Subgroups — #25097228

Order 11008.

718 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	#19143231	C₂	1	2	yes	maximal
2	C₂×#14582856	C₂	1	1	yes	maximal, characteristic
3	#19134768	C₂	1	2	yes	maximal, complemented
4	#19134731	C₂	1	2	yes	maximal, complemented
5	#19134636	C₂	1	2	yes	maximal
6	#19134630	C₂	1	2	yes	maximal
7	#19134501	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13063260	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#10621872	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
12	C₂×#13063282	C₂²	1	1	yes	characteristic
13	#14582961	—	2	4	no	—
14	#14582958	—	2	4	no	—
15	C₂×#13063261	C₂²	1	2	yes	—
16	C₂×#13063260	C₂²	1	4	yes	—
17	#14582922	C₂²	1	4	yes	—
18	#14582862	C₂²	1	4	yes	—
19	#14582856	C₄	1	8	yes	—
20	C₄×#10621877	C₂²	1	2	yes	—
21	C₂×#13063235	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
24	#14582664	C₂²	1	2	yes	—
25	#14582663	C₂²	1	4	yes	—
26	C₂×#108×C₄₃	C₂²	1	2	yes	—
27	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
28	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#341	C₂²	1	2	yes	—
30	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
31	C₂×#10621877	C₂³	1	4	yes	—
32	C₂×#10621872	—	2	4	no	—
33	C₂×#10621872	C₂×C₄	1	4	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₄₃	—	2	4	no	—
37	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
38	#13063282	C₂×C₄	1	4	yes	—
39	#13063282	D₄	1	4	yes	—
40	C₂×Dic₄₃⋊C₄	Q₈	1	2	yes	—
41	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
42	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
43	#13063261	—	2	4	no	—
44	#13063260	—	2	8	no	—
45	Dic₄₃⋊C₈	—	2	8	no	—
46	(C₂×D₄₃)⋊C₈	—	2	8	no	—
47	#13063236	—	2	2	no	—
48	#13063235	C₂×C₄	1	4	yes	—
49	C₄×C₄₃⋊C₈	—	2	2	no	—
50	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
51	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	#108×C₄₃	—	2	4	no	—
57	#97×C₄₃	—	2	2	no	—
58	#97×C₄₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₄₃	—	2	2	no	—
60	C₂³×D₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₄₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
65	C₂×C₄×D₄₃	—	2	2	no	—
66	C₂×C₄×D₄₃	—	2	2	no	—
67	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	2	4	no	—
73	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₃	#41	1	4	yes	—
76	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₃	C₂×D₄	1	2	yes	—
78	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
79	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
80	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
81	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
82	#10621877	—	2	4	no	—
83	#10621877	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	2	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
88	C₂×C₄₃⋊C₈	#41	1	4	yes	—
89	#10621872	—	2	8	no	—
90	C₈×D₄₃	—	2	8	no	—
91	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
92	C₂²×D₄₃	—	2	2	no	—
93	C₂²×D₄₃	—	4	2	no	—
94	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₃	—	4	2	no	—
96	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	4	2	no	—
103	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₄×D₄₃	—	2	2	no	—
106	C₄×D₄₃	—	2	2	no	—
107	C₄×D₄₃	—	2	4	no	—
108	C₈×C₄₃	—	2	2	no	—
109	C₈×C₄₃	—	4	4	no	—
110	C₈×C₄₃	#117	1	4	yes	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	4	4	no	—
113	C₄₃⋊C₈	#117	1	4	yes	—
114	#20829	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₄₃	#533	1	2	yes	—
118	C₂×D₄₃	—	2	4	no	—
119	C₂×D₄₃	—	2	2	no	—
120	C₂×D₄₃	—	4	4	no	—
121	C₄×C₄₃	—	2	4	no	—
122	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
124	C₄×C₄₃	#532	1	2	yes	—
125	Dic₄₃	—	2	4	no	—
126	Dic₄₃	—	2	2	no	—
127	#2921	—	43	2	no	—
128	C₂×#405	—	43	1	no	—
129	#1849	—	43	2	no	—
130	#1837	—	43	2	no	—
131	#1811	—	43	2	no	—
132	#1810	—	43	2	no	—
133	#1778	—	43	1	no	—
134	#1778	D₄₃	1	1	yes	characteristic, complemented
