Subgroups — #25084259

Order 11008.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145154	C₂	1	4	yes	maximal, complemented
2	#19145109	C₂	1	1	yes	maximal, characteristic, complemented
3	#19145023	C₂	1	1	yes	maximal, characteristic
4	#19134196	C₂	1	2	yes	maximal, complemented
5	#19134192	C₂	1	2	yes	maximal, complemented
6	#19134187	C₂	1	2	yes	maximal, complemented
7	#19134182	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	—	2	2	no	—
10	#14583456	C₂²	1	2	yes	—
11	#14583449	C₂²	1	2	yes	—
12	#14583304	C₂²	1	2	yes	—
13	#14583301	—	2	2	no	—
14	#14583297	C₂²	1	2	yes	—
15	#14583297	C₂²	1	4	yes	—
16	#14583283	—	2	2	no	—
17	#14583279	C₂²	1	1	yes	characteristic
18	#14583276	C₂²	1	1	yes	characteristic
19	#14583272	C₂²	1	2	yes	—
20	#14583272	C₂²	1	4	yes	—
21	#14583250	C₂²	1	4	yes	—
22	#14583246	C₂²	1	4	yes	—
23	#14583128	—	2	8	no	—
24	C₄×#10621873	—	2	8	no	—
25	#14582937	—	2	2	no	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
28	C₈²×C₄₃	C₂²	1	1	yes	characteristic
29	C₈²×C₄₃	C₂²	1	2	yes	—
30	C₈²×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×#10621898	—	2	2	no	—
32	#13063347	—	4	8	no	—
33	C₂×Dic₁₇₂	—	2	2	no	—
34	C₂×#10621873	—	2	8	no	—
35	C₂×#10621873	—	2	4	no	—
36	C₂×D₃₄₄	—	2	2	no	—
37	#13063325	C₂³	1	4	yes	—
38	#13063325	C₂³	1	2	yes	—
39	#13063322	C₂³	1	2	yes	—
40	#13063317	—	2	2	no	—
41	#13063313	—	2	2	no	—
42	#13063304	—	2	4	no	—
43	#13063299	—	2	4	no	—
44	C₄×Dic₈₆	—	2	4	no	—
45	C₄×D₁₇₂	—	2	4	no	—
46	#13063284	—	2	2	no	—
47	#13063259	—	2	4	no	—
48	#13063257	—	2	4	no	—
49	#13063257	—	2	8	no	—
50	#13063250	—	2	8	no	—
51	#13063247	—	2	8	no	—
52	C₂²×C₈×C₄₃	D₄	1	4	yes	—
53	C₂²×C₈×C₄₃	D₄	1	2	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₄₃	D₄	1	2	yes	—
56	C₄×C₈×C₄₃	D₄	1	4	yes	—
57	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂³	1	2	yes	—
59	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	—	4	2	no	—
64	#10621898	—	4	8	no	—
65	C₂×D₁₇₂	—	2	4	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×C₄×D₄₃	—	4	2	no	—
68	C₂×Dic₈₆	—	2	4	no	—
69	C₂×Dic₈₆	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	8	no	—
71	C₂×C₈×C₄₃	SD₁₆	1	8	yes	complemented
72	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
76	C₄²×C₄₃	C₂×D₄	1	2	yes	—
77	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₂²⋊Dic₄₃	—	4	2	no	—
80	(C₂×D₄₃)⋊C₄	—	4	4	no	—
81	C₄⋊Dic₄₃	—	2	4	no	—
82	C₄⋊Dic₄₃	—	2	4	no	—
83	C₄⋊Dic₄₃	—	2	4	no	—
84	Dic₄₃⋊C₄	—	4	4	no	—
85	C₄×Dic₄₃	—	4	2	no	—
86	Dic₁₇₂	—	4	4	no	—
87	D₃₄₄	—	4	4	no	—
88	#10621873	—	4	8	no	—
89	#10621873	—	4	16	no	—
90	C₂³×C₄₃	#127	1	1	yes	characteristic
91	C₂²×D₄₃	—	4	2	no	—
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	#127	1	2	yes	—
94	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
98	C₂²⋊D₄₃	—	8	4	no	—
99	C₂×Dic₄₃	—	4	4	no	—
100	C₂×Dic₄₃	—	4	2	no	—
101	D₁₇₂	—	4	4	no	—
102	D₁₇₂	—	4	8	no	—
103	C₄×D₄₃	—	8	4	no	—
104	Dic₈₆	—	4	4	no	—
105	Dic₈₆	—	4	8	no	—
106	C₈×C₄₃	—	2	4	no	—
107	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
108	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
109	C₈×C₄₃	#135	1	8	yes	complemented
110	#19596	—	43	1	no	maximal, Sylow
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	#492	1	2	yes	—
113	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
114	C₂×D₄₃	—	8	2	no	—
115	C₂×D₄₃	—	8	4	no	—
116	C₄×C₄₃	—	2	2	no	—
117	C₄×C₄₃	#492	1	2	yes	—
118	C₄×C₄₃	C₂²×SD₁₆	1	2	yes	—
119	C₄×C₄₃	C₂×#135	1	4	yes	—
120	Dic₄₃	—	8	2	no	—
121	Dic₄₃	—	8	4	no	—
122	#3167	—	43	4	no	—
123	#3160	—	43	1	no	—
124	#3147	—	43	1	no	—
125	#1708	—	43	2	no	—
126	#1707	—	43	2	no	—
127	#1706	—	43	2	no	—
128	#1705	—	43	2	no	—
129	C₂×C₈²	D₄₃	1	1	yes	characteristic, complemented
130	C₂×C₄₃	—	2	2	no	—
131	C₂×C₄₃	C₂×#492	1	1	yes	characteristic
132	C₂×C₄₃	#3147	1	2	yes	—
133	D₄₃	—	16	2	no	—
134	C₂×#135	—	86	2	no	—
135	#533	—	43	2	no	—
136	#532	—	43	2	no	—
137	#500	—	43	2	no	—
138	#499	—	86	2	no	—
139	#498	—	43	2	no	—
140	#498	—	43	4	no	—
141	#495	—	86	2	no	—
142	#494	—	43	1	no	—
