Subgroups — #19143649

Order 5504.

422 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583410	C₂	1	1	yes	maximal, characteristic
2	#14582970	C₂	1	2	yes	maximal, complemented
3	#14582967	C₂	1	2	yes	maximal, complemented
4	C₈×C₂²⋊D₄₃	C₂	1	2	yes	maximal, complemented
5	(C₂²⋊D₄₃)⋊C₈	C₂	1	2	yes	maximal, complemented
6	#14582957	C₂	1	2	yes	maximal, complemented
7	#14582864	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	C₂²	1	1	yes	characteristic
11	C₂×#10621872	C₂²	1	2	yes	complemented
12	C₂²×C₄₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₈×D₄₃	C₂²	1	2	yes	complemented
14	#13063308	C₄	1	2	yes	—
15	#13063304	C₄	1	1	yes	characteristic
16	#13063304	C₄	1	1	yes	characteristic
17	#13063299	C₄	1	1	yes	characteristic
18	#13063299	C₄	1	1	yes	characteristic
19	#13063294	C₄	1	2	yes	—
20	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
22	#13063284	C₂²	1	1	yes	characteristic
23	Dic₄₃⋊C₈	C₂²	1	2	yes	complemented
24	Dic₄₃⋊C₈	C₂²	1	2	yes	complemented
25	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
26	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
28	(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	complemented
29	(C₂×D₄₃)⋊C₈	C₂²	1	2	yes	complemented
30	#13063235	C₂²	1	2	yes	complemented
31	C₈×Dic₄₃	C₂²	1	2	yes	complemented
32	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	C₂²	1	2	yes	—
34	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
35	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₂⁴×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	—	4	2	no	—
39	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
48	#10621898	—	4	2	no	—
49	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₄₃	C₂³	1	2	yes	—
51	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₂×C₈×C₄₃	C₂³	1	2	yes	—
54	C₂×C₈×C₄₃	C₂³	1	2	yes	—
55	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
58	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
62	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₄₃	C₂³	1	2	yes	—
65	C₂×C₄₃⋊C₈	—	4	2	no	—
66	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
67	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
68	#10621872	—	4	2	no	—
69	C₈×D₄₃	—	4	2	no	—
70	C₂³×C₄₃	—	2	2	no	—
71	C₂³×C₄₃	—	2	2	no	—
72	C₂³×C₄₃	—	4	1	no	—
73	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
74	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	2	no	—
79	C₂×C₄×C₄₃	—	2	2	no	—
80	C₂×C₄×C₄₃	—	4	1	no	—
81	C₂×C₄×C₄₃	—	4	1	no	—
82	C₂×C₄×C₄₃	—	4	1	no	—
83	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₄₃	—	2	4	no	—
88	C₂²⋊D₄₃	—	2	4	no	—
89	C₂²⋊D₄₃	—	4	2	no	—
90	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
92	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
93	D₁₇₂	—	4	1	no	—
94	C₄×D₄₃	—	2	2	no	—
95	C₄×D₄₃	—	2	2	no	—
96	Dic₈₆	—	4	1	no	—
97	C₈×C₄₃	—	2	2	no	—
98	C₈×C₄₃	—	2	2	no	—
99	C₄₃⋊C₈	—	2	2	no	—
100	C₄₃⋊C₈	—	2	2	no	—
101	C₂²×C₄₃	—	2	1	no	—
102	C₂²×C₄₃	—	2	1	no	—
103	C₂²×C₄₃	—	2	1	no	—
104	C₂²×C₄₃	—	2	1	no	—
105	C₂²×C₄₃	—	2	1	no	—
106	C₂²×C₄₃	—	2	1	no	—
107	C₂²×C₄₃	—	2	1	no	—
108	C₂²×C₄₃	—	4	4	no	—
109	C₂²×C₄₃	#131	1	2	yes	—
110	C₂²×C₄₃	#131	1	2	yes	—
111	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₄₃	—	2	2	no	—
113	C₂×D₄₃	—	2	2	no	—
114	C₂×D₄₃	—	2	2	no	—
115	C₄×C₄₃	—	2	1	no	—
116	C₄×C₄₃	—	2	1	no	—
117	C₄×C₄₃	—	2	1	no	—
118	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
119	Dic₄₃	—	2	2	no	—
120	Dic₄₃	—	2	2	no	—
121	Dic₄₃	—	2	2	no	—
122	#2977	—	43	1	no	maximal, Sylow
123	C₂×C₄₃	—	2	2	no	—
124	C₂×C₄₃	—	2	2	no	—
125	C₂×C₄₃	—	4	1	no	—
126	C₂×C₄₃	#518	1	1	yes	characteristic
127	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
128	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
