Subgroups — C₂×#15689845 #20256936

Order 6016.

1112 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689845	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13094022	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13094019	C₂	1	2	yes	maximal, complemented
4	C₂×#13094018	C₂	1	2	yes	maximal, complemented
5	C₂×#13094013	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094012	C₂	1	1	yes	maximal, characteristic
7	C₂×#13094002	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13093989	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13093986	C₂	1	1	yes	maximal, characteristic
11	C₂×#13093981	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
15	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₄₇	C₂²	1	2	yes	—
17	#13094022	C₂²	1	8	yes	complemented
18	#13094019	C₂²	1	16	yes	complemented
19	#13094018	C₂²	1	16	yes	complemented
20	#13094013	C₂²	1	8	yes	—
21	#13094012	C₂²	1	8	yes	—
22	#13094002	C₂²	1	16	yes	complemented
23	C₄×Dic₉₄	C₂²	1	8	yes	complemented
24	#13093989	C₂²	1	8	yes	complemented
25	#13093986	C₂²	1	8	yes	—
26	#13093981	C₂²	1	16	yes	complemented
27	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
28	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
30	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
31	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×Dic₄₇⋊C₄	C₂²	1	2	yes	—
33	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
34	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
35	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
37	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
38	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
39	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
42	C₄×Q₈×C₄₇	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
45	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
46	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
47	C₂³×D₄₇	C₂³	1	2	yes	—
48	C₂×Q₈×C₄₇	—	2	4	no	—
49	C₂×Q₈×C₄₇	C₂³	1	4	yes	—
50	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
52	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
56	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
57	C₂²×Dic₄₇	C₂³	1	2	yes	—
58	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
59	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
60	C₂×D₁₈₈	—	2	4	no	—
61	C₂×D₁₈₈	C₂³	1	4	yes	—
62	C₂×C₄×D₄₇	—	2	8	no	—
63	C₂×C₄×D₄₇	C₂³	1	8	yes	—
64	C₂×Dic₉₄	—	2	4	no	—
65	C₂×Dic₉₄	—	2	4	no	—
66	C₂×Dic₉₄	C₂³	1	4	yes	—
67	C₂×Dic₉₄	C₂³	1	4	yes	—
68	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
69	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
70	C₄²×C₄₇	C₂³	1	8	yes	—
71	C₄²×C₄₇	C₂³	1	4	yes	—
72	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
74	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
75	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
76	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
77	C₄⋊Dic₄₇	C₂³	1	4	yes	—
78	Dic₄₇⋊C₄	C₂³	1	4	yes	—
79	Dic₄₇⋊C₄	C₂³	1	8	yes	—
80	Dic₄₇⋊C₄	C₂³	1	8	yes	—
81	Dic₄₇⋊C₄	C₂³	1	4	yes	—
82	Dic₄₇⋊C₄	C₂³	1	8	yes	—
83	C₄×Dic₄₇	C₂³	1	8	yes	—
