Subgroups — #26213757

Order 12032.

1054 conjugacy classes of subgroups.

The subgroup lattice has 218 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×#15688969	C₂	1	1	yes	maximal, characteristic
3	#20243141	C₂	1	4	yes	maximal, complemented
4	#20243139	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	C₂²	1	2	yes	—
9	C₄×#10625329	C₄	1	8	yes	complemented
10	C₂×#13093935	C₂²	1	2	yes	—
11	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
12	#15688969	C₄	1	4	yes	—
13	#15688969	C₂²	1	2	yes	—
14	#15688969	C₂²	1	2	yes	—
15	C₂×C₄₇⋊#95	C₄	1	4	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	C₂×C₄	1	4	yes	—
28	C₂×C₄×Dic₄₇	C₂³	1	4	yes	—
29	#13093935	C₂×C₄	1	8	yes	complemented
30	C₄×C₄₇⋊C₈	C₂×C₄	1	8	yes	complemented
31	C₄₇⋊#95	C₂×C₄	1	8	yes	—
32	C₄₇⋊#95	C₂×C₄	1	8	yes	—
33	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₇	—	2	8	no	—
36	C₂×M₁₆×C₄₇	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
38	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
43	#97×C₄₇	C₂×C₄	1	8	yes	complemented
44	C₄×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
45	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	—	4	4	no	—
47	C₂²×C₄×C₄₇	C₄²	1	8	yes	—
48	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₇	#41	1	8	yes	—
54	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₇	—	2	8	no	—
56	C₂²×Dic₄₇	—	4	8	no	—
57	C₂²×Dic₄₇	C₂²×C₄	1	8	yes	—
58	M₁₆×C₄₇	—	2	16	no	—
59	C₂×C₈×C₄₇	—	2	8	no	—
60	C₂×C₈×C₄₇	C₂²×C₄	1	8	yes	—
61	C₄²×C₄₇	C₄²	1	8	yes	—
62	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
64	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
65	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
66	C₄×Dic₄₇	—	2	8	no	—
67	C₄×Dic₄₇	—	2	8	no	—
68	#10625329	—	2	16	no	—
69	C₂×C₄₇⋊C₈	—	2	8	no	—
70	C₂×C₄₇⋊C₈	C₂²×C₄	1	8	yes	—
71	C₂³×C₄₇	—	2	4	no	—
72	C₂³×C₄₇	C₂×C₄²	1	2	yes	—
73	C₂³×C₄₇	#117	1	4	yes	—
74	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	—	2	4	no	—
76	C₂×C₄×C₄₇	—	2	4	no	—
77	C₂×C₄×C₄₇	—	4	8	no	—
78	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
79	C₂×C₄×C₄₇	C₂×C₄²	1	4	yes	—
80	C₂×C₄×C₄₇	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₄₇	#117	1	8	yes	—
82	C₂×C₄×C₄₇	#117	1	4	yes	—
83	C₂×C₄×C₄₇	#117	1	8	yes	—
84	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
87	C₂×Dic₄₇	—	2	8	no	—
88	C₂×Dic₄₇	—	2	16	no	—
89	C₂×Dic₄₇	—	4	16	no	—
90	C₈×C₄₇	—	2	8	no	—
91	C₄₇⋊C₈	—	2	8	no	—
92	#20365	—	47	1	no	maximal, Sylow
93	C₂²×C₄₇	—	2	4	no	—
94	C₂²×C₄₇	—	2	8	no	—
95	C₂²×C₄₇	#376	1	4	yes	—
96	C₂²×C₄₇	C₂²×C₄²	1	1	yes	characteristic
97	C₂²×C₄₇	C₂×#117	1	2	yes	—
98	C₂²×C₄₇	#568	1	4	yes	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	—	2	4	no	—
101	C₄×C₄₇	#376	1	4	yes	—
102	C₄×C₄₇	C₂×#117	1	2	yes	—
103	C₄×C₄₇	C₂×#117	1	2	yes	—
104	Dic₄₇	—	4	8	no	—
105	C₂×C₄×M₁₆	—	47	1	no	—
106	C₂×C₄×M₁₆	D₄₇	1	1	yes	characteristic
107	C₂×#376	—	47	1	no	—
108	#1746	—	47	4	no	—
109	#1746	—	47	4	no	—
110	#1728	—	47	4	no	—
111	C₂×C₄₇	—	2	4	no	—
112	C₂×C₄₇	C₂×#376	1	1	yes	characteristic
113	C₂×C₄₇	#2873	1	2	yes	—
114	C₂×C₄₇	#2921	1	4	yes	—
115	C₂²×M₁₆	—	47	2	no	—
116	C₂²×M₁₆	C₂×D₄₇	1	2	yes	—
117	C₂²×C₄²	—	47	2	no	—
118	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
119	C₄×M₁₆	—	47	8	no	—
120	C₄×M₁₆	Dic₄₇	1	8	yes	complemented
121	C₂×#97	—	47	2	no	—
122	C₂×#97	C₂×D₄₇	1	2	yes	—
123	C₂×C₄×C₈	—	47	2	no	—
124	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
