Subgroups — #19139183

Order 5504.

496 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	C₂×#13063318	C₂	1	2	yes	maximal, complemented
2	#14582824	C₂	1	2	yes	maximal, complemented
3	#14582822	C₂	1	2	yes	maximal, complemented
4	#14582802	C₂	1	1	yes	maximal, characteristic
5	#14582802	C₂	1	1	yes	maximal, characteristic
6	#14582795	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582792	C₂	1	2	yes	maximal, complemented
8	#14582769	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Dic₄₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#390	C₂	1	1	yes	maximal, characteristic
12	#13063318	—	2	8	no	—
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
25	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₄₃⋊#95	C₂²	1	1	yes	characteristic
28	C₄₃⋊#95	C₂²	1	2	yes	—
29	C₄₃⋊#95	C₂²	1	2	yes	—
30	C₄₃⋊#95	C₂²	1	1	yes	characteristic
31	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
35	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
36	C₂⁴×C₄₃	C₂³	1	2	yes	—
37	C₂×D₄×C₄₃	—	2	2	no	—
38	C₂×D₄×C₄₃	—	2	2	no	—
39	C₂×D₄×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₄₃	C₂³	1	2	yes	—
47	C₂²×Dic₄₃	C₂³	1	2	yes	—
48	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₂²⋊C₄×C₄₃	—	2	4	no	—
52	C₂²⋊C₄×C₄₃	—	2	4	no	—
53	C₄²×C₄₃	—	2	2	no	—
54	C₂²⋊Dic₄₃	—	2	4	no	—
55	C₂²⋊Dic₄₃	—	2	4	no	—
56	C₂²⋊Dic₄₃	—	4	2	no	—
57	C₂²⋊Dic₄₃	—	4	2	no	—
58	C₄⋊Dic₄₃	—	2	4	no	—
59	C₄⋊Dic₄₃	—	2	2	no	—
60	C₄⋊Dic₄₃	—	2	2	no	—
61	C₄⋊Dic₄₃	—	4	2	no	—
62	Dic₄₃⋊C₄	—	2	2	no	—
63	Dic₄₃⋊C₄	—	2	4	no	—
64	Dic₄₃⋊C₄	D₄	1	4	yes	complemented
65	C₄×Dic₄₃	—	2	2	no	—
66	C₄×Dic₄₃	—	2	2	no	—
67	C₄×Dic₄₃	—	2	4	no	—
68	C₂³×C₄₃	—	2	2	no	—
69	C₂³×C₄₃	—	2	2	no	—
70	C₂³×C₄₃	—	2	2	no	—
71	C₂³×C₄₃	—	2	2	no	—
72	C₂³×C₄₃	—	2	2	no	—
73	C₂³×C₄₃	—	2	2	no	—
74	C₂³×C₄₃	—	2	2	no	—
75	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
76	D₄×C₄₃	—	4	4	no	—
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₄₃	#41	1	2	yes	—
86	C₂×C₄×C₄₃	#41	1	2	yes	—
87	C₂×C₄×C₄₃	#41	1	2	yes	—
88	C₂×C₄×C₄₃	#41	1	2	yes	—
89	C₂×Dic₄₃	—	2	1	no	—
90	C₂×Dic₄₃	—	2	1	no	—
91	C₂×Dic₄₃	—	2	1	no	—
92	C₂×Dic₄₃	—	2	1	no	—
93	C₂×Dic₄₃	—	2	2	no	—
94	C₂×Dic₄₃	—	2	2	no	—
95	C₂×Dic₄₃	—	2	1	no	—
96	C₂×Dic₄₃	—	2	1	no	—
97	C₂×Dic₄₃	—	2	1	no	—
98	C₂×Dic₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	2	2	no	—
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	2	1	no	—
103	C₂×Dic₄₃	—	2	1	no	—
104	C₂×Dic₄₃	—	2	1	no	—
105	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
106	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
107	C₂²×C₄₃	—	4	2	no	—
108	C₂²×C₄₃	—	4	2	no	—
109	C₂²×C₄₃	—	4	2	no	—
110	C₂²×C₄₃	—	4	2	no	—
111	C₂²×C₄₃	—	4	2	no	—
112	C₂²×C₄₃	—	4	2	no	—
113	C₂²×C₄₃	—	4	2	no	—
114	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	—	2	2	no	—
123	C₄×C₄₃	—	4	2	no	—
124	C₄×C₄₃	#126	1	4	yes	complemented
125	Dic₄₃	—	2	2	no	—
126	Dic₄₃	—	2	2	no	—
127	Dic₄₃	—	4	1	no	—
128	Dic₄₃	—	4	2	no	—
129	Dic₄₃	—	4	2	no	—
130	Dic₄₃	—	4	1	no	—
131	#2512	—	43	1	no	maximal, Sylow
132	C₂×C₄₃	—	8	2	no	—
133	C₂×C₄₃	C₂×#126	1	1	yes	characteristic
134	C₂×C₄₃	#532	1	1	yes	characteristic
135	C₂×C₄₃	#542	1	1	yes	characteristic
136	C₂×C₄₃	#547	1	1	yes	characteristic
137	C₂×C₄₃	#550	1	1	yes	characteristic
138	C₂×C₄₃	#553	1	1	yes	characteristic
139	C₂×C₄₃	#556	1	1	yes	characteristic
140	C₂×#126	—	43	2	no	—
141	#397	—	43	2	no	—
142	#396	—	43	2	no	—
143	#390	—	43	1	no	—
144	#390	—	43	1	no	—
145	#390	D₄₃	1	1	yes	characteristic
146	#389	—	43	1	no	—
147	#388	—	43	2	no	—
148	#384	—	43	1	no	—
149	C₄×C₄⋊C₄	—	43	1	no	—
150	C₄×C₄⋊C₄	—	43	1	no	—
