Subgroups — C₂×#20245104 #26369165

Order 12032.

1406 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13093964	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15689161	C₂	1	1	yes	maximal, characteristic, complemented
3	#20245104	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688924	C₂	1	4	yes	maximal, complemented
5	C₂³×D₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093983	—	2	1	no	—
7	C₂×#13093964	—	2	4	no	—
8	C₂×#13093964	C₂²	1	4	yes	complemented
9	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	complemented
10	#15689161	—	2	2	no	—
11	#15689161	C₂²	1	4	yes	complemented
12	C₂×#13093958	C₂²	1	2	yes	complemented
13	C₂×#13093956	—	2	1	no	—
14	#15688924	C₂²	1	16	yes	complemented
15	C₂³×D₄×C₄₇	—	2	1	no	—
16	C₂²×D₈×C₄₇	—	2	1	no	—
17	C₂²×D₈×C₄₇	—	2	2	no	—
18	C₂²×D₈×C₄₇	—	2	4	no	—
19	C₂²×D₈×C₄₇	C₄	1	8	yes	complemented
20	C₂²×D₈×C₄₇	C₂²	1	2	yes	—
21	C₂²×D₈×C₄₇	C₂²	1	4	yes	—
22	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	#13093983	—	2	4	no	—
24	#13093983	—	4	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₂²⋊Dic₄₇	—	4	1	no	—
27	C₂×C₄×Dic₄₇	—	4	1	no	—
28	#13093964	—	2	8	no	—
29	C₂×C₄₇⋊C₁₆	—	2	4	no	—
30	C₂×C₄₇⋊C₁₆	C₂³	1	4	yes	complemented
31	#13093958	—	2	2	no	—
32	#13093958	C₂³	1	4	yes	complemented
33	#13093956	—	2	4	no	—
34	C₂⁵×C₄₇	—	4	1	no	—
35	C₂²×D₄×C₄₇	—	2	8	no	—
36	C₂²×D₄×C₄₇	—	2	2	no	—
37	C₂²×D₄×C₄₇	—	2	4	no	—
38	C₂²×D₄×C₄₇	—	4	1	no	—
39	C₂²×D₄×C₄₇	—	4	2	no	—
40	C₂²×D₄×C₄₇	—	4	4	no	—
41	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
42	C₂×D₈×C₄₇	—	2	16	no	—
43	C₂×D₈×C₄₇	—	2	8	no	—
44	C₂×D₈×C₄₇	—	2	4	no	—
45	C₂×D₈×C₄₇	—	2	2	no	—
46	C₂×D₈×C₄₇	—	2	4	no	—
47	C₂×D₈×C₄₇	—	2	8	no	—
48	C₂×D₈×C₄₇	C₂×C₄	1	8	yes	—
49	C₂×D₈×C₄₇	C₂×C₄	1	16	yes	complemented
50	C₂×D₈×C₄₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₄₇	D₄	1	2	yes	—
54	C₂²×C₈×C₄₇	D₄	1	4	yes	—
55	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₇	C₂³	1	2	yes	—
57	C₂⁴×C₄₇	—	4	8	no	—
58	C₂⁴×C₄₇	—	4	4	no	—
59	C₂⁴×C₄₇	—	4	2	no	—
60	C₂⁴×C₄₇	—	8	8	no	—
61	C₂⁴×C₄₇	D₈	1	1	yes	characteristic
62	C₂×D₄×C₄₇	—	2	8	no	—
63	C₂×D₄×C₄₇	—	2	16	no	—
64	C₂×D₄×C₄₇	—	2	4	no	—
65	C₂×D₄×C₄₇	—	4	16	no	—
66	C₂×D₄×C₄₇	—	4	8	no	—
67	C₂×D₄×C₄₇	—	4	4	no	—
68	C₂×D₄×C₄₇	—	4	2	no	—
69	C₂×D₄×C₄₇	—	4	8	no	—
70	C₂×D₄×C₄₇	—	4	4	no	—
71	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₄₇	D₈	1	2	yes	—
74	C₂²×C₄×C₄₇	SD₁₆	1	4	yes	—
75	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
77	C₂²×Dic₄₇	—	4	2	no	—
78	D₈×C₄₇	—	2	8	no	—
79	D₈×C₄₇	—	2	16	no	—
80	D₈×C₄₇	—	2	4	no	—
81	D₈×C₄₇	C₂²×C₄	1	8	yes	—
82	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
83	C₂×C₈×C₄₇	C₂²⋊C₄	1	8	yes	—
84	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₇	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₄₇	C₂×D₄	1	4	yes	—
90	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
91	C₂²⋊Dic₄₇	—	4	4	no	—
92	C₄⋊Dic₄₇	—	2	4	no	—
93	C₄⋊Dic₄₇	—	4	2	no	—
94	C₄×Dic₄₇	—	4	4	no	—
95	C₄₇⋊C₁₆	—	2	4	no	—
96	C₂³×C₄₇	—	2	4	no	—
97	C₂³×C₄₇	—	4	4	no	—
98	C₂³×C₄₇	—	4	16	no	—
99	C₂³×C₄₇	—	4	8	no	—
100	C₂³×C₄₇	—	8	16	no	—
101	C₂³×C₄₇	—	8	32	no	—
102	C₂³×C₄₇	—	8	8	no	—
103	C₂³×C₄₇	#102	1	4	yes	—
104	C₂³×C₄₇	C₂×D₈	1	2	yes	—
105	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
106	D₄×C₄₇	—	2	8	no	—
107	D₄×C₄₇	—	4	4	no	—
108	D₄×C₄₇	—	4	8	no	—
