Subgroups — #25250011

Order 11008.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 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	C₂×#14583017	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	#19136324	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	2	yes	—
9	C₂×#13063331	—	2	2	no	—
10	#14583342	C₄	1	4	yes	complemented
11	#14583337	C₄	1	4	yes	complemented
12	#14583331	C₂²	1	4	yes	—
13	#14583017	—	2	2	no	—
14	#14583017	—	2	2	no	—
15	#14583017	C₂²	1	4	yes	complemented
16	#14583017	C₂²	1	4	yes	complemented
17	C₂×#13063261	C₂²	1	2	yes	—
18	C₂×#13063253	—	2	2	no	—
19	#14582729	C₂²	1	4	yes	complemented
20	#14582707	C₂²	1	8	yes	complemented
21	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₄₃	C₂²	1	2	yes	complemented
26	C₂×#10621898	—	2	2	no	—
27	#13063347	—	2	2	no	—
28	#13063347	—	2	2	no	—
29	#13063347	C₂×C₄	1	4	yes	—
30	#13063347	C₂³	1	4	yes	—
31	C₂×Dic₁₇₂	C₂×C₄	1	2	yes	—
32	C₂×#10621873	—	2	2	no	—
33	C₂×D₃₄₄	C₂×C₄	1	2	yes	—
34	#13063331	—	2	4	no	—
35	#13063331	—	2	4	no	—
36	#13063331	—	4	2	no	—
37	#13063331	—	4	2	no	—
38	C₂×#10621877	—	2	2	no	—
39	C₂×#10621872	—	4	2	no	—
40	C₂×C₈×D₄₃	—	4	2	no	—
41	Dic₃₄₄	C₂×C₄	1	4	yes	complemented
42	#13063270	C₂×C₄	1	8	yes	complemented
43	D₆₈₈	C₂×C₄	1	4	yes	complemented
44	#13063261	—	2	2	no	—
45	#13063261	C₂³	1	4	yes	—
46	#13063253	—	2	4	no	—
47	#13063253	—	2	4	no	—
48	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
51	M₃₂×C₄₃	—	2	2	no	—
52	M₃₂×C₄₃	C₂×C₄	1	4	yes	—
53	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
54	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
55	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	1	no	—
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₃	—	4	2	no	—
62	#10621898	—	2	4	no	—
63	#10621898	—	2	4	no	—
64	#10621898	—	4	2	no	—
65	#10621898	—	4	2	no	—
66	C₂×D₁₇₂	—	2	2	no	—
67	C₂×C₄×D₄₃	—	4	2	no	—
68	C₂×Dic₈₆	—	2	2	no	—
69	C₂×C₈×C₄₃	D₈	1	2	yes	—
70	C₂×C₈×C₄₃	D₈	1	2	yes	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₄₃	#41	1	2	yes	—
76	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
77	C₄²×C₄₃	—	2	1	no	—
78	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
80	#10621877	—	2	4	no	—
81	#10621877	—	4	2	no	—
82	C₂×C₄₃⋊C₈	—	4	2	no	—
83	Dic₁₇₂	—	2	2	no	—
84	Dic₁₇₂	C₂²×C₄	1	4	yes	—
85	D₃₄₄	—	2	2	no	—
86	D₃₄₄	C₂²×C₄	1	4	yes	—
87	#10621873	—	2	4	no	—
88	#10621873	—	2	4	no	—
89	#10621872	—	4	4	no	—
90	#10621872	—	4	4	no	—
91	C₈×D₄₃	—	4	4	no	—
92	C₈×D₄₃	—	4	4	no	—
93	C₁₆×C₄₃	—	2	2	no	—
94	C₁₆×C₄₃	C₂²×C₄	1	4	yes	—
95	C₂³×C₄₃	#135	1	1	yes	characteristic
96	C₂²×D₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	—	2	1	no	—
98	C₂×C₄×C₄₃	—	2	1	no	—
99	C₂×C₄×C₄₃	—	4	2	no	—
100	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₄₃	—	4	4	no	—
105	C₂²⋊D₄₃	—	4	4	no	—
106	C₂×Dic₄₃	—	4	2	no	—
107	D₁₇₂	—	2	4	no	—
108	D₁₇₂	—	4	2	no	—
109	C₄×D₄₃	—	4	4	no	—
110	C₄×D₄₃	—	4	4	no	—
