Subgroups — #19143434

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×#13063355	C₂	1	1	yes	maximal, characteristic
2	#14583376	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583004	C₂	1	2	yes	maximal, complemented
4	#14582999	C₂	1	2	yes	maximal, complemented
5	C₄×#10621885	C₂	1	2	yes	maximal, complemented
6	C₄×#10621883	C₂	1	2	yes	maximal, complemented
7	#14582939	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582895	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063248	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582855	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#520	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
13	#13063355	C₄	1	8	yes	complemented
14	C₂×#10621885	C₂²	1	2	yes	—
15	C₂×#10621883	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×Dic₈₆	C₂²	1	2	yes	—
20	#13063290	—	2	1	no	—
21	#13063290	—	2	1	no	—
22	#13063284	C₂²	1	1	yes	characteristic
23	#13063281	—	2	2	no	—
24	#13063258	C₂²	1	2	yes	—
25	#13063256	C₂²	1	2	yes	—
26	#13063249	C₂²	1	1	yes	characteristic
27	#13063249	C₂²	1	1	yes	characteristic
28	#13063248	C₂²	1	2	yes	—
29	#13063248	C₂²	1	2	yes	—
30	#13063246	C₂²	1	1	yes	characteristic
31	#13063246	C₂²	1	1	yes	characteristic
32	#13063236	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
36	C₄×D₄×C₄₃	—	2	2	no	—
37	C₄×D₄×C₄₃	C₂²	1	2	yes	—
38	#117×C₄₃	—	2	1	no	—
39	#117×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
42	#41×C₄₃	—	2	2	no	—
43	#41×C₄₃	C₂×C₄	1	4	yes	—
44	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₃	—	2	1	no	—
46	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	—	2	1	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	—	2	1	no	—
50	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₃	—	2	1	no	—
52	C₂×Dic₈₆	—	2	1	no	—
53	C₂×Dic₈₆	—	2	2	no	—
54	C₂×Dic₈₆	C₂×C₄	1	4	yes	complemented
55	C₂×Dic₈₆	C₂³	1	2	yes	—
56	C₄⋊C₄×C₄₃	—	2	1	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	2	yes	—
61	C₂²⋊C₄×C₄₃	—	2	2	no	—
62	C₂²⋊C₄×C₄₃	D₄	1	2	yes	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	C₂³	1	2	yes	—
65	C₂²⋊Dic₄₃	—	2	1	no	—
66	#10621885	C₂×C₄	1	8	yes	complemented
67	#10621883	C₂×C₄	1	8	yes	complemented
68	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
70	Dic₄₃⋊C₄	—	2	2	no	—
71	Dic₄₃⋊C₄	—	2	2	no	—
72	C₄×Dic₄₃	—	2	2	no	—
73	C₄×Dic₄₃	—	2	2	no	—
74	C₄×Dic₄₃	—	2	1	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	—	2	1	no	—
81	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₄₃	—	2	1	no	—
83	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
84	D₄×C₄₃	—	2	4	no	—
85	D₄×C₄₃	—	2	1	no	—
86	D₄×C₄₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	2	4	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	8	2	no	—
91	C₂×C₄×C₄₃	—	8	1	no	—
92	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#41	1	2	yes	—
97	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₄₃	—	2	4	no	—
99	C₂×Dic₄₃	—	2	2	no	—
100	C₂×Dic₄₃	—	2	1	no	—
101	C₂×Dic₄₃	—	2	1	no	—
102	Dic₈₆	—	2	4	no	—
103	Dic₈₆	—	2	2	no	—
104	Dic₈₆	C₂²×C₄	1	4	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	—	2	2	no	—
111	C₂²×C₄₃	—	4	1	no	—
112	C₂²×C₄₃	—	4	1	no	—
113	C₂²×C₄₃	C₄×D₄	1	2	yes	—
