Subgroups — #25249682

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582853	C₂	1	1	yes	maximal, characteristic
2	#19136403	C₂	1	4	yes	maximal, complemented
3	#19136400	C₂	1	4	yes	maximal, complemented
4	#19136350	C₂	1	2	yes	maximal, complemented
5	#19136205	C₂	1	2	yes	maximal
6	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621872	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
10	C₂×#13063282	C₄	1	1	yes	characteristic
11	C₂×#13063265	C₂²	1	2	yes	—
12	#14582853	C₄	1	2	yes	—
13	#14582853	C₄	1	2	yes	—
14	#14582853	C₂²	1	2	yes	—
15	#14582853	C₂²	1	2	yes	—
16	C₂×#13063236	C₄	1	1	yes	characteristic
17	C₂×#13063235	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
19	Dic₄₃⋊C₁₆	C₂²	1	8	yes	complemented
20	(C₂×D₄₃)⋊C₁₆	C₂²	1	8	yes	—
21	#14582675	C₂²	1	2	yes	complemented
22	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#346	C₂²	1	2	yes	complemented
26	C₄×C₁₆×C₄₃	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#10621872	—	2	4	no	—
29	C₂×#10621872	C₂×C₄	1	2	yes	—
30	C₂×#10621872	C₂×C₄	1	2	yes	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₄₃	—	2	4	no	—
34	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₄₃	C₂³	1	2	yes	—
36	#13063282	C₈	1	4	yes	—
37	#13063282	C₂×C₄	1	2	yes	—
38	#13063282	C₂×C₄	1	2	yes	—
39	C₂×Dic₄₃⋊C₄	C₈	1	2	yes	—
40	C₂×(C₂×D₄₃)⋊C₄	C₈	1	2	yes	—
41	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
42	#13063265	—	2	4	no	—
43	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
44	#13063236	C₂×C₄	1	2	yes	—
45	#13063236	C₂×C₄	1	2	yes	—
46	#13063235	C₂×C₄	1	2	yes	—
47	#13063235	C₂×C₄	1	2	yes	—
48	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₄₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₄₃	—	2	4	no	—
54	C₂×C₁₆×C₄₃	C₂³	1	4	yes	—
55	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
59	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₄₃	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₄₃	—	2	2	no	—
65	C₂×C₄×D₄₃	—	2	2	no	—
66	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
67	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₄₃	M₁₆	1	4	yes	—
69	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₄₃	#41	1	4	yes	—
73	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₄²×C₄₃	C₂×C₈	1	2	yes	—
75	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
77	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
78	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
79	Dic₄₃⋊C₄	C₂×C₈	1	4	yes	—
80	Dic₄₃⋊C₄	C₂×C₈	1	4	yes	—
81	C₄×Dic₄₃	C₂×C₈	1	2	yes	—
82	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄₃⋊C₈	—	2	2	no	—
84	C₂×C₄₃⋊C₈	—	2	2	no	—
85	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
87	#10621872	—	2	4	no	—
88	#10621872	—	2	4	no	—
89	C₈×D₄₃	—	2	4	no	—
90	C₈×D₄₃	—	2	4	no	—
91	C₁₆×C₄₃	—	2	4	no	—
92	C₄₃⋊C₁₆	—	2	4	no	—
93	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₄₃	—	2	2	no	—
95	C₂²×D₄₃	—	4	2	no	—
96	C₂²×D₄₃	C₂²×C₈	1	2	yes	—
97	C₂×C₄×C₄₃	—	4	2	no	—
98	C₂×C₄×C₄₃	#117	1	2	yes	—
99	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
101	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₄₃	—	2	2	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
107	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
108	C₄×D₄₃	—	2	2	no	—
109	C₄×D₄₃	—	2	2	no	—
110	C₄×D₄₃	—	2	4	no	—
111	C₈×C₄₃	#117	1	2	yes	—
112	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
113	C₈×C₄₃	C₂×M₁₆	1	2	yes	—
114	C₈×C₄₃	C₂×#41	1	2	yes	—
115	C₄₃⋊C₈	—	2	2	no	—
116	C₄₃⋊C₈	—	2	2	no	—
117	#33419	—	43	1	no	maximal, Sylow
118	C₂²×C₄₃	—	2	2	no	—
119	C₂²×C₄₃	#431	1	2	yes	—
120	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
121	C₂×D₄₃	—	2	4	no	—
122	C₂×D₄₃	—	2	2	no	—
123	C₂×D₄₃	—	4	4	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	—	2	2	no	—
126	C₄×C₄₃	#431	1	2	yes	—
127	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
128	C₄×C₄₃	C₂²×M₁₆	1	1	yes	characteristic
129	Dic₄₃	—	2	4	no	—
130	Dic₄₃	—	2	2	no	—
131	C₂×#405	—	43	1	no	—
132	#2164	—	43	4	no	—
133	#2164	—	43	4	no	—
134	#2150	—	43	2	no	—
135	#2112	—	43	2	no	—
136	C₄×M₃₂	—	43	1	no	—
137	C₄×M₃₂	D₄₃	1	1	yes	characteristic, complemented
138	C₂×C₄₃	—	2	2	no	—
139	C₂×C₄₃	C₂×#431	1	1	yes	characteristic
