Subgroups — C₂²×C₄×C₂²⋊Dic₄₇ #26374451

Order 12032.

9534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
2	C₂³×C₄×Dic₄₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×C₂²⋊Dic₄₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄₇⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₄₇	C₂²	1	4	yes	—
7	C₂²×C₂²⋊Dic₄₇	C₄	1	32	yes	direct product factor
8	C₂²×C₂²⋊Dic₄₇	C₂²	1	24	yes	—
9	C₂²×C₄×Dic₄₇	—	2	16	no	—
10	C₂²×C₄×Dic₄₇	C₂²	1	24	yes	complemented
11	C₄×C₂²⋊Dic₄₇	C₂²	1	64	yes	direct product factor
12	C₂×C₄₇⋊#95	C₂²	1	24	yes	complemented
13	C₂⁶×C₄₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₄₇	—	2	16	no	—
15	C₂⁴×C₄×C₄₇	C₄	1	16	yes	complemented
16	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₄₇	C₂²	1	12	yes	—
18	C₂⁴×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂³×Dic₄₇	—	2	16	no	—
20	C₂³×Dic₄₇	—	2	16	no	—
21	C₂³×Dic₄₇	C₂×C₄	1	32	yes	complemented
22	C₂³×Dic₄₇	C₂³	1	24	yes	—
23	C₂×C₂²⋊Dic₄₇	C₂×C₄	1	192	yes	direct product factor
24	C₂×C₂²⋊Dic₄₇	C₂³	1	32	yes	—
25	C₂×C₄×Dic₄₇	—	2	96	no	—
26	C₂×C₄×Dic₄₇	C₂³	1	32	yes	complemented
27	C₄₇⋊#95	C₂³	1	32	yes	complemented
28	C₂⁵×C₄₇	—	2	8	no	—
29	C₂⁵×C₄₇	—	2	8	no	—
30	C₂⁵×C₄₇	C₂×C₄	1	16	yes	—
31	C₂⁵×C₄₇	C₂×C₄	1	8	yes	—
32	C₂⁵×C₄₇	C₂³	1	6	yes	—
33	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₄₇	—	2	8	no	—
35	C₂³×C₄×C₄₇	—	2	128	no	—
36	C₂³×C₄×C₄₇	—	2	96	no	—
37	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
38	C₂³×C₄×C₄₇	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₄₇	C₂×C₄	1	96	yes	complemented
40	C₂³×C₄×C₄₇	D₄	1	16	yes	—
41	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
42	C₂³×C₄×C₄₇	C₂³	1	16	yes	—
43	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
44	C₂⁴×C₄₇	—	2	32	no	—
45	C₂⁴×C₄₇	—	2	32	no	—
46	C₂⁴×C₄₇	—	2	128	no	—
47	C₂⁴×C₄₇	—	2	24	no	—
48	C₂⁴×C₄₇	—	2	24	no	—
49	C₂⁴×C₄₇	C₄²	1	64	yes	complemented
50	C₂⁴×C₄₇	C₂²×C₄	1	48	yes	—
51	C₂⁴×C₄₇	C₂²×C₄	1	24	yes	—
52	C₂⁴×C₄₇	C₂²×C₄	1	4	yes	—
53	C₂⁴×C₄₇	C₂²×C₄	1	8	yes	—
54	C₂⁴×C₄₇	C₂×D₄	1	8	yes	—
55	C₂⁴×C₄₇	#41	1	8	yes	—
56	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
57	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
58	C₂²×C₄×C₄₇	—	2	32	no	—
59	C₂²×C₄×C₄₇	—	2	24	no	—
60	C₂²×C₄×C₄₇	—	2	384	no	—
61	C₂²×C₄×C₄₇	—	2	64	no	—
62	C₂²×C₄×C₄₇	C₂²⋊C₄	1	64	yes	complemented
63	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₄₇	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₄₇	C₂²×C₄	1	64	yes	complemented
66	C₂²×C₄×C₄₇	C₂×D₄	1	48	yes	—
67	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
68	C₂²×C₄×C₄₇	C₂⁴	1	4	yes	—
69	C₂²×Dic₄₇	—	2	128	no	—