114	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₄₃	—	2	2	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	4	2	no	—
119	C₄×C₄₃	C₄×D₄	1	2	yes	—
120	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
122	Dic₄₃	—	2	4	no	—
123	Dic₄₃	—	4	1	no	—
124	Dic₄₃	—	4	1	no	—
125	#2947	—	43	1	no	maximal, Sylow
126	C₂×C₄₃	—	2	2	no	—
127	C₂×C₄₃	—	4	2	no	—
128	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×C₄₃	#577	1	1	yes	characteristic
130	C₂×C₄₃	#578	1	1	yes	characteristic
131	C₂×#137	—	43	1	no	—
132	#520	D₄₃	1	1	yes	characteristic
133	#519	—	43	1	no	—
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	C₂×D₄₃	1	1	yes	characteristic
144	C₂²×Q₈	—	43	1	no	—
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	—	43	2	no	—
151	C₄×Q₈	—	43	2	no	—
152	C₄×Q₈	—	86	1	no	—
153	C₄×Q₈	—	86	1	no	—
154	C₄×Q₈	C₂×D₄₃	1	2	yes	—
155	C₄×D₄	—	2	2	no	—
156	C₄×D₄	C₂×D₄₃	1	2	yes	—
157	#117	—	2	1	no	—
158	#117	—	43	1	no	—
159	#117	—	86	2	no	—
160	#117	C₂×D₄₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
163	#107	—	43	2	no	—
164	#106	—	43	2	no	—
165	#103	—	43	1	no	—
166	#103	—	43	1	no	—
167	#103	—	43	2	no	—
168	#103	—	43	2	no	—
169	#102	—	43	1	no	—
170	#102	—	43	1	no	—
171	#97	—	43	2	no	—
172	C₄×C₈	—	43	2	no	—
173	#41	—	2	2	no	—
174	#41	C₄×D₄₃	1	4	yes	—
175	C₂×Q₈	—	43	4	no	—
176	C₂×Q₈	—	43	2	no	—
177	C₂×Q₈	—	86	2	no	—
178	C₂×Q₈	—	86	1	no	—
179	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	86	1	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	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	43	1	no	—
197	C₄⋊C₄	—	43	1	no	—
198	C₄⋊C₄	—	86	2	no	—
199	C₄⋊C₄	—	86	2	no	—
200	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂²⋊D₄₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	86	1	no	—
205	C₂²⋊C₄	C₂²⋊D₄₃	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	86	1	no	—
208	C₄²	—	86	2	no	—
209	C₄²	—	86	2	no	—
210	C₄²	C₂²×D₄₃	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
213	Q₈	—	2	1	no	—
214	Q₈	—	43	4	no	—
215	Q₈	—	86	4	no	—
216	Q₈	—	86	2	no	—
217	Q₈	C₂×C₄×D₄₃	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂×C₄×D₄₃	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	86	4	no	—
227	C₂×C₄	—	86	2	no	—
228	C₂×C₄	—	86	1	no	—
229	C₂×C₄	—	86	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	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₄×C₂²⋊D₄₃	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	—	86	4	no	—
251	C₄	—	172	1	no	—
252	C₄	—	172	1	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	—	4	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
259	C₂	#14583710	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×#520	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#520	D₄₃
hypocenter	C₄₃	#2947
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #14583376	C₂	semidirect
3. #14583004	C₂	semidirect
4. #14582999	C₂	semidirect
5. C₄×#10621885	C₂	semidirect
6. C₄×#10621883	C₂	semidirect
7. #14582939	C₂	semidirect
9. C₂×#13063248	C₂	semidirect
10. #14582855	C₂	semidirect
13. #13063355	C₄	semidirect
54. C₂×Dic₈₆	C₂×C₄	semidirect
66. #10621885	C₂×C₄	semidirect
67. #10621883	C₂×C₄	semidirect
75. #10621877	C₂×C₄	semidirect
142. C₄₃	#2947	semidirect