84	C₄×Dic₄₇	C₂³	1	4	yes	—
85	C₄×Dic₄₇	C₂³	1	4	yes	—
86	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
87	C₂²×D₄₇	—	2	4	no	—
88	C₂²×D₄₇	—	2	4	no	—
89	C₂²×D₄₇	—	2	4	no	—
90	C₂²×D₄₇	C₂⁴	1	4	yes	—
91	Q₈×C₄₇	—	2	8	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	4	no	—
94	C₂×C₄×C₄₇	—	2	4	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	#41	1	4	yes	—
97	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
98	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
99	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
100	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
101	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
102	C₂×Dic₄₇	—	2	2	no	—
103	C₂×Dic₄₇	—	2	4	no	—
104	C₂×Dic₄₇	—	2	2	no	—
105	C₂×Dic₄₇	—	2	2	no	—
106	C₂×Dic₄₇	#41	1	4	yes	—
107	C₂×Dic₄₇	C₂⁴	1	2	yes	—
108	C₂×Dic₄₇	C₂⁴	1	2	yes	—
109	C₂×Dic₄₇	C₂⁴	1	4	yes	—
110	C₂×Dic₄₇	C₂⁴	1	2	yes	—
111	C₂×Dic₄₇	C₂⁴	1	2	yes	—
112	D₁₈₈	—	2	8	no	—
113	C₄×D₄₇	—	2	16	no	—
114	Dic₉₄	—	2	8	no	—
115	Dic₉₄	—	2	8	no	—
116	C₂²×C₄₇	C₂×#41	1	2	yes	—
117	C₂²×C₄₇	C₂×#41	1	2	yes	—
118	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
120	C₂×D₄₇	—	2	4	no	—
121	C₂×D₄₇	—	2	4	no	—
122	C₂×D₄₇	—	2	4	no	—
123	C₂×D₄₇	—	4	8	no	—
124	C₄×C₄₇	—	2	2	no	—
125	C₄×C₄₇	—	2	4	no	—
126	C₄×C₄₇	—	2	4	no	—
127	C₄×C₄₇	—	2	2	no	—
128	C₄×C₄₇	C₂×#41	1	4	yes	—
129	Dic₄₇	—	2	2	no	—
130	Dic₄₇	—	2	4	no	—
131	Dic₄₇	—	2	2	no	—
132	Dic₄₇	—	2	2	no	—
133	Dic₄₇	C₂×#41	1	4	yes	—
134	C₂×#556	—	47	1	no	maximal, Sylow
135	C₂×C₄₇	#556	1	4	yes	complemented
136	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
137	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
138	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
139	D₄₇	—	4	4	no	—
140	#556	—	47	16	no	—
141	C₂×C₄⋊Q₈	—	47	1	no	—
142	C₂×#126	—	47	2	no	—
143	C₂×#126	—	47	2	no	—
144	C₂×#124	—	47	1	no	—
145	C₂×#124	—	47	1	no	—
146	C₂×C₂²⋊Q₈	—	47	2	no	—
147	C₂×C₄×Q₈	—	47	1	no	—
148	C₂×C₄×Q₈	—	47	1	no	—
149	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic, complemented
150	C₂×C₄×D₄	—	47	1	no	—
151	C₂×#117	—	47	2	no	—
152	C₄₇	C₂×#556	1	1	yes	characteristic, complemented, Sylow
153	C₂²×Q₈	—	47	1	no	—
154	C₂²×Q₈	—	47	1	no	—
155	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
156	C₂²×D₄	—	47	1	no	—
157	C₂³×C₄	—	47	2	no	—
158	C₄⋊Q₈	—	47	8	no	—
159	#126	—	47	16	no	—
160	#126	—	47	16	no	—
161	#124	—	47	8	no	—
162	#124	—	47	8	no	—
163	C₂²⋊Q₈	—	47	16	no	—
164	C₄×Q₈	—	47	8	no	—
165	C₄×Q₈	—	47	8	no	—
166	C₄×Q₈	C₂×D₄₇	1	8	yes	complemented
167	C₄×D₄	—	47	8	no	—
168	#117	—	47	16	no	—
169	C₂×C₄⋊C₄	—	47	1	no	—
170	C₂×C₄⋊C₄	—	47	2	no	—
171	C₂×C₄⋊C₄	—	47	2	no	—
172	C₂×C₄⋊C₄	—	47	1	no	—
173	C₂×C₄⋊C₄	—	47	1	no	—
174	C₂×C₄⋊C₄	—	47	2	no	—
175	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
177	C₂×C₂²⋊C₄	—	47	2	no	—
178	C₂×C₂²⋊C₄	—	47	2	no	—
179	C₂×C₂²⋊C₄	—	47	2	no	—
180	C₂×C₂²⋊C₄	—	47	2	no	—
181	C₂×C₂²⋊C₄	—	47	2	no	—
182	C₂×C₄²	—	47	2	no	—
183	C₂×C₄²	—	47	1	no	—
184	C₂×C₄²	—	47	1	no	—
185	C₂×C₄²	C₂×D₄₇	1	2	yes	—
186	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
187	C₂⁴	—	47	2	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	47	4	no	—
