Subgroups — #26213753

Order 12032.

1286 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄²×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243136	C₂	1	4	yes	maximal, complemented
4	#20243133	C₂	1	4	yes	maximal, complemented
5	#20243081	C₂	1	4	yes	maximal
6	C₂×C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10625329	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₄₇	C₄	1	4	yes	—
9	C₂²×C₄×Dic₄₇	C₂²	1	2	yes	—
10	C₄×#10625329	C₄	1	8	yes	complemented
11	C₂×#13093935	C₂²	1	2	yes	—
12	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
13	C₄²×Dic₄₇	C₄	1	4	yes	—
14	C₄²×Dic₄₇	C₂²	1	2	yes	—
15	C₄²×Dic₄₇	C₂²	1	2	yes	—
16	#15688877	C₂²	1	16	yes	—
17	C₂²×M₁₆×C₄₇	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×M₁₆×C₄₇	C₄	1	8	yes	complemented
20	C₂×#97×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
22	C₂³×Dic₄₇	C₂×C₄	1	4	yes	—
23	C₂×#10625329	—	2	8	no	—
24	C₂×#10625329	C₂×C₄	1	8	yes	—
25	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
26	C₂×C₄×Dic₄₇	—	2	8	no	—
27	C₂×C₄×Dic₄₇	—	2	16	no	—
28	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₄₇	C₂×C₄	1	8	yes	—
31	C₂×C₄×Dic₄₇	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
33	#13093935	C₂×C₄	1	8	yes	complemented
34	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
35	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₇	—	2	8	no	—
38	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
39	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#97×C₄₇	C₂×C₄	1	8	yes	complemented
46	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
47	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₇	—	2	8	no	—
49	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₇	#41	1	8	yes	—
56	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
57	C₂²×Dic₄₇	—	2	8	no	—
58	C₂²×Dic₄₇	—	2	16	no	—
59	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
60	M₁₆×C₄₇	—	2	16	no	—
61	C₂×C₈×C₄₇	—	2	8	no	—
62	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
63	C₄²×C₄₇	C₄²	1	8	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
67	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
68	C₄×Dic₄₇	—	2	8	no	—
69	C₄×Dic₄₇	—	2	8	no	—
70	C₄×Dic₄₇	—	2	8	no	—
71	C₄×Dic₄₇	—	2	16	no	—
72	C₄×Dic₄₇	M₁₆	1	16	yes	complemented
73	#10625329	—	2	16	no	—
74	C₂×C₄₇⋊C₈	—	2	8	no	—
75	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
76	C₂³×C₄₇	—	2	4	no	—
77	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
78	C₂³×C₄₇	#117	1	4	yes	—
79	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₇	—	2	4	no	—
81	C₂×C₄×C₄₇	—	2	16	no	—
82	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
84	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₄₇	#117	1	8	yes	—
86	C₂×C₄×C₄₇	#117	1	4	yes	—
87	C₂×C₄×C₄₇	#117	1	8	yes	—
88	C₂×C₄×C₄₇	C₂×M₁₆	1	8	yes	—
89	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
92	C₂×Dic₄₇	—	2	32	no	—
93	C₂×Dic₄₇	—	2	8	no	—
94	C₂×Dic₄₇	—	2	8	no	—
95	C₂×Dic₄₇	C₂×M₁₆	1	16	yes	—
96	C₈×C₄₇	—	2	8	no	—
97	C₄₇⋊C₈	—	2	8	no	—
98	#20364	—	47	1	no	maximal, Sylow
99	C₂²×C₄₇	—	2	4	no	—
100	C₂²×C₄₇	—	2	8	no	—
101	C₂²×C₄₇	#376	1	4	yes	—
102	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
103	C₂²×C₄₇	C₂×#117	1	2	yes	—
104	C₂²×C₄₇	C₂²×M₁₆	1	4	yes	—
105	C₄×C₄₇	—	2	4	no	—
106	C₄×C₄₇	#376	1	4	yes	—
107	C₄×C₄₇	C₄×M₁₆	1	8	yes	—
108	C₄×C₄₇	C₂×#117	1	2	yes	—
109	C₄×C₄₇	C₂×#117	1	2	yes	—
110	Dic₄₇	—	2	16	no	—
111	C₂×C₄×M₁₆	—	47	1	no	—
112	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
113	C₂×C₄³	—	47	1	no	—
114	#1745	—	47	4	no	—
115	#1744	—	47	4	no	—
116	#1728	—	47	4	no	—
117	C₂×C₄₇	—	2	4	no	—
118	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
119	C₂×C₄₇	C₂×C₄×M₁₆	1	2	yes	—
120	C₂×C₄₇	#2920	1	4	yes	—
121	C₂²×M₁₆	—	47	2	no	—
122	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
123	C₂²×C₄²	—	47	4	no	—
124	C₂²×C₄²	—	47	2	no	—
125	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
126	C₄×M₁₆	—	47	8	no	—
