Subgroups — #19143645

Order 5504.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 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	#14582975	C₂	1	2	yes	maximal, complemented
3	#14582968	C₂	1	2	yes	maximal, complemented
4	#14582967	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	C₄₃×#407	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	1	yes	characteristic
13	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₄₃	C₂²	1	2	yes	complemented
15	#13063308	C₄	1	2	yes	—
16	#13063304	C₄	1	1	yes	characteristic
17	#13063304	C₄	1	1	yes	characteristic
18	#13063299	C₄	1	1	yes	characteristic
19	#13063299	C₄	1	1	yes	characteristic
20	#13063294	C₄	1	2	yes	—
21	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₄₃	C₂²	1	2	yes	—
23	#13063284	C₂²	1	1	yes	characteristic
24	Dic₄₃⋊C₈	C₂²	1	2	yes	complemented
25	Dic₄₃⋊C₈	C₂²	1	2	yes	complemented
26	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
27	C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
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	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
37	C₂⁴×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	—	4	1	no	—
39	C₂²×C₄×C₄₃	—	4	1	no	—
40	C₂²×C₄×C₄₃	C₂×C₄	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	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₄₃	C₂×C₄	1	2	yes	—
49	#10621898	—	4	1	no	—
50	#10621898	—	4	1	no	—
51	C₂×D₁₇₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₄₃	C₂³	1	2	yes	—
53	C₂×Dic₈₆	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₄₃	—	4	1	no	—
55	M₁₆×C₄₃	—	4	1	no	—
56	C₂×C₈×C₄₃	C₂³	1	2	yes	—
57	C₂×C₈×C₄₃	C₂³	1	2	yes	—
58	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
59	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
60	C₂²⋊Dic₄₃	C₂×C₄	1	2	yes	—
61	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
62	(C₂×D₄₃)⋊C₄	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
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	1	no	—
69	C₂×C₄₃⋊C₈	—	4	1	no	—
70	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
71	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
72	#10621872	—	4	2	no	—
73	C₈×D₄₃	—	4	2	no	—
74	C₂³×C₄₃	—	2	2	no	—
75	C₂³×C₄₃	—	4	1	no	—
76	C₂³×C₄₃	—	4	1	no	—
77	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	—	2	1	no	—
83	C₂×C₄×C₄₃	—	2	1	no	—
84	C₂×C₄×C₄₃	—	2	1	no	—
85	C₂×C₄×C₄₃	—	2	1	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	4	1	no	—
88	C₂×C₄×C₄₃	—	4	1	no	—
89	C₂×C₄×C₄₃	—	4	1	no	—
90	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₄₃	—	2	4	no	—
95	C₂²⋊D₄₃	—	4	2	no	—
96	C₂²⋊D₄₃	—	4	2	no	—
97	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
98	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
100	D₁₇₂	—	4	1	no	—
101	C₄×D₄₃	—	2	2	no	—
102	C₄×D₄₃	—	2	2	no	—
103	Dic₈₆	—	4	1	no	—
104	C₈×C₄₃	—	2	2	no	—
105	C₈×C₄₃	—	2	2	no	—
106	C₄₃⋊C₈	—	2	2	no	—
107	C₄₃⋊C₈	—	2	2	no	—
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	—	2	1	no	—
110	C₂²×C₄₃	—	2	1	no	—
111	C₂²×C₄₃	—	2	1	no	—
112	C₂²×C₄₃	—	2	1	no	—
113	C₂²×C₄₃	—	2	1	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	2	1	no	—
116	C₂²×C₄₃	—	4	4	no	—
117	C₂²×C₄₃	#131	1	2	yes	—
118	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₄₃	—	2	2	no	—
120	C₂×D₄₃	—	2	2	no	—
121	C₂×D₄₃	—	2	2	no	—
122	C₄×C₄₃	—	2	1	no	—
123	C₄×C₄₃	—	2	1	no	—
124	C₄×C₄₃	—	2	1	no	—
125	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₄₃	—	2	2	no	—
128	Dic₄₃	—	2	2	no	—
129	Dic₄₃	—	2	2	no	—
130	#2976	—	43	1	no	maximal, Sylow
131	C₂×C₄₃	—	2	2	no	—
132	C₂×C₄₃	—	4	1	no	—
133	C₂×C₄₃	—	4	1	no	—
134	C₂×C₄₃	#518	1	1	yes	characteristic
