Subgroups — #25110312

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×D₄×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
2	#19135390	C₂	1	2	yes	maximal, complemented
3	#19134888	C₂	1	8	yes	maximal, complemented
4	#19134712	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#127×C₄₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄₃⋊C₈	C₂²	1	4	yes	—
8	D₄×C₄₃⋊C₈	—	2	8	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
11	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	#14582654	C₂²	1	4	yes	—
15	#14582654	C₂²	1	4	yes	—
16	C₂×#127×C₄₃	C₄	1	1	yes	characteristic
17	C₂×C₄×D₄×C₄₃	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₄₃	C₂²	1	4	yes	—
19	C₄₃×#390	C₄	1	2	yes	—
20	C₄₃×#390	C₄	1	4	yes	—
21	C₄₃×#384	C₄	1	1	yes	characteristic
22	C₄³×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄₃⋊C₈	—	2	8	no	—
24	C₂²×C₄₃⋊C₈	—	4	8	no	—
25	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
26	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
27	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
28	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
29	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	4	8	no	—
31	C₄×C₄₃⋊C₈	—	2	2	no	—
32	C₄×C₄₃⋊C₈	D₄	1	4	yes	complemented
33	C₂²×D₄×C₄₃	C₂×C₄	1	2	yes	—
34	C₂²×D₄×C₄₃	C₂×C₄	1	4	yes	—
35	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
36	#127×C₄₃	C₈	1	8	yes	complemented
37	C₄×D₄×C₄₃	—	2	4	no	—
38	C₄×D₄×C₄₃	—	2	4	no	—
39	C₄×D₄×C₄₃	—	4	8	no	—
40	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
44	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
50	C₂×D₄×C₄₃	—	2	4	no	—
51	C₂×D₄×C₄₃	—	2	4	no	—
52	C₂×D₄×C₄₃	—	4	4	no	—
53	C₂×D₄×C₄₃	—	4	4	no	—
54	C₂×D₄×C₄₃	C₂×C₈	1	8	yes	complemented
55	C₂×D₄×C₄₃	C₂×C₈	1	16	yes	complemented
56	C₂²×C₄×C₄₃	—	2	4	no	—
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	—	2	8	no	—
59	C₂²×C₄×C₄₃	—	2	8	no	—
60	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₄₃	—	2	4	no	—
67	C₄⋊C₄×C₄₃	—	4	4	no	—
68	C₂²⋊C₄×C₄₃	—	2	8	no	—
69	C₂²⋊C₄×C₄₃	—	4	8	no	—
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×C₄₃⋊C₈	—	2	4	no	—
77	C₂×C₄₃⋊C₈	—	2	4	no	—
78	C₂×C₄₃⋊C₈	—	2	2	no	—
79	C₂×C₄₃⋊C₈	—	2	4	no	—
80	C₂×C₄₃⋊C₈	—	2	8	no	—
81	C₂×C₄₃⋊C₈	—	4	16	no	—
82	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
83	C₂³×C₄₃	—	2	4	no	—
84	C₂³×C₄₃	—	2	8	no	—
85	C₂³×C₄₃	—	2	4	no	—
86	C₂³×C₄₃	—	2	8	no	—
87	C₂³×C₄₃	C₂²×C₈	1	8	yes	—
88	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
89	D₄×C₄₃	—	2	16	no	—
90	D₄×C₄₃	—	4	16	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	4	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	2	8	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	4	4	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	8	no	—
102	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂²×C₈	1	4	yes	—
105	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₄₃	#131	1	2	yes	—
107	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
110	C₄₃⋊C₈	—	2	4	no	—
111	C₄₃⋊C₈	—	4	4	no	—
112	C₄₃⋊C₈	—	4	2	no	—
113	#22260	—	43	1	no	maximal, Sylow
114	C₂²×C₄₃	—	2	8	no	—
115	C₂²×C₄₃	—	2	16	no	—
116	C₂²×C₄₃	—	4	4	no	—
117	C₂²×C₄₃	—	4	4	no	—
118	C₂²×C₄₃	—	4	8	no	—
119	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₄₃	#518	1	2	yes	—
121	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
122	C₂²×C₄₃	C₂×#131	1	1	yes	characteristic
123	C₂²×C₄₃	#568	1	2	yes	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	—	2	4	no	—
126	C₄×C₄₃	—	4	4	no	—
127	C₄×C₄₃	—	4	2	no	—
128	C₄×C₄₃	C₈×D₄	1	4	yes	—
129	C₄×C₄₃	#535	1	2	yes	—
130	C₄×C₄₃	#540	1	2	yes	—
131	C₂×C₈×D₄	—	43	2	no	—
132	C₄×#127	D₄₃	1	1	yes	characteristic
133	#1954	—	43	2	no	—
134	#1869	—	43	8	no	—
135	#1831	—	43	1	no	—
136	#1744	—	43	1	no	—
137	C₂×C₄₃	—	4	8	no	—
138	C₂×C₄₃	#2328	1	1	yes	characteristic
139	C₂×C₄₃	C₂×C₈×D₄	1	1	yes	characteristic
