Subgroups — #25249981

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583331	C₂	1	1	yes	maximal, characteristic
2	#19143469	C₂	1	2	yes	maximal, complemented
3	#19136458	C₂	1	4	yes	maximal, complemented
4	#19136453	C₂	1	2	yes	maximal, complemented
5	#19136441	C₂	1	2	yes	maximal, complemented
6	#19136305	C₂	1	2	yes	maximal, complemented
7	#19136191	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063347	C₂²	1	2	yes	—
10	#14583380	—	2	2	no	—
11	#14583331	C₄	1	8	yes	complemented
12	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
13	C₂×#13063270	C₂²	1	2	yes	—
14	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
15	C₂×#13063265	C₂²	1	1	yes	characteristic
16	#14582998	—	2	4	no	—
17	#14582993	C₂²	1	4	yes	—
18	#14582981	C₂²	1	4	yes	—
19	#14582898	—	2	2	no	—
20	#14582854	C₂²	1	1	yes	characteristic
21	#14582702	C₂²	1	4	yes	—
22	#14582699	C₂²	1	4	yes	—
23	#14582674	C₂²	1	2	yes	—
24	C₁₆×Dic₄₃	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₄₃	C₂²	1	2	yes	—
27	C₄₃×#366	C₂²	1	2	yes	—
28	C₄₃×#366	C₂²	1	2	yes	—
29	C₂×#10621898	—	2	2	no	—
30	#13063347	—	2	4	no	—
31	#13063347	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₁₇₂	C₂³	1	2	yes	—
33	C₂×#10621873	—	2	2	no	—
34	C₂×D₃₄₄	C₂³	1	2	yes	—
35	C₂×#10621877	D₄	1	1	yes	characteristic
36	#13063290	—	2	4	no	—
37	#13063287	—	2	4	no	—
38	C₄×C₂²⋊D₄₃	—	4	4	no	—
39	#13063284	D₄	1	1	yes	characteristic
40	#13063283	—	4	2	no	—
41	C₄⋊C₄×D₄₃	—	4	2	no	—
42	Dic₃₄₄	C₂×C₄	1	4	yes	complemented
43	#13063270	C₂×C₄	1	8	yes	complemented
44	D₆₈₈	C₂×C₄	1	4	yes	complemented
45	#13063265	—	2	2	no	—
46	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
47	#13063248	—	2	4	no	—
48	#13063245	—	2	4	no	—
49	#13063235	D₄	1	2	yes	—
50	C₈×Dic₄₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
56	#107×C₄₃	—	2	2	no	—
57	#107×C₄₃	C₂³	1	4	yes	—
58	C₂²×C₄×C₄₃	—	4	2	no	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₃	—	4	2	no	—
61	#10621898	—	2	8	no	—
62	#10621898	—	4	4	no	—
63	C₂×D₁₇₂	—	2	2	no	—
64	C₂×C₄×D₄₃	—	4	2	no	—
65	C₂×C₄×D₄₃	—	4	4	no	—
66	C₂×Dic₈₆	—	2	2	no	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	#41	1	2	yes	—
70	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₄₃	—	2	4	no	—
72	C₄⋊C₄×C₄₃	—	4	2	no	—
73	C₂²⋊Dic₄₃	D₈	1	2	yes	—
74	(C₂×D₄₃)⋊C₄	—	4	4	no	—
75	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
76	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
77	Dic₄₃⋊C₄	—	4	4	no	—
78	C₄×Dic₄₃	—	4	4	no	—
79	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
80	#10621877	—	2	2	no	—
81	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
82	Dic₁₇₂	—	2	2	no	—
83	Dic₁₇₂	C₂²×C₄	1	4	yes	—
84	D₃₄₄	—	2	2	no	—
85	D₃₄₄	C₂²×C₄	1	4	yes	—
86	#10621873	—	2	8	no	—
87	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
89	C₄₃⋊C₁₆	—	2	2	no	—
90	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
