Subgroups — C₂²×#15689235 #27080323

Order 12032.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
2	C₂×#15689235	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093956	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13093945	C₂	1	2	yes	maximal, complemented
5	C₂²×#13093934	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	—	2	2	no	—
8	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×D₄×Dic₄₇	—	2	12	no	—
10	C₂×D₄×Dic₄₇	C₂²	1	24	yes	complemented
11	C₂²×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂²×C₂²⋊Dic₄₇	—	2	2	no	—
13	C₂²×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	#15689235	C₂²	1	64	yes	direct product factor
15	C₂×#13093956	C₂²	1	12	yes	complemented
16	C₂×#13093945	C₂²	1	24	yes	complemented
17	C₂×#13093934	C₂²	1	12	yes	complemented
18	C₂³×D₄×C₄₇	C₂²	1	2	yes	—
19	C₂²×D₈×C₄₇	C₄	1	16	yes	complemented
20	C₂²×D₈×C₄₇	C₂²	1	12	yes	—
21	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂³×Dic₄₇	—	2	2	no	—
23	C₂³×Dic₄₇	—	2	24	no	—
24	C₂³×Dic₄₇	—	4	16	no	—
25	C₂³×Dic₄₇	D₄	1	2	yes	—
26	C₂²×C₄₇⋊C₈	—	2	4	no	—
27	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
28	D₄×Dic₄₇	—	2	48	no	—
29	D₄×Dic₄₇	C₂³	1	32	yes	complemented
30	C₂×C₄⋊Dic₄₇	—	2	6	no	—
31	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
32	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
33	C₂×C₄×Dic₄₇	D₄	1	6	yes	—
34	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
35	#13093956	C₂³	1	16	yes	complemented
36	#13093945	C₂³	1	32	yes	complemented
37	#13093934	C₂³	1	16	yes	complemented
38	C₂⁵×C₄₇	—	2	2	no	—
39	C₂²×D₄×C₄₇	—	2	6	no	—
40	C₂²×D₄×C₄₇	—	2	8	no	—
41	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
42	C₂²×D₄×C₄₇	C₂³	1	12	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×D₈×C₄₇	C₂×C₄	1	96	yes	complemented
45	C₂×D₈×C₄₇	C₂³	1	16	yes	—
46	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
47	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
48	C₂⁴×C₄₇	—	2	16	no	—
49	C₂⁴×C₄₇	—	2	12	no	—
50	C₂⁴×C₄₇	—	4	8	no	—
51	C₂⁴×C₄₇	—	4	8	no	—
52	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×D₄×C₄₇	—	2	48	no	—
54	C₂×D₄×C₄₇	—	2	12	no	—
55	C₂×D₄×C₄₇	—	2	24	no	—
56	C₂×D₄×C₄₇	C₂²×C₄	1	48	yes	complemented
57	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
58	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₄₇	#41	1	4	yes	—
61	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
62	C₂²×Dic₄₇	—	2	8	no	—
63	C₂²×Dic₄₇	—	2	12	no	—
64	C₂²×Dic₄₇	—	2	32	no	—
65	C₂²×Dic₄₇	—	4	96	no	—
66	C₂²×Dic₄₇	—	4	8	no	—
67	C₂²×Dic₄₇	C₂×D₄	1	12	yes	—
68	D₈×C₄₇	C₂²×C₄	1	64	yes	complemented
69	C₂×C₈×C₄₇	C₂²×C₄	1	24	yes	complemented
70	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
71	C₂²⋊Dic₄₇	—	2	32	no	—
72	C₄⋊Dic₄₇	—	2	12	no	—
73	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
74	C₄×Dic₄₇	C₂×D₄	1	12	yes	—
75	C₄×Dic₄₇	C₂⁴	1	4	yes	—
76	C₂×C₄₇⋊C₈	—	2	12	no	—
77	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
78	C₂³×C₄₇	—	2	48	no	—
79	C₂³×C₄₇	—	2	8	no	—
80	C₂³×C₄₇	—	4	24	no	—
81	C₂³×C₄₇	—	4	24	no	—
82	C₂³×C₄₇	—	4	64	no	—
83	C₂³×C₄₇	C₄×D₄	1	4	yes	—
84	C₂³×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
85	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
86	D₄×C₄₇	—	2	48	no	—
