Subgroups — #25107236

Order 11008.

658 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143233	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582960	C₂	1	1	yes	maximal, characteristic
3	#19135368	C₂	1	4	yes	maximal, complemented
4	#19134800	C₂	1	4	yes	maximal
5	#19134769	C₂	1	1	yes	maximal, characteristic, complemented
6	#19134766	C₂	1	1	yes	maximal, characteristic, complemented
7	#19134735	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134732	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2905	C₂	1	1	yes	maximal, characteristic
10	#14582960	—	2	4	no	—
11	#14582960	C₄	1	8	yes	—
12	C₂×#13063260	C₂²	1	4	yes	—
13	C₂×#13063260	C₂²	1	4	yes	—
14	#14582923	C₂²	1	1	yes	characteristic
15	#14582923	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
17	#14582863	C₂²	1	2	yes	—
18	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
19	C₄×#10621877	C₂²	1	2	yes	—
20	C₂×#13063236	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
22	#14582663	C₂²	1	4	yes	—
23	#14582663	C₂²	1	4	yes	—
24	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
25	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#424	—	2	4	no	—
27	C₄₃×#423	C₂²	1	1	yes	characteristic
28	C₄₃×#423	C₂²	1	1	yes	characteristic
29	C₄₃×#423	C₂²	1	2	yes	—
30	C₂×#10621877	C₂³	1	2	yes	—
31	C₂×#10621877	C₂³	1	2	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	#13063260	—	4	4	no	—
35	#13063260	—	4	4	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
37	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
38	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
39	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
40	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
41	#13063236	—	2	2	no	—
42	#13063236	C₂×C₄	1	4	yes	—
43	C₄×C₄₃⋊C₈	—	2	2	no	—
44	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
45	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₄₃	—	2	2	no	—
47	C₂×M₁₆×C₄₃	—	4	4	no	—
48	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
49	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₃	—	2	2	no	—
51	#117×C₄₃	C₂×C₄	1	4	yes	—
52	#117×C₄₃	D₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₄₃	Q₈	1	2	yes	—
54	C₂×C₂²⋊C₄×C₄₃	Q₈	1	2	yes	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₄₃	—	2	4	no	—
58	C₄⋊C₈×C₄₃	—	2	2	no	—
59	C₄⋊C₈×C₄₃	—	2	2	no	—
60	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	4	4	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	4	4	no	—
75	C₄⋊C₄×C₄₃	C₄⋊C₄	1	8	yes	—
76	C₂²⋊C₄×C₄₃	C₄⋊C₄	1	8	yes	—
77	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₄₃	C₂×D₄	1	2	yes	—
80	C₄²×C₄₃	C₂×D₄	1	2	yes	—
81	#10621877	—	2	4	no	—
82	#10621877	—	2	4	no	—
83	C₂×C₄₃⋊C₈	—	2	2	no	—
84	C₂×C₄₃⋊C₈	—	2	2	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	2	2	no	—
87	C₂×C₄₃⋊C₈	—	2	2	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	—	4	2	no	—
93	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	2	4	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	—	4	2	no	—
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
107	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
108	C₈×C₄₃	—	4	2	no	—
109	C₈×C₄₃	—	4	2	no	—
110	C₄₃⋊C₈	—	2	4	no	—
111	C₄₃⋊C₈	—	2	4	no	—
112	C₄₃⋊C₈	—	4	2	no	—
113	C₄₃⋊C₈	—	4	2	no	—
114	#21995	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	2	4	no	—
118	C₂²×C₄₃	—	8	2	no	—
119	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₃	#544	1	2	yes	—
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	—	2	4	no	—
123	C₄×C₄₃	—	2	4	no	—
124	C₄×C₄₃	#520	1	2	yes	—
125	C₄×C₄₃	#535	1	1	yes	characteristic
126	C₄×C₄₃	#536	1	1	yes	characteristic
127	#2921	—	43	1	no	—
128	C₂×#433	—	43	1	no	—
129	#2905	D₄₃	1	1	yes	characteristic
130	#1950	—	43	4	no	—