111	Dic₈₆	—	2	4	no	—
112	Dic₈₆	—	4	2	no	—
113	C₈×C₄₃	C₄×D₄	1	2	yes	—
114	C₈×C₄₃	C₂×D₈	1	2	yes	—
115	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
116	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
117	C₈×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₄₃⋊C₈	—	4	4	no	—
120	#33463	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	—	4	1	no	—
122	C₂²×C₄₃	C₄×D₈	1	2	yes	—
123	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
124	C₂×D₄₃	—	4	4	no	—
125	C₂×D₄₃	—	8	4	no	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	4	1	no	—
128	C₄×C₄₃	C₄×D₈	1	2	yes	—
129	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
131	Dic₄₃	—	4	4	no	—
132	#3414	—	43	1	no	—
133	C₂×#443	—	43	2	no	—
134	#2179	—	43	4	no	—
135	#2178	—	43	4	no	—
136	#2161	—	43	1	no	—
137	#2143	—	43	2	no	—
138	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
139	C₂×C₄₃	—	2	1	no	—
140	C₂×C₄₃	—	2	2	no	—
141	C₂×C₄₃	C₂×C₄×D₈	1	1	yes	characteristic
142	D₄₃	—	8	4	no	—
143	C₂×#135	—	43	2	no	—
144	C₂×#131	—	86	2	no	—
145	#510	—	43	4	no	—
146	#509	—	43	4	no	—
147	#508	—	43	4	no	—
148	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
149	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
150	#443	—	43	4	no	—
151	#443	—	43	4	no	—
152	#443	—	86	2	no	—
153	#443	—	86	2	no	—
154	C₂×#108	—	43	2	no	—
155	C₂×#104	—	86	2	no	—
156	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
157	#368	—	43	4	no	—
158	#359	—	43	8	no	—
159	#346	C₂×D₄₃	1	2	yes	complemented
160	C₄×C₁₆	C₂×D₄₃	1	2	yes	complemented
161	C₄₃	#33463	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	86	2	no	—
163	#135	—	43	4	no	—
164	#135	—	43	4	no	—
165	#135	—	86	2	no	—
166	#135	—	86	2	no	—
167	C₂×Q₁₆	—	43	2	no	—
168	C₂×SD₁₆	—	86	2	no	—
169	C₂×D₈	—	43	2	no	—
170	#131	—	86	4	no	—
171	#131	—	86	4	no	—
172	#131	—	172	2	no	—
173	#131	—	172	2	no	—
174	C₂×M₁₆	—	86	2	no	—
175	C₂×M₁₆	—	172	2	no	—
176	C₂²×C₈	—	172	2	no	—
177	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
179	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
180	Q₃₂	—	43	4	no	—
181	SD₃₂	—	43	8	no	—
182	D₁₆	—	43	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	C₄×D₄₃	1	4	yes	—
185	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
186	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
187	#108	—	43	4	no	—
188	#108	—	86	2	no	—
189	#104	—	86	4	no	—
190	#104	—	86	4	no	—
191	C₄×C₈	D₁₇₂	1	1	yes	characteristic
192	C₄×C₈	D₁₇₂	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
195	#41	—	86	4	no	—
196	#41	—	86	4	no	—
197	#41	—	172	2	no	—
198	#41	—	172	2	no	—
199	C₂×Q₈	—	86	2	no	—
200	C₂×D₄	—	86	2	no	—
201	C₂×D₄	—	172	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	172	2	no	—
204	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
205	Q₁₆	—	43	4	no	—
206	Q₁₆	—	86	2	no	—
207	SD₁₆	—	86	4	no	—
208	SD₁₆	—	86	4	no	—
209	D₈	—	43	4	no	—
210	D₈	—	86	2	no	—