70	C₂²×Dic₄₇	—	2	48	no	—
71	C₂²×Dic₄₇	—	2	48	no	—
72	C₂²×Dic₄₇	C₂²×C₄	1	96	yes	complemented
73	C₂²×Dic₄₇	C₂⁴	1	16	yes	—
74	C₂²⋊Dic₄₇	C₂²×C₄	1	128	yes	direct product factor
75	C₄×Dic₄₇	—	2	64	no	—
76	C₂³×C₄₇	—	2	48	no	—
77	C₂³×C₄₇	—	2	48	no	—
78	C₂³×C₄₇	—	2	192	no	—
79	C₂³×C₄₇	—	2	8	no	—
80	C₂³×C₄₇	—	2	8	no	—
81	C₂³×C₄₇	—	2	256	no	—
82	C₂³×C₄₇	C₂×C₄²	1	16	yes	—
83	C₂³×C₄₇	C₂×C₄²	1	96	yes	complemented
84	C₂³×C₄₇	C₂×C₂²⋊C₄	1	16	yes	—
85	C₂³×C₄₇	#117	1	16	yes	—
86	C₂³×C₄₇	C₄×D₄	1	64	yes	—
87	C₂³×C₄₇	C₂³×C₄	1	16	yes	—
88	C₂³×C₄₇	C₂³×C₄	1	12	yes	—
89	C₂³×C₄₇	C₂³×C₄	1	8	yes	—
90	C₂³×C₄₇	C₂³×C₄	1	6	yes	—
91	C₂³×C₄₇	C₂²×D₄	1	12	yes	—
92	C₂³×C₄₇	C₂×#41	1	12	yes	—
93	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
94	C₂×C₄×C₄₇	—	2	48	no	—
95	C₂×C₄×C₄₇	—	2	8	no	—
96	C₂×C₄×C₄₇	—	2	128	no	—
97	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	96	yes	complemented
98	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
99	C₂×C₄×C₄₇	C₂³×C₄	1	8	yes	—
100	C₂×C₄×C₄₇	C₂²×D₄	1	16	yes	—
101	C₂×Dic₄₇	—	2	192	no	—
102	C₂×Dic₄₇	—	2	16	no	—
103	C₂×Dic₄₇	—	2	16	no	—
104	C₂×Dic₄₇	C₂³×C₄	1	32	yes	complemented
105	C₂²×C₄×C₂²⋊C₄	—	47	1	no	maximal, Sylow
106	C₂²×C₄₇	—	2	8	no	—
107	C₂²×C₄₇	—	2	8	no	—
108	C₂²×C₄₇	—	2	32	no	—
109	C₂²×C₄₇	—	2	192	no	—
110	C₂²×C₄₇	C₄×C₂²⋊C₄	1	64	yes	complemented
111	C₂²×C₄₇	C₂²×C₄²	1	12	yes	—
112	C₂²×C₄₇	C₂²×C₄²	1	16	yes	complemented
113	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	12	yes	—
114	C₂²×C₄₇	C₂×#117	1	12	yes	—
115	C₂²×C₄₇	C₂×C₄×D₄	1	48	yes	—
116	C₂²×C₄₇	C₂⁴×C₄	1	2	yes	—
117	C₂²×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
118	C₂²×C₄₇	C₂³×D₄	1	2	yes	—
119	C₂²×C₄₇	C₂²×#41	1	2	yes	—
120	C₄×C₄₇	—	2	8	no	—
121	C₄×C₄₇	C₂²×C₂²⋊C₄	1	16	yes	complemented
122	Dic₄₇	—	2	32	no	—
123	C₂⁵×C₄	D₄₇	1	1	yes	characteristic
124	C₂³×C₂²⋊C₄	—	47	2	no	—
125	C₂³×C₄²	—	47	2	no	—
126	C₂×C₄×C₂²⋊C₄	—	47	24	no	—
127	C₂²×#95	—	47	2	no	—
128	C₂×C₄₇	—	2	16	no	—
129	C₂×C₄₇	C₂×C₄×C₂²⋊C₄	1	24	yes	complemented
130	C₂×C₄₇	C₂³×C₄²	1	1	yes	characteristic
131	C₂×C₄₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄₇	C₂²×#117	1	1	yes	characteristic
133	C₂×C₄₇	C₂²×C₄×D₄	1	4	yes	—
134	C₂⁶	C₂×D₄₇	1	1	yes	characteristic
135	C₂⁴×C₄	—	2	16	no	—
136	C₂⁴×C₄	—	47	4	no	—
137	C₂⁴×C₄	Dic₄₇	1	16	yes	complemented
138	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
139	C₂⁴×C₄	C₂×D₄₇	1	12	yes	—
140	C₂⁴×C₄	C₂×D₄₇	1	1	yes	characteristic
141	C₂²×C₂²⋊C₄	—	47	32	no	—
142	C₂²×C₂²⋊C₄	—	47	24	no	—
143	C₂²×C₄²	—	47	24	no	—
144	C₂²×C₄²	—	94	16	no	—
145	C₄×C₂²⋊C₄	—	47	64	no	—
146	C₂×#95	—	47	24	no	—