125	#376	—	47	4	no	—
126	#376	—	47	2	no	—
127	#376	—	47	2	no	—
128	C₂×#95	—	47	4	no	—
129	#336	—	47	16	no	—
130	C₄₇	#20365	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	—	47	4	no	—
132	C₂³×C₄	C₄×D₄₇	1	2	yes	—
133	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
134	C₂×M₁₆	—	2	8	no	—
135	C₂×M₁₆	—	47	8	no	—
136	C₂×M₁₆	—	94	8	no	—
137	C₂×M₁₆	C₂×Dic₄₇	1	8	yes	—
138	C₂²×C₈	—	47	4	no	—
139	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
140	C₂×C₄²	—	47	4	no	—
141	C₂×C₄²	—	47	4	no	—
142	C₂×C₄²	—	94	8	no	—
143	C₂×C₄²	C₄×D₄₇	1	2	yes	—
144	C₂×C₄²	C₄×D₄₇	1	4	yes	—
145	C₂×C₄²	C₂×Dic₄₇	1	4	yes	—
146	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
148	#97	—	47	8	no	—
149	#97	C₂×Dic₄₇	1	8	yes	complemented
150	C₄×C₈	—	47	8	no	—
151	C₄×C₈	C₂×Dic₄₇	1	8	yes	complemented
152	#95	—	47	8	no	—
153	#95	—	47	8	no	—
154	C₂⁴	C₂×C₄×D₄₇	1	1	yes	characteristic
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	47	8	no	—
157	C₂²×C₄	—	94	8	no	—
158	C₂²×C₄	—	188	8	no	—
159	C₂²×C₄	C₄×Dic₄₇	1	8	yes	—
160	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
161	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
162	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₄₇	1	2	yes	—
164	C₂²×C₄	#10625350	1	8	yes	—
165	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
166	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
167	M₁₆	—	2	16	no	—
168	M₁₆	—	94	16	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	47	8	no	—
171	C₂×C₈	—	94	8	no	—
172	C₂×C₈	C₂²×Dic₄₇	1	8	yes	—
173	C₄²	—	94	8	no	—
174	C₄²	—	94	8	no	—
175	C₄²	C₄×Dic₄₇	1	8	yes	—
176	C₄²	C₂×C₄×D₄₇	1	2	yes	—
177	C₄²	C₂×C₄×D₄₇	1	2	yes	—
178	C₄²	C₂²×Dic₄₇	1	2	yes	—
179	C₄²	C₂²×Dic₄₇	1	2	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄×Dic₄₇	1	2	yes	—
182	C₂³	#13093981	1	4	yes	—
183	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	8	no	—
187	C₂×C₄	—	94	8	no	—
188	C₂×C₄	—	94	16	no	—
189	C₂×C₄	—	188	16	no	—
190	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
191	C₂×C₄	C₂×C₄×Dic₄₇	1	4	yes	—
192	C₂×C₄	C₂×C₄×Dic₄₇	1	2	yes	—
193	C₂×C₄	#13093981	1	4	yes	—
194	C₂×C₄	#13093981	1	8	yes	—
195	C₂×C₄	#13093983	1	8	yes	—
196	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
197	C₂×C₄	C₂³×Dic₄₇	1	1	yes	characteristic
198	C₂×C₄	C₂×#10625350	1	4	yes	—
199	C₈	—	2	8	no	—
200	C₈	—	94	8	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	4	no	—
203	C₂²	#15688969	1	4	yes	—
204	C₂²	C₂²×C₄×Dic₄₇	1	1	yes	characteristic
205	C₂²	C₂×#13093981	1	2	yes	—
206	C₂²	#15689889	1	4	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	188	8	no	—
210	C₄	#15688969	1	4	yes	—
211	C₄	C₂×#13093981	1	2	yes	—
212	C₄	C₂×#13093983	1	2	yes	—
213	C₂	—	2	4	no	—
214	C₂	C₂×#15688969	1	1	yes	characteristic
215	C₂	#20251806	1	2	yes	—
216	C₂	#20252024	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂²×C₄²
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	#20365
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20243141	C₂	semidirect
4. #20243139	C₂	semidirect
9. C₄×#10625329	C₄	semidirect
19. C₄×M₁₆×C₄₇	C₄	semidirect
29. #13093935	C₂×C₄	semidirect
30. C₄×C₄₇⋊C₈	C₂×C₄	semidirect
43. #97×C₄₇	C₂×C₄	semidirect
44. C₄×C₈×C₄₇	C₂×C₄	semidirect
120. C₄×M₁₆	Dic₄₇	semidirect
130. C₄₇	#20365	semidirect
149. #97	C₂×Dic₄₇	semidirect
151. 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×#15688969	2
#20243141	2
#20243139	2
#20243081	2
C₂×C₄×M₁₆×C₄₇	2
#20365	47