Subgroups — #25097210

Order 11008.

758 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143228	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#10621877	C₂	1	1	yes	maximal, characteristic
3	#19134769	C₂	1	2	yes	maximal, complemented
4	#19134735	C₂	1	2	yes	maximal, complemented
5	#19134635	C₂	1	4	yes	maximal
6	C₄×#13063260	C₂	1	4	yes	maximal, complemented
7	C₄₃×#2920	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10621877	C₂²	1	2	yes	—
9	#14582956	—	2	2	no	—
10	#14582952	—	2	2	no	—
11	C₂×#13063260	C₂²	1	8	yes	—
12	#14582923	C₂²	1	2	yes	—
13	#14582863	C₂²	1	2	yes	—
14	C₄×#10621877	C₄	1	8	yes	—
15	C₄×#10621877	C₂²	1	2	yes	—
16	C₂×#13063236	C₂²	1	1	yes	characteristic
17	C₂×#13063236	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
19	#14582663	C₂²	1	8	yes	—
20	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#432	—	2	2	no	—
22	C₄₃×#431	—	2	2	no	—
23	C₄₃×#423	C₂²	1	2	yes	—
24	C₄₃×#407	C₂²	1	2	yes	—
25	C₄×M₁₆×C₄₃	C₂²	1	2	yes	—
26	C₂×#10621877	—	2	8	no	—
27	C₂×#10621877	C₂×C₄	1	8	yes	—
28	C₂×#10621877	C₂³	1	4	yes	—
29	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
30	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
31	#13063260	—	2	16	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
33	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
34	#13063236	—	2	2	no	—
35	#13063236	C₂×C₄	1	4	yes	—
36	#13063236	C₂×C₄	1	4	yes	—
37	C₄×C₄₃⋊C₈	—	2	2	no	—
38	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₄₃	Q₈	1	2	yes	—
40	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₄₃	D₄	1	2	yes	—
44	C₂×C₄²×C₄₃	D₄	1	2	yes	—
45	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₃	—	2	4	no	—
49	C₂²⋊C₈×C₄₃	—	2	4	no	—
50	#97×C₄₃	—	2	2	no	—
51	C₄×C₈×C₄₃	—	2	2	no	—
52	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	—	2	4	no	—
54	C₂²×C₄×C₄₃	—	4	4	no	—
55	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₄₃	—	2	8	no	—
64	C₂×C₈×C₄₃	—	2	4	no	—
65	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
66	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
67	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
68	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₄₃	C₂×D₄	1	2	yes	—
71	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
72	#10621877	—	2	16	no	—
73	#10621877	—	2	8	no	—
74	C₂×C₄₃⋊C₈	—	2	4	no	—
75	C₂×C₄₃⋊C₈	—	2	4	no	—
76	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
78	C₂×C₄₃⋊C₈	#41	1	8	yes	—
79	C₂³×C₄₃	—	2	2	no	—
80	C₂³×C₄₃	—	4	2	no	—
81	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₄₃	—	2	2	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	—	2	4	no	—
86	C₂×C₄×C₄₃	—	2	8	no	—
87	C₂×C₄×C₄₃	—	2	4	no	—
88	C₂×C₄×C₄₃	—	4	4	no	—
89	C₂×C₄×C₄₃	—	4	2	no	—
90	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	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	4	yes	—
97	C₈×C₄₃	—	4	4	no	—
98	C₄₃⋊C₈	—	2	4	no	—
99	C₄₃⋊C₈	—	4	4	no	—
100	C₄₃⋊C₈	#117	1	8	yes	—
101	#20827	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	—	2	2	no	—
103	C₂²×C₄₃	—	2	4	no	—
104	C₂²×C₄₃	—	2	2	no	—
105	C₂²×C₄₃	—	4	4	no	—
106	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₄₃	#533	1	2	yes	—
108	C₄×C₄₃	—	2	4	no	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	2	2	no	—
111	C₄×C₄₃	#430	1	4	yes	—
112	C₄×C₄₃	C₂×#117	1	2	yes	—
113	C₄×C₄₃	#532	1	2	yes	—
114	#2920	—	43	1	no	—
115	#2920	D₄₃	1	1	yes	characteristic
116	C₂×C₄×M₁₆	—	43	1	no	—
