Subgroups — #24987076

Order 11008.

642 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#10621872	C₂	1	1	yes	maximal, characteristic
2	C₄×(C₂×D₄₃)⋊C₈	C₂	1	2	yes	maximal
3	C₂×#14582620	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₃⋊#1273	C₂	1	2	yes	maximal, complemented
5	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₄	1	2	yes	—
7	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
8	C₂×(C₂×D₄₃)⋊C₈	C₄	1	4	yes	—
9	C₄×#10621872	—	2	2	no	—
10	C₄×#10621872	—	2	2	no	—
11	C₂×#13063235	C₄	1	2	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
14	#14582653	C₄	1	4	yes	—
15	#14582620	—	2	2	no	—
16	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₈²×C₄₃	—	2	2	no	—
19	C₂²×C₄×D₄₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂×#10621872	—	2	4	no	—
22	C₂×#10621872	—	2	4	no	—
23	C₂×#10621872	C₈	1	8	yes	complemented
24	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
28	C₄²×D₄₃	—	2	1	no	—
29	C₄²×D₄₃	—	2	1	no	—
30	C₄²×D₄₃	—	2	2	no	—
31	(C₂×D₄₃)⋊C₈	—	2	8	no	—
32	#13063235	—	2	4	no	—
33	C₄×C₄₃⋊C₈	—	2	1	no	—
34	C₄×C₄₃⋊C₈	—	2	1	no	—
35	C₄×C₄₃⋊C₈	—	2	2	no	—
36	C₄×C₄₃⋊C₈	—	2	2	no	—
37	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₄₃	—	2	2	no	—
41	C₄×C₈×C₄₃	—	2	2	no	—
42	C₄×C₈×C₄₃	D₄	1	2	yes	—
43	C₄×C₈×C₄₃	D₄	1	2	yes	—
44	C₂³×D₄₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
47	C₂²×Dic₄₃	C₄²	1	2	yes	—
48	C₂²×Dic₄₃	C₄²	1	1	yes	characteristic
49	C₂×C₄×D₄₃	—	2	1	no	—
50	C₂×C₄×D₄₃	—	2	1	no	—
51	C₂×C₄×D₄₃	—	2	2	no	—
52	C₂×C₄×D₄₃	—	2	2	no	—
53	C₂×C₄×D₄₃	—	2	2	no	—
54	C₂×C₄×D₄₃	—	2	4	no	—
55	C₂×C₄×D₄₃	—	2	4	no	—
56	C₂×C₄×D₄₃	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₄₃	—	2	4	no	—
58	C₂×C₈×C₄₃	—	2	4	no	—
59	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
60	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
61	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
62	C₂×C₈×C₄₃	M₁₆	1	8	yes	complemented
63	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₄₃	#41	1	2	yes	—
66	C₄×Dic₄₃	—	2	1	no	—
67	C₄×Dic₄₃	—	2	1	no	—
68	C₄×Dic₄₃	—	2	2	no	—
69	C₄×Dic₄₃	—	2	2	no	—
70	C₂×C₄₃⋊C₈	—	2	2	no	—
71	C₂×C₄₃⋊C₈	—	2	2	no	—
72	C₂×C₄₃⋊C₈	—	2	4	no	—
73	C₂×C₄₃⋊C₈	—	2	4	no	—
74	C₂×C₄₃⋊C₈	—	2	4	no	—
75	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
76	#10621872	—	2	16	no	—
77	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
78	C₂²×D₄₃	—	2	1	no	—
79	C₂²×D₄₃	—	2	2	no	—
80	C₂²×D₄₃	—	2	1	no	—
81	C₂²×D₄₃	—	2	2	no	—
82	C₂²×D₄₃	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
89	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₄₃	#131	1	4	yes	—
91	C₂×Dic₄₃	—	2	1	no	—
92	C₂×Dic₄₃	—	2	1	no	—
93	C₂×Dic₄₃	—	2	2	no	—
