Subgroups — #25249190

Order 11008.

894 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583305	C₂	1	4	yes	maximal
2	#19142958	C₂	1	1	yes	maximal, characteristic
3	#19134761	C₂	1	4	yes	maximal
4	#19134754	C₂	1	2	yes	maximal
5	#19134751	C₂	1	2	yes	maximal
6	C₄×#13063236	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#2868	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063317	C₂²	1	2	yes	—
9	#14583305	—	2	16	no	—
10	C₂×#13063258	C₂²	1	8	yes	—
11	C₂×#13063256	C₂²	1	8	yes	—
12	C₂×#13063236	C₂²	1	2	yes	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
15	#14582797	—	2	2	no	—
16	#14582797	—	2	2	no	—
17	#14582769	C₂²	1	1	yes	characteristic
18	#14582769	C₂²	1	1	yes	characteristic
19	#14582765	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₄₃	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
22	C₄₃×#384	—	2	2	no	—
23	C₄₃×#384	C₂²	1	2	yes	—
24	C₄₃×#384	C₂²	1	2	yes	—
25	C₄³×C₄₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
27	#13063317	—	2	8	no	—
28	C₂×C₄⋊Dic₄₃	—	2	2	no	—
29	C₂×C₄⋊Dic₄₃	—	2	2	no	—
30	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
31	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
32	C₂×Dic₄₃⋊C₄	—	2	2	no	—
33	C₂×Dic₄₃⋊C₄	—	2	2	no	—
34	#13063258	—	2	16	no	—
35	#13063256	—	2	16	no	—
36	#13063236	D₄	1	8	yes	—
37	#13063236	Q₈	1	8	yes	complemented
38	C₄×C₄₃⋊C₈	Q₈	1	8	yes	complemented
39	C₄₃⋊#95	—	2	4	no	—
40	C₂²×Q₈×C₄₃	—	2	2	no	—
41	C₄⋊Q₈×C₄₃	—	2	8	no	—
42	C₄⋊Q₈×C₄₃	—	4	8	no	—
43	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
47	C₂×C₄²×C₄₃	D₄	1	2	yes	—
48	C₂×C₄²×C₄₃	D₄	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₄₃	—	4	8	no	—
53	C₂×Q₈×C₄₃	—	4	4	no	—
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	2	2	no	—
61	C₂²×Dic₄₃	—	2	2	no	—
62	C₄⋊C₄×C₄₃	—	2	8	no	—
63	C₄⋊C₄×C₄₃	—	4	8	no	—
64	C₄⋊C₄×C₄₃	—	4	4	no	—
65	C₄⋊C₄×C₄₃	—	4	8	no	—
66	C₄²×C₄₃	—	2	4	no	—
67	C₄²×C₄₃	—	2	4	no	—
68	C₄²×C₄₃	C₂×D₄	1	4	yes	—
69	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
70	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
71	C₄⋊Dic₄₃	—	2	4	no	—
72	C₄⋊Dic₄₃	—	2	4	no	—
73	C₄⋊Dic₄₃	—	4	4	no	—
74	C₄⋊Dic₄₃	—	4	4	no	—
75	Dic₄₃⋊C₄	—	4	4	no	—
76	Dic₄₃⋊C₄	—	4	4	no	—
77	C₂×C₄₃⋊C₈	C₂×D₄	1	8	yes	—
78	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
79	C₂×C₄₃⋊C₈	C₂×Q₈	1	8	yes	—
80	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
81	Q₈×C₄₃	—	8	8	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	4	4	no	—
87	C₂×C₄×C₄₃	—	4	8	no	—
88	C₂×C₄×C₄₃	#127	1	2	yes	—
89	C₂×C₄×C₄₃	#127	1	2	yes	—
90	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
91	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
92	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
93	C₂×C₄×C₄₃	#137	1	4	yes	—
94	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
97	C₂×Dic₄₃	—	4	2	no	—
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	—	4	4	no	—
100	C₂×Dic₄₃	—	4	4	no	—
101	C₄₃⋊C₈	C₄⋊Q₈	1	16	yes	complemented
102	#33316	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
104	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
106	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
107	C₂²×C₄₃	C₂×#137	1	2	yes	—
108	C₄×C₄₃	—	2	2	no	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	2	4	no	—
111	C₄×C₄₃	—	8	4	no	—
112	C₄×C₄₃	#501	1	8	yes	complemented
113	C₄×C₄₃	C₂×C₄⋊Q₈	1	4	yes	—
114	Dic₄₃	—	8	2	no	—
115	Dic₄₃	—	8	2	no	—
116	C₂×#501	—	43	4	no	—
117	#2868	D₄₃	1	1	yes	characteristic
118	#2859	—	43	1	no	—
119	#1847	—	43	4	no	—
120	#1845	—	43	2	no	—
121	#1844	—	43	2	no	—
122	C₄×#97	—	43	1	no	—
123	C₂×C₄₃	#2868	1	1	yes	characteristic
124	C₂×C₄₃	#3151	1	1	yes	characteristic
