Subgroups — C₂²×D₈×Dic₄₇ #27080280

Order 12032.

3694 conjugacy classes of subgroups.

The subgroup lattice has 256 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×D₈×Dic₄₇	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13093958	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13093945	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₄₇	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	D₈×Dic₄₇	C₂²	1	64	yes	direct product factor
15	C₂×#13093958	C₂²	1	12	yes	complemented
16	C₂×#13093945	C₂²	1	24	yes	complemented
17	C₂×C₈×Dic₄₇	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	1	yes	characteristic
26	C₂³×Dic₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₄₇⋊C₈	—	2	4	no	—
28	C₂²×C₄₇⋊C₈	C₂³	1	6	yes	—
29	D₄×Dic₄₇	—	2	48	no	—
30	D₄×Dic₄₇	C₂³	1	32	yes	complemented
31	C₂×C₄⋊Dic₄₇	—	2	6	no	—
32	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
33	C₂×C₂²⋊Dic₄₇	—	2	24	no	—
34	C₂×C₄×Dic₄₇	D₄	1	6	yes	—
35	C₂×C₄×Dic₄₇	C₂³	1	6	yes	—
36	#13093958	C₂³	1	16	yes	complemented
37	#13093945	C₂³	1	32	yes	complemented
38	C₈×Dic₄₇	C₂³	1	16	yes	complemented
39	C₂⁵×C₄₇	—	2	2	no	—
40	C₂²×D₄×C₄₇	—	2	6	no	—
41	C₂²×D₄×C₄₇	—	2	8	no	—
42	C₂²×D₄×C₄₇	C₂×C₄	1	16	yes	complemented
43	C₂²×D₄×C₄₇	C₂³	1	12	yes	—
44	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₄₇	C₂×C₄	1	96	yes	complemented
46	C₂×D₈×C₄₇	C₂³	1	16	yes	—
47	C₂²×C₈×C₄₇	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₄₇	C₂³	1	6	yes	—
49	C₂⁴×C₄₇	—	2	16	no	—
50	C₂⁴×C₄₇	—	2	12	no	—
51	C₂⁴×C₄₇	—	4	8	no	—
52	C₂⁴×C₄₇	—	4	8	no	—
53	C₂⁴×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×D₄×C₄₇	—	2	48	no	—
55	C₂×D₄×C₄₇	—	2	12	no	—
56	C₂×D₄×C₄₇	—	2	24	no	—
57	C₂×D₄×C₄₇	C₂²×C₄	1	48	yes	complemented
58	C₂×D₄×C₄₇	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₄₇	#41	1	4	yes	—
62	C₂²×C₄×C₄₇	C₂⁴	1	3	yes	—
63	C₂²×Dic₄₇	—	2	4	no	—
64	C₂²×Dic₄₇	—	2	12	no	—
65	C₂²×Dic₄₇	—	2	32	no	—
66	C₂²×Dic₄₇	—	4	8	no	—
67	C₂²×Dic₄₇	—	4	96	no	—
68	C₂²×Dic₄₇	D₈	1	8	yes	direct product factor
69	C₂²×Dic₄₇	C₂×D₄	1	6	yes	—
70	C₂²×Dic₄₇	C₂×D₄	1	6	yes	—
71	D₈×C₄₇	C₂²×C₄	1	64	yes	complemented
72	C₂×C₈×C₄₇	C₂²×C₄	1	24	yes	complemented
73	C₂×C₈×C₄₇	C₂⁴	1	4	yes	—
74	C₂²⋊Dic₄₇	—	2	32	no	—
75	C₄⋊Dic₄₇	—	2	12	no	—
76	C₄⋊Dic₄₇	C₂⁴	1	8	yes	—
77	C₄×Dic₄₇	C₂×D₄	1	12	yes	—
