Subgroups — #19144406

Order 5504.

416 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063324	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14583192	C₂	1	2	yes	maximal, complemented
4	#14583187	C₂	1	2	yes	maximal, complemented
5	#14583182	C₂	1	2	yes	maximal, complemented
6	#14583177	C₂	1	2	yes	maximal, complemented
7	#14582941	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582939	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582922	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582900	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#417	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₈₆	—	2	1	no	—
13	C₂×#10621877	C₂²	1	1	yes	characteristic
14	#13063324	—	2	1	no	—
15	#13063324	—	2	1	no	—
16	#13063324	C₂²	1	2	yes	—
17	#13063324	C₂²	1	2	yes	—
18	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
19	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
20	D₄×Dic₄₃	—	2	1	no	—
21	D₄×Dic₄₃	—	2	1	no	—
22	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
24	#13063284	—	2	1	no	—
25	#13063284	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
27	C₂×Dic₄₃⋊C₄	—	2	1	no	—
28	C₂×C₄×Dic₄₃	—	2	1	no	—
29	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	#13063259	C₂²	1	2	yes	—
31	#13063258	C₂²	1	2	yes	—
32	#13063257	C₂²	1	2	yes	—
33	#13063256	C₂²	1	2	yes	—
34	Dic₄₃⋊C₈	C₂²	1	2	yes	—
35	Dic₄₃⋊C₈	C₂²	1	2	yes	—
36	#13063249	C₂²	1	2	yes	—
37	#13063246	C₂²	1	2	yes	—
38	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
41	#103×C₄₃	C₂²	1	2	yes	—
42	#102×C₄₃	C₂²	1	2	yes	—
43	#41×C₄₃	—	2	2	no	—
44	#41×C₄₃	Q₈	1	4	yes	complemented
45	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₄₃	—	2	1	no	—
47	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	—	2	1	no	—
49	C₂²×C₄×C₄₃	—	2	1	no	—
50	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₃	—	2	1	no	—
52	C₂²×Dic₄₃	—	2	1	no	—
53	C₂²×Dic₄₃	—	2	1	no	—
54	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
55	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
56	C₂×Dic₈₆	—	2	2	no	—
57	C₂×Dic₈₆	—	2	2	no	—
58	C₂×Dic₈₆	—	4	4	no	—
59	M₁₆×C₄₃	—	2	2	no	—
60	C₂×C₈×C₄₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₂²⋊Dic₄₃	—	2	1	no	—
64	C₂²⋊Dic₄₃	—	2	1	no	—
65	C₂²⋊Dic₄₃	—	2	1	no	—
66	C₄⋊Dic₄₃	—	2	1	no	—
67	C₄⋊Dic₄₃	—	2	1	no	—
68	C₄⋊Dic₄₃	—	2	1	no	—
69	C₄⋊Dic₄₃	C₂³	1	2	yes	—
70	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
72	Dic₄₃⋊C₄	—	2	2	no	—
73	Dic₄₃⋊C₄	—	2	2	no	—
74	C₄×Dic₄₃	—	2	1	no	—
75	C₄×Dic₄₃	—	2	1	no	—
76	C₄×Dic₄₃	—	2	1	no	—
77	C₄×Dic₄₃	D₄	1	1	yes	characteristic
78	C₄×Dic₄₃	D₄	1	1	yes	characteristic
79	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
81	#10621877	—	2	2	no	—
82	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
83	C₂³×C₄₃	—	2	1	no	—
84	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₄₃	—	2	1	no	—
86	Q₈×C₄₃	C₂×Q₈	1	2	yes	complemented
87	D₄×C₄₃	—	2	4	no	—
88	D₄×C₄₃	—	2	1	no	—
89	D₄×C₄₃	C₂×Q₈	1	2	yes	complemented
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	4	2	no	—
94	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₄₃	#41	1	2	yes	—
97	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	2	2	no	—
100	C₂×Dic₄₃	—	2	1	no	—
101	C₂×Dic₄₃	—	2	1	no	—
102	C₂×Dic₄₃	—	2	2	no	—
103	C₂×Dic₄₃	—	4	2	no	—
104	C₂×Dic₄₃	—	4	2	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
110	Dic₈₆	—	4	4	no	—
111	Dic₈₆	—	4	4	no	—
112	C₈×C₄₃	—	2	2	no	—
113	C₄₃⋊C₈	—	2	2	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	4	1	no	—
118	C₂²×C₄₃	—	4	1	no	—
119	C₂²×C₄₃	C₂²⋊Q₈	1	2	yes	—
120	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	—	4	2	no	—
123	C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
126	Dic₄₃	—	2	1	no	—
127	Dic₄₃	—	2	1	no	—
128	Dic₄₃	—	4	2	no	—
129	Dic₄₃	—	4	1	no	—
130	Dic₄₃	—	4	1	no	—
131	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
132	Dic₄₃	#137	1	2	yes	—
133	#3079	—	43	1	no	maximal, Sylow
134	C₂×C₄₃	—	2	2	no	—
135	C₂×C₄₃	—	4	2	no	—
136	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
