Subgroups — #25259488

Order 11008.

1910 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137781	C₂	1	4	yes	maximal, complemented
2	#19137701	C₂	1	4	yes	maximal, complemented
3	#19137472	C₂	1	8	yes	maximal
4	#19137401	C₂	1	8	yes	maximal, complemented
5	#19137198	C₂	1	4	yes	maximal
6	#19136993	C₂	1	2	yes	maximal
7	C₂²×#125×C₄₃	C₂	1	1	yes	maximal, characteristic
8	#14582798	C₂²	1	16	yes	—
9	#14582785	C₂²	1	16	yes	—
10	#14582785	C₂²	1	16	yes	—
11	#14582785	C₂²	1	16	yes	—
12	#14582785	C₂²	1	16	yes	—
13	#14582767	C₂²	1	4	yes	—
14	#14582765	C₂²	1	4	yes	—
15	#14582762	C₂²	1	4	yes	—
16	#14582759	C₂²	1	8	yes	—
17	#14582759	C₂²	1	4	yes	—
18	C₄⋊C₄×Dic₄₃	C₂²	1	16	yes	—
19	C₄×C₄⋊Dic₄₃	C₂²	1	4	yes	—
20	C₄×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
21	C₄×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
22	#14582745	C₂²	1	4	yes	—
23	C₂×#125×C₄₃	C₄	1	16	yes	complemented
24	C₂×#125×C₄₃	C₂²	1	4	yes	—
25	C₂×#125×C₄₃	C₂²	1	2	yes	—
26	C₂×#125×C₄₃	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
29	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Dic₄₃	C₂³	1	16	yes	—
31	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
32	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
33	C₂×C₄×Dic₄₃	C₂³	1	16	yes	—
34	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
35	C₄₃⋊#95	C₂³	1	16	yes	—
36	C₄₃⋊#95	C₂³	1	16	yes	—
37	C₄₃⋊#95	C₂³	1	8	yes	—
38	C₄₃⋊#95	C₂³	1	8	yes	—
39	C₄₃⋊#95	C₂³	1	8	yes	—
40	C₄₃⋊#95	C₂³	1	8	yes	—
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
44	#125×C₄₃	C₂×C₄	1	32	yes	—
45	#125×C₄₃	C₂×C₄	1	16	yes	—
46	#125×C₄₃	C₂×C₄	1	16	yes	—
47	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	16	yes	—
48	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	32	yes	—
49	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
52	C₂×C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
53	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	—	4	4	no	—
60	C₂²×C₄×C₄₃	—	4	8	no	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	16	yes	—
64	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	8	yes	—
69	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
70	C₄⋊C₄×C₄₃	C₂²×C₄	1	16	yes	—
71	C₄⋊C₄×C₄₃	C₂²×C₄	1	16	yes	—
72	C₄⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
73	C₄⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
74	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
77	C₂²⋊Dic₄₃	—	2	16	no	—
78	C₄⋊Dic₄₃	—	4	16	no	—
79	C₄⋊Dic₄₃	—	4	8	no	—
80	C₄×Dic₄₃	—	4	8	no	—
81	C₄×Dic₄₃	—	4	16	no	—
82	C₂³×C₄₃	—	2	4	no	—
83	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
84	C₂³×C₄₃	2₋¹⁺⁴	1	4	yes	—
85	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
86	C₂×C₄×C₄₃	—	2	8	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	—	2	4	no	—
90	C₂×C₄×C₄₃	—	2	16	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	4	8	no	—
93	C₂×C₄×C₄₃	—	4	16	no	—
94	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
98	C₂×C₄×C₄₃	C₂³×C₄	1	16	yes	—
99	C₂×Dic₄₃	—	2	16	no	—
100	C₂×Dic₄₃	—	2	16	no	—
101	C₂×Dic₄₃	—	2	16	no	—
102	#34372	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	—	2	4	no	—
104	C₂²×C₄₃	—	2	8	no	—
105	C₂²×C₄₃	#519	1	4	yes	—
106	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
107	C₂²×C₄₃	C₂×2₋¹⁺⁴	1	2	yes	—
108	C₂²×C₄₃	#585	1	4	yes	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	2	2	no	—
111	C₄×C₄₃	—	2	4	no	—
112	C₄×C₄₃	—	2	16	no	—
113	C₄×C₄₃	—	2	4	no	—
114	Dic₄₃	—	4	16	no	—
115	C₂²×#125	D₄₃	1	1	yes	characteristic
116	#2382	—	43	4	no	—
117	#2371	—	43	4	no	—
118	#2345	—	43	8	no	—
119	#2336	—	43	8	no	—
120	#2313	—	43	4	no	—
121	#2283	—	43	2	no	—
122	C₂×C₄₃	—	2	4	no	—
123	C₂×C₄₃	C₂×#519	1	1	yes	characteristic
124	C₂×C₄₃	#3428	1	2	yes	—
125	C₂×C₄₃	#3553	1	4	yes	—
126	C₂×#125	Dic₄₃	1	16	yes	complemented
