Subgroups — C₂×#20256790 #27123119

Order 12032.

7022 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20256790	C₂	1	32	yes	maximal, direct product factor
2	C₂×#15689752	C₂	1	12	yes	maximal, complemented
3	C₂×#15689749	C₂	1	3	yes	maximal
4	C₂×#15689675	C₂	1	12	yes	maximal, complemented
5	C₂²×#13093980	C₂	1	3	yes	maximal
6	C₂×C₄₇×#543	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₄₇	C₂²	1	1	yes	characteristic
8	#15689752	C₂²	1	192	yes	complemented
9	#15689749	C₂²	1	48	yes	—
10	#15689675	C₂²	1	192	yes	complemented
11	C₂×#13094023	C₂²	1	12	yes	complemented
12	C₂×#13094022	C₂²	1	12	yes	complemented
13	C₂×#13094014	C₂²	1	24	yes	complemented
14	C₂×#13094003	C₂²	1	12	yes	—
15	C₂×#13094001	C₂²	1	24	yes	—
16	C₂×Q₈×Dic₄₇	C₂²	1	4	yes	complemented
17	C₂×#13093989	C₂²	1	12	yes	complemented
18	C₂×#13093980	C₂²	1	12	yes	—
19	C₂×#13093980	C₂²	1	48	yes	—
20	C₂×#13093980	C₂²	1	12	yes	—
21	C₂²×Dic₄₇⋊C₄	C₂²	1	6	yes	—
22	C₂²×C₂²⋊Dic₄₇	C₂²	1	3	yes	—
23	C₂²×C₄×Dic₄₇	C₂²	1	6	yes	—
24	C₄₇×#543	C₂²	1	16	yes	—
25	C₂×C₂²⋊Q₈×C₄₇	C₂²	1	12	yes	—
26	C₂²×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
27	C₂²×Dic₉₄	C₂³	1	12	yes	—
28	C₂³×Dic₄₇	Q₈	1	16	yes	complemented
29	C₂³×Dic₄₇	C₂³	1	6	yes	—
30	C₂³×Dic₄₇	C₂³	1	3	yes	—
31	C₂³×Dic₄₇	C₂³	1	3	yes	—
32	C₂³×Dic₄₇	C₂³	1	8	yes	—
33	#13094023	C₂³	1	96	yes	complemented
34	#13094022	C₂³	1	96	yes	complemented
35	#13094014	C₂³	1	192	yes	complemented
36	#13094003	C₂³	1	96	yes	—
37	#13094001	C₂³	1	192	yes	—
38	Q₈×Dic₄₇	C₂³	1	32	yes	complemented
39	#13093989	C₂³	1	96	yes	complemented
40	#13093980	C₂³	1	96	yes	—
41	#13093980	C₂³	1	96	yes	—
42	C₂×C₄⋊Dic₄₇	C₂³	1	12	yes	—
43	C₂×Dic₄₇⋊C₄	C₂³	1	48	yes	—
44	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
45	C₂×Dic₄₇⋊C₄	C₂³	1	24	yes	—
46	C₂×C₂²⋊Dic₄₇	C₂³	1	24	yes	—
47	C₂×C₂²⋊Dic₄₇	C₂³	1	12	yes	—
48	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
49	C₂×C₄×Dic₄₇	C₂³	1	12	yes	—
50	C₂×C₄×Dic₄₇	C₂³	1	48	yes	—
51	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₄₇	C₂³	1	4	yes	—
53	C₂³×C₄×C₄₇	C₂³	1	6	yes	—
54	C₂²⋊Q₈×C₄₇	C₂³	1	96	yes	—
55	C₂×C₄⋊C₄×C₄₇	C₂³	1	12	yes	—
56	C₂×C₂²⋊C₄×C₄₇	C₂³	1	24	yes	—
