Subgroups — #19137240

Order 5504.

444 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#14582806	C₂	1	1	yes	maximal, characteristic
3	#14582797	C₂	1	1	yes	maximal, characteristic
4	#14582795	C₂	1	1	yes	maximal, characteristic
5	#14582784	C₂	1	2	yes	maximal
6	#14582782	C₂	1	2	yes	maximal
7	#14582766	C₂	1	2	yes	maximal
8	#14582763	C₂	1	2	yes	maximal
9	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic
10	#14582745	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#391	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₄₃	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
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×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
22	C₄₃⋊#95	C₂²	1	1	yes	characteristic
23	C₄₃⋊#95	C₂²	1	2	yes	—
24	C₄₃⋊#95	C₂²	1	2	yes	—
25	C₄₃⋊#95	C₂²	1	2	yes	—
26	C₄₃⋊#95	C₂²	1	2	yes	—
27	C₄₃⋊#95	C₂²	1	1	yes	characteristic
28	C₄₃⋊#95	C₂²	1	1	yes	characteristic
29	C₄₃⋊#95	C₂²	1	1	yes	characteristic
30	C₄₃⋊#95	C₂²	1	2	yes	—
31	C₂²×Q₈×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	#95×C₄₃	C₂²	1	2	yes	—
35	#95×C₄₃	C₂²	1	2	yes	—
36	C₂×Q₈×C₄₃	—	2	2	no	—
37	C₂×Q₈×C₄₃	—	2	2	no	—
38	C₂×Q₈×C₄₃	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₄₃	C₂³	1	2	yes	—
47	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₃	C₂³	1	2	yes	—
50	C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₄²×C₄₃	—	2	2	no	—
52	C₄⋊Dic₄₃	—	2	2	no	—
53	C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
54	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
55	Dic₄₃⋊C₄	—	4	2	no	—
56	Dic₄₃⋊C₄	—	4	2	no	—
57	C₄×Dic₄₃	—	2	2	no	—
58	C₄×Dic₄₃	—	4	2	no	—
59	C₄×Dic₄₃	C₂×C₄	1	8	yes	—
60	C₄×Dic₄₃	C₂×C₄	1	4	yes	—
61	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
62	Q₈×C₄₃	—	2	8	no	—
63	C₂×C₄×C₄₃	—	2	2	no	—
64	C₂×C₄×C₄₃	—	2	2	no	—
65	C₂×C₄×C₄₃	—	2	1	no	—
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	1	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	2	1	no	—
73	C₂×C₄×C₄₃	—	2	1	no	—
74	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₄₃	#41	1	2	yes	—
77	C₂×C₄×C₄₃	#41	1	2	yes	—
78	C₂×Dic₄₃	—	2	1	no	—
79	C₂×Dic₄₃	—	2	2	no	—
80	C₂×Dic₄₃	—	2	1	no	—
81	C₂×Dic₄₃	—	2	1	no	—
82	C₂×Dic₄₃	—	2	1	no	—
83	C₂×Dic₄₃	—	2	2	no	—
84	C₂×Dic₄₃	—	2	1	no	—
85	C₂×Dic₄₃	—	2	1	no	—
86	C₂×Dic₄₃	—	2	2	no	—
87	C₂×Dic₄₃	—	2	2	no	—
88	C₂×Dic₄₃	—	2	1	no	—
89	C₂×Dic₄₃	—	2	1	no	—
90	C₂×Dic₄₃	—	2	1	no	—
91	C₂×Dic₄₃	—	2	1	no	—
92	C₂×Dic₄₃	—	2	2	no	—
93	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
95	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
96	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
98	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
101	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₄₃	—	2	4	no	—
104	C₄×C₄₃	—	4	1	no	—
105	C₄×C₄₃	—	4	2	no	—
106	C₄×C₄₃	—	4	1	no	—
107	C₄×C₄₃	#117	1	4	yes	—
108	Dic₄₃	—	2	2	no	—
109	Dic₄₃	—	2	2	no	—
110	Dic₄₃	—	2	4	no	—
111	Dic₄₃	—	4	1	no	—
112	Dic₄₃	—	4	1	no	—
113	Dic₄₃	—	4	2	no	—
114	#2320	—	43	1	no	maximal, Sylow
115	C₂×C₄₃	C₂×#117	1	1	yes	characteristic
116	C₂×C₄₃	#519	1	1	yes	characteristic
117	C₂×C₄₃	#520	1	1	yes	characteristic
118	C₂×C₄₃	#541	1	1	yes	characteristic
119	C₂×C₄₃	#541	1	1	yes	characteristic
120	C₂×C₄₃	#542	1	1	yes	characteristic
