Subgroups — #19143541

Order 5504.

356 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583452	C₂	1	1	yes	maximal, characteristic
2	#14583028	C₂	1	2	yes	maximal, complemented
3	#14583028	C₂	1	2	yes	maximal, complemented
4	#14583026	C₂	1	2	yes	maximal, complemented
5	#14583023	C₂	1	2	yes	maximal, complemented
6	#14582927	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#14582855	C₂	1	1	yes	maximal, characteristic
9	#14582854	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#424	C₂	1	1	yes	maximal, characteristic
12	C₂×#10621877	C₂²	1	1	yes	characteristic
13	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	#13063324	C₄	1	1	yes	characteristic
15	#13063324	C₄	1	1	yes	characteristic
16	#13063316	C₄	1	1	yes	characteristic
17	#13063316	C₄	1	1	yes	characteristic
18	#13063302	C₄	1	2	yes	—
19	#13063302	C₄	1	2	yes	—
20	C₄×Dic₈₆	C₂²	1	2	yes	—
21	C₄×Dic₈₆	C₂²	1	2	yes	—
22	#13063284	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
24	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₄₃⋊C₈	C₂²	1	2	yes	—
29	Dic₄₃⋊C₈	C₂²	1	2	yes	—
30	#13063236	C₂²	1	2	yes	—
31	#13063235	C₂²	1	2	yes	—
32	#13063235	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₄₃	C₂²	1	2	yes	—
35	C₈×Dic₄₃	C₂²	1	2	yes	—
36	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
38	#117×C₄₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
40	C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×Dic₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
45	C₂×Dic₈₆	C₂×C₄	1	2	yes	—
46	M₁₆×C₄₃	—	2	2	no	—
47	C₂×C₈×C₄₃	—	2	2	no	—
48	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₄₃	C₂³	1	2	yes	—
51	C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
54	C₄²×C₄₃	C₂³	1	2	yes	—
55	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
59	C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
60	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
62	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₄₃	C₂³	1	2	yes	—
68	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
69	#10621877	Q₈	1	4	yes	complemented
70	C₂×C₄₃⋊C₈	—	2	2	no	—
71	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
74	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₄₃	#41	1	2	yes	—
80	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₂×Dic₄₃	—	2	2	no	—
82	C₂×Dic₄₃	—	4	1	no	—
83	C₂×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
88	C₂×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
89	Dic₈₆	—	2	4	no	—
90	Dic₈₆	—	4	2	no	—
91	C₈×C₄₃	—	2	1	no	—
92	C₈×C₄₃	—	2	1	no	—
93	C₈×C₄₃	—	2	2	no	—
94	C₄₃⋊C₈	—	2	1	no	—
95	C₄₃⋊C₈	—	2	1	no	—
96	C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
97	C₂²×C₄₃	—	2	1	no	—
98	C₂²×C₄₃	—	2	1	no	—
99	C₂²×C₄₃	C₄×Q₈	1	2	yes	—
100	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₄₃	—	2	2	no	—
102	C₄×C₄₃	—	2	2	no	—
103	C₄×C₄₃	C₄×Q₈	1	2	yes	—
104	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
106	Dic₄₃	—	2	2	no	—
107	Dic₄₃	—	2	1	no	—
108	Dic₄₃	—	2	1	no	—
109	Dic₄₃	—	2	2	no	—
110	Dic₄₃	—	2	1	no	—
111	Dic₄₃	#131	1	2	yes	—
112	#2962	—	43	1	no	maximal, Sylow
113	C₂×C₄₃	—	2	2	no	—
114	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
116	C₂×C₄₃	#568	1	1	yes	characteristic
117	#533	—	43	1	no	—
118	#446	—	43	2	no	—