129	D₄₃	—	4	2	no	—
130	#525	—	43	1	no	—
131	#435	—	43	2	no	—
132	#435	—	43	2	no	—
133	C₈×D₄	—	43	2	no	—
134	C₈×D₄	—	43	2	no	—
135	#433	—	43	2	no	—
136	#408	—	43	2	no	—
137	C₂×C₂²⋊C₈	—	43	1	no	—
138	C₂×C₂²⋊C₈	D₄₃	1	1	yes	characteristic, complemented
139	C₄₃	#2977	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	43	1	no	—
141	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
142	C₂×M₁₆	—	43	2	no	—
143	C₂²×C₈	—	43	2	no	—
144	C₂²×C₈	—	43	2	no	—
145	C₂²×C₈	C₂×D₄₃	1	2	yes	—
146	#123	—	43	2	no	—
147	C₂²⋊Q₈	—	43	1	no	—
148	C₂²⋊Q₈	—	43	1	no	—
149	#121	—	43	1	no	—
150	#121	—	43	1	no	—
151	#120	—	43	2	no	—
152	C₄×D₄	—	43	2	no	—
153	C₄×D₄	—	43	2	no	—
154	#117	—	43	1	no	—
155	C₄⋊C₈	—	43	2	no	—
156	C₄⋊C₈	—	43	2	no	—
157	C₂²⋊C₈	—	43	2	no	—
158	C₂²⋊C₈	—	43	1	no	—
159	C₂²⋊C₈	—	43	1	no	—
160	C₂²⋊C₈	—	43	2	no	—
161	C₂²⋊C₈	—	43	2	no	—
162	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
163	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
164	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
165	#97	—	43	2	no	—
166	C₄×C₈	—	43	2	no	—
167	C₂⁴	C₄×D₄₃	1	1	yes	characteristic
168	#41	—	172	2	no	—
169	C₂×Q₈	—	43	1	no	—
170	C₂×D₄	—	43	2	no	—
171	C₂×D₄	—	43	2	no	—
172	C₂×D₄	—	43	2	no	—
173	C₂×D₄	—	43	1	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	43	2	no	—
176	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
177	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
180	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
182	M₁₆	—	172	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	43	2	no	—
185	C₂×C₈	—	43	2	no	—
186	C₂×C₈	—	172	2	no	—
187	C₂×C₈	—	172	2	no	—
188	C₂×C₈	C₂²×D₄₃	1	2	yes	—
189	C₂×C₈	C₂²×D₄₃	1	2	yes	—
190	C₄⋊C₄	—	43	1	no	—
191	C₄⋊C₄	—	43	1	no	—
192	C₄⋊C₄	—	43	2	no	—
193	C₄⋊C₄	—	43	2	no	—
194	C₂²⋊C₄	—	43	2	no	—
195	C₂²⋊C₄	—	43	2	no	—
196	C₂²⋊C₄	—	43	2	no	—
197	C₂²⋊C₄	—	43	2	no	—
198	C₂²⋊C₄	—	43	2	no	—
199	C₄²	—	43	2	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	2	no	—
202	C₂³	—	4	1	no	—
203	C₂³	—	43	2	no	—
204	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
205	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
207	Q₈	—	172	1	no	—
208	D₄	—	86	4	no	—
209	D₄	—	86	4	no	—
210	D₄	—	172	1	no	—
211	D₄	—	172	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	43	2	no	—
219	C₂×C₄	—	43	2	no	—
220	C₂×C₄	—	43	2	no	—
221	C₂×C₄	—	86	2	no	—
222	C₂×C₄	—	86	2	no	—
223	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	86	2	no	—
230	C₈	—	86	2	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	4	no	—
239	C₂²	—	86	2	no	—
240	C₂²	—	86	2	no	—
241	C₂²	—	86	2	no	—
242	C₂²	#13063331	1	2	yes	—
243	C₂²	#13063333	1	2	yes	—
244	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	86	2	no	—
249	C₄	—	86	2	no	—
250	C₄	—	86	2	no	—
251	C₄	#13063370	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	2	2	no	—
254	C₂	—	4	1	no	—
255	C₂	—	172	2	no	—
256	C₂	#14583372	1	1	yes	characteristic
257	C₂	C₂×#13063331	1	1	yes	characteristic
258	C₂	C₂×#13063333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×C₂²⋊C₈×C₄₃	C₂
hypercenter	C₂×C₂²⋊C₈	D₄₃
hypocenter	C₄₃	#2977
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #14582970	C₂	semidirect
3. #14582967	C₂	semidirect
4. C₈×C₂²⋊D₄₃	C₂	semidirect
5. (C₂²⋊D₄₃)⋊C₈	C₂	semidirect
6. #14582957	C₂	semidirect
7. #14582864	C₂	semidirect
8. C₂×C₂²⋊(C₄₃⋊C₈)	C₂	semidirect
9. C₂×C₂²⋊C₈×C₄₃	C₂	semidirect
11. C₂×#10621872	C₂²	semidirect
13. C₂×C₈×D₄₃	C₂²	semidirect
23. Dic₄₃⋊C₈	C₂²	semidirect
24. Dic₄₃⋊C₈	C₂²	semidirect
28. (C₂×D₄₃)⋊C₈	C₂²	semidirect
29. (C₂×D₄₃)⋊C₈	C₂²	semidirect
30. #13063235	C₂²	semidirect
31. C₈×Dic₄₃	C₂²	semidirect
138. C₂×C₂²⋊C₈	D₄₃	semidirect
139. C₄₃	#2977	semidirect