78	C₄×Dic₄₇	C₂⁴	1	4	yes	—
79	C₂×C₄₇⋊C₈	—	2	12	no	—
80	C₂×C₄₇⋊C₈	C₂⁴	1	4	yes	—
81	C₂³×C₄₇	—	2	48	no	—
82	C₂³×C₄₇	—	2	8	no	—
83	C₂³×C₄₇	—	4	24	no	—
84	C₂³×C₄₇	—	4	24	no	—
85	C₂³×C₄₇	—	4	64	no	—
86	C₂³×C₄₇	C₄×D₄	1	4	yes	—
87	C₂³×C₄₇	C₂×D₈	1	4	yes	—
88	C₂³×C₄₇	#135	1	4	yes	—
89	C₂³×C₄₇	C₂²×D₄	1	3	yes	—
90	D₄×C₄₇	—	2	48	no	—
91	D₄×C₄₇	—	2	8	no	—
92	D₄×C₄₇	C₂³×C₄	1	16	yes	complemented
93	C₂×C₄×C₄₇	C₄×D₄	1	12	yes	—
94	C₂×C₄×C₄₇	C₂³×C₄	1	6	yes	—
95	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
96	C₂×C₄×C₄₇	C₂×#41	1	6	yes	—
97	C₂×C₄×C₄₇	C₂⁵	1	1	yes	characteristic
98	C₂×Dic₄₇	—	2	12	no	—
99	C₂×Dic₄₇	—	2	8	no	—
100	C₂×Dic₄₇	—	4	24	no	—
101	C₂×Dic₄₇	—	4	64	no	—
102	C₂×Dic₄₇	C₂×D₈	1	24	yes	direct product factor
103	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
104	C₂×Dic₄₇	C₂²×D₄	1	4	yes	—
105	C₈×C₄₇	C₂³×C₄	1	8	yes	complemented
106	C₄₇⋊C₈	—	2	4	no	—
107	C₂²×C₄×D₈	—	47	1	no	maximal, Sylow
108	C₂²×C₄₇	—	2	16	no	—
109	C₂²×C₄₇	—	4	96	no	—
110	C₂²×C₄₇	—	4	8	no	—
111	C₂²×C₄₇	—	4	8	no	—
112	C₂²×C₄₇	C₄×D₈	1	16	yes	complemented
113	C₂²×C₄₇	C₂×C₄×D₄	1	6	yes	—
114	C₂²×C₄₇	C₂²×D₈	1	6	yes	—
115	C₂²×C₄₇	C₂×#135	1	6	yes	—
116	C₂²×C₄₇	C₂³×D₄	1	1	yes	characteristic
117	C₄×C₄₇	C₂×C₄×D₄	1	6	yes	—
118	C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
119	C₄×C₄₇	C₂²×#41	1	1	yes	characteristic
120	Dic₄₇	—	2	4	no	—
121	Dic₄₇	—	4	8	no	—
122	Dic₄₇	C₂²×D₈	1	8	yes	direct product factor
123	C₂³×D₈	D₄₇	1	1	yes	characteristic
124	C₂²×C₄×D₄	—	47	2	no	—
125	C₂×C₄×D₈	—	47	24	no	—
126	C₂²×#107	—	47	1	no	—
127	C₂²×#102	—	47	2	no	—
128	C₂²×C₄×C₈	—	47	1	no	—
129	C₂×C₄₇	—	4	16	no	—
130	C₂×C₄₇	C₂×C₄×D₈	1	12	yes	complemented
131	C₂×C₄₇	C₂²×C₄×D₄	1	1	yes	characteristic
132	C₂×C₄₇	C₂³×D₈	1	1	yes	characteristic
133	C₂×C₄₇	C₂²×#135	1	1	yes	characteristic
134	C₂³×D₄	C₂×D₄₇	1	2	yes	—
135	C₂⁴×C₄	—	94	2	no	—
136	C₂²×D₈	Dic₄₇	1	16	yes	direct product factor
137	C₂²×D₈	C₂×D₄₇	1	12	yes	—
138	C₂³×C₈	—	47	1	no	—
139	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
140	C₂×C₄×D₄	—	47	24	no	—
141	C₂×C₄×D₄	—	94	12	no	—
142	C₂²×C₄⋊C₄	—	47	2	no	—
143	C₂²×C₂²⋊C₄	—	94	2	no	—
144	C₂²×C₄²	—	47	1	no	—
145	C₄×D₈	—	47	64	no	—
146	C₂×#107	—	47	12	no	—
147	C₂×#102	—	47	24	no	—
148	C₂×C₄×C₈	—	47	12	no	—
149	C₄₇	C₂²×C₄×D₈	1	1	yes	characteristic, complemented, Sylow
