Subgroups — #19143438

Order 5504.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063356	C₂	1	1	yes	maximal, characteristic
2	#14583379	C₂	1	1	yes	maximal, characteristic
3	#14583004	C₂	1	2	yes	maximal, complemented
4	#14582997	C₂	1	2	yes	maximal, complemented
5	C₄×#10621885	C₂	1	2	yes	maximal, complemented
6	C₄×#10621884	C₂	1	2	yes	maximal, complemented
7	#14582942	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582893	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063249	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582855	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	#13063356	C₄	1	8	yes	complemented
14	C₂×#10621885	C₂²	1	2	yes	—
15	C₂×#10621884	C₂²	1	2	yes	—
16	C₂×#10621877	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Dic₈₆	C₂²	1	2	yes	—
19	C₄×D₁₇₂	C₂²	1	2	yes	—
20	(C₂²⋊D₄₃)⋊C₄	—	2	2	no	—
21	#13063283	—	2	1	no	—
22	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₄×D₄₃	—	2	1	no	—
24	#13063258	C₂²	1	2	yes	—
25	#13063256	C₂²	1	2	yes	—
26	#13063249	C₂²	1	2	yes	—
27	#13063249	C₂²	1	2	yes	—
28	#13063248	C₂²	1	1	yes	characteristic
29	#13063248	C₂²	1	1	yes	characteristic
30	#13063245	C₂²	1	1	yes	characteristic
31	#13063245	C₂²	1	1	yes	characteristic
32	#13063236	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
34	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₄₃	—	2	1	no	—
36	C₄×Q₈×C₄₃	—	2	1	no	—
37	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
38	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
39	#117×C₄₃	—	2	2	no	—
40	#117×C₄₃	C₂²	1	1	yes	characteristic
41	#117×C₄₃	C₂²	1	1	yes	characteristic
42	C₂×Q₈×C₄₃	—	2	1	no	—
43	C₂×Q₈×C₄₃	—	2	2	no	—
44	C₂×Q₈×C₄₃	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₃	—	2	1	no	—
47	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₄₃	—	2	1	no	—
49	C₂²×Dic₄₃	—	2	1	no	—
50	#10621898	—	2	2	no	—
51	#10621898	C₂×C₄	1	4	yes	—
52	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₄₃	—	2	1	no	—
54	C₂×C₄×D₄₃	—	2	2	no	—
55	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₄₃	—	2	2	no	—
57	C₄⋊C₄×C₄₃	—	2	2	no	—
58	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₄₃	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₄₃	—	2	1	no	—
63	C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	—	2	2	no	—
66	C₄²×C₄₃	—	2	1	no	—
67	C₄²×C₄₃	C₂³	1	2	yes	—
68	#10621885	C₂×C₄	1	8	yes	complemented
69	#10621884	C₂×C₄	1	8	yes	complemented
70	(C₂×D₄₃)⋊C₄	—	2	2	no	—
71	C₄⋊Dic₄₃	—	2	1	no	—
72	C₄⋊Dic₄₃	C₂³	1	2	yes	—
73	Dic₄₃⋊C₄	—	2	2	no	—
74	C₄×Dic₄₃	—	2	2	no	—
75	#10621877	C₂×C₄	1	4	yes	complemented
76	C₂×C₄₃⋊C₈	—	2	2	no	—
77	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
80	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₄₃	—	2	1	no	—
82	Q₈×C₄₃	—	2	4	no	—
83	Q₈×C₄₃	—	2	2	no	—
84	Q₈×C₄₃	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₄₃	—	2	4	no	—
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	2	1	no	—
89	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	#41	1	2	yes	—
94	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₄₃	—	2	4	no	—
96	C₂×Dic₄₃	—	2	1	no	—
97	C₂×Dic₄₃	—	2	2	no	—
98	C₂×Dic₄₃	—	8	1	no	—
99	D₁₇₂	—	2	1	no	—
100	D₁₇₂	C₂²×C₄	1	2	yes	—
101	C₄×D₄₃	—	2	4	no	—
102	C₄×D₄₃	—	8	2	no	—
103	Dic₈₆	—	2	1	no	—
104	Dic₈₆	C₂²×C₄	1	2	yes	—
105	C₄₃⋊C₈	—	2	1	no	—
106	C₄₃⋊C₈	—	2	1	no	—
107	C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	—	2	1	no	—
110	C₂²×C₄₃	C₄×D₄	1	2	yes	—
111	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×D₄₃	—	2	2	no	—
113	C₂×D₄₃	—	4	1	no	—
114	C₂×D₄₃	—	4	1	no	—
115	C₄×C₄₃	—	2	4	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	4	1	no	—
119	C₄×C₄₃	—	4	1	no	—
120	C₄×C₄₃	C₄×D₄	1	2	yes	—
121	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
122	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
123	Dic₄₃	—	2	2	no	—
124	Dic₄₃	—	4	2	no	—