143	#493	—	43	1	no	—
144	#492	—	43	2	no	—
145	#492	—	43	4	no	—
146	#487	—	43	4	no	—
147	#486	—	43	4	no	—
148	#465	—	86	8	no	—
149	C₄×SD₁₆	—	86	8	no	—
150	#427	—	86	2	no	—
151	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
153	C₈²	C₂×D₄₃	1	1	yes	characteristic
154	C₈²	C₂×D₄₃	1	2	yes	—
155	C₈²	C₂×D₄₃	1	1	yes	characteristic
156	C₄₃	#19596	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	—	86	2	no	—
158	#135	—	172	8	no	—
159	C₂×Q₁₆	—	86	2	no	—
160	C₂×SD₁₆	—	86	4	no	—
161	C₂×SD₁₆	—	86	8	no	—
162	C₂×D₈	—	86	2	no	—
163	C₂²×C₈	D₁₇₂	1	4	yes	—
164	C₂²×C₈	D₁₇₂	1	2	yes	—
165	C₄⋊Q₈	—	43	2	no	—
166	C₄⋊Q₈	—	43	4	no	—
167	#127	—	43	2	no	—
168	#125	—	86	2	no	—
169	#124	—	86	2	no	—
170	C₂²⋊Q₈	—	86	4	no	—
171	#121	—	86	4	no	—
172	C₄×Q₈	—	86	4	no	—
173	C₄×D₄	—	86	4	no	—
174	#117	—	86	2	no	—
175	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
176	#107	—	86	4	no	—
177	#106	—	86	4	no	—
178	#106	—	86	8	no	—
179	#103	—	86	8	no	—
180	#102	—	86	8	no	—
181	C₄×C₈	D₁₇₂	1	4	yes	—
182	C₄×C₈	D₁₇₂	1	2	yes	—
183	C₄×C₈	C₂²×D₄₃	1	2	yes	—
184	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₄₃	1	2	yes	—
186	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
187	#41	—	172	8	no	—
188	C₂×Q₈	—	86	4	no	—
189	C₂×Q₈	—	86	2	no	—
190	C₂×D₄	—	86	4	no	—
191	C₂×D₄	—	86	2	no	—
192	C₂×D₄	—	172	2	no	—
193	C₂²×C₄	—	172	2	no	—
194	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
195	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
196	Q₁₆	—	172	4	no	—
197	SD₁₆	—	172	8	no	—
198	SD₁₆	—	172	16	no	—
199	D₈	—	172	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	#10621873	1	8	yes	complemented
202	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
203	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
204	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
205	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
206	C₄⋊C₄	—	86	4	no	—
207	C₄⋊C₄	—	86	4	no	—
208	C₄⋊C₄	—	86	4	no	—
209	C₄⋊C₄	—	172	4	no	—
210	C₂²⋊C₄	—	172	2	no	—
211	C₂²⋊C₄	—	172	4	no	—
212	C₄²	—	172	2	no	—
213	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
214	C₄²	C₂×D₁₇₂	1	2	yes	—
215	C₄²	C₂³×D₄₃	1	1	yes	characteristic
216	C₂³	—	172	2	no	—
217	C₂³	#13063322	1	1	yes	characteristic
218	Q₈	—	172	4	no	—
219	Q₈	—	172	8	no	—
220	D₄	—	344	4	no	—
221	D₄	—	172	4	no	—
222	D₄	—	172	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	344	4	no	—
225	C₂×C₄	—	172	4	no	—
226	C₂×C₄	—	172	2	no	—
227	C₂×C₄	#13063322	1	2	yes	—
228	C₂×C₄	#13063322	1	1	yes	characteristic
229	C₂×C₄	C₂×#10621873	1	4	yes	—
230	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	C₂×#10621873	1	4	yes	—
234	C₈	C₂×#10621873	1	4	yes	—
235	C₈	#13063347	1	8	yes	complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	344	2	no	—
238	C₂²	—	344	4	no	—
239	C₂²	#14583272	1	2	yes	—
240	C₂²	C₂×#13063322	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	344	2	no	—
243	C₄	—	344	4	no	—
244	C₄	#14583272	1	2	yes	—
245	C₄	C₂²×#10621873	1	2	yes	—
246	C₄	C₂×#13063347	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	688	2	no	—
249	C₂	C₂×#14583272	1	1	yes	characteristic
250	C₂	#19145023	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063322
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₈²×C₄₃	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	#19145023	C₂
hypercenter	C₂×C₈²	D₄₃
hypocenter	C₄₃	#19596
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #19145154	C₂	semidirect
2. #19145109	C₂	semidirect
4. #19134196	C₂	semidirect
5. #19134192	C₂	semidirect
6. #19134187	C₂	semidirect
7. #19134182	C₂	semidirect
8. C₂×C₈²×C₄₃	C₂	semidirect
71. C₂×C₈×C₄₃	SD₁₆	semidirect
109. C₈×C₄₃	#135	semidirect
129. C₂×C₈²	D₄₃	semidirect
156. C₄₃	#19596	semidirect
201. C₂×C₈	#10621873	semidirect
235. C₈	#13063347	semidirect

Maximal subgroups

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

subgroup	index
#19145154	2
#19145109	2
#19145023	2
#19134196	2
#19134192	2
#19134187	2
#19134182	2
C₂×C₈²×C₄₃	2
#19596	43