150	C₂⁵	—	2	2	no	—
151	C₂²×D₄	—	2	6	no	—
152	C₂²×D₄	—	2	8	no	—
153	C₂²×D₄	C₂×Dic₄₇	1	16	yes	complemented
154	C₂²×D₄	C₂²×D₄₇	1	12	yes	—
155	C₂³×C₄	—	47	1	no	—
156	C₂³×C₄	—	47	1	no	—
157	C₂³×C₄	—	94	24	no	—
158	C₂³×C₄	—	94	2	no	—
159	C₂³×C₄	—	188	16	no	—
160	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
161	C₂×D₈	C₂×Dic₄₇	1	96	yes	direct product factor
162	C₂×D₈	C₂²×D₄₇	1	16	yes	—
163	C₂²×C₈	—	47	6	no	—
164	C₂²×C₈	—	94	4	no	—
165	C₂²×C₈	C₂×Dic₄₇	1	8	yes	complemented
166	C₂²×C₈	C₂²×D₄₇	1	6	yes	—
167	C₄×D₄	—	47	32	no	—
168	C₄×D₄	—	94	48	no	—
169	C₂×C₄⋊C₄	—	47	12	no	—
170	C₂×C₄⋊C₄	—	94	6	no	—
171	C₂×C₂²⋊C₄	—	94	24	no	—
172	C₂×C₄²	—	47	6	no	—
173	C₂×C₄²	—	47	6	no	—
174	#107	—	47	16	no	—
175	#102	—	47	32	no	—
176	C₄×C₈	—	47	16	no	—
177	C₂⁴	—	2	16	no	—
178	C₂⁴	—	2	12	no	—
179	C₂⁴	—	4	8	no	—
180	C₂⁴	—	4	8	no	—
181	C₂⁴	D₄×D₄₇	1	1	yes	characteristic
182	C₂×D₄	—	2	48	no	—
183	C₂×D₄	—	2	12	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	C₂²×Dic₄₇	1	48	yes	complemented
186	C₂×D₄	C₂³×D₄₇	1	8	yes	—
187	C₂²×C₄	—	47	8	no	—
188	C₂²×C₄	—	47	6	no	—
189	C₂²×C₄	—	47	6	no	—
190	C₂²×C₄	—	94	32	no	—
191	C₂²×C₄	—	94	4	no	—
192	C₂²×C₄	—	94	12	no	—
193	C₂²×C₄	—	188	96	no	—
194	C₂²×C₄	—	188	8	no	—
195	C₂²×C₄	D₄×D₄₇	1	3	yes	—
196	C₂²×C₄	#10625352	1	4	yes	—
197	C₂²×C₄	C₂²×Dic₄₇	1	4	yes	—
198	C₂²×C₄	C₂³×D₄₇	1	3	yes	—
199	D₈	C₂²×Dic₄₇	1	64	yes	direct product factor
200	C₂×C₈	—	47	4	no	—
201	C₂×C₈	—	94	12	no	—
202	C₂×C₈	C₂²×Dic₄₇	1	24	yes	complemented
203	C₂×C₈	C₂³×D₄₇	1	4	yes	—
204	C₄⋊C₄	—	47	8	no	—
205	C₄⋊C₄	—	94	12	no	—
206	C₂²⋊C₄	—	94	32	no	—
207	C₄²	—	47	12	no	—
208	C₄²	—	47	4	no	—
209	C₂³	—	2	48	no	—
210	C₂³	—	2	8	no	—
211	C₂³	—	4	24	no	—
212	C₂³	—	4	24	no	—
213	C₂³	—	4	64	no	—
214	C₂³	D₄×Dic₄₇	1	4	yes	—
215	C₂³	D₈×D₄₇	1	4	yes	—
216	C₂³	#13094043	1	4	yes	—
217	C₂³	C₂×D₄×D₄₇	1	3	yes	—
218	D₄	—	2	48	no	—
219	D₄	—	2	8	no	—
220	D₄	C₂³×Dic₄₇	1	16	yes	complemented
221	C₂×C₄	—	47	24	no	—
222	C₂×C₄	—	47	4	no	—
223	C₂×C₄	—	47	4	no	—
224	C₂×C₄	—	94	12	no	—
225	C₂×C₄	—	94	8	no	—
226	C₂×C₄	—	188	64	no	—
227	C₂×C₄	—	188	24	no	—
228	C₂×C₄	D₄×Dic₄₇	1	12	yes	—
229	C₂×C₄	C₂³×Dic₄₇	1	6	yes	—
230	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
231	C₂×C₄	C₂×#10625352	1	6	yes	—
232	C₂×C₄	C₂⁴×D₄₇	1	1	yes	characteristic