211	M₁₆	—	86	4	no	—
212	M₁₆	—	172	2	no	—
213	M₁₆	—	172	4	no	—
214	M₁₆	—	172	4	no	—
215	C₂×C₈	—	172	2	no	—
216	C₂×C₈	—	172	4	no	—
217	C₂×C₈	—	172	4	no	—
218	C₂×C₈	D₃₄₄	1	2	yes	—
219	C₂×C₈	D₃₄₄	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
221	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
222	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
223	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
224	C₂×C₈	#10621898	1	2	yes	—
225	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
226	C₄²	—	2	1	no	—
227	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
228	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
229	C₁₆	—	2	2	no	—
230	C₁₆	C₂×C₄×D₄₃	1	4	yes	—
231	C₂³	—	172	2	no	—
232	C₂³	#13063347	1	1	yes	characteristic
233	Q₈	—	86	4	no	—
234	Q₈	—	172	2	no	—
235	D₄	—	86	4	no	—
236	D₄	—	172	4	no	—
237	D₄	—	172	4	no	—
238	D₄	—	172	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	172	2	no	—
243	C₂×C₄	—	172	4	no	—
244	C₂×C₄	—	172	4	no	—
245	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
246	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
247	C₂×C₄	#13063347	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
249	C₈	—	172	4	no	—
250	C₈	C₄×D₁₇₂	1	2	yes	—
251	C₈	C₂×D₃₄₄	1	2	yes	—
252	C₈	C₂×D₃₄₄	1	1	yes	characteristic
253	C₈	C₂×D₃₄₄	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₄₃	1	1	yes	characteristic
255	C₈	C₂×#10621898	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	344	4	no	—
258	C₂²	—	172	4	no	—
259	C₂²	C₄×D₃₄₄	1	2	yes	—
260	C₂²	C₂×#13063347	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	172	4	no	—
264	C₄	C₄×D₃₄₄	1	2	yes	—
265	C₄	C₂×C₄×D₁₇₂	1	1	yes	characteristic
266	C₄	C₂²×D₃₄₄	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	344	4	no	—
270	C₂	C₂×C₄×D₃₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₃₄₄
commutator	C₈×C₄₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×M₃₂×C₄₃	C₂
socle	C₂×C₄₃	C₂×C₄×D₈
prime radical	#19147731	C₂
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#33463
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#14583017	C₂	semidirect
3. #19136482	C₂	semidirect
4. #19136477	C₂	semidirect
5. #19136390	C₂	semidirect
6. #19136324	C₂	semidirect
7. C₄×M₃₂×C₄₃	C₂	semidirect
10. #14583342	C₄	semidirect
11. #14583337	C₄	semidirect
15. #14583017	C₂²	semidirect
16. #14583017	C₂²	semidirect
19. #14582729	C₂²	semidirect
20. #14582707	C₂²	semidirect
24. C₄₃×#346	C₂²	semidirect
25. C₄×C₁₆×C₄₃	C₂²	semidirect
41. Dic₃₄₄	C₂×C₄	semidirect
42. #13063270	C₂×C₄	semidirect
43. D₆₈₈	C₂×C₄	semidirect
138. C₄×M₃₂	D₄₃	semidirect
159. #346	C₂×D₄₃	semidirect
160. C₄×C₁₆	C₂×D₄₃	semidirect
161. C₄₃	#33463	semidirect

Maximal subgroups

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

subgroup	index
#19147731	2
C₂×#14583017	2
#19136482	2
#19136477	2
#19136390	2
#19136324	2
C₄×M₃₂×C₄₃	2
#33463	43