Subgroups — #24983148

Order 11008.

622 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×#14582654	C₂	1	2	yes	maximal
2	C₄₃×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄₃⋊C₈	—	2	2	no	—
4	#14582654	—	2	2	no	—
5	#14582654	C₄	1	4	yes	complemented
6	C₂²×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
8	C₂×#95×C₄₃	—	2	2	no	—
9	C₂²×C₄₃⋊C₈	—	2	8	no	—
10	C₂²×C₄₃⋊C₈	—	2	4	no	—
11	C₂²×C₄₃⋊C₈	—	4	4	no	—
12	C₂²×C₄₃⋊C₈	—	4	2	no	—
13	C₂²×C₄₃⋊C₈	—	4	2	no	—
14	C₂³×C₄×C₄₃	—	2	2	no	—
15	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₄₃	D₄	1	2	yes	—
17	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
18	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
19	C₂×C₄⋊C₄×C₄₃	—	4	2	no	—
20	C₂×C₄⋊C₄×C₄₃	C₈	1	4	yes	complemented
21	C₂×C₄²×C₄₃	—	4	2	no	—
22	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	Q₈	1	1	yes	characteristic
25	#95×C₄₃	—	4	2	no	—
26	#95×C₄₃	—	4	2	no	—
27	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	—	2	4	no	—
29	C₂²×C₄×C₄₃	—	2	2	no	—
30	C₂²×C₄×C₄₃	—	2	4	no	—
31	C₂²×C₄×C₄₃	—	2	1	no	—
32	C₂²×C₄×C₄₃	—	2	1	no	—
33	C₂²×C₄×C₄₃	—	2	4	no	—
34	C₂²×C₄×C₄₃	—	4	2	no	—
35	C₂²×C₄×C₄₃	—	4	2	no	—
36	C₂²×C₄×C₄₃	—	4	4	no	—
37	C₂²×C₄×C₄₃	—	4	4	no	—
38	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
44	C₂²×C₄×C₄₃	D₈	1	2	yes	—
45	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
48	C₄⋊C₄×C₄₃	—	4	4	no	—
49	C₄⋊C₄×C₄₃	—	4	4	no	—
50	C₄²×C₄₃	—	4	2	no	—
51	C₄²×C₄₃	—	4	1	no	—
52	C₄²×C₄₃	—	4	1	no	—
53	C₂×C₄₃⋊C₈	—	2	8	no	—
54	C₂×C₄₃⋊C₈	—	4	8	no	—
55	C₂×C₄₃⋊C₈	—	4	8	no	—
56	C₂×C₄₃⋊C₈	—	4	4	no	—
57	C₂×C₄₃⋊C₈	—	4	4	no	—
58	C₂³×C₄₃	#95	1	1	yes	characteristic
59	C₂³×C₄₃	C₂²⋊C₈	1	2	yes	—
60	C₂³×C₄₃	#99	1	2	yes	—
61	C₂³×C₄₃	#100	1	1	yes	characteristic
62	C₂³×C₄₃	#101	1	1	yes	characteristic
63	C₂³×C₄₃	#102	1	2	yes	—
64	C₂³×C₄₃	#103	1	2	yes	—
65	C₂³×C₄₃	#104	1	4	yes	—
66	C₂×C₄×C₄₃	—	2	2	no	—
67	C₂×C₄×C₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	2	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	4	no	—
73	C₂×C₄×C₄₃	—	2	4	no	—
74	C₂×C₄×C₄₃	—	2	2	no	—
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	2	no	—
77	C₂×C₄×C₄₃	—	2	1	no	—
78	C₂×C₄×C₄₃	—	2	2	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	—	4	4	no	—
83	C₂×C₄×C₄₃	—	4	8	no	—
84	C₂×C₄×C₄₃	—	8	4	no	—
85	C₂×C₄×C₄₃	—	8	2	no	—
86	C₂×C₄×C₄₃	—	8	2	no	—
87	C₂×C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
88	C₂×C₄×C₄₃	#97	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#102	1	2	yes	—
90	C₂×C₄×C₄₃	#103	1	2	yes	—
91	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₄₃	#106	1	2	yes	—
93	C₂×C₄×C₄₃	#107	1	2	yes	—
94	C₄₃⋊C₈	—	4	8	no	—
95	#7025	—	43	1	no	maximal, Sylow
96	C₂²×C₄₃	#323	1	2	yes	—
97	C₂²×C₄₃	#324	1	2	yes	—
98	C₂²×C₄₃	D₄⋊C₈	1	4	yes	—
99	C₂²×C₄₃	Q₈⋊C₈	1	4	yes	—
100	C₂²×C₄₃	#327	1	4	yes	—
101	C₂²×C₄₃	#328	1	4	yes	—
102	C₂²×C₄₃	#329	1	2	yes	—
103	C₂²×C₄₃	#330	1	2	yes	—
104	C₂²×C₄₃	#331	1	1	yes	characteristic
105	C₂²×C₄₃	#332	1	2	yes	—
106	C₂²×C₄₃	#333	1	1	yes	characteristic
107	C₂²×C₄₃	#336	1	1	yes	characteristic
108	C₂²×C₄₃	#339	1	2	yes	—
109	C₂²×C₄₃	#340	1	2	yes	—
110	C₂²×C₄₃	#342	1	1	yes	characteristic
111	C₂²×C₄₃	#343	1	1	yes	characteristic
112	C₄×C₄₃	—	2	1	no	—
113	C₄×C₄₃	—	2	1	no	—
114	C₄×C₄₃	—	2	2	no	—
115	C₄×C₄₃	—	2	2	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	—	8	4	no	—
120	C₂×#336	—	43	2	no	—
121	C₄⋊#95	D₄₃	1	1	yes	characteristic
122	C₂×C₄₃	#1270	1	2	yes	—
123	C₂×C₄₃	#1271	1	4	yes	—
124	C₂×C₄₃	#1272	1	2	yes	—
125	C₂×C₄₃	#1279	1	2	yes	—
126	C₂×C₄₃	#1280	1	1	yes	characteristic