91	C₂²×D₄₃	—	4	2	no	—
92	C₂×C₄×C₄₃	—	4	4	no	—
93	C₂×C₄×C₄₃	—	16	2	no	—
94	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	#135	1	2	yes	—
97	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₄₃	—	4	8	no	—
99	C₂×Dic₄₃	—	4	2	no	—
100	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
101	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
102	D₁₇₂	—	2	4	no	—
103	D₁₇₂	—	4	2	no	—
104	C₄×D₄₃	—	4	8	no	—
105	C₄×D₄₃	—	16	4	no	—
106	Dic₈₆	—	2	4	no	—
107	Dic₈₆	—	4	2	no	—
108	C₈×C₄₃	C₄×D₄	1	2	yes	—
109	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
111	C₄₃⋊C₈	—	2	2	no	—
112	#33459	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	—	2	1	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	C₄×D₈	1	2	yes	—
116	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
117	C₂×D₄₃	—	4	4	no	—
118	C₂×D₄₃	—	8	2	no	—
119	C₂×D₄₃	—	8	2	no	—
120	C₄×C₄₃	—	8	4	no	—
121	C₄×C₄₃	C₄×D₈	1	2	yes	—
122	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
124	Dic₄₃	—	2	2	no	—
125	Dic₄₃	—	2	2	no	—
126	Dic₄₃	—	4	4	no	—
127	C₂×#508	—	43	1	no	—
128	#2951	—	43	2	no	—
129	#2175	—	43	4	no	—
130	C₄×Q₃₂	—	43	2	no	—
131	C₄×D₁₆	—	43	2	no	—
132	C₂×#366	D₄₃	1	1	yes	characteristic, complemented
133	#2139	—	43	2	no	—
134	#2108	—	43	1	no	—
135	C₂×C₄₃	—	2	2	no	—
136	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
137	C₂×C₄₃	#3415	1	1	yes	characteristic
138	C₂×C₄₃	#3417	1	1	yes	characteristic
139	D₄₃	—	8	4	no	—
140	C₂×#135	—	43	2	no	—
141	#520	—	86	2	no	—
142	#508	—	43	8	no	—
143	C₂×Q₃₂	—	43	1	no	—
144	C₂×SD₃₂	—	43	2	no	—
145	C₂×D₁₆	—	43	1	no	—
146	C₂×M₃₂	—	43	1	no	—
147	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
148	#440	—	86	4	no	—
149	C₄×Q₁₆	—	43	4	no	—
150	C₄×D₈	—	43	4	no	—
151	C₂×#107	C₂×D₄₃	1	2	yes	—
152	#417	—	86	2	no	—
153	#405	—	43	1	no	—
154	#366	C₂×D₄₃	1	2	yes	—
155	#366	C₂×D₄₃	1	2	yes	—
156	#358	—	43	4	no	—
157	#357	—	43	4	no	—
158	#346	—	43	2	no	—
159	C₄×C₁₆	—	43	2	no	—
160	C₄₃	#33459	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	86	2	no	—
162	#135	—	43	8	no	—
163	#135	—	86	4	no	—
164	C₂×Q₁₆	—	43	2	no	—
165	C₂×SD₁₆	—	86	2	no	—
166	C₂×D₈	—	43	2	no	—
167	C₂×M₁₆	—	43	1	no	—
168	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
169	C₄×Q₈	—	86	4	no	—
170	C₄×D₄	—	86	4	no	—
171	C₄×D₄	—	172	4	no	—
172	#117	—	43	1	no	—
173	#117	—	172	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	172	2	no	—
176	Q₃₂	—	43	4	no	—
177	SD₃₂	—	43	8	no	—
178	D₁₆	—	43	4	no	—
179	M₃₂	—	86	2	no	—
180	C₂×C₁₆	—	43	2	no	—
181	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
182	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
184	#107	—	2	2	no	—
185	#107	C₂²×D₄₃	1	4	yes	—
186	#103	—	86	4	no	—
187	#102	—	86	4	no	—
188	#97	—	43	2	no	—
189	C₄×C₈	—	43	2	no	—
190	#41	—	86	8	no	—
191	#41	—	172	4	no	—
192	C₂×Q₈	—	86	2	no	—
193	C₂×D₄	—	86	2	no	—
