Subgroups — #25110311

Order 11008.

1106 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×C₈×D₁₇₂	C₂	1	2	yes	maximal, complemented
2	C₄×#13063322	C₂	1	1	yes	maximal, characteristic
3	#19135389	C₂	1	2	yes	maximal, complemented
4	#19134886	C₂	1	8	yes	maximal, complemented
5	#19134712	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₄₃	C₂²	1	4	yes	—
8	C₂×#13063322	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₇₂	C₂²	1	2	yes	—
10	C₂×C₄×D₁₇₂	C₂²	1	4	yes	—
11	C₈×D₁₇₂	—	2	8	no	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
13	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	4	yes	—
14	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	8	yes	—
15	#14582801	C₄	1	2	yes	—
16	#14582801	C₄	1	4	yes	—
17	#14582769	C₄	1	1	yes	characteristic
18	#14582654	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
21	C₄³×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#336	C₂²	1	4	yes	—
23	C₂²×D₁₇₂	C₂×C₄	1	2	yes	—
24	C₂²×D₁₇₂	C₂×C₄	1	4	yes	—
25	C₂²×C₄×D₄₃	C₂³	1	4	yes	—
26	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
27	C₂×C₈×D₄₃	—	2	8	no	—
28	C₂×C₈×D₄₃	—	4	8	no	—
29	#13063322	C₈	1	8	yes	complemented
30	C₄×D₁₇₂	—	2	4	no	—
31	C₄×D₁₇₂	—	2	4	no	—
32	C₄×D₁₇₂	—	4	8	no	—
33	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	4	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	8	yes	—
37	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
38	(C₂×D₄₃)⋊C₈	—	2	8	no	—
39	(C₂×D₄₃)⋊C₈	—	4	8	no	—
40	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₄₃	—	2	2	no	—
48	C₄×C₈×C₄₃	D₄	1	4	yes	complemented
49	C₂³×D₄₃	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
56	C₂×D₁₇₂	—	2	4	no	—
57	C₂×D₁₇₂	—	2	4	no	—
58	C₂×D₁₇₂	—	4	4	no	—
59	C₂×D₁₇₂	—	4	4	no	—
60	C₂×D₁₇₂	C₂×C₈	1	8	yes	complemented
61	C₂×D₁₇₂	C₂×C₈	1	16	yes	complemented
62	C₂×C₄×D₄₃	—	2	4	no	—
63	C₂×C₄×D₄₃	—	2	4	no	—
64	C₂×C₄×D₄₃	—	2	8	no	—
65	C₂×C₄×D₄₃	—	2	8	no	—
66	C₂×C₈×C₄₃	—	2	4	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
70	C₄²×C₄₃	—	2	2	no	—
71	C₄²×C₄₃	—	2	4	no	—
72	C₄²×C₄₃	—	4	2	no	—
73	C₄²×C₄₃	C₂×C₈	1	4	yes	complemented
74	C₄²×C₄₃	C₂×D₄	1	2	yes	—
75	C₄²×C₄₃	#41	1	2	yes	—
76	(C₂×D₄₃)⋊C₄	—	2	8	no	—
77	(C₂×D₄₃)⋊C₄	—	4	8	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	4	4	no	—
80	C₂×C₄₃⋊C₈	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	2	8	no	—
82	C₈×D₄₃	—	4	16	no	—
83	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	4	no	—
85	C₂²×D₄₃	—	2	8	no	—
86	C₂²×D₄₃	—	2	4	no	—
87	C₂²×D₄₃	—	2	8	no	—
88	C₂²×D₄₃	C₂²×C₈	1	8	yes	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
98	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₄₃	#131	1	2	yes	—
100	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
103	C₂×Dic₄₃	—	2	4	no	—
104	C₂×Dic₄₃	—	2	8	no	—
105	D₁₇₂	—	2	16	no	—
106	D₁₇₂	—	4	16	no	—
107	C₄×D₄₃	—	4	4	no	—
108	C₄×D₄₃	—	4	4	no	—
109	C₄×D₄₃	—	4	8	no	—
110	C₈×C₄₃	—	2	4	no	—
111	C₈×C₄₃	—	4	2	no	—
112	C₄₃⋊C₈	—	4	4	no	—
113	#22260	—	43	1	no	maximal, Sylow
114	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂²×C₄₃	#518	1	2	yes	—
116	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
117	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
118	C₂²×C₄₃	#568	1	2	yes	—
119	C₂×D₄₃	—	2	8	no	—
120	C₂×D₄₃	—	2	16	no	—
121	C₂×D₄₃	—	4	4	no	—
122	C₂×D₄₃	—	4	4	no	—
123	C₂×D₄₃	—	4	8	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	—	2	4	no	—
126	C₄×C₄₃	—	4	2	no	—
127	C₄×C₄₃	C₈×D₄	1	4	yes	—
128	C₄×C₄₃	#535	1	2	yes	—
129	C₄×C₄₃	#540	1	2	yes	—
130	Dic₄₃	—	4	4	no	—
131	C₂×C₈×D₄	—	43	2	no	—
132	C₄×#127	—	43	1	no	—
133	#1954	—	43	2	no	—
134	#1869	—	43	8	no	—
135	#1831	—	43	1	no	—
136	#1744	D₄₃	1	1	yes	characteristic, complemented
137	C₂×C₄₃	#2328	1	1	yes	characteristic
138	C₂×C₄₃	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₄₃	#2932	1	1	yes	characteristic
140	C₂×C₄₃	#2972	1	2	yes	—
141	C₂×C₄₃	#2981	1	2	yes	—
142	D₄₃	—	4	8	no	—
143	C₂³×C₈	—	43	4	no	—
144	C₂×#127	—	43	1	no	—
145	C₂×C₄×D₄	—	43	2	no	—
146	C₂×C₄×D₄	—	43	4	no	—
147	C₈×D₄	—	86	8	no	—
148	C₂×C₄⋊C₈	—	43	2	no	—
