Subgroups — #25250018

Order 11008.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147731	C₂	1	1	yes	maximal, characteristic
2	#19143513	C₂	1	2	yes	maximal, complemented
3	#19136482	C₂	1	4	yes	maximal, complemented
4	#19136477	C₂	1	4	yes	maximal, complemented
5	#19136390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582707	C₂	1	2	yes	maximal, complemented
7	C₄₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	2	yes	—
9	#14583665	—	2	2	no	—
10	#14583342	C₄	1	4	yes	complemented
11	#14583337	C₄	1	4	yes	complemented
12	#14583331	C₂²	1	4	yes	—
13	#14583022	—	2	4	no	—
14	#14583017	C₂²	1	8	yes	complemented
15	C₂×#13063261	C₂²	1	2	yes	—
16	#14582917	—	2	2	no	—
17	#14582729	C₂²	1	4	yes	complemented
18	#14582707	C₂²	1	4	yes	complemented
19	#14582707	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#405	C₂²	1	1	yes	characteristic
23	C₄₃×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₄₃	C₂²	1	2	yes	complemented
25	C₂×#10621898	—	2	2	no	—
26	#13063347	—	2	2	no	—
27	#13063347	—	2	2	no	—
28	#13063347	C₂×C₄	1	4	yes	—
29	#13063347	C₂³	1	4	yes	—
30	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
31	C₂×#10621873	—	2	2	no	—
32	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
33	#13063333	—	4	4	no	—
34	#13063331	—	2	8	no	—
35	C₂×#10621877	—	2	2	no	—
36	M₁₆×D₄₃	—	4	4	no	—
37	Dic₃₄₄	C₂×C₄	1	4	yes	—
38	#13063270	C₂×C₄	1	8	yes	—
39	D₆₈₈	C₂×C₄	1	4	yes	—
40	#13063261	—	2	2	no	—
41	#13063261	C₂³	1	4	yes	—
42	#13063253	—	2	8	no	—
43	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
45	#117×C₄₃	D₄	1	1	yes	characteristic
46	M₃₂×C₄₃	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₄₃	—	2	2	no	—
48	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
51	#97×C₄₃	D₄	1	2	yes	—
52	C₄×C₈×C₄₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₄₃	—	4	2	no	—
55	#10621898	—	2	4	no	—
56	#10621898	—	2	4	no	—
57	#10621898	—	4	2	no	—
58	#10621898	—	4	2	no	—
59	C₂×D₁₇₂	—	2	2	no	—
60	C₂×C₄×D₄₃	—	4	2	no	—
61	C₂×Dic₈₆	—	2	2	no	—
62	M₁₆×C₄₃	D₈	1	4	yes	—
63	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₄₃	#41	1	2	yes	—
67	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₄₃	—	2	1	no	—
69	C₂²⋊C₄×C₄₃	—	2	1	no	—
70	C₄²×C₄₃	C₂×D₄	1	2	yes	—
71	#10621877	—	2	4	no	—
72	#10621877	—	4	2	no	—
73	C₂×C₄₃⋊C₈	—	4	2	no	—
74	Dic₁₇₂	—	2	2	no	—
75	Dic₁₇₂	C₂²×C₄	1	4	yes	—
76	D₃₄₄	—	2	2	no	—
77	D₃₄₄	C₂²×C₄	1	4	yes	—
78	#10621873	—	2	4	no	—
79	#10621873	—	2	4	no	—
80	#10621872	—	4	8	no	—
81	C₈×D₄₃	—	4	8	no	—
82	C₁₆×C₄₃	—	2	1	no	—
83	C₁₆×C₄₃	—	2	1	no	—
84	C₁₆×C₄₃	C₂²×C₄	1	4	yes	—
85	C₂³×C₄₃	#135	1	1	yes	characteristic
86	C₂²×D₄₃	—	4	2	no	—
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
92	C₂²⋊D₄₃	—	4	4	no	—
93	C₂²⋊D₄₃	—	4	4	no	—
94	C₂×Dic₄₃	—	4	2	no	—
95	D₁₇₂	—	2	4	no	—
96	D₁₇₂	—	4	2	no	—
97	C₄×D₄₃	—	4	4	no	—
98	C₄×D₄₃	—	4	4	no	—
99	Dic₈₆	—	2	4	no	—
100	Dic₈₆	—	4	2	no	—
101	C₈×C₄₃	C₄×D₄	1	2	yes	—
102	C₈×C₄₃	C₂×D₈	1	4	yes	—
103	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
105	C₄₃⋊C₈	—	4	4	no	—
106	#33464	—	43	1	no	maximal, Sylow
107	C₂²×C₄₃	—	4	1	no	—
108	C₂²×C₄₃	C₄×D₈	1	2	yes	—
109	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
110	C₂×D₄₃	—	4	4	no	—
111	C₂×D₄₃	—	16	2	no	—
112	C₄×C₄₃	—	4	2	no	—
113	C₄×C₄₃	C₄×D₈	1	2	yes	—
114	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
116	Dic₄₃	—	4	4	no	—
117	#3414	—	43	1	no	—
118	#2957	—	43	2	no	—
119	#2179	—	43	4	no	—
120	#2178	—	43	4	no	—
121	#2161	—	43	1	no	—
122	C₂×#359	—	43	2	no	—
123	#2108	D₄₃	1	1	yes	characteristic, complemented
124	C₂×C₄₃	—	2	1	no	—
125	C₂×C₄₃	—	2	2	no	—
126	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
127	D₄₃	—	8	4	no	—
128	C₂×#135	—	43	2	no	—
129	#568	—	86	2	no	—
130	#510	—	43	4	no	—
131	#509	—	43	4	no	—
132	#508	—	43	4	no	—
133	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
