Subgroups — #25249990

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 254 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	#19136470	C₂	1	2	yes	maximal, complemented
4	#19136465	C₂	1	2	yes	maximal, complemented
5	#19136446	C₂	1	4	yes	maximal, complemented
6	#19136305	C₂	1	2	yes	maximal, complemented
7	#19136191	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#367	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×#367	C₂²	1	2	yes	—
28	C₄₃×#367	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	2	yes	—
76	Dic₄₃⋊C₄	—	4	4	no	—
77	C₄×Dic₄₃	—	4	4	no	—
78	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
79	#10621877	—	2	2	no	—
80	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
81	Dic₁₇₂	—	2	2	no	—
82	Dic₁₇₂	C₂²×C₄	1	4	yes	—
83	D₃₄₄	—	2	2	no	—
84	D₃₄₄	C₂²×C₄	1	4	yes	—
85	#10621873	—	2	8	no	—
86	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
87	C₁₆×C₄₃	C₂²×C₄	1	2	yes	—
88	C₄₃⋊C₁₆	—	2	2	no	—
89	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
90	C₂²×D₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	—	4	4	no	—
92	C₂×C₄×C₄₃	—	16	2	no	—
93	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#135	1	2	yes	—
96	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₄₃	—	4	8	no	—
98	C₂×Dic₄₃	—	4	2	no	—
99	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
100	D₁₇₂	—	2	4	no	—
101	D₁₇₂	—	4	2	no	—
102	C₄×D₄₃	—	4	8	no	—
103	C₄×D₄₃	—	16	4	no	—
104	Dic₈₆	—	2	4	no	—
105	Dic₈₆	—	4	2	no	—
106	C₈×C₄₃	C₄×D₄	1	2	yes	—
107	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
109	C₄₃⋊C₈	—	2	2	no	—
110	#33460	—	43	1	no	maximal, Sylow
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	C₄×D₈	1	2	yes	—
113	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
114	C₂×D₄₃	—	4	4	no	—
115	C₂×D₄₃	—	8	2	no	—
116	C₂×D₄₃	—	8	2	no	—
117	C₄×C₄₃	—	8	4	no	—
118	C₄×C₄₃	C₄×D₈	1	2	yes	—
119	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
121	Dic₄₃	—	2	4	no	—
122	Dic₄₃	—	4	4	no	—
123	C₂×#508	—	43	1	no	—
124	#2951	—	43	2	no	—
125	#2177	—	43	2	no	—
126	#2176	—	43	2	no	—
127	C₄×SD₃₂	—	43	4	no	—
128	C₂×#367	D₄₃	1	1	yes	characteristic, complemented
129	#2139	—	43	2	no	—
130	#2108	—	43	1	no	—
131	C₂×C₄₃	—	2	2	no	—
132	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
133	C₂×C₄₃	#3416	1	2	yes	—
134	D₄₃	—	8	4	no	—
135	C₂×#135	—	43	2	no	—
136	#520	—	86	2	no	—
137	#508	—	43	8	no	—
138	C₂×Q₃₂	—	43	1	no	—
139	C₂×SD₃₂	—	43	2	no	—
140	C₂×D₁₆	—	43	1	no	—
141	C₂×M₃₂	—	43	1	no	—
142	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
143	#440	—	86	4	no	—
144	C₄×Q₁₆	—	43	4	no	—
145	C₄×D₈	—	43	4	no	—
146	C₂×#107	C₂×D₄₃	1	2	yes	—
147	#417	—	86	2	no	—
148	#405	—	43	1	no	—
149	#367	C₂×D₄₃	1	2	yes	—
150	#367	C₂×D₄₃	1	2	yes	—
151	#358	—	43	4	no	—
152	#357	—	43	4	no	—
153	#346	—	43	2	no	—
154	C₄×C₁₆	—	43	2	no	—
155	C₄₃	#33460	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	86	2	no	—
157	#135	—	43	8	no	—
158	#135	—	86	4	no	—
159	C₂×Q₁₆	—	43	2	no	—
160	C₂×SD₁₆	—	86	2	no	—
161	C₂×D₈	—	43	2	no	—
162	C₂×M₁₆	—	43	1	no	—
163	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
164	C₄×Q₈	—	86	4	no	—
165	C₄×D₄	—	86	4	no	—