149	C₂×C₂²⋊C₈	—	43	4	no	—
150	C₂×C₂²⋊C₈	—	43	8	no	—
151	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
153	#390	—	43	2	no	—
154	#390	—	43	4	no	—
155	#384	—	43	1	no	—
156	C₄³	C₂×D₄₃	1	1	yes	characteristic
157	#336	—	43	4	no	—
158	#336	C₂×D₄₃	1	4	yes	—
159	C₄₃	#22260	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	—	43	2	no	—
161	C₂²×D₄	—	43	4	no	—
162	C₂³×C₄	—	43	4	no	—
163	C₂²×C₈	—	43	4	no	—
164	C₂²×C₈	—	86	8	no	—
165	C₂²×C₈	—	172	8	no	—
166	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
167	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
168	#127	—	43	8	no	—
169	C₄×D₄	—	86	4	no	—
170	C₄×D₄	—	86	4	no	—
171	C₄×D₄	—	172	8	no	—
172	C₂×C₄⋊C₄	—	43	2	no	—
173	C₂×C₄⋊C₄	—	43	4	no	—
174	C₂×C₂²⋊C₄	—	43	4	no	—
175	C₂×C₂²⋊C₄	—	43	8	no	—
176	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
177	C₂×C₄²	C₄×D₄₃	1	2	yes	—
178	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
180	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
181	C₄⋊C₈	—	86	4	no	—
182	C₂²⋊C₈	—	86	8	no	—
183	C₂²⋊C₈	—	172	8	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	D₁₇₂	1	4	yes	complemented
186	C₂⁴	—	43	4	no	—
187	C₂×D₄	—	43	8	no	—
188	C₂×D₄	—	43	16	no	—
189	C₂×D₄	—	86	4	no	—
190	C₂×D₄	—	86	4	no	—
191	C₂×D₄	—	172	4	no	—
192	C₂×D₄	—	172	4	no	—
193	C₂²×C₄	—	43	4	no	—
194	C₂²×C₄	—	86	4	no	—
195	C₂²×C₄	—	86	4	no	—
196	C₂²×C₄	—	86	8	no	—
197	C₂²×C₄	—	86	8	no	—
198	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	86	4	no	—
207	C₂×C₈	—	86	8	no	—
208	C₂×C₈	—	172	16	no	—
209	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
210	C₄⋊C₄	—	86	4	no	—
211	C₄⋊C₄	—	172	4	no	—
212	C₂²⋊C₄	—	86	8	no	—
213	C₂²⋊C₄	—	172	8	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	2	no	—
217	C₄²	C₈×D₄₃	1	4	yes	complemented
218	C₄²	C₂×D₁₇₂	1	2	yes	—
219	C₄²	#10621898	1	2	yes	—
220	C₂³	—	43	8	no	—
221	C₂³	—	86	4	no	—
222	C₂³	—	86	8	no	—
223	C₂³	—	86	4	no	—
224	C₂³	—	86	8	no	—
225	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
226	D₄	—	86	16	no	—
227	D₄	—	172	16	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	86	4	no	—
235	C₂×C₄	—	86	8	no	—
236	C₂×C₄	—	172	4	no	—
237	C₂×C₄	—	172	4	no	—
238	C₂×C₄	—	172	8	no	—
239	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
240	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₄₃	1	4	yes	—
242	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
243	C₂×C₄	#13063331	1	2	yes	—
244	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
245	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
246	C₂×C₄	#13063371	1	4	yes	—
247	C₈	—	2	4	no	—
248	C₈	—	4	2	no	—
249	C₈	—	172	4	no	—
250	C₂²	—	86	8	no	—
251	C₂²	—	86	16	no	—
252	C₂²	—	172	4	no	—
253	C₂²	—	172	4	no	—
254	C₂²	—	172	8	no	—
255	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
256	C₂²	#14583373	1	2	yes	—
257	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
258	C₂²	C₂×#13063331	1	1	yes	characteristic
259	C₂²	#14583665	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	—	172	4	no	—
264	C₄	C₈×D₁₇₂	1	4	yes	—
265	C₄	#14583468	1	2	yes	—
266	C₄	#14583504	1	2	yes	—
267	C₂	—	172	8	no	—
268	C₂	#19137320	1	1	yes	characteristic
269	C₂	C₂×C₈×D₁₇₂	1	1	yes	characteristic
270	C₂	#19143314	1	1	yes	characteristic
271	C₂	#19143618	1	2	yes	—
272	C₂	#19143670	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#1744	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂×#13063322	C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#22260
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₁₇₂	C₂	semidirect
3. #19135389	C₂	semidirect
4. #19134886	C₂	semidirect
5. #19134712	C₂	semidirect
6. C₄₃×#1744	C₂	semidirect
29. #13063322	C₈	semidirect
48. C₄×C₈×C₄₃	D₄	semidirect
60. C₂×D₁₇₂	C₂×C₈	semidirect
61. C₂×D₁₇₂	C₂×C₈	semidirect
73. C₄²×C₄₃	C₂×C₈	semidirect
136. #1744	D₄₃	semidirect
159. C₄₃	#22260	semidirect
185. C₄×C₈	D₁₇₂	semidirect
217. C₄²	C₈×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₁₇₂	2
C₄×#13063322	2
#19135389	2
#19134886	2
#19134712	2
C₄₃×#1744	2
#22260	43