87	D₄×C₄₇	—	2	8	no	—
88	D₄×C₄₇	C₂³×C₄	1	16	yes	complemented
89	C₂×C₄×C₄₇	C₄×D₄	1	12	yes	—
90	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
91	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
92	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
93	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
94	C₂×Dic₄₇	—	2	24	no	—
95	C₂×Dic₄₇	—	2	8	no	—
96	C₂×Dic₄₇	—	4	64	no	—
97	C₂×Dic₄₇	—	4	24	no	—
98	C₂×Dic₄₇	C₂²×D₄	1	8	yes	—
99	C₈×C₄₇	C₂³×C₄	1	8	yes	complemented
100	C₄₇⋊C₈	—	2	4	no	—
101	C₂²×#442	—	47	1	no	maximal, Sylow
102	C₂²×C₄₇	—	2	16	no	—
103	C₂²×C₄₇	—	4	96	no	—
104	C₂²×C₄₇	—	4	8	no	—
105	C₂²×C₄₇	—	4	8	no	—
106	C₂²×C₄₇	#442	1	16	yes	complemented
107	C₂²×C₄₇	C₂×C₄×D₄	1	6	yes	—
108	C₂²×C₄₇	C₂×AGL(1,ℤ/8)	1	12	yes	—
109	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
110	C₄×C₄₇	C₂×C₄×D₄	1	6	yes	—
111	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
112	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
113	Dic₄₇	—	2	8	no	—
114	Dic₄₇	—	4	8	no	—
115	C₂³×D₈	D₄₇	1	1	yes	characteristic
116	C₂²×C₄×D₄	—	47	2	no	—
117	C₂×#442	—	47	24	no	—
118	C₂²×#106	—	47	1	no	—
119	C₂²×#102	—	47	2	no	—
120	C₂²×#97	—	47	1	no	—
121	C₂×C₄₇	—	4	16	no	—
122	C₂×C₄₇	C₂×#442	1	12	yes	complemented
123	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
124	C₂×C₄₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
125	C₂³×D₄	C₂×D₄₇	1	2	yes	—
126	C₂⁴×C₄	—	94	2	no	—
127	C₂²×D₈	Dic₄₇	1	16	yes	complemented
128	C₂²×D₈	C₂×D₄₇	1	12	yes	—
129	C₂³×C₈	—	47	1	no	—
130	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
131	C₂×C₄×D₄	—	47	24	no	—
132	C₂×C₄×D₄	—	94	12	no	—
133	C₂²×C₄⋊C₄	—	47	2	no	—
134	C₂²×C₂²⋊C₄	—	94	2	no	—
135	C₂²×C₄²	—	47	1	no	—
136	#442	—	47	64	no	—
137	C₂×#106	—	47	12	no	—
138	C₂×#102	—	47	24	no	—
139	C₂×#97	—	47	12	no	—
140	C₄₇	C₂²×#442	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	2	2	no	—
142	C₂²×D₄	—	2	6	no	—
143	C₂²×D₄	—	2	8	no	—
144	C₂²×D₄	C₂×Dic₄₇	1	16	yes	complemented
145	C₂²×D₄	C₂²×D₄₇	1	12	yes	—
146	C₂³×C₄	—	47	2	no	—
147	C₂³×C₄	—	94	24	no	—
148	C₂³×C₄	—	94	2	no	—
149	C₂³×C₄	—	188	16	no	—
150	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	C₂×D₈	C₂×Dic₄₇	1	96	yes	complemented
152	C₂×D₈	C₂²×D₄₇	1	16	yes	—
153	C₂²×C₈	—	47	6	no	—
154	C₂²×C₈	—	94	4	no	—
155	C₂²×C₈	C₂×Dic₄₇	1	8	yes	complemented
156	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
157	C₄×D₄	—	47	32	no	—
158	C₄×D₄	—	94	48	no	—
159	C₂×C₄⋊C₄	—	47	12	no	—
160	C₂×C₄⋊C₄	—	94	6	no	—
161	C₂×C₂²⋊C₄	—	94	24	no	—
162	C₂×C₄²	—	47	6	no	—
163	C₂×C₄²	—	47	6	no	—
164	#106	—	47	16	no	—
165	#102	—	47	32	no	—
166	#97	—	47	16	no	—
167	C₂⁴	—	2	16	no	—
168	C₂⁴	—	2	12	no	—
169	C₂⁴	—	4	8	no	—
170	C₂⁴	—	4	8	no	—
171	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
172	C₂×D₄	—	2	48	no	—
173	C₂×D₄	—	2	12	no	—
174	C₂×D₄	—	2	24	no	—
175	C₂×D₄	C₂²×Dic₄₇	1	48	yes	complemented
176	C₂×D₄	C₂³×D₄₇	1	8	yes	—
177	C₂²×C₄	—	47	12	no	—
178	C₂²×C₄	—	94	32	no	—
179	C₂²×C₄	—	94	8	no	—
180	C₂²×C₄	—	94	12	no	—
181	C₂²×C₄	—	188	96	no	—
182	C₂²×C₄	—	188	8	no	—
183	C₂²×C₄	D₄×D₄₇	1	3	yes	—
184	C₂²×C₄	#10625352	1	4	yes	—
185	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
186	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