135	C₂×C₄₃	C₂×#131	1	1	yes	characteristic
136	C₂×C₄₃	#568	1	1	yes	characteristic
137	D₄₃	—	4	2	no	—
138	#525	—	43	1	no	—
139	#436	—	43	2	no	—
140	#435	—	43	2	no	—
141	#435	—	43	2	no	—
142	C₈×D₄	—	43	2	no	—
143	#433	—	43	2	no	—
144	#408	—	43	2	no	—
145	#407	D₄₃	1	1	yes	characteristic, complemented
146	C₂×C₂²⋊C₈	—	43	1	no	—
147	C₄₃	#2976	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	43	1	no	—
149	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
150	C₂×M₁₆	—	43	2	no	—
151	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
152	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
153	C₂²×C₈	—	43	1	no	—
154	C₂²×C₈	—	43	1	no	—
155	C₂²×C₈	—	43	2	no	—
156	#123	—	43	2	no	—
157	C₂²⋊Q₈	—	43	1	no	—
158	C₂²⋊Q₈	—	43	1	no	—
159	#121	—	43	1	no	—
160	#121	—	43	1	no	—
161	#120	—	43	2	no	—
162	C₄×D₄	—	43	2	no	—
163	C₄×D₄	—	43	2	no	—
164	#117	—	43	1	no	—
165	C₄⋊C₈	—	43	2	no	—
166	C₄⋊C₈	—	43	2	no	—
167	C₂²⋊C₈	—	43	2	no	—
168	C₂²⋊C₈	—	43	2	no	—
169	C₂²⋊C₈	—	43	2	no	—
170	C₂²⋊C₈	—	43	2	no	—
171	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
172	C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
173	#97	—	43	2	no	—
174	C₄×C₈	—	43	2	no	—
175	C₂⁴	C₄×D₄₃	1	1	yes	characteristic
176	#41	—	172	1	no	—
177	#41	—	172	1	no	—
178	C₂×Q₈	—	43	1	no	—
179	C₂×D₄	—	43	2	no	—
180	C₂×D₄	—	43	2	no	—
181	C₂×D₄	—	43	2	no	—
182	C₂×D₄	—	43	1	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	43	2	no	—
186	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
188	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	1	no	—
194	M₁₆	—	172	2	no	—
195	C₂×C₈	—	43	2	no	—
196	C₂×C₈	—	43	2	no	—
197	C₂×C₈	—	172	2	no	—
198	C₂×C₈	—	172	1	no	—
199	C₂×C₈	—	172	1	no	—
200	C₂×C₈	C₂²×D₄₃	1	2	yes	—
201	C₂×C₈	C₂²×D₄₃	1	2	yes	—
202	C₄⋊C₄	—	43	1	no	—
203	C₄⋊C₄	—	43	1	no	—
204	C₄⋊C₄	—	43	2	no	—
205	C₄⋊C₄	—	43	2	no	—
206	C₂²⋊C₄	—	43	2	no	—
207	C₂²⋊C₄	—	43	2	no	—
208	C₂²⋊C₄	—	43	2	no	—
209	C₂²⋊C₄	—	43	2	no	—
210	C₂²⋊C₄	—	43	2	no	—
211	C₄²	—	43	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	43	2	no	—
216	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
217	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂³	C₂×C₄×D₄₃	1	1	yes	characteristic
219	Q₈	—	172	1	no	—
220	D₄	—	86	4	no	—
221	D₄	—	172	1	no	—
222	D₄	—	172	2	no	—
223	D₄	—	172	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	43	2	no	—
234	C₂×C₄	—	43	2	no	—
235	C₂×C₄	—	43	2	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	86	2	no	—
238	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	86	2	no	—
245	C₈	—	86	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	86	2	no	—
256	C₂²	—	86	2	no	—
257	C₂²	—	86	2	no	—
258	C₂²	#13063333	1	2	yes	—
259	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	86	2	no	—
264	C₄	—	86	2	no	—
265	C₄	—	86	2	no	—
266	C₄	#13063370	1	1	yes	characteristic
267	C₄	#13063370	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	172	2	no	—
272	C₂	#14583372	1	1	yes	characteristic
273	C₂	C₂×#13063333	1	1	yes	characteristic
274	C₂	#14583665	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#407	C₂
hypercenter	#407	D₄₃
hypocenter	C₄₃	#2976
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #14582975	C₂	semidirect
3. #14582968	C₂	semidirect
4. #14582967	C₂	semidirect
5. (C₂²⋊D₄₃)⋊C₈	C₂	semidirect
6. #14582957	C₂	semidirect
7. #14582864	C₂	semidirect
8. C₂×C₂²⋊(C₄₃⋊C₈)	C₂	semidirect
9. C₄₃×#407	C₂	semidirect
11. C₂×#10621872	C₂²	semidirect
14. C₂×C₈×D₄₃	C₂²	semidirect
24. Dic₄₃⋊C₈	C₂²	semidirect
25. 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
145. #407	D₄₃	semidirect
147. C₄₃	#2976	semidirect