194	C₂×D₄	—	172	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	172	4	no	—
197	C₂²×C₄	—	172	2	no	—
198	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
199	Q₁₆	—	43	4	no	—
200	Q₁₆	—	86	2	no	—
201	SD₁₆	—	86	8	no	—
202	D₈	—	43	4	no	—
203	D₈	—	86	2	no	—
204	M₁₆	—	86	2	no	—
205	C₂×C₈	—	43	2	no	—
206	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
207	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
208	C₂×C₈	#10621902	1	2	yes	—
209	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	43	1	no	—
213	C₄⋊C₄	—	43	1	no	—
214	C₄⋊C₄	—	172	4	no	—
215	C₂²⋊C₄	—	43	2	no	—
216	C₂²⋊C₄	—	172	4	no	—
217	C₄²	—	43	2	no	—
218	C₄²	—	172	4	no	—
219	C₁₆	—	86	2	no	—
220	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
221	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
222	C₂³	—	172	2	no	—
223	C₂³	D₈×D₄₃	1	1	yes	characteristic
224	Q₈	—	86	4	no	—
225	Q₈	—	172	2	no	—
226	D₄	—	86	4	no	—
227	D₄	—	172	8	no	—
228	D₄	—	172	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	16	2	no	—
231	C₂×C₄	—	43	2	no	—
232	C₂×C₄	—	43	2	no	—
233	C₂×C₄	—	172	2	no	—
234	C₂×C₄	—	172	8	no	—
235	C₂×C₄	—	688	4	no	—
236	C₂×C₄	#13063287	1	2	yes	—
237	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
238	C₂×C₄	#13063348	1	2	yes	—
239	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
240	C₈	—	86	2	no	—
241	C₈	#13063287	1	2	yes	—
242	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
243	C₈	C₂×#10621902	1	1	yes	characteristic
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	344	2	no	—
247	C₂²	—	344	2	no	—
248	C₂²	—	172	4	no	—
249	C₂²	#14582981	1	2	yes	—
250	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
251	C₄	—	8	4	no	—
252	C₄	—	86	2	no	—
253	C₄	—	86	2	no	—
254	C₄	—	172	4	no	—
255	C₄	#14582981	1	2	yes	—
256	C₄	C₂×#13063287	1	1	yes	characteristic
257	C₄	C₂×#13063348	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	344	4	no	—
260	C₂	C₂×#14582981	1	1	yes	characteristic
261	C₂	#19147732	1	1	yes	characteristic
262	C₂	#19147744	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#366	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#14583331	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#366	D₄₃
hypocenter	C₄₃	#33459
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #19143469	C₂	semidirect
3. #19136458	C₂	semidirect
4. #19136453	C₂	semidirect
5. #19136441	C₂	semidirect
6. #19136305	C₂	semidirect
7. #19136191	C₂	semidirect
8. C₂×C₄₃×#366	C₂	semidirect
11. #14583331	C₄	semidirect
31. #13063347	C₂×C₄	semidirect
42. Dic₃₄₄	C₂×C₄	semidirect
43. #13063270	C₂×C₄	semidirect
44. D₆₈₈	C₂×C₄	semidirect
53. C₂×C₁₆×C₄₃	C₂×C₄	semidirect
132. C₂×#366	D₄₃	semidirect
160. C₄₃	#33459	semidirect
181. C₂×C₁₆	C₄×D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583331	2
#19143469	2
#19136458	2
#19136453	2
#19136441	2
#19136305	2
#19136191	2
C₂×C₄₃×#366	2
#33459	43