190	C₂×Q₈	—	47	4	no	—
191	C₂×Q₈	—	94	4	no	—
192	C₂×Q₈	—	94	4	no	—
193	C₂×Q₈	C₂²×D₄₇	1	4	yes	—
194	C₂×D₄	—	47	4	no	—
195	C₂×D₄	—	94	4	no	—
196	C₂²×C₄	—	47	1	no	—
197	C₂²×C₄	—	47	1	no	—
198	C₂²×C₄	—	47	1	no	—
199	C₂²×C₄	—	47	1	no	—
200	C₂²×C₄	—	47	2	no	—
201	C₂²×C₄	—	47	8	no	—
202	C₂²×C₄	—	94	8	no	—
203	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
204	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
206	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
208	C₄⋊C₄	—	47	4	no	—
209	C₄⋊C₄	—	47	8	no	—
210	C₄⋊C₄	—	47	8	no	—
211	C₄⋊C₄	—	47	4	no	—
212	C₄⋊C₄	—	47	8	no	—
213	C₄⋊C₄	—	47	4	no	—
214	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
215	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
216	C₂²⋊C₄	—	47	8	no	—
217	C₂²⋊C₄	—	47	8	no	—
218	C₂²⋊C₄	—	47	8	no	—
219	C₂²⋊C₄	—	47	8	no	—
220	C₂²⋊C₄	—	47	8	no	—
221	C₄²	—	47	8	no	—
222	C₄²	—	47	4	no	—
223	C₄²	—	47	4	no	—
224	C₄²	C₂²×D₄₇	1	8	yes	—
225	C₄²	C₂²×D₄₇	1	4	yes	—
226	C₂³	—	47	4	no	—
227	C₂³	—	94	4	no	—
228	C₂³	—	94	4	no	—
229	C₂³	—	94	4	no	—
230	C₂³	C₂³×D₄₇	1	1	yes	characteristic
231	Q₈	—	2	8	no	—
232	Q₈	—	94	8	no	—
233	Q₈	—	94	8	no	—
234	D₄	—	94	8	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	47	4	no	—
240	C₂×C₄	—	47	2	no	—
241	C₂×C₄	—	47	2	no	—
242	C₂×C₄	—	47	2	no	—
243	C₂×C₄	—	47	2	no	—
244	C₂×C₄	—	47	4	no	—
245	C₂×C₄	—	94	16	no	—
246	C₂×C₄	—	94	2	no	—
247	C₂×C₄	—	94	4	no	—
248	C₂×C₄	—	94	2	no	—
249	C₂×C₄	—	94	2	no	—
250	C₂×C₄	#10625350	1	4	yes	—
251	C₂×C₄	C₂³×D₄₇	1	4	yes	—
252	C₂×C₄	C₂³×D₄₇	1	2	yes	—
253	C₂×C₄	C₂³×D₄₇	1	4	yes	—
254	C₂×C₄	C₂³×D₄₇	1	2	yes	—
255	C₂×C₄	C₂³×D₄₇	1	2	yes	—
256	C₂²	—	94	4	no	—
257	C₂²	—	94	4	no	—
258	C₂²	—	94	4	no	—
259	C₂²	—	188	8	no	—
260	C₂²	#41×D₄₇	1	2	yes	—
261	C₂²	C₂×#10625350	1	2	yes	—
262	C₂²	#13094072	1	2	yes	—
263	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	4	no	—
267	C₄	—	2	2	no	—
268	C₄	—	47	4	no	—
269	C₄	—	94	2	no	—
270	C₄	—	94	4	no	—
271	C₄	—	94	2	no	—
272	C₄	—	94	2	no	—
273	C₄	C₂×#10625350	1	4	yes	—
274	C₂	—	188	4	no	—
275	C₂	#15689845	1	4	yes	direct product factor
276	C₂	C₂×#41×D₄₇	1	1	yes	characteristic
277	C₂	C₂²×#10625350	1	1	yes	characteristic
278	C₂	C₂×#13094072	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	C₂×#556
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #15689845	C₂	direct (N × H)
2. C₂×#13094022	C₂	semidirect
3. C₂×#13094019	C₂	semidirect
4. C₂×#13094018	C₂	semidirect
7. C₂×#13094002	C₂	semidirect
8. C₂×C₄×Dic₉₄	C₂	semidirect
9. C₂×#13093989	C₂	semidirect
11. C₂×#13093981	C₂	semidirect
12. C₂×C₄×Q₈×C₄₇	C₂	semidirect
17. #13094022	C₂²	semidirect
18. #13094019	C₂²	semidirect
19. #13094018	C₂²	semidirect
22. #13094002	C₂²	semidirect
23. C₄×Dic₉₄	C₂²	semidirect
24. #13093989	C₂²	semidirect
26. #13093981	C₂²	semidirect
42. C₄×Q₈×C₄₇	C₂²	semidirect
135. C₂×C₄₇	#556	semidirect
149. C₂×C₄×Q₈	D₄₇	semidirect
152. C₄₇	C₂×#556	semidirect
166. C₄×Q₈	C₂×D₄₇	semidirect
275. C₂	#15689845	direct (N × H)

Subgroups — C2×#15689845 #20256936

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689845 #20256936