Subgroups — #19143757

Order 5504.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583433	C₂	1	1	yes	maximal, characteristic
2	#14583029	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582978	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582967	C₂	1	2	yes	maximal, complemented
6	C₈×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₄₃)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#14582959	C₂	1	2	yes	maximal, complemented
9	#14582924	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621872	C₂²	1	2	yes	—
13	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×D₄₃	C₂²	1	2	yes	—
15	#13063319	C₄	1	2	yes	—
16	#13063317	C₄	1	1	yes	characteristic
17	#13063313	C₄	1	1	yes	characteristic
18	#13063308	C₄	1	2	yes	—
19	#13063304	C₄	1	1	yes	characteristic
20	#13063299	C₄	1	1	yes	characteristic
21	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
22	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
24	#13063282	C₂²	1	2	yes	—
25	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
28	Dic₄₃⋊C₈	C₂²	1	2	yes	—
29	Dic₄₃⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	—
32	#13063235	C₂²	1	2	yes	—
33	C₈×Dic₄₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
45	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₄₃	C₂³	1	2	yes	—
47	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
57	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
63	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
64	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
65	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
67	C₄×Dic₄₃	C₂³	1	2	yes	—
68	C₂×C₄₃⋊C₈	—	4	2	no	—
69	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
70	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
71	#10621872	—	4	2	no	—
72	C₈×D₄₃	—	4	2	no	—
73	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₄₃	—	2	4	no	—
86	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
89	D₁₇₂	—	2	2	no	—
90	C₄×D₄₃	—	2	2	no	—
91	C₄×D₄₃	—	2	2	no	—
92	Dic₈₆	—	2	2	no	—
93	C₈×C₄₃	—	2	1	no	—
94	C₈×C₄₃	—	2	1	no	—
95	C₈×C₄₃	—	2	1	no	—
96	C₈×C₄₃	—	2	1	no	—
97	C₄₃⋊C₈	—	2	2	no	—
98	C₄₃⋊C₈	—	2	2	no	—
99	C₂²×C₄₃	—	2	1	no	—
100	C₂²×C₄₃	—	2	1	no	—
101	C₂²×C₄₃	#131	1	2	yes	—
102	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₄₃	—	2	2	no	—
104	C₂×D₄₃	—	2	2	no	—
105	C₂×D₄₃	—	2	2	no	—
106	C₄×C₄₃	—	2	1	no	—
107	C₄×C₄₃	—	2	1	no	—
108	C₄×C₄₃	—	2	1	no	—
109	C₄×C₄₃	—	2	1	no	—
110	C₄×C₄₃	#131	1	2	yes	—
111	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₄₃	—	2	2	no	—
114	Dic₄₃	—	2	2	no	—
115	Dic₄₃	—	2	2	no	—
116	#2994	—	43	1	no	maximal, Sylow
117	C₂×C₄₃	—	2	2	no	—
118	C₂×C₄₃	#520	1	1	yes	characteristic
119	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
120	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
121	D₄₃	—	4	2	no	—
122	#529	—	43	1	no	—
123	#446	—	43	1	no	—
124	C₈×Q₈	—	43	1	no	—
125	#436	—	43	1	no	—
126	#435	—	43	2	no	—
127	C₈×D₄	—	43	2	no	—
128	C₈×D₄	—	43	1	no	—
129	#433	—	43	2	no	—
130	#431	D₄₃	1	1	yes	characteristic, complemented
