Subgroups — #25062317

Order 11008.

814 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	#19143561	C₂	1	2	yes	maximal
2	C₄×#13063331	C₂	1	1	yes	maximal, characteristic
3	#19132897	C₂	1	2	yes	maximal, complemented
4	#19132889	C₂	1	2	yes	maximal, complemented
5	#19132884	C₂	1	2	yes	maximal, complemented
6	#19132878	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#10621872	C₂	1	4	yes	maximal, complemented
8	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063331	C₄	1	2	yes	—
10	C₄×#10621898	C₂²	1	1	yes	characteristic
11	#14583029	C₄	1	4	yes	—
12	#14582978	C₄	1	4	yes	—
13	#14582970	C₄	1	8	yes	—
14	#14582952	C₂²	1	2	yes	—
15	#14582950	C₂²	1	4	yes	—
16	#14582854	C₄	1	2	yes	—
17	#14582853	C₄	1	4	yes	—
18	C₄×#10621877	C₂²	1	1	yes	characteristic
19	C₄×#10621872	C₂²	1	2	yes	—
20	C₄×#10621872	C₂²	1	4	yes	—
21	C₄×C₈×D₄₃	C₂²	1	2	yes	—
22	#14582621	C₂²	1	4	yes	—
23	#14582620	C₂²	1	4	yes	—
24	#14582620	C₂²	1	2	yes	—
25	C₈×C₄₃⋊C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
28	C₈²×C₄₃	C₂²	1	2	yes	—
29	C₈²×C₄₃	C₂²	1	1	yes	characteristic
30	C₈²×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×#10621898	C₂×C₄	1	1	yes	characteristic
32	#13063331	—	2	8	no	—
33	C₂×#10621877	C₂×C₄	1	2	yes	—
34	C₂×#10621872	C₂×C₄	1	4	yes	—
35	C₂×#10621872	C₂×C₄	1	8	yes	—
36	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
37	C₄×Dic₈₆	C₂×C₄	1	2	yes	—
38	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
39	C₄×C₂²⋊D₄₃	C₂×C₄	1	4	yes	—
40	#13063284	C₂×C₄	1	1	yes	characteristic
41	#13063282	C₂×C₄	1	2	yes	—
42	C₄²×D₄₃	C₂³	1	2	yes	—
43	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
44	Dic₄₃⋊C₈	C₂×C₄	1	8	yes	—
45	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
46	(C₂×D₄₃)⋊C₈	C₂×C₄	1	8	yes	—
47	#13063236	C₂×C₄	1	2	yes	—
48	#13063235	C₂×C₄	1	4	yes	—
49	#13063235	C₂×C₄	1	8	yes	—
50	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
51	C₄×C₄₃⋊C₈	C₂³	1	4	yes	—
52	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₄₃	C₂³	1	2	yes	—
60	C₄×C₈×C₄₃	C₂³	1	2	yes	—
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	C₄²	1	2	yes	—
65	#10621898	—	2	4	no	—
66	C₂×D₁₇₂	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₄₃	C₂²×C₄	1	4	yes	—
68	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
69	C₂×Dic₈₆	C₄²	1	1	yes	characteristic
70	C₂×C₈×C₄₃	—	2	8	no	—
71	C₂×C₈×C₄₃	M₁₆	1	8	yes	complemented
72	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊Dic₄₃	C₄²	1	2	yes	—
80	(C₂×D₄₃)⋊C₄	C₄²	1	4	yes	—
81	C₄⋊Dic₄₃	C₄²	1	2	yes	—
82	Dic₄₃⋊C₄	C₄²	1	4	yes	—
83	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
84	C₄×Dic₄₃	C₂²×C₄	1	4	yes	—
85	#10621877	—	2	4	no	—
