Subgroups — #19137484

Order 5504.

504 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063283	C₂	1	1	yes	maximal, characteristic
2	#14582797	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582795	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582790	C₂	1	2	yes	maximal
5	#14582788	C₂	1	2	yes	maximal
6	#14582766	C₂	1	2	yes	maximal, complemented
7	#14582756	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×Dic₄₃⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₄×(C₂×D₄₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
13	#13063283	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
18	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
21	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
25	C₄₃⋊#95	C₂²	1	2	yes	—
26	C₄₃⋊#95	C₂²	1	2	yes	—
27	C₄₃⋊#95	C₂²	1	2	yes	—
28	C₄₃⋊#95	C₂²	1	1	yes	characteristic
29	C₄₃⋊#95	C₂²	1	1	yes	characteristic
30	C₄₃⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
35	#95×C₄₃	C₂²	1	2	yes	—
36	C₂³×D₄₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₄₃	C₂³	1	2	yes	—
44	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₄₃	C₂³	1	2	yes	—
46	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₄₃	—	2	2	no	—
48	C₂×C₄×D₄₃	—	2	2	no	—
49	C₂×C₄×D₄₃	C₂×C₄	1	4	yes	—
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
52	C₄²×C₄₃	—	2	2	no	—
53	C₄²×C₄₃	—	4	2	no	—
54	(C₂×D₄₃)⋊C₄	—	2	2	no	—
55	(C₂×D₄₃)⋊C₄	—	2	2	no	—
56	(C₂×D₄₃)⋊C₄	—	4	2	no	—
57	(C₂×D₄₃)⋊C₄	C₂×C₄	1	8	yes	—
58	C₄⋊Dic₄₃	—	2	2	no	—
59	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
60	Dic₄₃⋊C₄	—	2	4	no	—
61	Dic₄₃⋊C₄	—	2	4	no	—
62	C₄×Dic₄₃	—	2	2	no	—
63	C₄×Dic₄₃	—	4	2	no	—
64	C₄×Dic₄₃	C₂×C₄	1	8	yes	—
65	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₂²×D₄₃	—	2	1	no	—
67	C₂²×D₄₃	—	2	1	no	—
68	C₂²×D₄₃	—	2	1	no	—
69	C₂²×D₄₃	—	2	1	no	—
70	C₂²×D₄₃	—	2	1	no	—
71	C₂²×D₄₃	—	2	1	no	—
72	C₂²×D₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₄₃	—	2	1	no	—
74	C₂×C₄×C₄₃	—	2	2	no	—
75	C₂×C₄×C₄₃	—	2	1	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	2	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	1	no	—
83	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₄₃	#41	1	2	yes	—
86	C₂×C₄×C₄₃	#41	1	2	yes	—
87	C₂×C₄×C₄₃	#41	1	2	yes	—
88	C₂×Dic₄₃	—	2	1	no	—
89	C₂×Dic₄₃	—	2	2	no	—
90	C₂×Dic₄₃	—	2	2	no	—
91	C₂×Dic₄₃	—	2	1	no	—
92	C₂×Dic₄₃	—	2	1	no	—
93	C₂×Dic₄₃	—	2	1	no	—
94	C₂×Dic₄₃	—	2	1	no	—
95	C₂×Dic₄₃	—	2	2	no	—
96	C₂×Dic₄₃	—	2	2	no	—
97	C₂×Dic₄₃	—	2	1	no	—
98	C₂×Dic₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	2	1	no	—
100	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
101	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
102	C₂×Dic₄₃	#41	1	2	yes	—
103	C₄×D₄₃	—	2	8	no	—
104	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
106	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
107	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
108	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×D₄₃	—	2	2	no	—
112	C₂×D₄₃	—	2	2	no	—
113	C₂×D₄₃	—	2	2	no	—
114	C₂×D₄₃	—	4	1	no	—
115	C₂×D₄₃	—	4	1	no	—
116	C₂×D₄₃	—	4	1	no	—
117	C₂×D₄₃	—	4	1	no	—
118	C₄×C₄₃	—	2	4	no	—
119	C₄×C₄₃	—	2	2	no	—
120	C₄×C₄₃	—	4	2	no	—
121	C₄×C₄₃	—	4	1	no	—
122	C₄×C₄₃	#117	1	4	yes	—
123	Dic₄₃	—	2	2	no	—
124	Dic₄₃	—	2	4	no	—
125	Dic₄₃	—	2	2	no	—
126	Dic₄₃	—	4	2	no	—
127	Dic₄₃	—	4	1	no	—
128	#2347	—	43	1	no	maximal, Sylow
129	C₂×C₄₃	C₂×#117	1	1	yes	characteristic
130	C₂×C₄₃	C₄×#41	1	1	yes	characteristic
131	C₂×C₄₃	#520	1	1	yes	characteristic
132	C₂×C₄₃	#553	1	1	yes	characteristic
133	C₂×C₄₃	#553	1	1	yes	characteristic
134	C₂×C₄₃	#556	1	1	yes	characteristic
135	C₂×C₄₃	#556	1	1	yes	characteristic
