Subgroups — #25094054

Order 11008.

1996 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583666	C₂	1	1	yes	maximal, characteristic
2	#19147798	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582670	C₂	1	12	yes	maximal, complemented
4	C₂×C₄₃×#568	C₂	1	1	yes	maximal, characteristic
5	#14583666	—	2	4	no	—
6	#14583666	C₄	1	8	yes	complemented
7	C₂×#13063332	—	2	4	no	—
8	C₂²×#10621877	C₂²	1	6	yes	—
9	#14583382	—	2	4	no	—
10	#14583377	C₄	1	2	yes	—
11	#14583374	C₄	1	6	yes	—
12	#41×Dic₄₃	—	2	4	no	—
13	C₂×#13063284	C₂²	1	6	yes	—
14	#14582670	C₂²	1	16	yes	—
15	C₂²×#41×C₄₃	C₂²	1	1	yes	characteristic
16	C₄₃×#568	—	2	4	no	—
17	C₄₃×#568	—	2	4	no	—
18	C₂×#131×C₄₃	C₄	1	8	yes	complemented
19	C₂²×M₁₆×C₄₃	C₂²	1	6	yes	—
20	#13063332	—	2	32	no	—
21	C₂×#10621877	—	2	6	no	—
22	C₂×#10621877	—	2	12	no	—
23	C₂×#10621877	C₂×C₄	1	24	yes	complemented
24	C₂×#10621877	C₂³	1	4	yes	—
25	C₂²×C₄₃⋊C₈	—	2	6	no	—
26	Q₈×Dic₄₃	—	2	8	no	—
27	D₄×Dic₄₃	—	2	24	no	—
28	#13063284	—	2	6	no	—
29	#13063284	—	2	6	no	—
30	#13063284	C₂×C₄	1	12	yes	—
31	#13063284	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₄₃	—	2	6	no	—
33	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	12	yes	—
34	C₂×C₄×Dic₄₃	—	2	6	no	—
35	C₂×#41×C₄₃	C₂×C₄	1	4	yes	—
36	C₂×#41×C₄₃	C₂×C₄	1	4	yes	—
37	C₂×#41×C₄₃	D₄	1	4	yes	—
38	C₂×#41×C₄₃	D₄	1	4	yes	—
39	C₂×#41×C₄₃	D₄	1	4	yes	—
40	C₂×#41×C₄₃	Q₈	1	4	yes	—
41	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×D₄×C₄₃	C₂×C₄	1	3	yes	—
43	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
44	#131×C₄₃	—	2	32	no	—
45	C₂×M₁₆×C₄₃	—	2	12	no	—
46	C₂×M₁₆×C₄₃	—	2	12	no	—
47	C₂×M₁₆×C₄₃	C₂×C₄	1	12	yes	complemented
48	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
49	C₂²×C₈×C₄₃	C₂×C₄	1	12	yes	complemented
50	C₂⁴×C₄₃	C₂²×C₄	1	3	yes	—
51	#41×C₄₃	C₂²⋊C₄	1	16	yes	—
52	#41×C₄₃	C₂²⋊C₄	1	16	yes	—
53	#41×C₄₃	C₄⋊C₄	1	16	yes	—
54	#41×C₄₃	C₄⋊C₄	1	16	yes	—
55	C₂×Q₈×C₄₃	C₄²	1	4	yes	—
56	C₂×Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
57	C₂×Q₈×C₄₃	C₄⋊C₄	1	4	yes	—
58	C₂×D₄×C₄₃	C₄²	1	12	yes	—
59	C₂×D₄×C₄₃	C₂²⋊C₄	1	12	yes	—
60	C₂×D₄×C₄₃	C₄⋊C₄	1	12	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	3	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₄₃	C₂×Q₈	1	6	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	12	no	—
70	M₁₆×C₄₃	—	2	24	no	—
71	M₁₆×C₄₃	—	2	4	no	—
72	M₁₆×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	—	2	24	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	8	yes	—
75	C₂²⋊Dic₄₃	—	2	12	no	—
76	C₂²⋊Dic₄₃	C₂²×C₄	1	8	yes	—
77	C₄⋊Dic₄₃	—	2	12	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄×Dic₄₃	—	2	12	no	—
80	C₄×Dic₄₃	—	2	4	no	—
81	#10621877	—	2	24	no	—
82	#10621877	—	2	4	no	—
83	#10621877	C₂²×C₄	1	8	yes	—
84	C₂×C₄₃⋊C₈	—	2	24	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂³×C₄₃	—	8	3	no	—
87	C₂³×C₄₃	C₂×C₄²	1	6	yes	—
88	C₂³×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂³×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