94	C₂×Dic₄₃	—	2	2	no	—
95	C₂×Dic₄₃	—	2	2	no	—
96	C₂×Dic₄₃	C₄×C₈	1	2	yes	—
97	C₄×D₄₃	—	2	4	no	—
98	C₄×D₄₃	—	2	4	no	—
99	C₄×D₄₃	—	4	4	no	—
100	C₄×D₄₃	—	4	4	no	—
101	C₈×C₄₃	—	2	8	no	—
102	C₈×C₄₃	C₂²⋊C₈	1	8	yes	complemented
103	C₄₃⋊C₈	—	2	4	no	—
104	C₄₃⋊C₈	—	4	4	no	—
105	#8146	—	43	1	no	maximal, Sylow
106	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
109	C₂²×C₄₃	C₄×M₁₆	1	1	yes	characteristic
110	C₂²×C₄₃	#405	1	2	yes	—
111	C₂×D₄₃	—	2	4	no	—
112	C₂×D₄₃	—	2	2	no	—
113	C₂×D₄₃	—	4	1	no	—
114	C₂×D₄₃	—	4	1	no	—
115	C₂×D₄₃	—	4	2	no	—
116	C₄×C₄₃	C₂×C₂²⋊C₈	1	2	yes	—
117	C₄×C₄₃	#431	1	2	yes	—
118	C₄×C₄₃	#435	1	4	yes	—
119	C₄×C₄₃	#436	1	4	yes	—
120	Dic₄₃	—	2	2	no	—
121	Dic₄₃	—	4	2	no	—
122	C₂×C₄×M₁₆	—	43	1	no	—
123	C₄×C₂²⋊C₈	—	43	2	no	—
124	C₂×#322	—	43	1	no	—
125	C₂×C₈²	D₄₃	1	1	yes	characteristic, complemented
126	#1273	—	43	2	no	—
127	C₂×C₄₃	C₈×M₁₆	1	2	yes	—
128	C₂×C₄₃	#1453	1	1	yes	characteristic
129	C₂×C₄₃	#1455	1	1	yes	characteristic
130	C₂×C₄₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄₃	#1753	1	2	yes	—
132	D₄₃	—	4	2	no	—
133	C₂²×M₁₆	—	43	2	no	—
134	C₂²×C₄²	—	43	1	no	—
135	C₂×C₂²⋊C₈	—	43	4	no	—
136	C₄×M₁₆	—	86	2	no	—
137	C₄×M₁₆	—	86	2	no	—
138	C₂×#97	—	43	2	no	—
139	C₂×C₄×C₈	—	43	1	no	—
140	C₂×C₄×C₈	—	43	2	no	—
141	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
142	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
143	#336	—	43	4	no	—
144	#322	—	86	2	no	—
145	C₈²	—	2	2	no	—
146	C₄₃	#8146	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	43	1	no	—
148	C₂³×C₄	—	43	2	no	—
149	C₂×M₁₆	—	43	8	no	—
150	C₂×M₁₆	—	86	4	no	—
151	C₂×M₁₆	—	86	4	no	—
152	C₂²×C₈	—	43	2	no	—
153	C₂²×C₈	—	43	4	no	—
154	C₂²×C₈	C₄×D₄₃	1	4	yes	—
155	C₂²×C₈	C₄×D₄₃	1	2	yes	—
156	C₂×C₄²	—	43	2	no	—
157	C₂×C₄²	—	43	1	no	—
158	C₂×C₄²	—	86	1	no	—
159	C₂×C₄²	—	86	2	no	—
160	C₂×C₄²	—	86	1	no	—
161	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
162	C₂²⋊C₈	—	86	8	no	—
163	#97	—	86	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	86	2	no	—
167	C₄×C₈	—	86	1	no	—
168	C₄×C₈	—	86	2	no	—
169	C₄×C₈	—	86	1	no	—
170	C₄×C₈	D₁₇₂	1	2	yes	—
171	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
172	C₂⁴	—	43	1	no	—
173	C₂²×C₄	—	43	1	no	—
174	C₂²×C₄	—	43	4	no	—
175	C₂²×C₄	—	43	2	no	—
176	C₂²×C₄	—	86	1	no	—
177	C₂²×C₄	—	86	1	no	—
178	C₂²×C₄	—	86	2	no	—
179	C₂²×C₄	—	86	2	no	—
180	C₂²×C₄	—	86	2	no	—
181	C₂²×C₄	—	86	4	no	—
182	C₂²×C₄	—	86	4	no	—
183	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
185	M₁₆	—	86	16	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	43	4	no	—
189	C₂×C₈	—	86	2	no	—
190	C₂×C₈	—	86	2	no	—
191	C₂×C₈	—	86	4	no	—