131	#1855	—	43	4	no	—
132	#1849	—	43	1	no	—
133	#1848	—	43	1	no	—
134	#1838	—	43	1	no	—
135	#1837	—	43	1	no	—
136	C₂×C₄₃	—	2	2	no	—
137	C₂×C₄₃	—	4	2	no	—
138	C₂×C₄₃	#2317	1	1	yes	characteristic
139	C₂×C₄₃	#2916	1	2	yes	—
140	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
141	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
142	#433	—	43	8	no	—
143	#433	—	86	4	no	—
144	C₂×#108	—	43	4	no	—
145	C₂×#108	—	43	4	no	—
146	#424	—	2	4	no	—
147	#423	—	43	1	no	—
148	#423	—	43	1	no	—
149	#423	C₂×D₄₃	1	1	yes	characteristic
150	#423	C₂×D₄₃	1	1	yes	characteristic
151	#423	C₂×D₄₃	1	2	yes	—
152	C₂×C₄⋊C₈	—	43	2	no	—
153	#407	—	43	2	no	—
154	C₂×C₂²⋊C₈	—	43	2	no	—
155	C₄×M₁₆	—	43	2	no	—
156	C₂×#97	—	43	1	no	—
157	C₂×C₄×C₈	—	43	1	no	—
158	#341	—	43	4	no	—
159	#341	—	43	4	no	—
160	C₄₃	#21995	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	4	4	no	—
164	C₂×M₁₆	—	43	2	no	—
165	C₂×M₁₆	—	43	2	no	—
166	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
167	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	43	2	no	—
170	C₂²×C₈	—	43	2	no	—
171	#117	C₄×D₄₃	1	4	yes	—
172	#117	C₂²⋊D₄₃	1	4	yes	—
173	C₂×C₄⋊C₄	Dic₈₆	1	2	yes	—
174	C₂×C₂²⋊C₄	Dic₈₆	1	2	yes	—
175	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
177	#108	—	172	4	no	—
178	#108	—	172	4	no	—
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	43	8	no	—
183	C₄⋊C₈	—	86	2	no	—
184	C₄⋊C₈	—	86	2	no	—
185	C₂²⋊C₈	—	43	8	no	—
186	C₂²⋊C₈	—	86	4	no	—
187	#97	—	43	4	no	—
188	#97	—	86	2	no	—
189	C₄×C₈	—	43	4	no	—
190	C₄×C₈	—	86	2	no	—
191	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
195	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
196	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
198	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
199	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	M₁₆	—	86	4	no	—
204	M₁₆	—	86	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	4	no	—
208	C₂×C₈	—	43	4	no	—
209	C₂×C₈	—	43	4	no	—
210	C₂×C₈	—	86	2	no	—
211	C₂×C₈	—	86	2	no	—
212	C₂×C₈	—	86	2	no	—
213	C₂×C₈	—	86	2	no	—
214	C₂×C₈	—	86	2	no	—
215	C₂×C₈	—	86	2	no	—
216	C₄⋊C₄	Dic₄₃⋊C₄	1	8	yes	—
217	C₂²⋊C₄	Dic₄₃⋊C₄	1	8	yes	—
218	C₄²	C₂×C₄×D₄₃	1	2	yes	—
219	C₄²	C₂×C₄×D₄₃	1	2	yes	—
220	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
221	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₂×Dic₄₃⋊C₄	1	2	yes	—
225	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
233	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
234	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
237	C₂×C₄	#13063370	1	2	yes	—
238	C₂×C₄	#13063373	1	2	yes	—
239	C₈	—	4	2	no	—
240	C₈	—	4	2	no	—
241	C₈	—	86	4	no	—
242	C₈	—	86	4	no	—
243	C₈	—	172	2	no	—
244	C₈	—	172	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
250	C₂²	#14583530	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	2	no	—
254	C₄	#14583380	1	2	yes	—
255	C₄	#14583466	1	1	yes	characteristic
256	C₄	#14583473	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	#19137219	1	1	yes	characteristic
260	C₂	#19143204	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2905	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2905	D₄₃
hypocenter	C₄₃	#21995
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ #2905

Complements

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

N	H ≅ G/N	type
3. #19135368	C₂	semidirect
5. #19134769	C₂	semidirect
6. #19134766	C₂	semidirect
7. #19134735	C₂	semidirect
8. #19134732	C₂	semidirect
160. C₄₃	#21995	semidirect

Maximal subgroups

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

subgroup	index
#19143233	2
C₂×#14582960	2
#19135368	2
#19134800	2
#19134769	2
#19134766	2
#19134735	2
#19134732	2
C₄₃×#2905	2
#21995	43