Subgroups — #25097226

Order 11008.

718 conjugacy classes of subgroups.

The subgroup lattice has 274 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×#14582853	C₂	1	1	yes	maximal, characteristic
3	#19134768	C₂	1	2	yes	maximal, complemented
4	#19134731	C₂	1	2	yes	maximal, complemented
5	#19134634	C₂	1	2	yes	maximal
6	#19134632	C₂	1	2	yes	maximal
7	#19134502	C₂	1	2	yes	maximal, complemented
8	C₄×#13063261	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#108×C₄₃	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	#14582853	C₄	1	8	yes	—
20	C₄×#10621877	C₂²	1	2	yes	—
21	C₂×#13063236	C₂²	1	1	yes	characteristic
22	C₂×#13063235	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×C₄×C₈×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	2	yes	—
40	#13063282	D₄	1	2	yes	—
41	C₂×Dic₄₃⋊C₄	Q₈	1	2	yes	—
42	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
44	#13063261	—	2	4	no	—
45	#13063260	—	2	8	no	—
46	Dic₄₃⋊C₈	—	2	8	no	—
47	(C₂×D₄₃)⋊C₈	—	2	8	no	—
48	#13063236	—	2	2	no	—
49	#13063236	C₂×C₄	1	4	yes	—
50	#13063235	C₂×C₄	1	4	yes	—
51	C₄×C₄₃⋊C₈	—	2	2	no	—
52	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
54	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	#108×C₄₃	—	2	4	no	—
58	#97×C₄₃	—	2	2	no	—
59	C₄×C₈×C₄₃	—	2	2	no	—
60	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
61	C₂³×D₄₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
66	C₂×C₄×D₄₃	—	2	2	no	—
67	C₂×C₄×D₄₃	—	2	2	no	—
68	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
70	M₁₆×C₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₃	#41	1	4	yes	—
77	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
80	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	4	yes	—
81	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	4	yes	—
82	Dic₄₃⋊C₄	C₄⋊C₄	1	4	yes	—
83	Dic₄₃⋊C₄	C₄⋊C₄	1	4	yes	—
84	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
85	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
86	#10621877	—	2	4	no	—
87	#10621877	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	2	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₄₃⋊C₈	#41	1	4	yes	—
93	#10621872	—	2	8	no	—
94	C₈×D₄₃	—	2	8	no	—
95	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×D₄₃	—	2	2	no	—
97	C₂²×D₄₃	—	4	2	no	—
98	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
106	C₂×Dic₄₃	—	2	2	no	—
107	C₂×Dic₄₃	—	4	2	no	—
108	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₄×D₄₃	—	2	2	no	—
111	C₄×D₄₃	—	2	2	no	—
112	C₄×D₄₃	—	2	4	no	—
113	C₈×C₄₃	—	2	2	no	—
114	C₈×C₄₃	—	4	4	no	—
115	C₈×C₄₃	#117	1	4	yes	—
116	C₄₃⋊C₈	—	2	2	no	—
117	C₄₃⋊C₈	—	4	4	no	—
118	C₄₃⋊C₈	#117	1	4	yes	—
119	#20829	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄₃	#533	1	2	yes	—
123	C₂×D₄₃	—	2	4	no	—
124	C₂×D₄₃	—	2	2	no	—
125	C₂×D₄₃	—	4	4	no	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
129	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
130	C₄×C₄₃	#532	1	2	yes	—
131	Dic₄₃	—	2	4	no	—
132	Dic₄₃	—	2	2	no	—
133	#2921	—	43	2	no	—
134	C₂×#405	—	43	1	no	—
135	#1849	—	43	2	no	—
136	#1837	—	43	2	no	—
137	#1811	—	43	2	no	—
138	#1810	—	43	2	no	—
139	#1778	—	43	2	no	—
140	C₄×#108	—	43	1	no	—
141	C₄×#108	D₄₃	1	1	yes	characteristic, complemented
142	C₂×C₄₃	—	2	2	no	—
143	C₂×C₄₃	#2285	1	1	yes	characteristic
144	C₂×C₄₃	#2916	1	2	yes	—
145	D₄₃	—	4	2	no	—
146	C₂²×M₁₆	—	43	1	no	—
147	C₂³×C₈	—	43	1	no	—
148	C₂×#117	—	43	1	no	—
149	#433	—	86	4	no	—
150	#433	—	86	4	no	—
151	C₂×#108	—	43	4	no	—
152	C₂×#108	—	43	2	no	—
153	C₂×#108	C₂×D₄₃	1	2	yes	—
154	#423	—	43	4	no	—
155	#407	—	43	4	no	—
156	#405	—	43	8	no	—
157	C₄×M₁₆	—	43	2	no	—
158	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