127	C₂×C₄₃	#1294	1	2	yes	—
128	C₂×C₄₃	#1295	1	1	yes	characteristic
129	C₂×C₄₃	#1296	1	1	yes	characteristic
130	C₂³×C₈	—	86	2	no	—
131	C₂²×C₄⋊C₄	Dic₄₃	1	2	yes	—
132	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
133	C₂×#95	—	2	2	no	—
134	#336	—	43	4	no	—
135	#336	—	86	2	no	—
136	C₄₃	#7025	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	2	2	no	—
138	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
139	C₂³×C₄	C₂²⋊D₄₃	1	2	yes	—
140	C₂²×C₈	—	86	8	no	—
141	C₂²×C₈	—	86	4	no	—
142	C₂²×C₈	—	172	4	no	—
143	C₂²×C₈	—	172	2	no	—
144	C₂²×C₈	—	172	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄⋊C₄	—	4	2	no	—
148	C₂×C₄⋊C₄	C₄₃⋊C₈	1	4	yes	complemented
149	C₂×C₄²	—	4	2	no	—
150	C₂×C₄²	Dic₈₆	1	1	yes	characteristic
151	C₂×C₄²	C₄×D₄₃	1	2	yes	—
152	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
153	#95	—	4	2	no	—
154	#95	—	4	2	no	—
155	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
167	C₂²×C₄	#10621877	1	2	yes	—
168	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
169	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
170	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
171	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
172	C₂²×C₄	#10621882	1	2	yes	—
173	C₂²×C₄	#10621883	1	2	yes	—
174	C₂²×C₄	#10621884	1	2	yes	—
175	C₂²×C₄	#10621885	1	2	yes	—
176	C₂×C₈	—	86	8	no	—
177	C₂×C₈	—	172	8	no	—
178	C₂×C₈	—	172	8	no	—
179	C₂×C₈	—	172	4	no	—
180	C₂×C₈	—	172	4	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	—	4	1	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	4	1	no	—
186	C₂³	C₄₃⋊#95	1	1	yes	characteristic
187	C₂³	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
188	C₂³	#13063240	1	2	yes	—
189	C₂³	#13063242	1	1	yes	characteristic
190	C₂³	#13063244	1	1	yes	characteristic
191	C₂³	#13063246	1	2	yes	—
192	C₂³	#13063249	1	2	yes	—
193	C₂³	#13063252	1	4	yes	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
216	C₂×C₄	#13063236	1	1	yes	characteristic
217	C₂×C₄	#13063245	1	2	yes	—
218	C₂×C₄	#13063248	1	2	yes	—
219	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
220	C₂×C₄	#13063256	1	2	yes	—
221	C₂×C₄	#13063258	1	2	yes	—
222	C₈	—	172	8	no	—
223	C₂²	#14582623	1	2	yes	—
224	C₂²	#14582625	1	2	yes	—
225	C₂²	D₄⋊(C₄₃⋊C₈)	1	4	yes	—
226	C₂²	Q₈⋊(C₄₃⋊C₈)	1	4	yes	—
227	C₂²	#14582633	1	4	yes	—
228	C₂²	#14582636	1	4	yes	—
229	C₂²	#14582639	1	2	yes	—
230	C₂²	#14582641	1	2	yes	—
231	C₂²	#14582643	1	1	yes	characteristic
232	C₂²	#14582645	1	2	yes	—
233	C₂²	#14582648	1	1	yes	characteristic
234	C₂²	#14582654	1	1	yes	characteristic
235	C₂²	#14582658	1	2	yes	—
236	C₂²	#14582661	1	2	yes	—
237	C₂²	#14582666	1	1	yes	characteristic
238	C₂²	#14582668	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	8	4	no	—
247	C₂	#19132416	1	2	yes	—
248	C₂	#19132418	1	4	yes	—
249	C₂	#19132421	1	2	yes	—
250	C₂	#19132436	1	2	yes	—
251	C₂	#19132438	1	1	yes	characteristic
252	C₂	#19132473	1	2	yes	—
253	C₂	#19132476	1	1	yes	characteristic
254	C₂	#19132478	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₄₃
Fitting	C₄₃×C₄⋊#95	C₂
socle, prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₄₃	C₂²
hypercenter	C₄⋊#95	D₄₃
hypocenter	C₄₃	#7025
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₄⋊#95

Complements

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

N	H ≅ G/N	type
5. #14582654	C₄	semidirect
20. C₂×C₄⋊C₄×C₄₃	C₈	semidirect
136. C₄₃	#7025	semidirect
148. C₂×C₄⋊C₄	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×#14582654	2
C₄₃×C₄⋊#95	2
#7025	43