Subgroups — #25097229

Order 11008.

718 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	#19143232	C₂	1	2	yes	maximal
2	C₂×#14582853	C₂	1	1	yes	maximal, characteristic
3	#19134769	C₂	1	2	yes	maximal, complemented
4	#19134733	C₂	1	1	yes	maximal, characteristic, complemented
5	#19134636	C₂	1	2	yes	maximal
6	#19134632	C₂	1	2	yes	maximal
7	#19134500	C₂	1	2	yes	maximal, complemented
8	#19134496	C₂	1	2	yes	maximal, complemented
9	C₄₃×#1837	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	#14582959	—	2	8	no	—
14	C₂×#13063260	C₂²	1	4	yes	—
15	C₂×#13063260	C₂²	1	4	yes	—
16	#14582923	C₂²	1	2	yes	—
17	#14582862	C₂²	1	4	yes	—
18	#14582853	C₄	1	8	yes	—
19	M₁₆×Dic₄₃	C₂²	1	4	yes	—
20	C₂×#13063236	C₂²	1	1	yes	characteristic
21	C₂×#13063235	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
23	#14582664	C₂²	1	2	yes	—
24	#14582664	C₂²	1	2	yes	—
25	C₄₃×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄₃×#341	C₂²	1	2	yes	—
28	C₄₃×#341	C₂²	1	2	yes	—
29	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
30	C₂×#10621877	C₂³	1	4	yes	—
31	C₂×#10621872	—	2	4	no	—
32	C₂×#10621872	C₂×C₄	1	4	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂×C₈×D₄₃	—	2	4	no	—
36	C₂×C₈×D₄₃	C₂×C₄	1	4	yes	—
37	#13063282	C₂×C₄	1	4	yes	—
38	#13063282	D₄	1	4	yes	—
39	C₂×Dic₄₃⋊C₄	Q₈	1	2	yes	—
40	C₂×(C₂×D₄₃)⋊C₄	Q₈	1	2	yes	—
41	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
42	#13063260	—	2	8	no	—
43	#13063260	—	2	8	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
45	(C₂×D₄₃)⋊C₈	—	2	8	no	—
46	#13063236	C₂×C₄	1	4	yes	—
47	#13063235	—	2	4	no	—
48	#13063235	C₂×C₄	1	4	yes	—
49	C₈×Dic₄₃	—	2	4	no	—
50	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
52	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	—	2	4	no	—
56	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
57	C₂³×D₄₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×Dic₄₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×D₄₃	—	2	2	no	—
63	C₂×C₄×D₄₃	—	2	2	no	—
64	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
66	M₁₆×C₄₃	—	2	4	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₄₃	C₂×D₄	1	2	yes	—
75	(C₂×D₄₃)⋊C₄	C₄⋊C₄	1	8	yes	—
76	Dic₄₃⋊C₄	C₄⋊C₄	1	8	yes	—
77	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
78	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
79	#10621877	—	2	4	no	—
80	#10621877	—	2	4	no	—
81	C₂×C₄₃⋊C₈	—	2	4	no	—
82	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₄₃⋊C₈	#41	1	4	yes	—
85	C₂×C₄₃⋊C₈	#41	1	4	yes	—
86	#10621872	—	2	8	no	—
87	C₈×D₄₃	—	2	8	no	—
88	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
89	C₂²×D₄₃	—	2	2	no	—
90	C₂²×D₄₃	—	4	2	no	—
91	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
98	C₂×Dic₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₄×D₄₃	—	2	2	no	—
103	C₄×D₄₃	—	2	2	no	—
104	C₄×D₄₃	—	2	4	no	—
105	C₈×C₄₃	—	2	2	no	—
106	C₈×C₄₃	—	2	2	no	—
107	C₈×C₄₃	—	4	4	no	—
108	C₄₃⋊C₈	—	4	4	no	—
109	C₄₃⋊C₈	#117	1	4	yes	—
110	C₄₃⋊C₈	#117	1	4	yes	—
111	#20829	—	43	1	no	maximal, Sylow
112	C₂²×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₄₃	#533	1	2	yes	—
115	C₂×D₄₃	—	2	4	no	—
116	C₂×D₄₃	—	2	2	no	—
117	C₂×D₄₃	—	4	4	no	—
118	C₄×C₄₃	—	2	4	no	—
119	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
120	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
121	C₄×C₄₃	#532	1	2	yes	—
122	Dic₄₃	—	2	4	no	—
123	Dic₄₃	—	2	2	no	—
124	#2921	—	43	2	no	—
125	C₂×#405	—	43	1	no	—
126	#1849	—	43	2	no	—
127	#1837	—	43	1	no	—
128	#1837	D₄₃	1	1	yes	characteristic, complemented
129	#1811	—	43	2	no	—
130	#1810	—	43	2	no	—
131	#1778	—	43	2	no	—
132	C₄×#108	—	43	2	no	—
133	C₂×C₄₃	—	2	2	no	—
134	C₂×C₄₃	#2285	1	1	yes	characteristic
135	C₂×C₄₃	#2916	1	2	yes	—
136	D₄₃	—	4	2	no	—