147	C₄₇	C₂²×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
148	C₂⁵	—	2	8	no	—
149	C₂⁵	—	2	8	no	—
150	C₂⁵	C₄×D₄₇	1	16	yes	—
151	C₂⁵	C₂×Dic₄₇	1	8	yes	—
152	C₂⁵	C₂²×D₄₇	1	6	yes	—
153	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	128	no	—
156	C₂³×C₄	—	2	96	no	—
157	C₂³×C₄	—	47	32	no	—
158	C₂³×C₄	—	47	24	no	—
159	C₂³×C₄	—	94	16	no	—
160	C₂³×C₄	—	94	16	no	—
161	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
162	C₂³×C₄	C₂×Dic₄₇	1	8	yes	—
163	C₂³×C₄	C₂×Dic₄₇	1	96	yes	complemented
164	C₂³×C₄	C₂²⋊D₄₇	1	16	yes	—
165	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
166	C₂³×C₄	C₂²×D₄₇	1	16	yes	—
167	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
168	C₂×C₂²⋊C₄	—	47	192	no	—
169	C₂×C₂²⋊C₄	—	47	32	no	—
170	C₂×C₄²	—	47	32	no	—
171	C₂×C₄²	—	94	96	no	—
172	#95	—	47	32	no	—
173	C₂⁴	—	2	32	no	—
174	C₂⁴	—	2	32	no	—
175	C₂⁴	—	2	128	no	—
176	C₂⁴	—	2	24	no	—
177	C₂⁴	—	2	24	no	—
178	C₂⁴	C₄×Dic₄₇	1	64	yes	complemented
179	C₂⁴	C₂×C₄×D₄₇	1	48	yes	—
180	C₂⁴	C₂×C₄×D₄₇	1	8	yes	—
181	C₂⁴	#10625350	1	8	yes	—
182	C₂⁴	C₂²×Dic₄₇	1	24	yes	—
183	C₂⁴	C₂²×Dic₄₇	1	4	yes	—
184	C₂⁴	C₂×C₂²⋊D₄₇	1	8	yes	—
185	C₂⁴	C₂³×D₄₇	1	4	yes	—
186	C₂⁴	C₂³×D₄₇	1	3	yes	—
187	C₂²×C₄	—	2	32	no	—
188	C₂²×C₄	—	2	24	no	—
189	C₂²×C₄	—	2	384	no	—
190	C₂²×C₄	—	2	64	no	—
191	C₂²×C₄	—	47	96	no	—
192	C₂²×C₄	—	47	16	no	—
193	C₂²×C₄	—	94	128	no	—
194	C₂²×C₄	—	94	48	no	—
195	C₂²×C₄	—	94	48	no	—
196	C₂²×C₄	C₂²⋊Dic₄₇	1	64	yes	direct product factor
197	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
198	C₂²×C₄	C₂²×Dic₄₇	1	24	yes	—
199	C₂²×C₄	C₂²×Dic₄₇	1	64	yes	complemented
200	C₂²×C₄	C₂×C₂²⋊D₄₇	1	48	yes	—
201	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	4	yes	—
203	C₂²⋊C₄	—	47	128	no	—
204	C₄²	—	94	64	no	—
205	C₂³	—	2	256	no	—
206	C₂³	—	2	48	no	—
207	C₂³	—	2	48	no	—
208	C₂³	—	2	192	no	—
209	C₂³	—	2	8	no	—
210	C₂³	—	2	8	no	—
211	C₂³	C₂×C₄×Dic₄₇	1	16	yes	—
212	C₂³	C₂×C₄×Dic₄₇	1	96	yes	complemented
213	C₂³	C₂×C₂²⋊Dic₄₇	1	16	yes	—
214	C₂³	#13093983	1	16	yes	—
215	C₂³	C₄×C₂²⋊D₄₇	1	64	yes	—
216	C₂³	C₂²×C₄×D₄₇	1	16	yes	—
217	C₂³	C₂²×C₄×D₄₇	1	12	yes	—
218	C₂³	C₂³×Dic₄₇	1	8	yes	—
219	C₂³	C₂³×Dic₄₇	1	6	yes	—
220	C₂³	C₂²×C₂²⋊D₄₇	1	12	yes	—
221	C₂³	C₂×#10625350	1	12	yes	—
222	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
223	C₂×C₄	—	2	48	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	128	no	—
226	C₂×C₄	—	47	32	no	—
227	C₂×C₄	—	94	192	no	—
228	C₂×C₄	—	94	16	no	—
229	C₂×C₄	—	94	16	no	—
230	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	96	yes	direct product factor
231	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
232	C₂×C₄	C₂³×Dic₄₇	1	8	yes	—