57	C₂×C₂²⋊C₄×C₄₇	C₂³	1	12	yes	—
58	C₂⁴×C₄₇	—	4	4	no	—
59	C₂⁴×C₄₇	C₂×Q₈	1	8	yes	—
60	C₂⁴×C₄₇	C₂⁴	1	3	yes	—
61	C₂⁴×C₄₇	C₂⁴	1	4	yes	—
62	C₂×Q₈×C₄₇	—	4	8	no	—
63	C₂×Q₈×C₄₇	C₂⁴	1	16	yes	—
64	C₂²×C₄×C₄₇	—	4	12	no	—
65	C₂²×C₄×C₄₇	C₂⁴	1	12	yes	—
66	C₂²×C₄×C₄₇	C₂⁴	1	24	yes	—
67	C₂²×Dic₄₇	—	4	12	no	—
68	C₂²×Dic₄₇	C₂×Q₈	1	24	yes	—
69	C₂²×Dic₄₇	C₂×Q₈	1	24	yes	—
70	C₂²×Dic₄₇	C₂×Q₈	1	64	yes	complemented
71	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
72	C₂²×Dic₄₇	C₂⁴	1	24	yes	—
73	C₂²×Dic₄₇	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₄₇	C₂⁴	1	3	yes	—
75	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
76	C₂²×Dic₄₇	C₂⁴	1	12	yes	—
77	C₂×Dic₉₄	—	4	24	no	—
78	C₂×Dic₉₄	C₂⁴	1	48	yes	—
79	C₄⋊C₄×C₄₇	C₂⁴	1	48	yes	—
80	C₂²⋊C₄×C₄₇	C₂⁴	1	48	yes	—
81	C₂²⋊Dic₄₇	C₂⁴	1	48	yes	—
82	C₄⋊Dic₄₇	C₂⁴	1	48	yes	—
83	Dic₄₇⋊C₄	C₂⁴	1	96	yes	—
84	Dic₄₇⋊C₄	C₂⁴	1	96	yes	—
85	C₄×Dic₄₇	C₂⁴	1	48	yes	—
86	C₄×Dic₄₇	C₂⁴	1	48	yes	—
87	C₂³×C₄₇	—	2	12	no	—
88	C₂³×C₄₇	—	4	16	no	—
89	C₂³×C₄₇	—	4	8	no	—
90	C₂³×C₄₇	C₂²×Q₈	1	12	yes	—
91	C₂³×C₄₇	C₂²×Q₈	1	16	yes	—
92	C₂³×C₄₇	C₂⁵	1	6	yes	—
93	C₂³×C₄₇	C₂⁵	1	1	yes	characteristic
94	Q₈×C₄₇	—	4	16	no	—
95	C₂×C₄×C₄₇	—	2	24	no	—
96	C₂×C₄×C₄₇	—	4	24	no	—
97	C₂×C₄×C₄₇	C₂⁵	1	24	yes	—
98	C₂×Dic₄₇	—	2	24	no	—
99	C₂×Dic₄₇	—	4	24	no	—
100	C₂×Dic₄₇	C₂²×Q₈	1	4	yes	—
101	C₂×Dic₄₇	C₂²×Q₈	1	12	yes	—
102	C₂×Dic₄₇	C₂²×Q₈	1	48	yes	—
103	C₂×Dic₄₇	C₂²×Q₈	1	48	yes	—
104	C₂×Dic₄₇	C₂⁵	1	24	yes	—
105	C₂×Dic₄₇	C₂⁵	1	2	yes	—
106	C₂×Dic₄₇	C₂⁵	1	6	yes	—
107	Dic₉₄	—	4	48	no	—
108	C₂×#3458	—	47	1	no	maximal, Sylow
109	C₂²×C₄₇	—	2	12	no	—
110	C₂²×C₄₇	—	2	24	no	—
111	C₂²×C₄₇	—	4	16	no	—
112	C₂²×C₄₇	C₂³×Q₈	1	12	yes	—
113	C₂²×C₄₇	C₂³×Q₈	1	2	yes	—
114	C₂²×C₄₇	#585	1	4	yes	—
115	C₂²×C₄₇	C₂⁶	1	1	yes	characteristic
116	C₄×C₄₇	—	2	24	no	—
117	Dic₄₇	—	2	24	no	—
118	Dic₄₇	C₂³×Q₈	1	4	yes	—
119	Dic₄₇	C₂³×Q₈	1	12	yes	—
120	#3458	—	47	32	no	—
121	C₂×#544	—	47	12	no	—
122	C₂×#543	—	47	3	no	—
123	C₂×#543	D₄₇	1	1	yes	characteristic
124	C₂×#533	—	47	12	no	—
125	C₂²×#117	—	47	3	no	—
126	C₂×C₄₇	—	2	12	no	—
127	C₂×C₄₇	#3458	1	4	yes	complemented
128	C₂×C₄₇	C₂⁴×Q₈	1	1	yes	characteristic