127	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
128	C₂×#97	—	47	2	no	—
129	C₂×#97	C₂×D₄₇	1	2	yes	—
130	C₂×C₄×C₈	—	47	2	no	—
131	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
132	C₄³	—	47	4	no	—
133	C₄³	—	47	2	no	—
134	C₄³	—	47	2	no	—
135	#336	—	47	16	no	—
136	C₄₇	#20364	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	47	4	no	—
138	C₂³×C₄	C₄×D₄₇	1	2	yes	—
139	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
140	C₂×M₁₆	—	2	8	no	—
141	C₂×M₁₆	—	47	8	no	—
142	C₂×M₁₆	—	94	8	no	—
143	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
144	C₂²×C₈	—	47	4	no	—
145	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
146	C₂×C₄²	—	47	4	no	—
147	C₂×C₄²	—	47	8	no	—
148	C₂×C₄²	—	47	4	no	—
149	C₂×C₄²	—	47	4	no	—
150	C₂×C₄²	—	47	4	no	—
151	C₂×C₄²	—	94	16	no	—
152	C₂×C₄²	—	94	8	no	—
153	C₂×C₄²	C₄×D₄₇	1	2	yes	—
154	C₂×C₄²	C₄×D₄₇	1	4	yes	—
155	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
156	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
158	#97	—	47	8	no	—
159	#97	C₂×Dic₄₇	1	8	yes	complemented
160	C₄×C₈	—	47	8	no	—
161	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
162	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	47	8	no	—
165	C₂²×C₄	—	94	16	no	—
166	C₂²×C₄	—	94	8	no	—
167	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
168	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
169	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
172	C₂²×C₄	#10625350	1	8	yes	—
173	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
174	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
175	M₁₆	—	2	16	no	—
176	M₁₆	—	94	16	no	—
177	C₂×C₈	—	2	8	no	—
178	C₂×C₈	—	47	8	no	—
179	C₂×C₈	—	94	8	no	—
180	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
181	C₄²	—	47	16	no	—
182	C₄²	—	94	16	no	—
183	C₄²	—	94	8	no	—
184	C₄²	—	94	8	no	—
185	C₄²	—	94	8	no	—
186	C₄²	C₄×Dic₄₇	1	8	yes	—
187	C₄²	C₂×C₄×D₄₇	1	2	yes	—
188	C₄²	C₂×C₄×D₄₇	1	2	yes	—
189	C₄²	C₂²×Dic₄₇	1	2	yes	—
190	C₄²	C₂²×Dic₄₇	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
193	C₂³	#13093981	1	4	yes	—
194	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	16	no	—
197	C₂×C₄	—	47	16	no	—
198	C₂×C₄	—	94	32	no	—
199	C₂×C₄	—	94	8	no	—
200	C₂×C₄	—	94	8	no	—
201	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
202	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
204	C₂×C₄	#13093981	1	4	yes	—
205	C₂×C₄	#13093981	1	8	yes	—
206	C₂×C₄	#13093983	1	8	yes	—
207	C₂×C₄	M₁₆×D₄₇	1	8	yes	—
208	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
209	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
210	C₂×C₄	C₂×#10625350	1	4	yes	—
211	C₈	—	2	8	no	—
212	C₈	—	94	8	no	—
213	C₂²	—	2	8	no	—
214	C₂²	—	2	4	no	—
215	C₂²	#15688969	1	4	yes	—
216	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
217	C₂²	C₂×#13093981	1	2	yes	—
218	C₂²	C₂×M₁₆×D₄₇	1	4	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	94	16	no	—
221	C₄	#15688969	1	4	yes	—
222	C₄	M₁₆×Dic₄₇	1	8	yes	—
223	C₄	C₂×#13093981	1	2	yes	—
224	C₄	C₂×#13093983	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	C₂×#15688969	1	1	yes	characteristic
227	C₂	C₂×M₁₆×Dic₄₇	1	2	yes	—
228	C₂	#20252019	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₂×C₄×M₁₆×C₄₇	C₂
socle	C₂³×C₄₇	C₂³×C₄
prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₇
hypocenter	C₄₇	#20364
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20243136	C₂	semidirect
4. #20243133	C₂	semidirect
10. C₄×#10625329	C₄	semidirect
19. C₄×M₁₆×C₄₇	C₄	semidirect
33. #13093935	C₂×C₄	semidirect
34. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
45. #97×C₄₇	C₂×C₄	semidirect
46. C₄×C₈×C₄₇	C₂×C₄	semidirect
72. C₄×Dic₄₇	M₁₆	semidirect
127. C₄×M₁₆	Dic₄₇	semidirect
136. C₄₇	#20364	semidirect
159. #97	C₂×Dic₄₇	semidirect
161. 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
C₂×C₄×#10625329	2
C₂×C₄²×Dic₄₇	2
#20243136	2
#20243133	2
#20243081	2
C₂×C₄×M₁₆×C₄₇	2
#20364	47