86	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
88	#10621872	—	2	8	no	—
89	#10621872	—	2	16	no	—
90	C₈×D₄₃	—	2	8	no	—
91	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
92	C₂²×D₄₃	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₄₃	—	2	4	no	—
94	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
97	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
99	C₂²⋊D₄₃	—	2	4	no	—
100	C₂×Dic₄₃	C₂×C₄²	1	2	yes	—
101	C₂×Dic₄₃	C₂×C₄²	1	4	yes	—
102	D₁₇₂	—	2	2	no	—
103	C₄×D₄₃	—	2	4	no	—
104	C₄×D₄₃	—	2	8	no	—
105	Dic₈₆	—	2	2	no	—
106	C₈×C₄₃	—	2	4	no	—
107	C₈×C₄₃	C₂×M₁₆	1	4	yes	—
108	C₈×C₄₃	C₂×M₁₆	1	4	yes	—
109	C₈×C₄₃	#131	1	8	yes	complemented
110	C₄₃⋊C₈	—	2	4	no	—
111	C₄₃⋊C₈	—	2	8	no	—
112	#17335	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
115	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
116	C₂×D₄₃	—	2	2	no	—
117	C₂×D₄₃	—	2	4	no	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	C₄×M₁₆	1	2	yes	—
120	C₄×C₄₃	C₂²×M₁₆	1	2	yes	—
121	C₄×C₄₃	C₂×#131	1	4	yes	—
122	Dic₄₃	—	2	2	no	—
123	Dic₄₃	—	2	4	no	—
124	#2965	—	43	2	no	—
125	C₄×#131	—	43	1	no	—
126	#1455	—	43	2	no	—
127	#1453	—	43	2	no	—
128	#1451	—	43	2	no	—
129	#1450	—	43	1	no	—
130	C₈×M₁₆	—	43	4	no	—
131	C₂×C₈²	D₄₃	1	1	yes	characteristic, complemented
132	C₂×C₄₃	—	2	2	no	—
133	C₂×C₄₃	C₂×C₄×M₁₆	1	1	yes	characteristic
134	C₂×C₄₃	C₄×#131	1	2	yes	—
135	D₄₃	—	4	2	no	—
136	C₂×#131	—	43	2	no	—
137	C₄×#41	—	43	1	no	—
138	#446	—	43	4	no	—
139	#436	—	43	4	no	—
140	#435	—	43	8	no	—
141	#431	—	43	2	no	—
142	#431	—	43	4	no	—
143	#405	—	43	2	no	—
144	#405	—	43	4	no	—
145	C₄×M₁₆	—	43	1	no	—
146	C₄×M₁₆	—	43	2	no	—
147	C₄×M₁₆	—	43	4	no	—
148	C₂×C₄×C₈	—	43	2	no	—
149	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
151	#322	—	43	4	no	—
152	#322	—	43	2	no	—
153	#322	—	43	4	no	—
154	C₈²	—	43	2	no	—
155	C₈²	C₂×D₄₃	1	1	yes	characteristic
156	C₈²	C₂×D₄₃	1	1	yes	characteristic
157	C₈²	C₂×D₄₃	1	2	yes	—
158	C₄₃	#17335	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	43	1	no	—
160	#131	—	86	8	no	—
161	C₂×M₁₆	—	43	2	no	—
162	C₂×M₁₆	—	43	4	no	—
163	C₂×M₁₆	—	43	8	no	—
164	C₂²×C₈	—	43	4	no	—
165	C₂²×C₈	C₄×D₄₃	1	4	yes	—
166	C₂²×C₈	C₄×D₄₃	1	2	yes	—
167	C₄×Q₈	—	43	2	no	—
168	C₄×D₄	—	43	4	no	—
169	C₄×D₄	—	43	2	no	—
170	#117	—	43	1	no	—
171	#117	—	43	2	no	—
172	C₂×C₄²	—	43	2	no	—
173	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
174	C₄⋊C₈	—	43	8	no	—
175	C₄⋊C₈	—	43	4	no	—
176	C₂²⋊C₈	—	43	4	no	—
177	C₂²⋊C₈	—	43	8	no	—
178	#97	—	43	4	no	—
179	#97	—	43	8	no	—
180	#97	—	43	2	no	—
181	C₄×C₈	—	43	4	no	—
182	C₄×C₈	—	43	4	no	—
183	C₄×C₈	—	43	2	no	—
184	C₄×C₈	C₄×D₄₃	1	2	yes	—
185	C₄×C₈	C₄×D₄₃	1	4	yes	—
186	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₄₃	1	2	yes	—