129	C₂×C₄₇	C₂×#585	1	2	yes	—
130	C₂⁴×C₄	—	47	1	no	—
131	#544	—	47	192	no	—
132	#543	—	47	48	no	—
133	#543	C₂×D₄₇	1	16	yes	—
134	#533	—	47	192	no	—
135	C₂×C₄⋊Q₈	—	47	12	no	—
136	C₂×C₄⋊Q₈	—	47	12	no	—
137	C₂×#125	—	47	24	no	—
138	C₂×C₂²⋊Q₈	—	47	12	no	—
139	C₂×C₂²⋊Q₈	—	47	24	no	—
140	C₂×C₂²⋊Q₈	C₂×D₄₇	1	12	yes	—
141	C₂×C₄×Q₈	—	47	4	no	—
142	C₂×C₄×Q₈	—	47	12	no	—
143	C₂×#117	—	47	12	no	—
144	C₂×#117	—	47	12	no	—
145	C₂×#117	—	47	48	no	—
146	C₂²×C₄⋊C₄	—	47	6	no	—
147	C₂²×C₂²⋊C₄	—	47	3	no	—
148	C₂²×C₂²⋊C₄	C₂×D₄₇	1	3	yes	—
149	C₂²×C₄²	—	47	6	no	—
150	C₄₇	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
151	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
152	C₂²×Q₈	—	47	12	no	—
153	C₂²×Q₈	C₂²×D₄₇	1	4	yes	—
154	C₂³×C₄	—	47	16	no	—
155	C₂³×C₄	—	47	3	no	—
156	C₂³×C₄	—	47	8	no	—
157	C₂³×C₄	—	47	3	no	—
158	C₂³×C₄	—	47	6	no	—
159	C₂³×C₄	C₂²×D₄₇	1	6	yes	—
160	C₄⋊Q₈	—	47	96	no	—
161	C₄⋊Q₈	—	47	96	no	—
162	#125	—	47	192	no	—
163	C₂²⋊Q₈	—	47	96	no	—
164	C₂²⋊Q₈	—	47	192	no	—
165	C₂²⋊Q₈	C₂²×D₄₇	1	96	yes	—
166	C₄×Q₈	—	47	32	no	—
167	C₄×Q₈	—	47	96	no	—
168	#117	—	47	96	no	—
169	#117	—	47	96	no	—
170	C₂×C₄⋊C₄	—	47	24	no	—
171	C₂×C₄⋊C₄	—	47	48	no	—
172	C₂×C₄⋊C₄	—	47	24	no	—
173	C₂×C₄⋊C₄	—	47	12	no	—
174	C₂×C₄⋊C₄	C₂²×D₄₇	1	12	yes	—
175	C₂×C₂²⋊C₄	—	47	12	no	—
176	C₂×C₂²⋊C₄	—	47	24	no	—
177	C₂×C₂²⋊C₄	C₂²×D₄₇	1	12	yes	—
178	C₂×C₂²⋊C₄	C₂²×D₄₇	1	24	yes	—
179	C₂×C₄²	—	47	12	no	—
180	C₂×C₄²	—	47	12	no	—
181	C₂×C₄²	—	47	48	no	—
182	C₂⁴	—	4	4	no	—
183	C₂⁴	Q₈×D₄₇	1	8	yes	—
184	C₂⁴	C₂³×D₄₇	1	3	yes	—
185	C₂⁴	C₂³×D₄₇	1	4	yes	—
186	C₂×Q₈	—	4	8	no	—
187	C₂×Q₈	—	47	48	no	—
188	C₂×Q₈	—	188	24	no	—
189	C₂×Q₈	C₂³×D₄₇	1	16	yes	—
190	C₂²×C₄	—	4	12	no	—
191	C₂²×C₄	—	47	24	no	—
192	C₂²×C₄	—	47	24	no	—
193	C₂²×C₄	—	47	64	no	—
194	C₂²×C₄	—	47	12	no	—
195	C₂²×C₄	—	47	12	no	—
196	C₂²×C₄	—	47	1	no	—
197	C₂²×C₄	—	47	3	no	—
198	C₂²×C₄	—	47	12	no	—
199	C₂²×C₄	—	47	24	no	—
200	C₂²×C₄	—	188	12	no	—
201	C₂²×C₄	C₂³×D₄₇	1	12	yes	—
202	C₂²×C₄	C₂³×D₄₇	1	24	yes	—
203	C₄⋊C₄	—	47	96	no	—
204	C₄⋊C₄	—	47	96	no	—
205	C₄⋊C₄	—	47	48	no	—
206	C₄⋊C₄	C₂³×D₄₇	1	48	yes	—
207	C₂²⋊C₄	—	47	48	no	—
208	C₂²⋊C₄	C₂³×D₄₇	1	48	yes	—