90	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
91	Q₈×C₄₃	#95	1	16	yes	—
92	D₄×C₄₃	#95	1	48	yes	—
93	C₂×C₄×C₄₃	C₂×C₄²	1	6	yes	—
94	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
95	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
96	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	12	yes	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
98	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	6	yes	—
99	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
100	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×Dic₄₃	—	2	8	no	—
107	C₂×Dic₄₃	—	8	12	no	—
108	C₈×C₄₃	—	2	8	no	—
109	C₄₃⋊C₈	—	2	8	no	—
110	#20360	—	43	1	no	maximal, Sylow
111	C₂²×C₄₃	—	4	1	no	—
112	C₂²×C₄₃	—	4	6	no	—
113	C₂²×C₄₃	—	4	6	no	—
114	C₂²×C₄₃	—	4	6	no	—
115	C₂²×C₄₃	C₂×#95	1	12	yes	—
116	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
117	C₂²×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₄₃	C₂×#95	1	12	yes	—
120	C₄×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	Dic₄₃	—	4	8	no	—
125	C₂×#568	—	43	1	no	—
126	C₂×#568	D₄₃	1	1	yes	characteristic
127	#3428	—	43	1	no	—
128	C₂×#344	—	43	12	no	—
129	C₂×C₄₃	—	2	1	no	—
130	C₂×C₄₃	—	2	1	no	—
131	C₂×C₄₃	—	2	1	no	—
132	C₂×C₄₃	—	2	12	no	—
133	C₂×C₄₃	C₂²×#95	1	1	yes	characteristic
134	C₂²×#41	C₂×D₄₃	1	1	yes	characteristic
135	#568	—	2	4	no	—
136	#568	—	2	4	no	—
137	#568	—	43	8	no	—
138	#568	—	86	4	no	—
139	C₂×#131	—	86	4	no	—
140	C₂×#131	Dic₄₃	1	8	yes	complemented
141	C₂²×M₁₆	—	43	6	no	—
142	C₂²×M₁₆	C₂×D₄₃	1	6	yes	—
143	#520	—	86	4	no	—
144	#519	—	43	2	no	—
145	#518	—	43	6	no	—
146	C₄×#41	—	86	4	no	—
147	C₂×#117	—	43	6	no	—
148	#344	—	43	16	no	—
149	C₄₃	#20360	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	Dic₈₆	1	4	yes	—
151	C₂×#41	C₄×D₄₃	1	4	yes	—
152	C₂×#41	D₁₇₂	1	4	yes	—
153	C₂×#41	C₂×Dic₄₃	1	4	yes	—
154	C₂×#41	C₂²⋊D₄₃	1	4	yes	—
155	C₂×#41	C₂²⋊D₄₃	1	4	yes	—
156	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
157	C₂²×D₄	C₄×D₄₃	1	3	yes	—
158	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
159	#131	—	2	32	no	—
160	#131	—	86	32	no	—
161	C₂×M₁₆	—	2	12	no	—
162	C₂×M₁₆	—	2	12	no	—
163	C₂×M₁₆	—	43	24	no	—
164	C₂×M₁₆	—	43	4	no	—
165	C₂×M₁₆	—	86	12	no	—
166	C₂×M₁₆	—	86	6	no	—
167	C₂×M₁₆	C₂×Dic₄₃	1	12	yes	complemented
168	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
169	C₂²×C₈	—	86	6	no	—
170	C₂²×C₈	C₂×Dic₄₃	1	12	yes	complemented
171	C₄×Q₈	—	86	8	no	—
172	C₄×D₄	—	86	24	no	—
173	#117	—	43	12	no	—
174	#117	—	43	4	no	—
175	#117	—	86	6	no	—
176	#117	—	86	6	no	—
177	C₂×C₄⋊C₄	—	86	6	no	—
178	C₂×C₂²⋊C₄	—	43	12	no	—
179	C₂×C₄²	—	86	6	no	—
180	C₂⁴	C₂×C₄×D₄₃	1	3	yes	—
181	#41	Dic₄₃⋊C₄	1	16	yes	—
182	#41	C₄⋊Dic₄₃	1	16	yes	—
183	#41	(C₂×D₄₃)⋊C₄	1	16	yes	—
184	#41	C₂²⋊Dic₄₃	1	16	yes	—
185	C₂×Q₈	C₄×Dic₄₃	1	4	yes	—
186	C₂×Q₈	Dic₄₃⋊C₄	1	4	yes	—
187	C₂×Q₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
188	C₂×D₄	C₄×Dic₄₃	1	12	yes	—