117	#1849	—	43	2	no	—
118	#1838	—	43	2	no	—
119	#1811	—	43	4	no	—
120	C₄×#108	—	43	4	no	—
121	C₂×C₄₃	—	2	2	no	—
122	C₂×C₄₃	—	4	2	no	—
123	C₂×C₄₃	#2285	1	1	yes	characteristic
124	C₂×C₄₃	C₂×#430	1	2	yes	—
125	C₂²×M₁₆	—	43	2	no	—
126	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
127	#432	—	2	2	no	—
128	#432	—	86	2	no	—
129	#431	—	2	2	no	—
130	#431	—	86	2	no	—
131	C₂×#108	—	43	8	no	—
132	#423	—	43	2	no	—
133	#423	C₂×D₄₃	1	2	yes	—
134	#407	—	43	2	no	—
135	#407	C₂×D₄₃	1	2	yes	—
136	C₄×M₁₆	—	43	8	no	—
137	C₄×M₁₆	—	43	2	no	—
138	C₄×M₁₆	C₂×D₄₃	1	2	yes	—
139	C₂×#97	—	43	1	no	—
140	C₂×#97	—	43	1	no	—
141	C₂×C₄×C₈	—	43	2	no	—
142	#341	—	43	8	no	—
143	C₄₃	#20827	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	Dic₈₆	1	2	yes	—
145	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
146	C₂×M₁₆	—	43	8	no	—
147	C₂×M₁₆	—	43	4	no	—
148	C₂×M₁₆	—	86	8	no	—
149	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
150	C₂²×C₈	—	43	2	no	—
151	C₂²×C₈	—	43	2	no	—
152	C₂×C₄²	Dic₈₆	1	2	yes	—
153	C₂×C₄²	C₄×D₄₃	1	4	yes	—
154	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
156	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
158	#108	—	86	16	no	—
159	C₄⋊C₈	—	2	4	no	—
160	C₄⋊C₈	—	86	4	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	C₂²⋊C₈	—	86	4	no	—
163	#97	—	2	2	no	—
164	#97	—	43	4	no	—
165	#97	—	43	4	no	—
166	#97	—	86	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	43	8	no	—
169	C₄×C₈	—	86	2	no	—
170	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
174	C₂²×C₄	Dic₄₃⋊C₄	1	4	yes	—
175	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
176	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
177	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
180	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
181	M₁₆	—	2	8	no	—
182	M₁₆	—	86	16	no	—
183	M₁₆	—	86	8	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	43	4	no	—
186	C₂×C₈	—	43	8	no	—
187	C₂×C₈	—	43	4	no	—
188	C₂×C₈	—	86	4	no	—
189	C₂×C₈	—	86	4	no	—
190	C₄²	Dic₄₃⋊C₄	1	4	yes	—
191	C₄²	Dic₄₃⋊C₄	1	4	yes	—
192	C₄²	C₂×C₄×D₄₃	1	2	yes	—
193	C₄²	C₂×C₄×D₄₃	1	2	yes	—
194	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
196	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×Dic₄₃⋊C₄	1	2	yes	—
200	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
210	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
211	C₂×C₄	C₂×Dic₄₃⋊C₄	1	2	yes	—
212	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
214	C₂×C₄	#41×D₄₃	1	4	yes	—
215	C₈	—	4	4	no	—
216	C₈	—	43	8	no	—
217	C₈	—	86	4	no	—
218	C₈	—	172	4	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
224	C₂²	#14583452	1	2	yes	—
225	C₄	—	2	4	no	—
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	#14582947	1	4	yes	—
229	C₄	#117×D₄₃	1	2	yes	—
230	C₄	#14583445	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	#19136998	1	1	yes	characteristic
234	C₂	C₂×#14582947	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#2920	C₂
socle	C₂²×C₄₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2920	D₄₃
hypocenter	C₄₃	#20827
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #19134769	C₂	semidirect
4. #19134735	C₂	semidirect
6. C₄×#13063260	C₂	semidirect
143. C₄₃	#20827	semidirect

Maximal subgroups

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

subgroup	index
#19143228	2
C₂×C₄×#10621877	2
#19134769	2
#19134735	2
#19134635	2
C₄×#13063260	2
C₄₃×#2920	2
#20827	43