166	C₄×D₄	—	172	4	no	—
167	#117	—	43	1	no	—
168	#117	—	172	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	172	2	no	—
171	Q₃₂	—	43	4	no	—
172	SD₃₂	—	43	8	no	—
173	D₁₆	—	43	4	no	—
174	M₃₂	—	86	2	no	—
175	C₂×C₁₆	—	43	2	no	—
176	C₂×C₁₆	C₄×D₄₃	1	4	yes	complemented
177	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
178	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
179	#107	—	2	2	no	—
180	#107	C₂²×D₄₃	1	4	yes	—
181	#103	—	86	4	no	—
182	#102	—	86	4	no	—
183	#97	—	43	2	no	—
184	C₄×C₈	—	43	2	no	—
185	#41	—	86	8	no	—
186	#41	—	172	4	no	—
187	C₂×Q₈	—	86	2	no	—
188	C₂×D₄	—	86	2	no	—
189	C₂×D₄	—	172	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	172	4	no	—
192	C₂²×C₄	—	172	2	no	—
193	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
194	Q₁₆	—	43	4	no	—
195	Q₁₆	—	86	2	no	—
196	SD₁₆	—	86	8	no	—
197	D₈	—	43	4	no	—
198	D₈	—	86	2	no	—
199	M₁₆	—	86	2	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
202	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
203	C₂×C₈	#10621902	1	2	yes	—
204	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	43	2	no	—
208	C₄⋊C₄	—	172	4	no	—
209	C₂²⋊C₄	—	43	2	no	—
210	C₂²⋊C₄	—	172	4	no	—
211	C₄²	—	43	2	no	—
212	C₄²	—	172	4	no	—
213	C₁₆	—	86	2	no	—
214	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
215	C₁₆	C₂×C₄×D₄₃	1	2	yes	—
216	C₂³	—	172	2	no	—
217	C₂³	D₈×D₄₃	1	1	yes	characteristic
218	Q₈	—	86	4	no	—
219	Q₈	—	172	2	no	—
220	D₄	—	86	4	no	—
221	D₄	—	172	8	no	—
222	D₄	—	172	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	16	2	no	—
225	C₂×C₄	—	43	4	no	—
226	C₂×C₄	—	172	2	no	—
227	C₂×C₄	—	172	8	no	—
228	C₂×C₄	—	688	4	no	—
229	C₂×C₄	#13063287	1	2	yes	—
230	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
231	C₂×C₄	#13063348	1	2	yes	—
232	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
233	C₈	—	86	2	no	—
234	C₈	#13063287	1	2	yes	—
235	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
236	C₈	C₂×#10621902	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	344	2	no	—
239	C₂²	—	344	2	no	—
240	C₂²	—	172	4	no	—
241	C₂²	#14582981	1	2	yes	—
242	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
243	C₄	—	8	4	no	—
244	C₄	—	86	4	no	—
245	C₄	—	172	4	no	—
246	C₄	#14582981	1	2	yes	—
247	C₄	C₂×#13063287	1	1	yes	characteristic
248	C₄	C₂×#13063348	1	1	yes	characteristic
249	C₂	—	2	2	no	—
250	C₂	—	344	4	no	—
251	C₂	C₂×#14582981	1	1	yes	characteristic
252	C₂	#19147737	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#367	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂×#14583331	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₂×#367	D₄₃
hypocenter	C₄₃	#33460
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
2. #19143469	C₂	semidirect
3. #19136470	C₂	semidirect
4. #19136465	C₂	semidirect
5. #19136446	C₂	semidirect
6. #19136305	C₂	semidirect
7. #19136191	C₂	semidirect
8. C₂×C₄₃×#367	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
128. C₂×#367	D₄₃	semidirect
155. C₄₃	#33460	semidirect
176. 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
#19136470	2
#19136465	2
#19136446	2
#19136305	2
#19136191	2
C₂×C₄₃×#367	2
#33460	43