135	C₄×#108	—	43	2	no	—
136	C₂×C₄₃	—	2	2	no	—
137	C₂×C₄₃	#2285	1	1	yes	characteristic
138	C₂×C₄₃	#2916	1	2	yes	—
139	D₄₃	—	4	2	no	—
140	C₂²×M₁₆	—	43	1	no	—
141	C₂³×C₈	—	43	1	no	—
142	C₂×#117	—	43	1	no	—
143	#433	—	86	4	no	—
144	#433	—	86	4	no	—
145	C₂×#108	—	43	4	no	—
146	C₂×#108	—	43	2	no	—
147	C₂×#108	C₂×D₄₃	1	2	yes	—
148	#423	—	43	4	no	—
149	#407	—	43	4	no	—
150	#405	—	43	8	no	—
151	C₄×M₁₆	—	43	2	no	—
152	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
153	C₂×#97	—	43	1	no	—
154	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
155	C₂×C₄×C₈	—	43	1	no	—
156	C₂×C₄×C₈	—	43	1	no	—
157	#341	—	43	4	no	—
158	#341	—	43	2	no	—
159	#341	C₂×D₄₃	1	2	yes	—
160	C₄₃	#20829	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	—	43	1	no	—
162	C₂×M₁₆	—	43	4	no	—
163	C₂×M₁₆	—	43	4	no	—
164	C₂×M₁₆	—	86	4	no	—
165	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
166	C₂²×C₈	—	43	1	no	—
167	C₂²×C₈	—	43	1	no	—
168	C₂²×C₈	—	43	4	no	—
169	C₂²×C₈	—	86	4	no	—
170	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
172	#117	—	43	4	no	—
173	#117	—	43	4	no	—
174	C₂×C₄⋊C₄	—	43	2	no	—
175	C₂×C₂²⋊C₄	—	43	2	no	—
176	C₂×C₄²	—	43	1	no	—
177	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
178	#108	—	2	4	no	—
179	#108	—	86	8	no	—
180	#108	—	86	4	no	—
181	C₄⋊C₈	—	86	8	no	—
182	C₂²⋊C₈	—	86	8	no	—
183	#97	—	2	2	no	—
184	#97	—	43	4	no	—
185	#97	—	86	2	no	—
186	#97	C₄×D₄₃	1	4	yes	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	43	4	no	—
189	C₄×C₈	—	43	4	no	—
190	C₄×C₈	—	86	2	no	—
191	C₂⁴	—	43	1	no	—
192	C₂²×C₄	—	43	1	no	—
193	C₂²×C₄	—	43	2	no	—
194	C₂²×C₄	—	43	2	no	—
195	C₂²×C₄	—	43	2	no	—
196	C₂²×C₄	—	86	2	no	—
197	C₂²×C₄	—	86	2	no	—
198	C₂²×C₄	Q₈×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	—	86	8	no	—
203	M₁₆	—	86	4	no	—
204	M₁₆	—	86	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	43	2	no	—
208	C₂×C₈	—	43	4	no	—
209	C₂×C₈	—	43	2	no	—
210	C₂×C₈	—	86	2	no	—
211	C₂×C₈	—	86	8	no	—
212	C₂×C₈	—	86	4	no	—
213	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
215	C₂×C₈	#10621898	1	4	yes	—
216	C₄⋊C₄	—	43	8	no	—
217	C₂²⋊C₄	—	43	8	no	—
218	C₄²	—	43	2	no	—
219	C₄²	—	43	2	no	—
220	C₄²	C₂×C₄×D₄₃	1	2	yes	—
221	C₄²	D₄×D₄₃	1	2	yes	—
222	C₂³	—	43	2	no	—
223	C₂³	—	86	2	no	—
224	C₂³	—	172	2	no	—
225	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	43	4	no	—
228	C₂×C₄	—	43	2	no	—
229	C₂×C₄	—	86	2	no	—
230	C₂×C₄	—	86	2	no	—
231	C₂×C₄	—	86	2	no	—
232	C₂×C₄	—	86	4	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
235	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
237	C₂×C₄	#41×D₄₃	1	2	yes	—
238	C₂×C₄	C₂×#10621898	1	2	yes	—
239	C₈	—	2	2	no	—
240	C₈	—	4	4	no	—
241	C₈	—	43	4	no	—
242	C₈	—	86	2	no	—
243	C₈	—	172	4	no	—
244	C₈	#13063282	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	86	4	no	—
247	C₂²	—	86	2	no	—
248	C₂²	—	172	4	no	—
249	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
250	C₂²	#14583453	1	2	yes	—
251	C₄	—	2	4	no	—
252	C₄	—	86	4	no	—
253	C₄	—	86	2	no	—
254	C₄	#117×D₄₃	1	1	yes	characteristic
255	C₄	C₂×#13063282	1	1	yes	characteristic
256	C₄	#14583446	1	2	yes	—
257	C₂	—	2	2	no	—
258	C₂	—	172	2	no	—
259	C₂	#19136999	1	1	yes	characteristic
260	C₂	#19143205	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1778	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1778	D₄₃
hypocenter	C₄₃	#20829
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
3. #19134768	C₂	semidirect
4. #19134731	C₂	semidirect
7. #19134501	C₂	semidirect
8. C₄×#13063260	C₂	semidirect
9. C₄₃×#1778	C₂	semidirect
134. #1778	D₄₃	semidirect
160. C₄₃	#20829	semidirect

Maximal subgroups

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

subgroup	index
#19143231	2
C₂×#14582856	2
#19134768	2
#19134731	2
#19134636	2
#19134630	2
#19134501	2
C₄×#13063260	2
C₄₃×#1778	2
#20829	43