Subgroups — #19143300

Order 5504.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063331	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#10621872	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583411	C₂	1	1	yes	maximal, characteristic
4	#14582978	C₂	1	4	yes	maximal, complemented
5	#14582972	C₂	1	4	yes	maximal, complemented
6	#14582926	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582867	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582862	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
12	#13063331	—	2	4	no	—
13	C₂×#10621877	C₂²	1	1	yes	characteristic
14	C₂×#10621872	—	2	4	no	—
15	C₂×#10621872	C₂²	1	2	yes	—
16	C₂×#10621872	C₂²	1	2	yes	—
17	C₂×#10621872	C₂²	1	2	yes	complemented
18	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₄₃	C₂²	1	2	yes	complemented
20	#13063309	C₄	1	2	yes	—
21	#13063305	C₄	1	2	yes	—
22	#13063300	C₄	1	2	yes	—
23	#13063295	C₄	1	2	yes	—
24	C₄×D₁₇₂	C₂²	1	4	yes	—
25	C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	complemented
26	(C₂×D₄₃)⋊C₈	C₂²	1	4	yes	—
27	(C₂×D₄₃)⋊C₈	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	#117×C₄₃	C₂²	1	1	yes	characteristic
31	#97×C₄₃	C₂²	1	2	yes	complemented
32	C₄×C₈×C₄₃	C₂²	1	2	yes	complemented
33	C₂³×D₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
36	C₂²×Dic₄₃	C₂×C₄	1	1	yes	characteristic
37	#10621898	—	2	2	no	—
38	#10621898	—	2	2	no	—
39	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
40	C₂×D₁₇₂	C₂×C₄	1	4	yes	—
41	C₂×C₄×D₄₃	—	2	2	no	—
42	C₂×C₄×D₄₃	—	2	2	no	—
43	C₂×C₄×D₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₄₃	C₂³	1	2	yes	—
45	C₂×C₄×D₄₃	C₂³	1	2	yes	—
46	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
47	M₁₆×C₄₃	—	2	2	no	—
48	C₂×C₈×C₄₃	D₄	1	4	yes	complemented
49	C₂×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
54	C₄²×C₄₃	C₂³	1	2	yes	—
55	(C₂×D₄₃)⋊C₄	C₂×C₄	1	4	yes	—
56	(C₂×D₄₃)⋊C₄	C₂×C₄	1	4	yes	—
57	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
58	#10621877	—	2	2	no	—
59	C₂×C₄₃⋊C₈	—	2	2	no	—
60	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
61	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
62	#10621872	—	2	4	no	—
63	#10621872	—	2	4	no	—
64	#10621872	—	2	4	no	—
65	C₈×D₄₃	—	2	4	no	—
66	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₄₃	—	2	2	no	—
68	C₂²×D₄₃	—	4	2	no	—
69	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
70	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₄₃	#41	1	2	yes	—
75	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
76	C₂²⋊D₄₃	—	2	4	no	—
77	C₂×Dic₄₃	—	2	2	no	—
78	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
79	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
80	D₁₇₂	—	2	2	no	—
81	D₁₇₂	—	4	4	no	—
82	C₄×D₄₃	—	2	4	no	—
83	C₄×D₄₃	—	2	2	no	—
84	C₄×D₄₃	—	2	2	no	—
85	C₄×D₄₃	—	2	2	no	—
86	C₄×D₄₃	—	2	2	no	—
87	Dic₈₆	—	2	2	no	—
88	C₈×C₄₃	—	2	2	no	—
89	C₈×C₄₃	C₂×D₄	1	2	yes	—
90	C₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₄₃⋊C₈	—	2	2	no	—
92	C₄₃⋊C₈	—	2	2	no	—
93	C₂²×C₄₃	—	2	2	no	—
94	C₂²×C₄₃	C₄×D₄	1	2	yes	—
95	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₄₃	—	2	2	no	—
97	C₂×D₄₃	—	2	2	no	—
98	C₂×D₄₃	—	2	2	no	—
99	C₂×D₄₃	—	2	2	no	—
100	C₂×D₄₃	—	2	4	no	—
101	C₂×D₄₃	—	4	4	no	—
102	C₄×C₄₃	—	2	4	no	—
103	C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
106	Dic₄₃	—	2	2	no	—
107	Dic₄₃	—	2	2	no	—
108	#2930	—	43	1	no	maximal, Sylow
109	C₂×C₄₃	—	2	2	no	—
110	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₂×C₄₃	#568	1	2	yes	—
112	D₄₃	—	4	2	no	—
113	D₄₃	—	4	2	no	—
114	C₂×#131	—	43	1	no	—
115	C₂²×M₁₆	—	43	1	no	—
