165	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
167	#97	C₂×Dic₄₇	1	8	yes	complemented
168	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
169	#95	—	94	4	no	—
170	#95	—	94	4	no	—
171	#95	—	94	4	no	—
172	C₂⁴	C₂×D₁₈₈	1	1	yes	characteristic
173	C₂²×C₄	—	4	4	no	—
174	C₂²×C₄	—	94	4	no	—
175	C₂²×C₄	—	94	4	no	—
176	C₂²×C₄	—	188	8	no	—
177	C₂²×C₄	—	188	4	no	—
178	C₂²×C₄	C₄⋊Dic₄₇	1	8	yes	—
179	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
180	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
181	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
182	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
183	C₂²×C₄	#10625350	1	4	yes	—
184	C₂²×C₄	#10625350	1	4	yes	—
185	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
187	M₁₆	—	2	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
191	C₄⋊C₄	—	94	8	no	—
192	C₄⋊C₄	—	94	8	no	—
193	C₄⋊C₄	—	188	8	no	—
194	C₄⋊C₄	—	188	4	no	—
195	C₂²⋊C₄	—	188	4	no	—
196	C₄²	—	188	4	no	—
197	C₄²	C₄⋊Dic₄₇	1	8	yes	—
198	C₄²	C₂×Dic₉₄	1	2	yes	—
199	C₄²	C₂×Dic₉₄	1	2	yes	—
200	C₄²	C₂²×Dic₄₇	1	2	yes	—
201	C₄²	C₂²×Dic₄₇	1	2	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	C₂×C₄⋊Dic₄₇	1	2	yes	—
204	C₂³	#13094012	1	2	yes	—
205	C₂³	#13094012	1	2	yes	—
206	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	376	8	no	—
211	C₂×C₄	—	188	4	no	—
212	C₂×C₄	—	188	4	no	—
213	C₂×C₄	—	188	8	no	—
214	C₂×C₄	—	188	8	no	—
215	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
216	C₂×C₄	C₂×C₄⋊Dic₄₇	1	4	yes	—
217	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
218	C₂×C₄	#13093983	1	8	yes	—
219	C₂×C₄	#13094012	1	2	yes	—
220	C₂×C₄	#13094012	1	2	yes	—
221	C₂×C₄	#13094016	1	8	yes	—
222	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
223	C₂×C₄	C₂²×Dic₉₄	1	1	yes	characteristic
224	C₂×C₄	C₂×#10625350	1	2	yes	—
225	C₂×C₄	C₂×#10625350	1	2	yes	—
226	C₈	—	2	8	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	#15688989	1	4	yes	—
231	C₂²	C₂²×C₄⋊Dic₄₇	1	1	yes	characteristic
232	C₂²	C₂×#13094012	1	1	yes	characteristic
233	C₂²	C₂×#13094012	1	1	yes	characteristic
234	C₂²	#15689923	1	4	yes	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	376	4	no	—
238	C₄	—	376	4	no	—
239	C₄	#15688989	1	4	yes	—
240	C₄	C₂×#13093983	1	2	yes	—
241	C₄	C₂×#13094016	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×#15688989	1	1	yes	characteristic
244	C₂	#20251978	1	2	yes	—
245	C₂	#20253775	1	2	yes	—
246	C₂	#20253789	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	#20765
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. #20243519	C₂	semidirect
20. C₄×M₁₆×C₄₇	C₄	semidirect
46. #97×C₄₇	C₂×C₄	semidirect
47. C₄×C₈×C₄₇	C₂×C₄	semidirect
134. C₄×M₁₆	Dic₄₇	semidirect
146. C₄₇	#20765	semidirect
167. #97	C₂×Dic₄₇	semidirect
168. 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×#15688989	2
#20243522	2
#20243519	2
#20243108	2
C₂×C₄×M₁₆×C₄₇	2
#20765	47