233	C₈	—	94	4	no	—
234	C₈	C₂³×Dic₄₇	1	8	yes	complemented
235	C₂²	—	2	16	no	—
236	C₂²	—	4	8	no	—
237	C₂²	—	4	8	no	—
238	C₂²	—	4	96	no	—
239	C₂²	D₈×Dic₄₇	1	16	yes	direct product factor
240	C₂²	C₂×D₄×Dic₄₇	1	6	yes	—
241	C₂²	C₂×D₈×D₄₇	1	6	yes	—
242	C₂²	C₂×#13094043	1	6	yes	—
243	C₂²	C₂²×D₄×D₄₇	1	1	yes	characteristic
244	C₄	—	47	8	no	—
245	C₄	—	94	4	no	—
246	C₄	—	188	8	no	—
247	C₄	C₂×D₄×Dic₄₇	1	6	yes	—
248	C₄	C₂⁴×Dic₄₇	1	1	yes	characteristic
249	C₄	C₂²×#10625352	1	1	yes	characteristic
250	C₂	—	4	16	no	—
251	C₂	C₂×D₈×Dic₄₇	1	12	yes	direct product factor
252	C₂	C₂²×D₄×Dic₄₇	1	1	yes	characteristic
253	C₂	C₂²×D₈×D₄₇	1	1	yes	characteristic
254	C₂	C₂²×#13094043	1	1	yes	characteristic

	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×C₄×D₈
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×D₈×Dic₄₇	C₂	direct (N × H)
3. C₂²×#13093958	C₂	semidirect
4. C₂²×#13093945	C₂	semidirect
5. C₂²×C₈×Dic₄₇	C₂	semidirect
10. C₂×D₄×Dic₄₇	C₂²	semidirect
14. D₈×Dic₄₇	C₂²	direct (N × H)
15. C₂×#13093958	C₂²	semidirect
16. C₂×#13093945	C₂²	semidirect
17. C₂×C₈×Dic₄₇	C₂²	semidirect
19. C₂²×D₈×C₄₇	C₄	semidirect
30. D₄×Dic₄₇	C₂³	semidirect
36. #13093958	C₂³	semidirect
37. #13093945	C₂³	semidirect
38. C₈×Dic₄₇	C₂³	semidirect
42. C₂²×D₄×C₄₇	C₂×C₄	semidirect
45. C₂×D₈×C₄₇	C₂×C₄	semidirect
47. C₂²×C₈×C₄₇	C₂×C₄	semidirect
57. C₂×D₄×C₄₇	C₂²×C₄	semidirect
68. C₂²×Dic₄₇	D₈	direct (N × H)
71. D₈×C₄₇	C₂²×C₄	semidirect
72. C₂×C₈×C₄₇	C₂²×C₄	semidirect
92. D₄×C₄₇	C₂³×C₄	semidirect
102. C₂×Dic₄₇	C₂×D₈	direct (N × H)
105. C₈×C₄₇	C₂³×C₄	semidirect
112. C₂²×C₄₇	C₄×D₈	semidirect
122. Dic₄₇	C₂²×D₈	direct (N × H)
130. C₂×C₄₇	C₂×C₄×D₈	semidirect
136. C₂²×D₈	Dic₄₇	direct (N × H)
149. C₄₇	C₂²×C₄×D₈	semidirect
153. C₂²×D₄	C₂×Dic₄₇	semidirect
161. C₂×D₈	C₂×Dic₄₇	direct (N × H)
165. C₂²×C₈	C₂×Dic₄₇	semidirect
185. C₂×D₄	C₂²×Dic₄₇	semidirect
199. D₈	C₂²×Dic₄₇	direct (N × H)
202. C₂×C₈	C₂²×Dic₄₇	semidirect
220. D₄	C₂³×Dic₄₇	semidirect
234. C₈	C₂³×Dic₄₇	semidirect
239. C₂²	D₈×Dic₄₇	direct (N × H)
251. C₂	C₂×D₈×Dic₄₇	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×D₈×Dic₄₇	2
C₂²×#13093958	2
C₂²×#13093945	2
C₂²×C₈×Dic₄₇	2
C₂³×D₈×C₄₇	2
C₂²×C₄×D₈	47

Subgroups — C22×D8×Dic47 #27080280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₈×Dic₄₇ #27080280