119	#446	—	43	2	no	—
120	#446	—	43	2	no	—
121	C₈×Q₈	—	43	2	no	—
122	#424	—	43	1	no	—
123	#424	D₄₃	1	1	yes	characteristic
124	C₂×C₄⋊C₈	—	43	1	no	—
125	#405	—	43	1	no	—
126	#405	—	43	1	no	—
127	C₄×M₁₆	—	43	1	no	—
128	C₄₃	#2962	1	1	yes	characteristic, complemented, Sylow
129	C₂×M₁₆	—	43	1	no	—
130	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
131	C₂²×C₈	—	43	1	no	—
132	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	43	1	no	—
134	C₄⋊Q₈	—	43	1	no	—
135	#125	—	43	1	no	—
136	#125	—	43	1	no	—
137	C₂²⋊Q₈	—	43	2	no	—
138	C₂²⋊Q₈	—	43	2	no	—
139	C₄×Q₈	—	43	2	no	—
140	C₄×Q₈	—	43	2	no	—
141	#117	—	43	1	no	—
142	#117	C₂×D₄₃	1	1	yes	characteristic
143	C₂×C₄²	—	43	1	no	—
144	C₄⋊C₈	—	43	1	no	—
145	C₄⋊C₈	—	43	1	no	—
146	C₄⋊C₈	—	43	1	no	—
147	C₄⋊C₈	—	43	1	no	—
148	C₄⋊C₈	—	43	2	no	—
149	C₄⋊C₈	—	43	2	no	—
150	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
151	C₄⋊C₈	C₂×D₄₃	1	2	yes	—
152	#97	—	43	2	no	—
153	#97	—	43	2	no	—
154	#97	—	43	2	no	—
155	C₄×C₈	—	43	2	no	—
156	C₄×C₈	—	43	2	no	—
157	C₄×C₈	—	43	2	no	—
158	C₂×Q₈	—	43	2	no	—
159	C₂×Q₈	—	43	2	no	—
160	C₂²×C₄	—	43	1	no	—
161	C₂²×C₄	—	43	1	no	—
162	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	43	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	43	2	no	—
167	C₂×C₈	—	43	1	no	—
168	C₂×C₈	—	43	1	no	—
169	C₂×C₈	—	86	2	no	—
170	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₄₃	1	2	yes	—
172	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	C₄⋊C₄	—	43	1	no	—
174	C₄⋊C₄	—	43	1	no	—
175	C₄⋊C₄	—	43	2	no	—
176	C₄⋊C₄	—	43	2	no	—
177	C₄⋊C₄	—	43	2	no	—
178	C₄⋊C₄	—	43	2	no	—
179	C₄⋊C₄	—	43	2	no	—
180	C₄⋊C₄	—	43	2	no	—
181	C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
182	C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
183	C₂²⋊C₄	—	43	2	no	—
184	C₂²⋊C₄	C₄×D₄₃	1	2	yes	—
185	C₄²	—	43	1	no	—
186	C₄²	—	43	1	no	—
187	C₄²	—	43	1	no	—
188	C₄²	—	43	2	no	—
189	C₄²	—	43	1	no	—
190	C₄²	C₂²×D₄₃	1	2	yes	—
191	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
192	Q₈	—	86	4	no	—
193	Q₈	—	172	2	no	—
194	C₂×C₄	—	43	1	no	—
195	C₂×C₄	—	43	1	no	—
196	C₂×C₄	—	43	1	no	—
197	C₂×C₄	—	43	1	no	—
198	C₂×C₄	—	43	2	no	—
199	C₂×C₄	—	43	2	no	—
200	C₂×C₄	—	86	2	no	—
201	C₂×C₄	—	172	1	no	—
202	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
205	C₂×C₄	#10621900	1	2	yes	—
206	C₂×C₄	Q₈×D₄₃	1	2	yes	—
207	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	43	4	no	—
212	C₈	—	86	1	no	—
213	C₈	—	86	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	#13063290	1	2	yes	—
217	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	43	2	no	—
221	C₄	—	86	1	no	—
222	C₄	—	86	2	no	—
223	C₄	—	86	1	no	—
224	C₄	—	86	1	no	—
225	C₄	—	86	2	no	—
226	C₄	#13063290	1	2	yes	—
227	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
228	C₄	C₂×#10621900	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	C₂×#13063290	1	1	yes	characteristic
231	C₂	#131×D₄₃	1	1	yes	characteristic
232	C₂	#14583664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#424	C₂
hypercenter	#424	D₄₃
hypocenter	C₄₃	#2962
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
2. #14583028	C₂	semidirect
3. #14583028	C₂	semidirect
4. #14583026	C₂	semidirect
5. #14583023	C₂	semidirect
69. #10621877	Q₈	semidirect
128. C₄₃	#2962	semidirect