209	C₄²	—	47	48	no	—
210	C₄²	—	47	48	no	—
211	C₂³	—	2	12	no	—
212	C₂³	—	4	16	no	—
213	C₂³	—	4	8	no	—
214	C₂³	C₂×Q₈×D₄₇	1	12	yes	—
215	C₂³	C₂×Q₈×D₄₇	1	16	yes	—
216	C₂³	C₂⁴×D₄₇	1	6	yes	—
217	C₂³	C₂⁴×D₄₇	1	1	yes	characteristic
218	Q₈	—	4	16	no	—
219	Q₈	—	188	48	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	47	48	no	—
223	C₂×C₄	—	47	48	no	—
224	C₂×C₄	—	47	4	no	—
225	C₂×C₄	—	47	12	no	—
226	C₂×C₄	—	47	2	no	—
227	C₂×C₄	—	47	6	no	—
228	C₂×C₄	—	47	24	no	—
229	C₂×C₄	—	94	24	no	—
230	C₂×C₄	—	188	24	no	—
231	C₂×C₄	C₂⁴×D₄₇	1	24	yes	—
232	C₂²	—	2	24	no	—
233	C₂²	—	2	12	no	—
234	C₂²	—	4	16	no	—
235	C₂²	C₂²×Q₈×D₄₇	1	12	yes	—
236	C₂²	C₂²×Q₈×D₄₇	1	2	yes	—
237	C₂²	#15689958	1	4	yes	—
238	C₂²	C₂⁵×D₄₇	1	1	yes	characteristic
239	C₄	—	2	24	no	—
240	C₄	—	47	4	no	—
241	C₄	—	47	12	no	—
242	C₄	—	94	24	no	—
243	C₂	—	2	12	no	—
244	C₂	#20256790	1	4	yes	direct product factor
245	C₂	C₂³×Q₈×D₄₇	1	1	yes	characteristic
246	C₂	C₂×#15689958	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₄₇
commutator, cosocle	C₂²×C₄₇	C₂⁶
Frattini	C₂²	C₂⁵×D₄₇
Fitting	C₂×C₄₇×#543	C₂
socle	C₂³×C₄₇	C₂⁵
prime radical	C₂⁵×C₄₇	C₂³
hypercenter	C₂×#543	D₄₇
hypocenter	C₄₇	C₂×#3458
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. #20256790	C₂	direct (N × H)
2. C₂×#15689752	C₂	semidirect
4. C₂×#15689675	C₂	semidirect
8. #15689752	C₂²	semidirect
10. #15689675	C₂²	semidirect
11. C₂×#13094023	C₂²	semidirect
12. C₂×#13094022	C₂²	semidirect
13. C₂×#13094014	C₂²	semidirect
16. C₂×Q₈×Dic₄₇	C₂²	semidirect
17. C₂×#13093989	C₂²	semidirect
28. C₂³×Dic₄₇	Q₈	semidirect
33. #13094023	C₂³	semidirect
34. #13094022	C₂³	semidirect
35. #13094014	C₂³	semidirect
38. Q₈×Dic₄₇	C₂³	semidirect
39. #13093989	C₂³	semidirect
70. C₂²×Dic₄₇	C₂×Q₈	semidirect
127. C₂×C₄₇	#3458	semidirect
150. C₄₇	C₂×#3458	semidirect
244. C₂	#20256790	direct (N × H)

Maximal subgroups

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

subgroup	index
#20256790	2
C₂×#15689752	2
C₂×#15689749	2
C₂×#15689675	2
C₂²×#13093980	2
C₂×C₄₇×#543	2
C₂×#3458	47

Subgroups — C2×#20256790 #27123119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20256790 #27123119