121	C₂×C₄₃	#542	1	1	yes	characteristic
122	C₂×C₄×Q₈	—	43	1	no	—
123	#391	—	43	1	no	—
124	#391	D₄₃	1	1	yes	characteristic
125	#389	—	43	1	no	—
126	#389	—	43	1	no	—
127	#387	—	43	2	no	—
128	#387	—	43	2	no	—
129	#383	—	43	2	no	—
130	#382	—	43	2	no	—
131	C₄×C₄⋊C₄	—	43	1	no	—
132	#376	—	43	1	no	—
133	C₄₃	#2320	1	1	yes	characteristic, complemented, Sylow
134	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
135	C₄×Q₈	—	43	8	no	—
136	C₂×C₄⋊C₄	—	43	2	no	—
137	C₂×C₄⋊C₄	—	43	1	no	—
138	C₂×C₄⋊C₄	—	43	1	no	—
139	C₂×C₄⋊C₄	—	43	2	no	—
140	C₂×C₄⋊C₄	—	43	2	no	—
141	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
142	C₂×C₄²	—	43	1	no	—
143	C₂×C₄²	—	43	2	no	—
144	C₂×C₄²	—	43	1	no	—
145	C₂×C₄²	—	43	2	no	—
146	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
147	#95	—	43	1	no	—
148	#95	—	43	1	no	—
149	#95	—	43	1	no	—
150	#95	—	43	1	no	—
151	#95	—	43	2	no	—
152	#95	—	43	2	no	—
153	#95	—	43	2	no	—
154	#95	—	43	2	no	—
155	#95	—	43	2	no	—
156	#95	C₂×D₄₃	1	2	yes	—
157	#95	C₂×D₄₃	1	2	yes	—
158	C₂×Q₈	—	2	2	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	C₄×D₄₃	1	4	yes	—
161	C₂²×C₄	—	43	1	no	—
162	C₂²×C₄	—	43	1	no	—
163	C₂²×C₄	—	43	2	no	—
164	C₂²×C₄	—	43	1	no	—
165	C₂²×C₄	—	43	1	no	—
166	C₂²×C₄	—	43	2	no	—
167	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
171	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	43	8	no	—
174	C₄⋊C₄	—	43	4	no	—
175	C₄⋊C₄	—	86	2	no	—
176	C₄⋊C₄	—	172	2	no	—
177	C₄⋊C₄	—	172	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	43	8	no	—
180	C₄²	—	43	4	no	—
181	C₄²	—	86	2	no	—
182	C₄²	—	172	2	no	—
183	C₂³	C₂³×D₄₃	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	43	2	no	—
197	C₂×C₄	—	43	2	no	—
198	C₂×C₄	—	43	4	no	—
199	C₂×C₄	—	86	1	no	—
200	C₂×C₄	—	86	2	no	—
201	C₂×C₄	—	86	1	no	—
202	C₂×C₄	—	86	1	no	—
203	C₂×C₄	—	86	1	no	—
204	C₂×C₄	—	86	2	no	—
205	C₂×C₄	—	86	1	no	—
206	C₂×C₄	—	86	1	no	—
207	C₂×C₄	—	86	2	no	—
208	C₂×C₄	—	86	2	no	—
209	C₂×C₄	—	86	1	no	—
210	C₂×C₄	—	86	1	no	—
211	C₂×C₄	—	86	1	no	—
212	C₂×C₄	—	86	1	no	—
213	C₂×C₄	—	86	2	no	—
214	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
215	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
216	C₂×C₄	#10621900	1	2	yes	—
217	C₂×C₄	#10621900	1	2	yes	—
218	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
219	C₂²	C₂×#10621900	1	1	yes	characteristic
220	C₂²	C₂×#10621900	1	1	yes	characteristic
221	C₂²	#13063371	1	1	yes	characteristic
222	C₂²	#13063372	1	1	yes	characteristic
223	C₂²	#13063373	1	1	yes	characteristic
224	C₂²	#13063373	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	1	no	—
229	C₄	—	86	2	no	—
230	C₄	—	86	2	no	—
231	C₄	—	86	4	no	—
232	C₄	—	172	1	no	—
233	C₄	—	172	1	no	—
234	C₄	—	172	2	no	—
235	C₄	#13063281	1	4	yes	—
236	C₂	C₂×#13063281	1	1	yes	characteristic
237	C₂	#14583375	1	1	yes	characteristic
238	C₂	#14583381	1	1	yes	characteristic
239	C₂	#14583509	1	1	yes	characteristic
240	C₂	#14583510	1	1	yes	characteristic
241	C₂	#14583518	1	1	yes	characteristic
242	C₂	#14583522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₄₃×#391	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	#391	D₄₃
hypocenter	C₄₃	#2320
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
12. Q₈×Dic₄₃	C₄	semidirect
133. C₄₃	#2320	semidirect