189	C₂×D₄	Dic₄₃⋊C₄	1	12	yes	—
190	C₂×D₄	(C₂×D₄₃)⋊C₄	1	12	yes	—
191	C₂²×C₄	—	86	12	no	—
192	C₂²×C₄	C₂×Dic₈₆	1	6	yes	—
193	C₂²×C₄	C₂×C₄×D₄₃	1	3	yes	—
194	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
195	C₂²×C₄	C₂×D₁₇₂	1	6	yes	—
196	C₂²×C₄	C₂²×Dic₄₃	1	6	yes	—
197	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
199	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
200	M₁₆	—	2	24	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	43	8	no	—
204	M₁₆	—	86	24	no	—
205	M₁₆	—	86	4	no	—
206	C₂×C₈	—	2	24	no	—
207	C₂×C₈	—	86	24	no	—
208	C₂×C₈	—	86	4	no	—
209	C₂×C₈	C₂²×Dic₄₃	1	8	yes	—
210	C₄⋊C₄	—	86	12	no	—
211	C₄⋊C₄	—	86	4	no	—
212	C₂²⋊C₄	—	43	8	no	—
213	C₂²⋊C₄	—	86	12	no	—
214	C₄²	—	86	12	no	—
215	C₄²	—	86	4	no	—
216	C₂³	—	8	3	no	—
217	C₂³	C₂×C₄×Dic₄₃	1	6	yes	—
218	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
219	C₂³	C₂×Dic₄₃⋊C₄	1	6	yes	—
220	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
221	Q₈	C₄₃⋊#95	1	16	yes	—
222	D₄	C₄₃⋊#95	1	48	yes	—
223	C₂×C₄	—	86	8	no	—
224	C₂×C₄	—	344	12	no	—
225	C₂×C₄	C₂×C₄×Dic₄₃	1	6	yes	—
226	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	12	yes	—
227	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
228	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	12	yes	—
229	C₂×C₄	C₂×Dic₄₃⋊C₄	1	12	yes	—
230	C₂×C₄	C₂×Dic₄₃⋊C₄	1	6	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₄₃	1	12	yes	—
232	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
233	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
234	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
238	C₈	—	2	8	no	—
239	C₈	—	86	8	no	—
240	C₂²	—	4	6	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	6	no	—
243	C₂²	—	4	6	no	—
244	C₂²	C₂×C₄₃⋊#95	1	12	yes	—
245	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
246	C₂²	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
247	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
248	C₄	—	172	8	no	—
249	C₄	C₂×C₄₃⋊#95	1	12	yes	—
250	C₄	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
251	C₄	C₂²×C₂²⋊Dic₄₃	1	1	yes	characteristic
252	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
253	C₄	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	2	12	no	—
258	C₂	C₂²×C₄₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊Dic₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#568	C₂
socle	C₂×C₄₃	C₂²×#95
prime radical	C₂²×#41×C₄₃	C₂²
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#568	D₄₃
hypocenter	C₄₃	#20360
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. C₂×#14582670	C₂	semidirect
6. #14583666	C₄	semidirect
18. C₂×#131×C₄₃	C₄	semidirect
23. C₂×#10621877	C₂×C₄	semidirect
47. C₂×M₁₆×C₄₃	C₂×C₄	semidirect
49. C₂²×C₈×C₄₃	C₂×C₄	semidirect
140. C₂×#131	Dic₄₃	semidirect
149. C₄₃	#20360	semidirect
167. C₂×M₁₆	C₂×Dic₄₃	semidirect
170. C₂²×C₈	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583666	2
#19147798	2
C₂×#14582670	2
C₂×C₄₃×#568	2
#20360	43