187	D₈	C₂²×Dic₄₇	1	64	yes	complemented
188	C₂×C₈	—	47	4	no	—
189	C₂×C₈	—	94	12	no	—
190	C₂×C₈	C₂²×Dic₄₇	1	24	yes	complemented
191	C₂×C₈	C₂³×D₄₇	1	4	yes	—
192	C₄⋊C₄	—	47	8	no	—
193	C₄⋊C₄	—	94	12	no	—
194	C₂²⋊C₄	—	94	32	no	—
195	C₄²	—	47	12	no	—
196	C₄²	—	47	4	no	—
197	C₂³	—	2	48	no	—
198	C₂³	—	2	8	no	—
199	C₂³	—	4	64	no	—
200	C₂³	—	4	24	no	—
201	C₂³	—	4	24	no	—
202	C₂³	D₄×Dic₄₇	1	4	yes	—
203	C₂³	#13094048	1	8	yes	—
204	C₂³	C₂×D₄×D₄₇	1	3	yes	—
205	D₄	—	2	48	no	—
206	D₄	—	2	8	no	—
207	D₄	C₂³×Dic₄₇	1	16	yes	complemented
208	C₂×C₄	—	47	8	no	—
209	C₂×C₄	—	94	24	no	—
210	C₂×C₄	—	94	8	no	—
211	C₂×C₄	—	188	64	no	—
212	C₂×C₄	—	188	24	no	—
213	C₂×C₄	D₄×Dic₄₇	1	12	yes	—
214	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
215	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
216	C₂×C₄	C₂×#10625352	1	6	yes	—
217	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
218	C₈	—	94	4	no	—
219	C₈	C₂³×Dic₄₇	1	8	yes	complemented
220	C₂²	—	2	16	no	—
221	C₂²	—	4	96	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	8	no	—
224	C₂²	#15689235	1	16	yes	direct product factor
225	C₂²	C₂×D₄×Dic₄₇	1	6	yes	—
226	C₂²	C₂×#13094048	1	12	yes	—
227	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
228	C₄	—	94	8	no	—
229	C₄	—	188	8	no	—
230	C₄	C₂×D₄×Dic₄₇	1	6	yes	—
231	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
232	C₄	C₂²×#10625352	1	1	yes	characteristic
233	C₂	—	4	16	no	—
234	C₂	C₂×#15689235	1	12	yes	direct product factor
235	C₂	C₂²×D₄×Dic₄₇	1	1	yes	characteristic
236	C₂	C₂²×#13094048	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₄₇
commutator	C₄×C₄₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₄₇
Fitting, prime radical	C₂³×D₈×C₄₇	C₂
socle	C₂⁴×C₄₇	C₂×D₄
cosocle	C₂×C₄×C₄₇	C₂⁵
hypercenter	C₂³×D₈	D₄₇
hypocenter	C₄₇	C₂²×#442
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×Dic₄₇	C₂	semidirect
2. C₂×#15689235	C₂	direct (N × H)
3. C₂²×#13093956	C₂	semidirect
4. C₂²×#13093945	C₂	semidirect
5. C₂²×#13093934	C₂	semidirect
10. C₂×D₄×Dic₄₇	C₂²	semidirect
14. #15689235	C₂²	direct (N × H)
15. C₂×#13093956	C₂²	semidirect
16. C₂×#13093945	C₂²	semidirect
17. C₂×#13093934	C₂²	semidirect
19. C₂²×D₈×C₄₇	C₄	semidirect
29. D₄×Dic₄₇	C₂³	semidirect
35. #13093956	C₂³	semidirect
36. #13093945	C₂³	semidirect
37. #13093934	C₂³	semidirect
41. C₂²×D₄×C₄₇	C₂×C₄	semidirect
44. C₂×D₈×C₄₇	C₂×C₄	semidirect
46. C₂²×C₈×C₄₇	C₂×C₄	semidirect
56. C₂×D₄×C₄₇	C₂²×C₄	semidirect
68. D₈×C₄₇	C₂²×C₄	semidirect
69. C₂×C₈×C₄₇	C₂²×C₄	semidirect
88. D₄×C₄₇	C₂³×C₄	semidirect
99. C₈×C₄₇	C₂³×C₄	semidirect
106. C₂²×C₄₇	#442	semidirect
122. C₂×C₄₇	C₂×#442	semidirect
127. C₂²×D₈	Dic₄₇	semidirect
140. C₄₇	C₂²×#442	semidirect
144. C₂²×D₄	C₂×Dic₄₇	semidirect
151. C₂×D₈	C₂×Dic₄₇	semidirect
155. C₂²×C₈	C₂×Dic₄₇	semidirect
175. C₂×D₄	C₂²×Dic₄₇	semidirect
187. D₈	C₂²×Dic₄₇	semidirect
190. C₂×C₈	C₂²×Dic₄₇	semidirect
207. D₄	C₂³×Dic₄₇	semidirect
219. C₈	C₂³×Dic₄₇	semidirect
224. C₂²	#15689235	direct (N × H)
234. C₂	C₂×#15689235	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₄×Dic₄₇	2
C₂×#15689235	2
C₂²×#13093956	2
C₂²×#13093945	2
C₂²×#13093934	2
C₂³×D₈×C₄₇	2
C₂²×#442	47

Subgroups — C22×#15689235 #27080323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15689235 #27080323