233	C₂×C₄	C₂²×C₂²⋊D₄₇	1	16	yes	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	32	no	—
237	C₂²	—	2	192	no	—
238	C₂²	C₄×C₂²⋊Dic₄₇	1	64	yes	direct product factor
239	C₂²	C₂²×C₄×Dic₄₇	1	12	yes	—
240	C₂²	C₂²×C₄×Dic₄₇	1	16	yes	complemented
241	C₂²	C₂²×C₂²⋊Dic₄₇	1	12	yes	—
242	C₂²	C₂×#13093983	1	12	yes	—
243	C₂²	C₂×C₄×C₂²⋊D₄₇	1	48	yes	—
244	C₂²	C₂³×C₄×D₄₇	1	2	yes	—
245	C₂²	C₂⁴×Dic₄₇	1	1	yes	characteristic
246	C₂²	C₂³×C₂²⋊D₄₇	1	2	yes	—
247	C₂²	C₂²×#10625350	1	2	yes	—
248	C₄	—	2	8	no	—
249	C₄	—	94	32	no	—
250	C₄	C₂²×C₂²⋊Dic₄₇	1	16	yes	direct product factor
251	C₂	—	2	16	no	—
252	C₂	C₂×C₄×C₂²⋊Dic₄₇	1	24	yes	direct product factor
253	C₂	C₂³×C₄×Dic₄₇	1	1	yes	characteristic
254	C₂	C₂³×C₂²⋊Dic₄₇	1	1	yes	characteristic
255	C₂	C₂²×#13093983	1	1	yes	characteristic
256	C₂	C₂²×C₄×C₂²⋊D₄₇	1	4	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₄₇
commutator	C₂×C₄₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₄₇
Fitting	C₂⁵×C₄×C₄₇	C₂
socle	C₂⁵×C₄₇	C₂³
prime radical	C₂⁶×C₄₇	C₂²
cosocle	C₂³×C₄₇	C₂⁵
hypercenter	C₂⁵×C₄	D₄₇
hypocenter	C₄₇	C₂²×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×Dic₄₇	C₂	semidirect
3. C₂×C₄×C₂²⋊Dic₄₇	C₂	direct (N × H)
4. C₂²×C₄₇⋊#95	C₂	semidirect
7. C₂²×C₂²⋊Dic₄₇	C₄	direct (N × H)
10. C₂²×C₄×Dic₄₇	C₂²	semidirect
11. C₄×C₂²⋊Dic₄₇	C₂²	direct (N × H)
12. C₂×C₄₇⋊#95	C₂²	semidirect
15. C₂⁴×C₄×C₄₇	C₄	semidirect
21. C₂³×Dic₄₇	C₂×C₄	semidirect
23. C₂×C₂²⋊Dic₄₇	C₂×C₄	direct (N × H)
26. C₂×C₄×Dic₄₇	C₂³	semidirect
27. C₄₇⋊#95	C₂³	semidirect
39. C₂³×C₄×C₄₇	C₂×C₄	semidirect
49. C₂⁴×C₄₇	C₄²	semidirect
62. C₂²×C₄×C₄₇	C₂²⋊C₄	semidirect
65. C₂²×C₄×C₄₇	C₂²×C₄	semidirect
72. C₂²×Dic₄₇	C₂²×C₄	semidirect
74. C₂²⋊Dic₄₇	C₂²×C₄	direct (N × H)
83. C₂³×C₄₇	C₂×C₄²	semidirect
97. C₂×C₄×C₄₇	C₂×C₂²⋊C₄	semidirect
104. C₂×Dic₄₇	C₂³×C₄	semidirect
110. C₂²×C₄₇	C₄×C₂²⋊C₄	semidirect
112. C₂²×C₄₇	C₂²×C₄²	semidirect
121. C₄×C₄₇	C₂²×C₂²⋊C₄	semidirect
129. C₂×C₄₇	C₂×C₄×C₂²⋊C₄	semidirect
137. C₂⁴×C₄	Dic₄₇	semidirect
147. C₄₇	C₂²×C₄×C₂²⋊C₄	semidirect
163. C₂³×C₄	C₂×Dic₄₇	semidirect
178. C₂⁴	C₄×Dic₄₇	semidirect
196. C₂²×C₄	C₂²⋊Dic₄₇	direct (N × H)
199. C₂²×C₄	C₂²×Dic₄₇	semidirect
212. C₂³	C₂×C₄×Dic₄₇	semidirect
230. C₂×C₄	C₂×C₂²⋊Dic₄₇	direct (N × H)
238. C₂²	C₄×C₂²⋊Dic₄₇	direct (N × H)
240. C₂²	C₂²×C₄×Dic₄₇	semidirect
250. C₄	C₂²×C₂²⋊Dic₄₇	direct (N × H)
252. C₂	C₂×C₄×C₂²⋊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
C₂³×C₂²⋊Dic₄₇	2
C₂³×C₄×Dic₄₇	2
C₂×C₄×C₂²⋊Dic₄₇	2
C₂²×C₄₇⋊#95	2
C₂⁵×C₄×C₄₇	2
C₂²×C₄×C₂²⋊C₄	47

Subgroups — C22×C4×C22⋊Dic47 #26374451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₂²⋊Dic₄₇ #26374451