188	C₄×C₈	C₂²×D₄₃	1	2	yes	—
189	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
190	#41	—	86	4	no	—
191	C₂×Q₈	—	43	1	no	—
192	C₂×D₄	—	43	2	no	—
193	C₂×D₄	—	43	1	no	—
194	C₂²×C₄	—	43	4	no	—
195	C₂²×C₄	—	43	2	no	—
196	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
198	M₁₆	—	86	4	no	—
199	M₁₆	—	86	8	no	—
200	M₁₆	—	86	16	no	—
201	C₂×C₈	—	2	8	no	—
202	C₂×C₈	—	43	4	no	—
203	C₂×C₈	—	43	8	no	—
204	C₂×C₈	—	86	8	no	—
205	C₂×C₈	#10621872	1	8	yes	complemented
206	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
207	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
210	C₄⋊C₄	—	43	4	no	—
211	C₄⋊C₄	—	43	2	no	—
212	C₂²⋊C₄	—	43	2	no	—
213	C₂²⋊C₄	—	43	4	no	—
214	C₄²	—	43	2	no	—
215	C₄²	—	43	4	no	—
216	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
217	C₄²	C₂×C₄×D₄₃	1	2	yes	—
218	C₄²	C₂³×D₄₃	1	1	yes	characteristic
219	C₂³	—	43	2	no	—
220	C₂³	C₄²×D₄₃	1	1	yes	characteristic
221	Q₈	—	86	2	no	—
222	D₄	—	86	4	no	—
223	D₄	—	86	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	43	4	no	—
226	C₂×C₄	—	43	2	no	—
227	C₂×C₄	—	86	4	no	—
228	C₂×C₄	—	86	8	no	—
229	C₂×C₄	C₄²×D₄₃	1	2	yes	—
230	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
231	C₂×C₄	C₂×#10621872	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	86	4	no	—
236	C₈	—	86	8	no	—
237	C₈	C₂×#10621872	1	4	yes	—
238	C₈	C₂×#10621872	1	4	yes	—
239	C₈	#13063331	1	8	yes	complemented
240	C₂²	—	2	2	no	—
241	C₂²	—	86	2	no	—
242	C₂²	—	86	4	no	—
243	C₂²	C₄×#10621872	1	2	yes	—
244	C₂²	C₂×C₄²×D₄₃	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	86	2	no	—
247	C₄	—	86	4	no	—
248	C₄	C₄×#10621872	1	2	yes	—
249	C₄	C₂²×#10621872	1	2	yes	—
250	C₄	C₂×#13063331	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	172	2	no	—
253	C₂	C₂×C₄×#10621872	1	1	yes	characteristic
254	C₂	C₄×#13063331	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₄₃
commutator, socle	C₂²×C₄₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
prime radical	C₂×#10621898	C₂×C₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#17335
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #19132897	C₂	semidirect
4. #19132889	C₂	semidirect
5. #19132884	C₂	semidirect
6. #19132878	C₂	semidirect
7. C₈×#10621872	C₂	semidirect
8. C₂×C₈²×C₄₃	C₂	semidirect
71. C₂×C₈×C₄₃	M₁₆	semidirect
109. C₈×C₄₃	#131	semidirect
131. C₂×C₈²	D₄₃	semidirect
158. C₄₃	#17335	semidirect
205. C₂×C₈	#10621872	semidirect
239. C₈	#13063331	semidirect

Maximal subgroups

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

subgroup	index
#19143561	2
C₄×#13063331	2
#19132897	2
#19132889	2
#19132884	2
#19132878	2
C₈×#10621872	2
C₂×C₈²×C₄₃	2
#17335	43