134	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
135	#444	—	86	4	no	—
136	#443	—	43	8	no	—
137	C₂×#108	—	43	2	no	—
138	#421	—	86	2	no	—
139	#405	C₂×D₄₃	1	1	yes	characteristic
140	#368	—	43	4	no	—
141	#359	—	43	4	no	—
142	#359	—	43	4	no	—
143	#346	C₂×D₄₃	1	2	yes	complemented
144	C₄×C₁₆	C₂×D₄₃	1	2	yes	complemented
145	C₄₃	#33464	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	86	2	no	—
147	#135	—	43	4	no	—
148	#135	—	43	4	no	—
149	#135	—	86	2	no	—
150	#135	—	86	2	no	—
151	C₂×Q₁₆	—	43	2	no	—
152	C₂×SD₁₆	—	86	2	no	—
153	C₂×D₈	—	43	2	no	—
154	#131	—	86	8	no	—
155	#131	—	172	4	no	—
156	C₂×M₁₆	—	86	2	no	—
157	C₂×M₁₆	—	172	4	no	—
158	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
160	#117	D₁₇₂	1	1	yes	characteristic
161	Q₃₂	—	43	4	no	—
162	SD₃₂	—	43	8	no	—
163	D₁₆	—	43	4	no	—
164	M₃₂	C₄×D₄₃	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
169	#108	—	43	4	no	—
170	#108	—	86	2	no	—
171	#104	—	86	8	no	—
172	#97	D₁₇₂	1	2	yes	—
173	C₄×C₈	C₂²×D₄₃	1	2	yes	—
174	#41	—	86	4	no	—
175	#41	—	86	4	no	—
176	#41	—	172	2	no	—
177	#41	—	172	2	no	—
178	C₂×Q₈	—	86	2	no	—
179	C₂×D₄	—	86	2	no	—
180	C₂×D₄	—	172	2	no	—
181	C₂²×C₄	—	172	2	no	—
182	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
183	Q₁₆	—	43	4	no	—
184	Q₁₆	—	86	2	no	—
185	SD₁₆	—	86	4	no	—
186	SD₁₆	—	86	4	no	—
187	D₈	—	43	4	no	—
188	D₈	—	86	2	no	—
189	M₁₆	—	86	4	no	—
190	M₁₆	—	172	2	no	—
191	M₁₆	—	172	8	no	—
192	M₁₆	D₃₄₄	1	4	yes	—
193	C₂×C₈	—	172	2	no	—
194	C₂×C₈	—	172	8	no	—
195	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
196	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
197	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
198	C₂×C₈	#10621898	1	2	yes	—
199	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₁₇₂	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₄₃	1	4	yes	—
206	C₂³	—	172	2	no	—
207	C₂³	#13063347	1	1	yes	characteristic
208	Q₈	—	86	4	no	—
209	Q₈	—	172	2	no	—
210	D₄	—	86	4	no	—
211	D₄	—	172	4	no	—
212	D₄	—	172	4	no	—
213	D₄	—	172	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	172	2	no	—
216	C₂×C₄	—	172	4	no	—
217	C₂×C₄	—	172	4	no	—
218	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
219	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
220	C₂×C₄	#13063347	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
222	C₈	—	172	4	no	—
223	C₈	C₄×D₁₇₂	1	2	yes	—
224	C₈	C₂×D₃₄₄	1	4	yes	—
225	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
226	C₈	C₂×#10621898	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	172	4	no	—
229	C₂²	—	688	2	no	—
230	C₂²	C₄×D₃₄₄	1	2	yes	—
231	C₂²	C₂×#13063347	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	172	4	no	—
234	C₄	C₄×D₃₄₄	1	2	yes	—
235	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
236	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	344	4	no	—
240	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2108	C₂
socle	C₂×C₄₃	C₂×C₄×D₈
prime radical	#19147731	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2108	D₄₃
hypocenter	C₄₃	#33464
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #19143513	C₂	semidirect
3. #19136482	C₂	semidirect
4. #19136477	C₂	semidirect
5. #19136390	C₂	semidirect
6. C₂×#14582707	C₂	semidirect
7. C₄₃×#2108	C₂	semidirect
10. #14583342	C₄	semidirect
11. #14583337	C₄	semidirect
14. #14583017	C₂²	semidirect
17. #14582729	C₂²	semidirect
18. #14582707	C₂²	semidirect
19. #14582707	C₂²	semidirect
23. C₄₃×#346	C₂²	semidirect
24. C₄×C₁₆×C₄₃	C₂²	semidirect
123. #2108	D₄₃	semidirect
143. #346	C₂×D₄₃	semidirect
144. C₄×C₁₆	C₂×D₄₃	semidirect
145. C₄₃	#33464	semidirect

Maximal subgroups

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

subgroup	index
#19147731	2
#19143513	2
#19136482	2
#19136477	2
#19136390	2
C₂×#14582707	2
C₄₃×#2108	2
#33464	43