151	C₄₃	#2512	1	1	yes	characteristic, complemented, Sylow
152	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
153	#126	—	86	8	no	—
154	C₂×C₄⋊C₄	—	43	1	no	—
155	C₂×C₄⋊C₄	—	43	2	no	—
156	C₂×C₄⋊C₄	—	43	2	no	—
157	C₂×C₄⋊C₄	—	43	1	no	—
158	C₂×C₄⋊C₄	—	43	1	no	—
159	C₂×C₄⋊C₄	—	43	1	no	—
160	C₂×C₄⋊C₄	—	43	2	no	—
161	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	—	43	2	no	—
163	C₂×C₂²⋊C₄	—	43	2	no	—
164	C₂×C₂²⋊C₄	—	43	2	no	—
165	C₂×C₂²⋊C₄	—	43	2	no	—
166	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
168	C₂×C₄²	—	43	2	no	—
169	C₂×C₄²	—	43	1	no	—
170	C₂×C₄²	—	43	1	no	—
171	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
172	#95	—	43	1	no	—
173	#95	—	43	2	no	—
174	#95	—	43	1	no	—
175	#95	—	43	2	no	—
176	C₂⁴	C₂²×D₄₃	1	2	yes	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂²×C₄	—	43	1	no	—
181	C₂²×C₄	—	43	1	no	—
182	C₂²×C₄	—	43	1	no	—
183	C₂²×C₄	—	43	1	no	—
184	C₂²×C₄	—	43	2	no	—
185	C₂²×C₄	—	43	2	no	—
186	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
187	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
188	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	43	4	no	—
192	C₄⋊C₄	—	86	4	no	—
193	C₄⋊C₄	—	86	2	no	—
194	C₄⋊C₄	—	86	2	no	—
195	C₄⋊C₄	—	86	4	no	—
196	C₄⋊C₄	—	86	2	no	—
197	C₄⋊C₄	—	172	2	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	86	4	no	—
201	C₂²⋊C₄	—	86	4	no	—
202	C₂²⋊C₄	—	172	2	no	—
203	C₂²⋊C₄	—	172	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	86	4	no	—
206	C₄²	—	86	2	no	—
207	C₄²	—	86	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	C₂³×D₄₃	1	1	yes	characteristic
216	D₄	—	4	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	43	2	no	—
225	C₂×C₄	—	43	2	no	—
226	C₂×C₄	—	86	1	no	—
227	C₂×C₄	—	86	1	no	—
228	C₂×C₄	—	86	1	no	—
229	C₂×C₄	—	86	1	no	—
230	C₂×C₄	—	86	2	no	—
231	C₂×C₄	—	86	2	no	—
232	C₂×C₄	—	86	1	no	—
233	C₂×C₄	—	86	1	no	—
234	C₂×C₄	—	86	1	no	—
235	C₂×C₄	—	86	2	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	86	1	no	—
238	C₂×C₄	—	86	1	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	86	2	no	—
241	C₂×C₄	—	86	1	no	—
242	C₂×C₄	#10621898	1	2	yes	—
243	C₂×C₄	D₄×D₄₃	1	2	yes	—
244	C₂×C₄	#10621900	1	2	yes	—
245	C₂×C₄	#10621900	1	2	yes	—
246	C₂×C₄	#10621900	1	2	yes	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
255	C₂²	C₂×#10621900	1	1	yes	characteristic
256	C₂²	C₂×#10621900	1	1	yes	characteristic
257	C₂²	C₂×#10621900	1	1	yes	characteristic
258	C₂²	C₂×#10621898	1	1	yes	characteristic
259	C₂²	#13063371	1	1	yes	characteristic
260	C₂²	#13063372	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	86	2	no	—
265	C₄	—	86	2	no	—
266	C₄	—	172	2	no	—
267	C₄	—	172	2	no	—
268	C₄	—	172	1	no	—
269	C₄	—	172	1	no	—
270	C₄	#13063318	1	4	yes	complemented
271	C₂	—	8	2	no	—
272	C₂	C₂×#13063318	1	1	yes	characteristic
273	C₂	#14583447	1	1	yes	characteristic
274	C₂	#14583518	1	1	yes	characteristic
275	C₂	#14583549	1	1	yes	characteristic
276	C₂	#14583570	1	1	yes	characteristic
277	C₂	#14583601	1	1	yes	characteristic
278	C₂	#14583619	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄₃×#390	C₂
hypercenter	#390	D₄₃
hypocenter	C₄₃	#2512
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #390

Complements

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

N	H ≅ G/N	type
1. C₂×#13063318	C₂	semidirect
2. #14582824	C₂	semidirect
3. #14582822	C₂	semidirect
6. #14582795	C₂	semidirect
7. #14582792	C₂	semidirect
8. #14582769	C₂	semidirect
9. C₄⋊C₄×Dic₄₃	C₂	semidirect
10. C₄×Dic₄₃⋊C₄	C₂	semidirect
64. Dic₄₃⋊C₄	D₄	semidirect
124. C₄×C₄₃	#126	semidirect
151. C₄₃	#2512	semidirect
270. C₄	#13063318	semidirect