140	C₂×C₄₃	#3407	1	2	yes	—
141	D₄₃	—	4	2	no	—
142	C₂²×M₁₆	—	43	1	no	—
143	C₂³×C₈	—	43	1	no	—
144	C₂×#117	—	43	1	no	—
145	C₂×M₃₂	—	43	2	no	—
146	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
147	#405	—	43	2	no	—
148	#405	—	43	2	no	—
149	#405	—	43	2	no	—
150	#405	—	43	2	no	—
151	C₂×#97	—	43	1	no	—
152	C₂×#97	—	43	1	no	—
153	C₂×C₄×C₈	—	43	1	no	—
154	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
155	C₄⋊C₁₆	—	43	8	no	—
156	C₂²⋊C₁₆	—	43	8	no	—
157	#346	—	43	2	no	—
158	#346	C₂×D₄₃	1	2	yes	complemented
159	C₄×C₁₆	—	43	2	no	—
160	C₄×C₁₆	C₂×D₄₃	1	2	yes	complemented
161	C₄₃	#33419	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₄	—	43	1	no	—
163	C₂×M₁₆	—	43	2	no	—
164	C₂×M₁₆	—	43	2	no	—
165	C₂×M₁₆	—	86	4	no	—
166	C₂²×C₈	—	43	1	no	—
167	C₂²×C₈	—	43	1	no	—
168	C₂²×C₈	—	43	2	no	—
169	C₂²×C₈	—	43	2	no	—
170	C₂²×C₈	—	86	4	no	—
171	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	#117	—	43	4	no	—
174	#117	—	43	2	no	—
175	#117	—	43	2	no	—
176	C₂×C₄⋊C₄	—	43	2	no	—
177	C₂×C₂²⋊C₄	—	43	2	no	—
178	C₂×C₄²	—	43	1	no	—
179	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
180	M₃₂	—	2	4	no	—
181	M₃₂	—	86	4	no	—
182	C₂×C₁₆	—	43	4	no	—
183	C₂×C₁₆	C₂²×D₄₃	1	4	yes	—
184	#97	—	43	2	no	—
185	#97	—	43	2	no	—
186	#97	—	43	2	no	—
187	#97	—	43	2	no	—
188	C₄×C₈	—	43	2	no	—
189	C₄×C₈	—	43	2	no	—
190	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
191	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	C₂⁴	—	43	1	no	—
195	C₂²×C₄	—	43	1	no	—
196	C₂²×C₄	—	43	2	no	—
197	C₂²×C₄	—	43	2	no	—
198	C₂²×C₄	—	43	2	no	—
199	C₂²×C₄	—	86	2	no	—
200	C₂²×C₄	—	86	2	no	—
201	C₂²×C₄	C₈×D₄₃	1	2	yes	—
202	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
203	M₁₆	—	86	4	no	—
204	M₁₆	—	86	4	no	—
205	C₂×C₈	—	43	2	no	—
206	C₂×C₈	—	43	2	no	—
207	C₂×C₈	—	86	2	no	—
208	C₂×C₈	—	86	2	no	—
209	C₂×C₈	—	86	4	no	—
210	C₂×C₈	—	86	4	no	—
211	C₂×C₈	#10621872	1	4	yes	—
212	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
215	C₂×C₈	#10621898	1	4	yes	—
216	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
217	C₄⋊C₄	—	43	4	no	—
218	C₄⋊C₄	—	43	4	no	—
219	C₂²⋊C₄	—	43	4	no	—
220	C₂²⋊C₄	—	43	4	no	—
221	C₄²	—	43	2	no	—
222	C₄²	—	43	2	no	—
223	C₄²	C₈×D₄₃	1	2	yes	—
224	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
225	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
226	C₁₆	—	2	4	no	—
227	C₁₆	—	86	4	no	—
228	C₂³	—	43	2	no	—
229	C₂³	—	86	2	no	—
230	C₂³	—	172	2	no	—
231	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	43	4	no	—
234	C₂×C₄	—	43	2	no	—
235	C₂×C₄	—	86	2	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	86	2	no	—
238	C₂×C₄	—	86	4	no	—
239	C₂×C₄	—	172	2	no	—
240	C₂×C₄	#13063282	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#10621872	1	2	yes	—
245	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
246	C₈	—	86	2	no	—
247	C₈	—	86	2	no	—
248	C₈	#13063282	1	2	yes	—
249	C₈	C₂×#10621872	1	2	yes	—
250	C₈	C₂×#10621872	1	2	yes	—
251	C₈	C₂×#10621898	1	2	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	86	2	no	—
254	C₂²	—	86	4	no	—
255	C₂²	—	172	4	no	—
256	C₂²	#14582950	1	2	yes	—
257	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	86	2	no	—
261	C₄	—	86	4	no	—
262	C₄	#14582950	1	2	yes	—
263	C₄	C₂×#13063282	1	1	yes	characteristic
264	C₄	C₂²×#10621872	1	1	yes	characteristic
265	C₂	—	2	2	no	—
266	C₂	—	172	2	no	—
267	C₂	C₂×#14582950	1	1	yes	characteristic
268	C₂	#19147697	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₄×M₃₂×C₄₃	C₂
prime radical	C₂³×D₄₃	C₂×C₈
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	C₄×M₃₂	D₄₃
hypocenter	C₄₃	#33419
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #19136403	C₂	semidirect
3. #19136400	C₂	semidirect
4. #19136350	C₂	semidirect
6. C₄×#13063265	C₂	semidirect
7. C₄×M₃₂×C₄₃	C₂	semidirect
19. Dic₄₃⋊C₁₆	C₂²	semidirect
21. #14582675	C₂²	semidirect
22. C₄×C₄₃⋊C₁₆	C₂²	semidirect
25. C₄₃×#346	C₂²	semidirect
26. C₄×C₁₆×C₄₃	C₂²	semidirect
137. C₄×M₃₂	D₄₃	semidirect
158. #346	C₂×D₄₃	semidirect
160. C₄×C₁₆	C₂×D₄₃	semidirect
161. C₄₃	#33419	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582853	2
#19136403	2
#19136400	2
#19136350	2
#19136205	2
C₄×#13063265	2
C₄×M₃₂×C₄₃	2
#33419	43