136	D₄₃	—	4	2	no	—
137	C₂×#117	—	43	1	no	—
138	#389	—	43	1	no	—
139	#389	—	43	1	no	—
140	#388	—	43	2	no	—
141	#388	—	43	2	no	—
142	#383	—	43	2	no	—
143	#380	—	43	1	no	—
144	C₄×C₄⋊C₄	—	43	1	no	—
145	C₄×C₄⋊C₄	—	43	2	no	—
146	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
147	C₄×C₂²⋊C₄	—	43	1	no	—
148	C₄₃	#2347	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	—	43	1	no	—
150	#117	—	43	8	no	—
151	C₂×C₄⋊C₄	—	43	1	no	—
152	C₂×C₄⋊C₄	—	43	2	no	—
153	C₂×C₄⋊C₄	—	43	2	no	—
154	C₂×C₄⋊C₄	—	43	1	no	—
155	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	43	2	no	—
158	C₂×C₂²⋊C₄	—	43	1	no	—
159	C₂×C₂²⋊C₄	—	43	1	no	—
160	C₂×C₂²⋊C₄	—	43	2	no	—
161	C₂×C₄²	—	43	2	no	—
162	C₂×C₄²	—	43	1	no	—
163	C₂×C₄²	—	43	2	no	—
164	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
165	C₂×C₄²	C₂×D₄₃	1	2	yes	—
166	#95	—	43	1	no	—
167	#95	—	43	1	no	—
168	#95	—	43	2	no	—
169	#95	—	43	2	no	—
170	#95	—	43	2	no	—
171	#95	—	43	2	no	—
172	#95	C₂×D₄₃	1	2	yes	—
173	C₂⁴	—	43	1	no	—
174	C₂²×C₄	—	43	1	no	—
175	C₂²×C₄	—	43	2	no	—
176	C₂²×C₄	—	43	1	no	—
177	C₂²×C₄	—	43	4	no	—
178	C₂²×C₄	—	43	1	no	—
179	C₂²×C₄	—	43	2	no	—
180	C₂²×C₄	—	86	2	no	—
181	C₂²×C₄	—	86	2	no	—
182	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
185	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	43	4	no	—
189	C₄⋊C₄	—	86	4	no	—
190	C₄⋊C₄	—	86	2	no	—
191	C₄⋊C₄	—	86	4	no	—
192	C₄⋊C₄	C₄×D₄₃	1	4	yes	—
193	C₂²⋊C₄	—	43	8	no	—
194	C₂²⋊C₄	—	86	2	no	—
195	C₂²⋊C₄	—	86	2	no	—
196	C₂²⋊C₄	—	172	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	43	8	no	—
200	C₄²	—	86	2	no	—
201	C₄²	—	172	2	no	—
202	C₂³	—	43	2	no	—
203	C₂³	—	86	1	no	—
204	C₂³	—	86	1	no	—
205	C₂³	—	86	1	no	—
206	C₂³	—	86	1	no	—
207	C₂³	—	86	1	no	—
208	C₂³	—	86	1	no	—
209	C₂³	C₂³×D₄₃	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	43	2	no	—
221	C₂×C₄	—	43	4	no	—
222	C₂×C₄	—	43	2	no	—
223	C₂×C₄	—	86	2	no	—
224	C₂×C₄	—	86	1	no	—
225	C₂×C₄	—	86	1	no	—
226	C₂×C₄	—	86	1	no	—
227	C₂×C₄	—	86	2	no	—
228	C₂×C₄	—	86	1	no	—
229	C₂×C₄	—	86	8	no	—
230	C₂×C₄	—	86	1	no	—
231	C₂×C₄	—	86	2	no	—
232	C₂×C₄	—	86	1	no	—
233	C₂×C₄	—	86	2	no	—
234	C₂×C₄	—	86	2	no	—
235	C₂×C₄	—	86	1	no	—
236	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
237	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
238	C₂×C₄	#10621898	1	2	yes	—
239	C₂×C₄	#10621898	1	2	yes	—
240	C₂×C₄	#10621902	1	2	yes	—
241	C₂²	—	86	2	no	—
242	C₂²	—	86	2	no	—
243	C₂²	—	86	2	no	—
244	C₂²	—	172	1	no	—
245	C₂²	—	172	1	no	—
246	C₂²	—	172	1	no	—
247	C₂²	—	172	1	no	—
248	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
249	C₂²	#41×D₄₃	1	1	yes	characteristic
250	C₂²	C₂×#10621898	1	1	yes	characteristic
251	C₂²	C₂×#10621898	1	1	yes	characteristic
252	C₂²	C₂×#10621902	1	1	yes	characteristic
253	C₂²	#13063370	1	1	yes	characteristic
254	C₂²	#13063372	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	86	2	no	—
260	C₄	—	86	2	no	—
261	C₄	—	86	4	no	—
262	C₄	—	172	2	no	—
263	C₄	—	172	1	no	—
264	C₄	#13063282	1	4	yes	—
265	C₂	—	172	2	no	—
266	C₂	C₂×#13063282	1	1	yes	characteristic
267	C₂	C₄×#10621902	1	1	yes	characteristic
268	C₂	#14583378	1	1	yes	characteristic
269	C₂	#14583597	1	1	yes	characteristic
270	C₂	#14583599	1	1	yes	characteristic
271	C₂	#14583617	1	1	yes	characteristic
272	C₂	#14583622	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2347
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #14582797	C₂	semidirect
3. #14582795	C₂	semidirect
6. #14582766	C₂	semidirect
8. C₄⋊C₄×Dic₄₃	C₂	semidirect
9. C₄×Dic₄₃⋊C₄	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₃	C₂	semidirect
13. #13063283	C₄	semidirect
146. C₄×C₄⋊C₄	D₄₃	semidirect
148. C₄₃	#2347	semidirect