127	C₂×#125	C₂×D₄₃	1	4	yes	—
128	C₂×#125	C₂×D₄₃	1	2	yes	—
129	C₂×#125	C₂×D₄₃	1	2	yes	—
130	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
131	C₂²×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
132	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
133	#389	—	43	16	no	—
134	#387	—	43	16	no	—
135	#387	—	43	16	no	—
136	#387	—	43	16	no	—
137	#387	—	43	16	no	—
138	#383	—	43	4	no	—
139	#382	—	43	4	no	—
140	#381	—	43	4	no	—
141	#380	—	43	8	no	—
142	#380	—	43	4	no	—
143	C₄×C₄⋊C₄	—	43	16	no	—
144	C₄×C₄⋊C₄	—	43	4	no	—
145	C₄×C₂²⋊C₄	—	43	8	no	—
146	C₄×C₂²⋊C₄	—	43	4	no	—
147	#376	—	43	4	no	—
148	C₄₃	#34372	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
150	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
151	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
152	#125	C₂×Dic₄₃	1	32	yes	—
153	#125	C₂×Dic₄₃	1	16	yes	—
154	#125	C₂×Dic₄₃	1	16	yes	—
155	C₂×C₄⋊C₄	—	43	8	no	—
156	C₂×C₄⋊C₄	—	43	16	no	—
157	C₂×C₄⋊C₄	C₂×Dic₄₃	1	16	yes	—
158	C₂×C₄⋊C₄	C₂×Dic₄₃	1	32	yes	—
159	C₂×C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
160	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
161	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
162	C₂×C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
163	C₂×C₂²⋊C₄	—	43	8	no	—
164	C₂×C₄²	—	43	8	no	—
165	C₂×C₄²	—	43	16	no	—
166	C₂×C₄²	C₂×Dic₄₃	1	8	yes	—
167	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
169	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
170	#95	—	43	8	no	—
171	#95	—	43	8	no	—
172	#95	—	43	8	no	—
173	#95	—	43	16	no	—
174	#95	—	43	16	no	—
175	#95	—	43	8	no	—
176	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	4	8	no	—
180	C₂²×C₄	—	43	16	no	—
181	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
183	C₂²×C₄	C₂²×Dic₄₃	1	16	yes	—
184	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
185	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
187	C₂²×C₄	C₂³×D₄₃	1	8	yes	—
188	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
189	C₄⋊C₄	—	172	16	no	—
190	C₄⋊C₄	—	172	8	no	—
191	C₄⋊C₄	C₂²×Dic₄₃	1	32	yes	—
192	C₄⋊C₄	C₂²×Dic₄₃	1	16	yes	—
193	C₄⋊C₄	C₂²×Dic₄₃	1	16	yes	—
194	C₄⋊C₄	C₂²×Dic₄₃	1	32	yes	—
195	C₂²⋊C₄	—	86	16	no	—
196	C₄²	—	172	8	no	—
197	C₄²	—	172	16	no	—
198	C₄²	C₂²×Dic₄₃	1	4	yes	—
199	C₄²	C₂²×Dic₄₃	1	8	yes	—
200	C₄²	C₂²×Dic₄₃	1	4	yes	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂³×Dic₄₃	1	2	yes	—
203	C₂³	#13063373	1	4	yes	—
204	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	4	16	no	—
213	C₂×C₄	—	86	16	no	—
214	C₂×C₄	—	86	16	no	—
215	C₂×C₄	—	86	16	no	—
216	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
217	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
218	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
219	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
220	C₂×C₄	C₂³×Dic₄₃	1	16	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	#14583377	1	4	yes	—
224	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
225	C₂²	C₂×#13063373	1	2	yes	—
226	C₂²	#14583735	1	4	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	—	2	16	no	—
231	C₄	—	2	4	no	—
232	C₄	—	172	16	no	—
233	C₂	—	2	4	no	—
234	C₂	C₂×#14583377	1	1	yes	characteristic
235	C₂	#19147798	1	2	yes	—
236	C₂	#19149051	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Fitting	C₂²×#125×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂⁴×C₄₃	C₂⁴
hypercenter	C₂²×#125	D₄₃
hypocenter	C₄₃	#34372
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
1. #19137781	C₂	semidirect
2. #19137701	C₂	semidirect
4. #19137401	C₂	semidirect
23. C₂×#125×C₄₃	C₄	semidirect
126. C₂×#125	Dic₄₃	semidirect
148. C₄₃	#34372	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19137781	2
#19137701	2
#19137472	2
#19137401	2
#19137198	2
#19136993	2
C₂²×#125×C₄₃	2
#34372	43