109	D₄×C₄₇	—	4	16	no	—
110	C₂×C₄×C₄₇	#102	1	8	yes	—
111	C₂×C₄×C₄₇	#102	1	4	yes	—
112	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
113	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
114	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
117	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
118	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₄₇	—	4	4	no	—
120	C₂×Dic₄₇	—	8	4	no	—
121	C₈×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
122	C₈×C₄₇	C₂×C₂²⋊C₄	1	4	yes	—
123	C₈×C₄₇	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₄₇	C₂²×D₄	1	1	yes	characteristic
125	C₂×#2140	—	47	1	no	maximal, Sylow
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	8	no	—
128	C₂²×C₄₇	—	4	8	no	—
129	C₂²×C₄₇	—	8	32	no	—
130	C₂²×C₄₇	—	8	8	no	—
131	C₂²×C₄₇	—	8	16	no	—
132	C₂²×C₄₇	C₂×#102	1	4	yes	—
133	C₂²×C₄₇	C₂×#102	1	2	yes	—
134	C₂²×C₄₇	#509	1	4	yes	—
135	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₄₇	C₂×#102	1	4	yes	—
137	C₄×C₄₇	C₂×#102	1	2	yes	—
138	C₄×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₄₇	C₂²×SD₁₆	1	1	yes	characteristic
140	Dic₄₇	—	8	4	no	—
141	C₂³×D₈	D₄₇	1	1	yes	characteristic
142	C₂²×M₃₂	—	47	1	no	—
143	C₂×#427	—	47	1	no	—
144	#2140	—	47	8	no	—
145	C₂×#357	—	47	4	no	—
146	C₂×C₄₇	—	2	4	no	—
147	C₂×C₄₇	—	8	8	no	—
148	C₂×C₄₇	#2140	1	4	yes	complemented
149	C₂×C₄₇	C₂²×#102	1	1	yes	characteristic
150	C₂×C₄₇	C₂×#509	1	2	yes	—
151	C₂³×D₄	—	2	1	no	—
152	C₂²×D₈	—	2	1	no	—
153	C₂²×D₈	—	2	2	no	—
154	C₂²×D₈	—	2	4	no	—
155	C₂²×D₈	Dic₄₇	1	8	yes	complemented
156	C₂²×D₈	C₂×D₄₇	1	2	yes	—
157	C₂²×D₈	C₂×D₄₇	1	4	yes	—
158	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
159	C₂×#117	—	94	1	no	—
160	C₂×M₃₂	—	47	4	no	—
161	C₂×M₃₂	—	94	4	no	—
162	C₂²×C₁₆	—	47	2	no	—
163	#427	—	47	4	no	—
164	#427	—	94	2	no	—
165	C₂×#107	—	47	2	no	—
166	C₂×#106	—	94	1	no	—
167	#357	—	47	16	no	—
168	C₄₇	C₂×#2140	1	1	yes	characteristic, complemented, Sylow
169	C₂⁵	—	4	1	no	—
170	C₂²×D₄	—	2	8	no	—
171	C₂²×D₄	—	2	4	no	—
172	C₂²×D₄	—	2	2	no	—
173	C₂²×D₄	—	4	2	no	—
174	C₂²×D₄	—	4	1	no	—
175	C₂²×D₄	—	4	4	no	—
176	C₂³×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
177	C₂×D₈	—	2	16	no	—
178	C₂×D₈	—	2	8	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	—	2	8	no	—
183	C₂×D₈	C₂×Dic₄₇	1	8	yes	—
184	C₂×D₈	C₂×Dic₄₇	1	16	yes	complemented
185	C₂×D₈	C₂²×D₄₇	1	4	yes	—
186	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
187	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
189	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
190	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₄₇	1	2	yes	—
192	#117	—	94	4	no	—
193	#117	—	188	2	no	—
194	C₂×C₄⋊C₄	—	94	2	no	—
195	C₂×C₂²⋊C₄	—	188	1	no	—
196	C₂×C₄²	—	188	1	no	—
197	M₃₂	—	94	8	no	—
198	C₂×C₁₆	—	47	4	no	—
199	C₂×C₁₆	—	94	4	no	—
200	#107	—	47	4	no	—
201	#107	—	94	2	no	—
202	#106	—	94	4	no	—
203	C₂⁴	—	4	8	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	—	4	2	no	—
206	C₂⁴	—	8	8	no	—
207	C₂⁴	#10625334	1	1	yes	characteristic
208	C₂×D₄	—	2	8	no	—
209	C₂×D₄	—	2	16	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	4	16	no	—
212	C₂×D₄	—	4	8	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂²×C₄	—	188	2	no	—
218	C₂²×C₄	#10625334	1	2	yes	—
219	C₂²×C₄	#10625334	1	1	yes	characteristic
220	C₂²×C₄	#10625335	1	4	yes	—