116	#525	—	43	1	no	—
117	#436	—	43	4	no	—
118	#435	—	43	4	no	—
119	#424	—	43	1	no	—
120	#408	—	43	1	no	—
121	#407	—	43	1	no	—
122	#405	D₄₃	1	1	yes	characteristic, complemented
123	C₄₃	#2930	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	43	1	no	—
125	C₂³×C₄	—	43	1	no	—
126	#131	—	86	4	no	—
127	C₂×M₁₆	—	43	1	no	—
128	C₂×M₁₆	—	43	2	no	—
129	C₂×M₁₆	—	43	2	no	—
130	C₂×M₁₆	—	43	2	no	—
131	C₂×M₁₆	—	86	4	no	—
132	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
133	C₂²×C₈	—	43	1	no	—
134	C₂²×C₈	—	43	2	no	—
135	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
136	#123	—	43	2	no	—
137	C₂²⋊Q₈	—	43	2	no	—
138	#121	—	43	2	no	—
139	#120	—	43	2	no	—
140	C₄×D₄	—	43	4	no	—
141	#117	C₂×D₄₃	1	1	yes	characteristic
142	C₄⋊C₈	—	43	4	no	—
143	C₂²⋊C₈	—	43	4	no	—
144	C₂²⋊C₈	—	43	4	no	—
145	#97	C₂×D₄₃	1	2	yes	complemented
146	C₄×C₈	C₂×D₄₃	1	2	yes	complemented
147	C₂⁴	—	43	1	no	—
148	#41	—	86	2	no	—
149	#41	—	86	2	no	—
150	C₂×Q₈	—	43	1	no	—
151	C₂×D₄	—	43	2	no	—
152	C₂×D₄	—	43	1	no	—
153	C₂×D₄	—	43	4	no	—
154	C₂²×C₄	—	43	1	no	—
155	C₂²×C₄	—	43	2	no	—
156	C₂²×C₄	—	43	2	no	—
157	C₂²×C₄	—	43	2	no	—
158	C₂²×C₄	—	86	2	no	—
159	C₂²×C₄	—	86	2	no	—
160	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	86	2	no	—
163	M₁₆	—	86	4	no	—
164	M₁₆	—	86	4	no	—
165	M₁₆	—	86	4	no	—
166	C₂×C₈	—	43	2	no	—
167	C₂×C₈	—	43	2	no	—
168	C₂×C₈	—	86	2	no	—
169	C₂×C₈	—	86	4	no	—
170	C₂×C₈	D₁₇₂	1	4	yes	complemented
171	C₂×C₈	C₂²×D₄₃	1	2	yes	—
172	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
174	C₄⋊C₄	—	43	4	no	—
175	C₄⋊C₄	C₄×D₄₃	1	2	yes	—
176	C₂²⋊C₄	—	43	4	no	—
177	C₂²⋊C₄	—	43	4	no	—
178	C₂²⋊C₄	C₄×D₄₃	1	2	yes	—
179	C₄²	C₂²×D₄₃	1	2	yes	—
180	C₂³	—	43	2	no	—
181	C₂³	—	43	2	no	—
182	C₂³	—	86	2	no	—
183	C₂³	—	172	2	no	—
184	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
185	Q₈	—	86	2	no	—
186	D₄	—	86	4	no	—
187	D₄	—	86	2	no	—
188	D₄	—	172	4	no	—
189	C₂×C₄	—	43	2	no	—
190	C₂×C₄	—	43	2	no	—
191	C₂×C₄	—	86	2	no	—
192	C₂×C₄	—	86	4	no	—
193	C₂×C₄	—	86	2	no	—
194	C₂×C₄	—	86	2	no	—
195	C₂×C₄	—	86	2	no	—
196	C₂×C₄	—	86	2	no	—
197	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₁₇₂	1	2	yes	—
200	C₂×C₄	#10621898	1	2	yes	—
201	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	86	2	no	—
204	C₈	—	86	2	no	—
205	C₈	C₂×D₁₇₂	1	2	yes	—
206	C₈	C₂×D₁₇₂	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	86	2	no	—
209	C₂²	—	86	2	no	—
210	C₂²	—	86	2	no	—
211	C₂²	—	86	4	no	—
212	C₂²	—	86	2	no	—
213	C₂²	—	172	4	no	—
214	C₂²	C₄×D₁₇₂	1	2	yes	—
215	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	86	2	no	—
218	C₄	—	86	2	no	—
219	C₄	C₄×D₁₇₂	1	2	yes	—
220	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
221	C₄	C₂×#10621898	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	172	2	no	—
224	C₂	—	172	2	no	—
225	C₂	C₂×C₄×D₁₇₂	1	1	yes	characteristic
226	C₂	#14583664	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#405	C₂
hypercenter	#405	D₄₃
hypocenter	C₄₃	#2930
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#13063331	C₂	semidirect
2. C₂²×#10621872	C₂	semidirect
4. #14582978	C₂	semidirect
5. #14582972	C₂	semidirect
6. #14582926	C₂	semidirect
7. #14582867	C₂	semidirect
8. #14582862	C₂	semidirect
9. C₄₃×#405	C₂	semidirect
17. C₂×#10621872	C₂²	semidirect
19. C₂×C₈×D₄₃	C₂²	semidirect
25. C₄⋊(C₄₃⋊C₈)	C₂²	semidirect
27. (C₂×D₄₃)⋊C₈	C₂²	semidirect
31. #97×C₄₃	C₂²	semidirect
32. C₄×C₈×C₄₃	C₂²	semidirect
48. C₂×C₈×C₄₃	D₄	semidirect
122. #405	D₄₃	semidirect
123. C₄₃	#2930	semidirect
145. #97	C₂×D₄₃	semidirect
146. C₄×C₈	C₂×D₄₃	semidirect
170. C₂×C₈	D₁₇₂	semidirect