139	C₂×C₄⋊Q₈	—	43	1	no	—
140	C₄×#41	—	43	1	no	—
141	#477	—	43	2	no	—
142	#476	—	43	2	no	—
143	#475	—	43	2	no	—
144	#474	—	43	2	no	—
145	#428	—	43	1	no	—
146	#428	—	43	1	no	—
147	#423	—	43	1	no	—
148	#417	—	43	1	no	—
149	#417	D₄₃	1	1	yes	characteristic
150	C₄₃	#3079	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
152	C₂²×Q₈	—	86	1	no	—
153	C₂×M₁₆	—	43	1	no	—
154	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
155	C₄⋊Q₈	—	43	2	no	—
156	C₄⋊Q₈	—	43	2	no	—
157	C₄⋊Q₈	—	86	1	no	—
158	C₄⋊Q₈	—	86	1	no	—
159	C₄×Q₈	—	43	1	no	—
160	C₄×Q₈	—	43	1	no	—
161	C₄×D₄	—	43	1	no	—
162	C₄×D₄	—	43	1	no	—
163	C₄×D₄	—	86	1	no	—
164	C₄×D₄	—	86	1	no	—
165	#117	—	43	1	no	—
166	#117	—	86	1	no	—
167	C₂×C₄⋊C₄	—	43	1	no	—
168	C₂×C₄⋊C₄	—	86	1	no	—
169	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
170	C₂×C₄²	—	43	1	no	—
171	C₂×C₄²	—	86	1	no	—
172	#107	—	43	2	no	—
173	#107	—	43	2	no	—
174	#106	—	43	2	no	—
175	#106	—	43	2	no	—
176	C₄⋊C₈	—	43	2	no	—
177	C₄⋊C₈	—	43	2	no	—
178	#103	—	43	2	no	—
179	#103	C₂×D₄₃	1	2	yes	—
180	#102	—	43	2	no	—
181	#102	C₂×D₄₃	1	2	yes	—
182	#41	—	2	2	no	—
183	#41	Dic₈₆	1	4	yes	complemented
184	C₂×Q₈	—	86	2	no	—
185	C₂×Q₈	—	86	2	no	—
186	C₂×Q₈	—	172	4	no	—
187	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	43	1	no	—
193	C₂²×C₄	—	43	1	no	—
194	C₂²×C₄	—	86	1	no	—
195	C₂²×C₄	—	86	1	no	—
196	C₂²×C₄	—	86	1	no	—
197	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	M₁₆	—	2	2	no	—
199	M₁₆	—	86	2	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	C₂²×D₄₃	1	2	yes	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	43	2	no	—
204	C₄⋊C₄	—	43	1	no	—
205	C₄⋊C₄	—	43	1	no	—
206	C₄⋊C₄	—	86	2	no	—
207	C₄⋊C₄	—	86	2	no	—
208	C₄⋊C₄	—	86	1	no	—
209	C₄⋊C₄	—	86	1	no	—
210	C₄⋊C₄	—	86	1	no	—
211	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
212	C₂²⋊C₄	—	86	1	no	—
213	C₂²⋊C₄	—	86	1	no	—
214	C₂²⋊C₄	—	86	1	no	—
215	C₄²	—	43	1	no	—
216	C₄²	—	43	1	no	—
217	C₄²	—	43	1	no	—
218	C₄²	—	43	1	no	—
219	C₄²	—	86	1	no	—
220	C₄²	—	86	1	no	—
221	C₄²	—	86	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	D₄×D₄₃	1	1	yes	characteristic
224	Q₈	—	2	1	no	—
225	Q₈	—	172	4	no	—
226	Q₈	—	172	4	no	—
227	Q₈	C₂×Dic₈₆	1	2	yes	complemented
228	D₄	—	2	4	no	—
229	D₄	—	2	1	no	—
230	D₄	C₂×Dic₈₆	1	2	yes	complemented
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	43	1	no	—
236	C₂×C₄	—	43	1	no	—
237	C₂×C₄	—	43	1	no	—
238	C₂×C₄	—	43	1	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	86	2	no	—
241	C₂×C₄	—	86	1	no	—
242	C₂×C₄	—	86	1	no	—
243	C₂×C₄	—	86	2	no	—
244	C₂×C₄	—	172	2	no	—
245	C₂×C₄	—	172	2	no	—
246	C₂×C₄	—	172	2	no	—
247	C₂×C₄	C₂×Dic₈₆	1	2	yes	—
248	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
249	C₂×C₄	#10621900	1	2	yes	—
250	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	86	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	#13063302	1	2	yes	—
259	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	43	2	no	—
263	C₄	—	43	2	no	—
264	C₄	—	86	1	no	—
265	C₄	—	86	1	no	—
266	C₄	—	172	2	no	—
267	C₄	—	172	1	no	—
268	C₄	—	172	1	no	—
269	C₄	#13063302	1	2	yes	—
270	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
271	C₄	C₂×#10621900	1	1	yes	characteristic
272	C₂	—	2	2	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#13063302	1	1	yes	characteristic
275	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
276	C₂	#137×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#417	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#417	D₄₃
hypocenter	C₄₃	#3079
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
1. C₂×#13063324	C₂	semidirect
3. #14583192	C₂	semidirect
4. #14583187	C₂	semidirect
5. #14583182	C₂	semidirect
6. #14583177	C₂	semidirect
7. #14582941	C₂	semidirect
8. #14582939	C₂	semidirect
9. #14582922	C₂	semidirect
44. #41×C₄₃	Q₈	semidirect
86. Q₈×C₄₃	C₂×Q₈	semidirect
89. D₄×C₄₃	C₂×Q₈	semidirect
150. C₄₃	#3079	semidirect
183. #41	Dic₈₆	semidirect
227. Q₈	C₂×Dic₈₆	semidirect
230. D₄	C₂×Dic₈₆	semidirect