137	C₂²×M₁₆	—	43	1	no	—
138	C₂³×C₈	—	43	1	no	—
139	C₂×#117	—	43	1	no	—
140	#433	—	86	8	no	—
141	C₂×#108	—	43	4	no	—
142	C₂×#108	—	43	4	no	—
143	#423	—	43	2	no	—
144	#423	C₂×D₄₃	1	2	yes	—
145	#407	—	43	4	no	—
146	#405	—	43	8	no	—
147	C₄×M₁₆	—	43	4	no	—
148	C₂×#97	—	43	1	no	—
149	C₂×#97	—	43	1	no	—
150	C₂×C₄×C₈	—	43	1	no	—
151	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
152	#341	—	43	2	no	—
153	#341	—	43	2	no	—
154	#341	C₂×D₄₃	1	2	yes	—
155	#341	C₂×D₄₃	1	2	yes	—
156	C₄₃	#20829	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	—	43	1	no	—
158	C₂×M₁₆	—	43	4	no	—
159	C₂×M₁₆	—	43	4	no	—
160	C₂×M₁₆	—	86	4	no	—
161	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
162	C₂²×C₈	—	43	1	no	—
163	C₂²×C₈	—	43	1	no	—
164	C₂²×C₈	—	43	4	no	—
165	C₂²×C₈	—	86	4	no	—
166	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
167	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
168	#117	—	43	4	no	—
169	#117	—	43	4	no	—
170	C₂×C₄⋊C₄	—	43	2	no	—
171	C₂×C₂²⋊C₄	—	43	2	no	—
172	C₂×C₄²	—	43	1	no	—
173	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
174	#108	—	86	8	no	—
175	#108	—	86	8	no	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	—	86	4	no	—
178	C₂²⋊C₈	—	86	8	no	—
179	#97	—	43	4	no	—
180	#97	—	43	4	no	—
181	#97	—	86	4	no	—
182	C₄×C₈	—	43	4	no	—
183	C₄×C₈	—	86	4	no	—
184	C₄×C₈	C₄×D₄₃	1	4	yes	—
185	C₂⁴	—	43	1	no	—
186	C₂²×C₄	—	43	1	no	—
187	C₂²×C₄	—	43	2	no	—
188	C₂²×C₄	—	43	2	no	—
189	C₂²×C₄	—	43	2	no	—
190	C₂²×C₄	—	86	2	no	—
191	C₂²×C₄	—	86	2	no	—
192	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
193	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	86	8	no	—
197	M₁₆	—	86	4	no	—
198	M₁₆	—	86	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	43	2	no	—
203	C₂×C₈	—	43	2	no	—
204	C₂×C₈	—	43	4	no	—
205	C₂×C₈	—	43	4	no	—
206	C₂×C₈	—	86	8	no	—
207	C₂×C₈	—	86	4	no	—
208	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
210	C₄⋊C₄	—	43	8	no	—
211	C₂²⋊C₄	—	43	8	no	—
212	C₄²	—	43	2	no	—
213	C₄²	—	43	2	no	—
214	C₄²	C₂×C₄×D₄₃	1	2	yes	—
215	C₄²	D₄×D₄₃	1	2	yes	—
216	C₂³	—	43	2	no	—
217	C₂³	—	86	2	no	—
218	C₂³	—	172	2	no	—
219	C₂³	C₂×Q₈×D₄₃	1	1	yes	characteristic
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	43	4	no	—
222	C₂×C₄	—	43	2	no	—
223	C₂×C₄	—	86	2	no	—
224	C₂×C₄	—	86	2	no	—
225	C₂×C₄	—	86	2	no	—
226	C₂×C₄	—	86	4	no	—
227	C₂×C₄	—	172	2	no	—
228	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
229	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
231	C₂×C₄	#41×D₄₃	1	2	yes	—
232	C₂×C₄	#41×D₄₃	1	2	yes	—
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	—	4	4	no	—
236	C₈	—	43	4	no	—
237	C₈	—	43	4	no	—
238	C₈	—	172	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	86	4	no	—
241	C₂²	—	86	2	no	—
242	C₂²	—	172	4	no	—
243	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
244	C₂²	#14583455	1	2	yes	—
245	C₄	—	2	4	no	—
246	C₄	—	86	4	no	—
247	C₄	—	86	2	no	—
248	C₄	#117×D₄₃	1	1	yes	characteristic
249	C₄	#117×D₄₃	1	1	yes	characteristic
250	C₄	#14583448	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	172	2	no	—
253	C₂	#19137001	1	1	yes	characteristic
254	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	C₄₃×#1837	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1837	D₄₃
hypocenter	C₄₃	#20829
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
3. #19134769	C₂	semidirect
4. #19134733	C₂	semidirect
7. #19134500	C₂	semidirect
8. #19134496	C₂	semidirect
9. C₄₃×#1837	C₂	semidirect
128. #1837	D₄₃	semidirect
156. C₄₃	#20829	semidirect

Maximal subgroups

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

subgroup	index
#19143232	2
C₂×#14582853	2
#19134769	2
#19134733	2
#19134636	2
#19134632	2
#19134500	2
#19134496	2
C₄₃×#1837	2
#20829	43