131	#424	—	43	2	no	—
132	C₂×C₄⋊C₈	—	43	1	no	—
133	C₄₃	#2994	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	43	2	no	—
135	C₂²×C₈	—	43	2	no	—
136	C₂²×C₈	—	43	2	no	—
137	#126	—	43	2	no	—
138	#125	—	43	1	no	—
139	#124	—	43	1	no	—
140	#123	—	43	2	no	—
141	C₂²⋊Q₈	—	43	1	no	—
142	#121	—	43	1	no	—
143	C₄×Q₈	—	43	1	no	—
144	C₄×D₄	—	43	2	no	—
145	C₄×D₄	—	43	1	no	—
146	#117	—	43	2	no	—
147	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
148	C₄⋊C₈	—	43	2	no	—
149	C₄⋊C₈	—	43	2	no	—
150	C₄⋊C₈	—	43	2	no	—
151	C₄⋊C₈	—	43	1	no	—
152	C₄⋊C₈	—	43	1	no	—
153	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
155	C₂²⋊C₈	—	43	2	no	—
156	C₂²⋊C₈	—	43	2	no	—
157	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
159	#97	—	43	2	no	—
160	C₄×C₈	—	43	2	no	—
161	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
162	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
163	C₂×Q₈	—	43	1	no	—
164	C₂×D₄	—	43	2	no	—
165	C₂×D₄	—	43	1	no	—
166	C₂²×C₄	—	43	2	no	—
167	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	M₁₆	—	172	2	no	—
171	C₂×C₈	—	43	2	no	—
172	C₂×C₈	—	43	2	no	—
173	C₂×C₈	—	172	2	no	—
174	C₂×C₈	—	172	2	no	—
175	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
179	C₄⋊C₄	—	43	2	no	—
180	C₄⋊C₄	—	43	2	no	—
181	C₄⋊C₄	—	43	1	no	—
182	C₄⋊C₄	—	43	2	no	—
183	C₄⋊C₄	—	43	1	no	—
184	C₄⋊C₄	—	43	2	no	—
185	C₂²⋊C₄	—	43	2	no	—
186	C₂²⋊C₄	—	43	2	no	—
187	C₂²⋊C₄	—	43	2	no	—
188	C₂²⋊C₄	—	43	2	no	—
189	C₂²⋊C₄	—	43	2	no	—
190	C₄²	—	43	2	no	—
191	C₄²	C₄×D₄₃	1	1	yes	characteristic
192	C₄²	C₄×D₄₃	1	1	yes	characteristic
193	C₄²	C₂²×D₄₃	1	1	yes	characteristic
194	C₄²	C₂²×D₄₃	1	1	yes	characteristic
195	C₂³	—	43	2	no	—
196	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
197	Q₈	—	86	2	no	—
198	D₄	—	86	4	no	—
199	D₄	—	86	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	43	2	no	—
205	C₂×C₄	—	43	2	no	—
206	C₂×C₄	—	43	2	no	—
207	C₂×C₄	—	86	2	no	—
208	C₂×C₄	—	86	2	no	—
209	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	86	2	no	—
220	C₈	—	86	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	86	2	no	—
224	C₂²	—	86	2	no	—
225	C₂²	—	86	2	no	—
226	C₂²	#13063331	1	2	yes	—
227	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	86	2	no	—
233	C₄	—	86	2	no	—
234	C₄	—	86	2	no	—
235	C₄	#13063333	1	2	yes	—
236	C₄	#13063371	1	1	yes	characteristic
237	C₄	#13063372	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	172	2	no	—
240	C₂	#14583379	1	1	yes	characteristic
241	C₂	C₂×#13063331	1	1	yes	characteristic
242	C₂	C₂×#13063333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#431	C₂
hypercenter	#431	D₄₃
hypocenter	C₄₃	#2994
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #14583029	C₂	semidirect
3. C₈×Dic₈₆	C₂	semidirect
4. #14582978	C₂	semidirect
5. #14582967	C₂	semidirect
6. C₈×D₁₇₂	C₂	semidirect
7. (C₂²⋊D₄₃)⋊C₈	C₂	semidirect
8. #14582959	C₂	semidirect
9. #14582924	C₂	semidirect
10. C₂×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
11. C₄₃×#431	C₂	semidirect
130. #431	D₄₃	semidirect
133. C₄₃	#2994	semidirect