140	C₂×C₄₃	#2932	1	1	yes	characteristic
141	C₂×C₄₃	#2972	1	2	yes	—
142	C₂×C₄₃	#2981	1	2	yes	—
143	C₂³×C₈	—	43	4	no	—
144	C₂×#127	Dic₄₃	1	1	yes	characteristic
145	C₂×C₄×D₄	C₂×D₄₃	1	2	yes	—
146	C₂×C₄×D₄	C₂×D₄₃	1	4	yes	—
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	—	43	1	no	—
152	C₂×C₄×C₈	—	43	1	no	—
153	#390	Dic₄₃	1	2	yes	—
154	#390	Dic₄₃	1	4	yes	—
155	#384	Dic₄₃	1	1	yes	characteristic
156	C₄³	C₂×D₄₃	1	1	yes	characteristic
157	#336	—	43	4	no	—
158	#336	—	43	4	no	—
159	C₄₃	#22260	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	C₂×Dic₄₃	1	2	yes	—
161	C₂²×D₄	C₂×Dic₄₃	1	4	yes	—
162	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
163	C₂²×C₈	—	43	2	no	—
164	C₂²×C₈	—	43	4	no	—
165	C₂²×C₈	—	43	2	no	—
166	C₂²×C₈	—	86	8	no	—
167	C₂²×C₈	—	172	8	no	—
168	#127	C₄₃⋊C₈	1	8	yes	complemented
169	C₄×D₄	—	2	4	no	—
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	—	4	8	no	—
172	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
175	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
176	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
178	C₂×C₄²	C₂×Dic₄₃	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	—	43	4	no	—
185	C₄×C₈	—	86	2	no	—
186	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	4	4	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	C₂×C₄₃⋊C₈	1	8	yes	complemented
192	C₂×D₄	C₂×C₄₃⋊C₈	1	16	yes	complemented
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
199	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
203	C₂×C₈	—	43	4	no	—
204	C₂×C₈	—	86	4	no	—
205	C₂×C₈	—	86	4	no	—
206	C₂×C₈	—	86	2	no	—
207	C₂×C₈	—	86	4	no	—
208	C₂×C₈	—	86	8	no	—
209	C₂×C₈	—	172	16	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	4	8	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	2	no	—
217	C₄²	C₂×C₄₃⋊C₈	1	4	yes	complemented
218	C₄²	D₄×D₄₃	1	2	yes	—
219	C₄²	#10621900	1	2	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	8	no	—
224	C₂³	C₂²×C₄₃⋊C₈	1	8	yes	—
225	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
226	D₄	—	2	16	no	—
227	D₄	—	4	16	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
240	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
241	C₂×C₄	C₂²×C₄₃⋊C₈	1	4	yes	—
242	C₂×C₄	C₂²×C₄₃⋊C₈	1	2	yes	—
243	C₂×C₄	#13063332	1	2	yes	—
244	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
246	C₂×C₄	#13063370	1	4	yes	—
247	C₈	—	86	4	no	—
248	C₈	—	172	4	no	—
249	C₈	—	172	2	no	—
250	C₂²	—	2	8	no	—
251	C₂²	—	2	16	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	8	no	—
255	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
256	C₂²	#14583374	1	2	yes	—
257	C₂²	C₂³×C₄₃⋊C₈	1	1	yes	characteristic
258	C₂²	C₂×#13063332	1	1	yes	characteristic
259	C₂²	#14583666	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	2	no	—
264	C₄	D₄×C₄₃⋊C₈	1	4	yes	—
265	C₄	#14583462	1	2	yes	—
266	C₄	#14583503	1	2	yes	—
267	C₂	—	4	8	no	—
268	C₂	#19137321	1	1	yes	characteristic
269	C₂	C₂×D₄×C₄₃⋊C₈	1	1	yes	characteristic
270	C₂	#19143313	1	1	yes	characteristic
271	C₂	#19143619	1	2	yes	—
272	C₂	#19143671	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₈
Fitting	C₄×#127×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂×#127×C₄₃	C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#127	D₄₃
hypocenter	C₄₃	#22260
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₄₃⋊C₈	C₂	semidirect
2. #19135390	C₂	semidirect
3. #19134888	C₂	semidirect
4. #19134712	C₂	semidirect
5. #19134341	C₂	semidirect
32. C₄×C₄₃⋊C₈	D₄	semidirect
36. #127×C₄₃	C₈	semidirect
54. C₂×D₄×C₄₃	C₂×C₈	semidirect
55. C₂×D₄×C₄₃	C₂×C₈	semidirect
73. C₄²×C₄₃	C₂×C₈	semidirect
159. C₄₃	#22260	semidirect
168. #127	C₄₃⋊C₈	semidirect
191. C₂×D₄	C₂×C₄₃⋊C₈	semidirect
192. C₂×D₄	C₂×C₄₃⋊C₈	semidirect
217. C₄²	C₂×C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₄₃⋊C₈	2
#19135390	2
#19134888	2
#19134712	2
#19134341	2
C₄×#127×C₄₃	2
#22260	43