221	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
224	D₈	—	2	8	no	—
225	D₈	—	2	16	no	—
226	D₈	—	2	4	no	—
227	D₈	C₂²×Dic₄₇	1	8	yes	—
228	C₂×C₈	C₂²⋊Dic₄₇	1	4	yes	—
229	C₂×C₈	C₂²⋊Dic₄₇	1	8	yes	—
230	C₂×C₈	C₂²×Dic₄₇	1	2	yes	—
231	C₂×C₈	C₂²×Dic₄₇	1	4	yes	—
232	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
233	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
234	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
235	C₂×C₈	C₂×C₂²⋊D₄₇	1	4	yes	—
236	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
237	C₄⋊C₄	—	94	4	no	—
238	C₄⋊C₄	—	188	2	no	—
239	C₂²⋊C₄	—	188	4	no	—
240	C₄²	—	188	4	no	—
241	C₁₆	—	94	4	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	16	no	—
244	C₂³	—	4	8	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	8	32	no	—
247	C₂³	—	8	16	no	—
248	C₂³	—	8	8	no	—
249	C₂³	#13093945	1	4	yes	—
250	C₂³	C₂×#10625334	1	2	yes	—
251	C₂³	C₂×#10625334	1	1	yes	characteristic
252	D₄	—	2	8	no	—
253	D₄	—	4	8	no	—
254	D₄	—	4	16	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	376	4	no	—
257	C₂×C₄	—	188	4	no	—
258	C₂×C₄	#13093945	1	8	yes	—
259	C₂×C₄	#13093945	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	4	yes	—
261	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	2	yes	—
262	C₂×C₄	C₂×#10625334	1	2	yes	—
263	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
264	C₂×C₄	C₂×#10625335	1	4	yes	—
265	C₂×C₄	C₂×#10625335	1	2	yes	—
266	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
267	C₈	C₂×C₂²⋊Dic₄₇	1	2	yes	—
268	C₈	C₂×C₂²⋊Dic₄₇	1	4	yes	—
269	C₈	C₂³×Dic₄₇	1	1	yes	characteristic
270	C₈	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
271	C₂²	—	2	8	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	8	32	no	—
275	C₂²	—	8	16	no	—
276	C₂²	—	8	8	no	—
277	C₂²	C₂×#13093945	1	4	yes	—
278	C₂²	C₂×#13093945	1	2	yes	—
279	C₂²	#15689559	1	4	yes	—
280	C₂²	C₂²×#10625334	1	1	yes	characteristic
281	C₄	—	376	4	no	—
282	C₄	C₂×#13093945	1	4	yes	—
283	C₄	C₂×#13093945	1	2	yes	—
284	C₄	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
285	C₄	C₂²×#10625335	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	—	8	8	no	—
288	C₂	#20245104	1	4	yes	direct product factor
289	C₂	C₂²×#13093945	1	1	yes	characteristic
290	C₂	C₂×#15689559	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₈×C₄₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₂³×D₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₈
cosocle	C₂×C₈×C₄₇	C₂⁴
hypercenter	C₂³×D₈	D₄₇
hypocenter	C₄₇	C₂×#2140
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#13093964	C₂	semidirect
2. C₂×#15689161	C₂	semidirect
3. #20245104	C₂	direct (N × H)
4. C₂×#15688924	C₂	semidirect
8. C₂×#13093964	C₂²	semidirect
9. C₂²×C₄₇⋊C₁₆	C₂²	semidirect
11. #15689161	C₂²	semidirect
12. C₂×#13093958	C₂²	semidirect
14. #15688924	C₂²	semidirect
19. C₂²×D₈×C₄₇	C₄	semidirect
30. C₂×C₄₇⋊C₁₆	C₂³	semidirect
32. #13093958	C₂³	semidirect
49. C₂×D₈×C₄₇	C₂×C₄	semidirect
148. C₂×C₄₇	#2140	semidirect
155. C₂²×D₈	Dic₄₇	semidirect
168. C₄₇	C₂×#2140	semidirect
184. C₂×D₈	C₂×Dic₄₇	semidirect
288. C₂	#20245104	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×#13093964	2
C₂×#15689161	2
#20245104	2
C₂×#15688924	2
C₂³×D₈×C₄₇	2
C₂×#2140	47

Subgroups — C2×#20245104 #26369165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20245104 #26369165