125	C₂×C₄₃	#3152	1	1	yes	characteristic
126	C₂×C₄₃	C₂×#501	1	2	yes	—
127	C₂×C₄₃	#3165	1	2	yes	—
128	C₂×C₄⋊Q₈	—	2	2	no	—
129	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
130	C₂×#125	—	43	2	no	—
131	#501	—	86	16	no	—
132	C₂×#107	—	43	8	no	—
133	C₂×#106	—	43	8	no	—
134	C₂×#97	—	43	2	no	—
135	C₂×#97	—	43	2	no	—
136	C₂×C₄×C₈	—	43	2	no	—
137	#389	—	86	2	no	—
138	#389	—	86	2	no	—
139	#384	—	2	2	no	—
140	#384	—	43	1	no	—
141	#384	—	43	1	no	—
142	#384	C₂×D₄₃	1	2	yes	—
143	#384	C₂×D₄₃	1	2	yes	—
144	#382	—	43	2	no	—
145	C₄³	C₂×D₄₃	1	1	yes	characteristic
146	C₄₃	#33316	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×C₈	—	43	4	no	—
149	C₄⋊Q₈	—	2	8	no	—
150	C₄⋊Q₈	—	4	8	no	—
151	#125	—	86	8	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	43	2	no	—
156	C₂×C₄⋊C₄	—	43	2	no	—
157	C₂×C₄⋊C₄	—	86	2	no	—
158	C₂×C₄⋊C₄	—	86	2	no	—
159	C₂×C₄⋊C₄	—	86	2	no	—
160	C₂×C₄⋊C₄	—	86	2	no	—
161	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
162	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
164	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
166	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
167	#107	—	86	16	no	—
168	#106	—	86	16	no	—
169	#97	—	43	8	no	—
170	#97	—	43	8	no	—
171	C₄×C₈	—	43	8	no	—
172	#95	—	86	4	no	—
173	C₂×Q₈	—	4	8	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	86	2	no	—
177	C₂²×C₄	—	86	2	no	—
178	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
182	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
183	C₂×C₈	—	43	8	no	—
184	C₂×C₈	—	43	8	no	—
185	C₂×C₈	—	43	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	8	no	—
190	C₄⋊C₄	—	86	4	no	—
191	C₄⋊C₄	—	86	4	no	—
192	C₄⋊C₄	—	172	4	no	—
193	C₄⋊C₄	—	172	4	no	—
194	C₄⋊C₄	—	172	4	no	—
195	C₄⋊C₄	—	172	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	D₄×D₄₃	1	4	yes	—
199	C₄²	Q₈×D₄₃	1	4	yes	—
200	C₄²	Q₈×D₄₃	1	4	yes	—
201	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
202	Q₈	—	8	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	172	4	no	—
210	C₂×C₄	—	172	4	no	—
211	C₂×C₄	—	172	2	no	—
212	C₂×C₄	—	172	2	no	—
213	C₂×C₄	#13063321	1	2	yes	—
214	C₂×C₄	#13063321	1	2	yes	—
215	C₂×C₄	#13063323	1	4	yes	—
216	C₂×C₄	#13063323	1	4	yes	—
217	C₂×C₄	#13063351	1	4	yes	—
218	C₂×C₄	#13063356	1	4	yes	—
219	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
220	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
221	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
222	C₈	—	43	16	no	—
223	C₂²	C₂×#13063321	1	1	yes	characteristic
224	C₂²	C₂×#13063323	1	1	yes	characteristic
225	C₂²	C₂×#13063323	1	1	yes	characteristic
226	C₂²	C₂×#13063351	1	2	yes	—
227	C₂²	C₂×#13063356	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	4	no	—
231	C₄	—	8	4	no	—
232	C₄	—	344	2	no	—
233	C₄	—	344	2	no	—
234	C₄	#14583305	1	8	yes	complemented
235	C₄	C₄⋊Q₈×D₄₃	1	4	yes	—
236	C₂	#19142991	1	1	yes	characteristic
237	C₂	#19145045	1	1	yes	characteristic
238	C₂	#19145052	1	1	yes	characteristic
239	C₂	C₂×#14583305	1	2	yes	—
240	C₂	#19145137	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₄₃
commutator, cosocle	C₂²×C₄×C₄₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2868	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
hypercenter	#2868	D₄₃
hypocenter	C₄₃	#33316
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
37. #13063236	Q₈	semidirect
38. C₄×C₄₃⋊C₈	Q₈	semidirect
101. C₄₃⋊C₈	C₄⋊Q₈	semidirect
112. C₄×C₄₃	#501	semidirect
146. C₄₃	#33316	semidirect
234. C₄	#14583305	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583305	2
#19142958	2
#19134761	2
#19134754	2
#19134751	2
C₄×#13063236	2
C₄₃×#2868	2
#33316	43