159	C₂×#97	—	43	1	no	—
160	C₂×#97	—	43	1	no	—
161	C₂×C₄×C₈	—	43	1	no	—
162	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
163	#341	—	43	4	no	—
164	#341	—	43	2	no	—
165	#341	C₂×D₄₃	1	2	yes	—
166	C₄₃	#20829	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	—	43	1	no	—
168	C₂×M₁₆	—	43	4	no	—
169	C₂×M₁₆	—	43	4	no	—
170	C₂×M₁₆	—	86	4	no	—
171	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
172	C₂²×C₈	—	43	1	no	—
173	C₂²×C₈	—	43	1	no	—
174	C₂²×C₈	—	43	4	no	—
175	C₂²×C₈	—	86	4	no	—
176	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
178	#117	—	43	4	no	—
179	#117	—	43	2	no	—
180	#117	—	43	2	no	—
181	C₂×C₄⋊C₄	—	43	2	no	—
182	C₂×C₂²⋊C₄	—	43	2	no	—
183	C₂×C₄²	—	43	1	no	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	#108	—	2	4	no	—
186	#108	—	86	8	no	—
187	#108	—	86	4	no	—
188	C₄⋊C₈	—	86	8	no	—
189	C₂²⋊C₈	—	86	8	no	—
190	#97	—	2	2	no	—
191	#97	—	43	4	no	—
192	#97	—	43	4	no	—
193	#97	—	86	2	no	—
194	C₄×C₈	—	2	2	no	—
195	C₄×C₈	—	43	4	no	—
196	C₄×C₈	—	86	2	no	—
197	C₄×C₈	C₄×D₄₃	1	4	yes	—
198	C₂⁴	—	43	1	no	—
199	C₂²×C₄	—	43	1	no	—
200	C₂²×C₄	—	43	2	no	—
201	C₂²×C₄	—	43	2	no	—
202	C₂²×C₄	—	43	2	no	—
203	C₂²×C₄	—	86	2	no	—
204	C₂²×C₄	—	86	2	no	—
205	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
206	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	86	8	no	—
210	M₁₆	—	86	4	no	—
211	M₁₆	—	86	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	43	2	no	—
215	C₂×C₈	—	43	4	no	—
216	C₂×C₈	—	43	2	no	—
217	C₂×C₈	—	86	8	no	—
218	C₂×C₈	—	86	2	no	—
219	C₂×C₈	—	86	4	no	—
220	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
221	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
222	C₂×C₈	#10621898	1	4	yes	—
223	C₄⋊C₄	—	43	4	no	—
224	C₄⋊C₄	—	43	4	no	—
225	C₂²⋊C₄	—	43	4	no	—
226	C₂²⋊C₄	—	43	4	no	—
227	C₄²	—	43	2	no	—
228	C₄²	—	43	2	no	—
229	C₄²	C₂×C₄×D₄₃	1	2	yes	—
230	C₄²	D₄×D₄₃	1	1	yes	characteristic
231	C₄²	D₄×D₄₃	1	1	yes	characteristic
232	C₂³	—	43	2	no	—
233	C₂³	—	86	2	no	—
234	C₂³	—	172	2	no	—
235	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	43	4	no	—
238	C₂×C₄	—	43	2	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	86	2	no	—
241	C₂×C₄	—	86	2	no	—
242	C₂×C₄	—	86	4	no	—
243	C₂×C₄	—	172	2	no	—
244	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
245	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
246	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
248	C₂×C₄	#41×D₄₃	1	2	yes	—
249	C₂×C₄	C₂×#10621898	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	4	4	no	—
252	C₈	—	43	4	no	—
253	C₈	—	86	2	no	—
254	C₈	—	172	4	no	—
255	C₈	#13063282	1	4	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	86	4	no	—
258	C₂²	—	86	2	no	—
259	C₂²	—	172	4	no	—
260	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
261	C₂²	#14583453	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	86	4	no	—
265	C₄	—	86	2	no	—
266	C₄	#117×D₄₃	1	1	yes	characteristic
267	C₄	C₂×#13063282	1	1	yes	characteristic
268	C₄	#14583446	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	172	2	no	—
271	C₂	#19136999	1	1	yes	characteristic
272	C₂	#19143203	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#108×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#108	D₄₃
hypocenter	C₄₃	#20829
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

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. #19134502	C₂	semidirect
8. C₄×#13063261	C₂	semidirect
9. C₄×#108×C₄₃	C₂	semidirect
141. C₄×#108	D₄₃	semidirect
166. 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×#14582853	2
#19134768	2
#19134731	2
#19134634	2
#19134632	2
#19134502	2
C₄×#13063261	2
C₄×#108×C₄₃	2
#20829	43