192	C₂×C₈	—	86	4	no	—
193	C₂×C₈	—	86	4	no	—
194	C₂×C₈	C₈×D₄₃	1	4	yes	—
195	C₂×C₈	#10621872	1	4	yes	—
196	C₂×C₈	#10621872	1	8	yes	complemented
197	C₂×C₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
198	C₄²	—	86	1	no	—
199	C₄²	—	86	1	no	—
200	C₄²	—	86	2	no	—
201	C₄²	—	86	2	no	—
202	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
203	C₄²	#10621898	1	2	yes	—
204	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
205	C₂³	—	43	2	no	—
206	C₂³	—	86	1	no	—
207	C₂³	—	86	2	no	—
208	C₂³	—	86	2	no	—
209	C₂³	—	86	1	no	—
210	C₂³	C₄²×D₄₃	1	1	yes	characteristic
211	C₂×C₄	—	43	2	no	—
212	C₂×C₄	—	86	1	no	—
213	C₂×C₄	—	86	1	no	—
214	C₂×C₄	—	86	4	no	—
215	C₂×C₄	—	86	4	no	—
216	C₂×C₄	—	86	2	no	—
217	C₂×C₄	—	86	2	no	—
218	C₂×C₄	—	86	2	no	—
219	C₂×C₄	—	172	4	no	—
220	C₂×C₄	—	172	4	no	—
221	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#13063282	1	1	yes	characteristic
223	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
224	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
225	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
226	C₂×C₄	C₂×#10621872	1	2	yes	—
227	C₂×C₄	C₂×#10621872	1	4	yes	—
228	C₂×C₄	#13063331	1	4	yes	—
229	C₈	—	2	8	no	—
230	C₈	—	86	4	no	—
231	C₈	—	172	4	no	—
232	C₈	(C₂×D₄₃)⋊C₈	1	8	yes	complemented
233	C₂²	—	86	2	no	—
234	C₂²	—	86	4	no	—
235	C₂²	—	172	2	no	—
236	C₂²	—	172	1	no	—
237	C₂²	—	172	1	no	—
238	C₂²	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
239	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
240	C₂²	C₄×#10621872	1	1	yes	characteristic
241	C₂²	C₄×#10621872	1	2	yes	—
242	C₂²	#14582853	1	2	yes	—
243	C₄	—	86	2	no	—
244	C₄	—	172	2	no	—
245	C₄	C₂×(C₂×D₄₃)⋊C₈	1	2	yes	—
246	C₄	#14582950	1	2	yes	—
247	C₄	#14582970	1	4	yes	—
248	C₄	#14582978	1	4	yes	—
249	C₂	—	172	2	no	—
250	C₂	C₈×#10621872	1	2	yes	—
251	C₂	#19132889	1	1	yes	characteristic
252	C₂	#19132897	1	1	yes	characteristic
253	C₂	C₄×(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
254	C₂	#19134378	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₂×C₈²×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₄²
prime radical	C₂³×D₄₃	C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#8146
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. C₂×#14582620	C₂	semidirect
4. C₄₃⋊#1273	C₂	semidirect
5. C₂×C₈²×C₄₃	C₂	semidirect
23. C₂×#10621872	C₈	semidirect
62. C₂×C₈×C₄₃	M₁₆	semidirect
102. C₈×C₄₃	C₂²⋊C₈	semidirect
125. C₂×C₈²	D₄₃	semidirect
146. C₄₃	#8146	semidirect
196. C₂×C₈	#10621872	semidirect
232. C₈	(C₂×D₄₃)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#10621872	2
C₄×(C₂×D₄₃)⋊C₈	2
C₂×#14582620	2
C₄₃⋊#1273	2
C₂×C₈²×C₄₃	2
#8146	43