125	#2947	—	43	1	no	maximal, Sylow
126	C₂×C₄₃	—	2	2	no	—
127	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₄₃	#577	1	1	yes	characteristic
129	C₂×C₄₃	#578	1	1	yes	characteristic
130	D₄₃	—	4	2	no	—
131	C₂×#137	—	43	1	no	—
132	#520	—	43	1	no	—
133	#519	D₄₃	1	1	yes	characteristic, complemented
134	#441	—	43	2	no	—
135	#440	—	43	2	no	—
136	C₄×Q₁₆	—	43	2	no	—
137	C₄×SD₁₆	—	43	2	no	—
138	#428	—	43	1	no	—
139	#416	—	43	1	no	—
140	C₂×#103	—	43	1	no	—
141	#405	—	43	1	no	—
142	C₄₃	#2947	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	—	43	1	no	—
144	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
145	#137	—	43	8	no	—
146	C₂×Q₁₆	—	43	2	no	—
147	C₂×SD₁₆	—	43	2	no	—
148	C₂×M₁₆	—	43	1	no	—
149	C₂²×C₈	—	43	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	2	1	no	—
152	C₄×Q₈	—	43	2	no	—
153	C₄×Q₈	C₂×D₄₃	1	2	yes	—
154	C₄×Q₈	C₂×D₄₃	1	2	yes	—
155	C₄×D₄	—	43	2	no	—
156	C₄×D₄	—	86	2	no	—
157	#117	—	2	2	no	—
158	#117	—	86	1	no	—
159	#117	C₂×D₄₃	1	1	yes	characteristic
160	#117	C₂×D₄₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	43	1	no	—
162	C₂×C₄⋊C₄	—	86	1	no	—
163	#107	—	43	2	no	—
164	#106	—	43	2	no	—
165	#103	—	43	2	no	—
166	#103	—	43	2	no	—
167	#103	—	43	1	no	—
168	#103	—	43	1	no	—
169	#102	—	43	1	no	—
170	#102	—	43	1	no	—
171	#97	—	43	2	no	—
172	C₄×C₈	—	43	2	no	—
173	#41	—	43	4	no	—
174	#41	—	86	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	43	1	no	—
178	C₂×Q₈	C₄×D₄₃	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
180	C₂×D₄	—	43	1	no	—
181	C₂×D₄	—	86	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	86	1	no	—
184	C₂²×C₄	—	86	1	no	—
185	C₂²×C₄	—	86	2	no	—
186	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
187	Q₁₆	—	43	8	no	—
188	SD₁₆	—	43	8	no	—
189	M₁₆	—	43	4	no	—
190	C₂×C₈	—	43	2	no	—
191	C₂×C₈	—	43	1	no	—
192	C₂×C₈	—	43	1	no	—
193	C₂×C₈	—	86	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	43	2	no	—
197	C₄⋊C₄	—	86	1	no	—
198	C₄⋊C₄	—	86	2	no	—
199	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	86	2	no	—
205	C₂²⋊C₄	C₂²⋊D₄₃	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	86	2	no	—
210	C₄²	C₂²×D₄₃	1	2	yes	—
211	C₂³	—	86	1	no	—
212	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	43	2	no	—
216	Q₈	—	86	1	no	—
217	Q₈	C₂×C₄×D₄₃	1	4	yes	—
218	D₄	—	43	2	no	—
219	D₄	—	86	4	no	—
220	D₄	—	86	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	86	1	no	—
226	C₂×C₄	—	86	4	no	—
227	C₂×C₄	—	86	2	no	—
228	C₂×C₄	—	344	2	no	—
229	C₂×C₄	—	344	1	no	—
230	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
231	C₂×C₄	#10621898	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	C₈	—	43	4	no	—
237	C₈	—	86	1	no	—
238	C₈	—	86	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	86	2	no	—
242	C₂²	—	172	1	no	—
243	C₂²	—	172	1	no	—
244	C₂²	C₄×C₂²⋊D₄₃	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	86	2	no	—
252	C₄	—	172	2	no	—
253	C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
254	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
255	C₄	C₂×#10621898	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	172	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
259	C₂	#14583707	1	1	yes	characteristic
260	C₂	#14583714	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#519	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#519	D₄₃
hypocenter	C₄₃	#2947
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #14583004	C₂	semidirect
4. #14582997	C₂	semidirect
5. C₄×#10621885	C₂	semidirect
6. C₄×#10621884	C₂	semidirect
7. #14582942	C₂	semidirect
9. C₂×#13063249	C₂	semidirect
10. #14582855	C₂	semidirect
11. C₄₃×#519	C₂	semidirect
13. #13063356	C₄	semidirect
44. C₂×Q₈×C₄₃	C₂×C₄	semidirect
68. #10621885	C₂×C₄	semidirect
69. #10621884	C₂×C₄	semidirect
75. #10621877	C₂×C₄	semidirect
133. #519	D₄₃	semidirect
142. C₄₃	#2947	semidirect
178. C₂×Q₈	C₄×D₄₃	semidirect