Subgroups — #25961006

Order 11008.

1810 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19148000	C₂	1	1	yes	maximal, characteristic
2	#19143539	C₂	1	16	yes	maximal, complemented
3	#19143535	C₂	1	4	yes	maximal, complemented
4	#19143529	C₂	1	4	yes	maximal, complemented
5	#19143159	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582925	C₂	1	2	yes	maximal
7	#19143013	C₂	1	2	yes	maximal
8	C₂×C₄₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	2	yes	—
10	#14583530	C₄	1	4	yes	—
11	#14583527	C₄	1	8	yes	—
12	#14583525	C₄	1	4	yes	—
13	#14583456	C₂²	1	4	yes	—
14	#14583452	C₂²	1	8	yes	—
15	C₂×#13063284	C₂²	1	2	yes	—
16	#14583029	C₂²	1	16	yes	complemented
17	#14583027	C₂²	1	32	yes	complemented
18	C₈×Dic₈₆	C₂²	1	16	yes	complemented
19	#14582926	C₂²	1	8	yes	—
20	#14582925	C₂²	1	4	yes	—
21	#14582925	C₂²	1	4	yes	—
22	#14582925	C₂²	1	8	yes	—
23	#14582923	C₂²	1	4	yes	—
24	C₂×Dic₄₃⋊C₈	C₂²	1	8	yes	—
25	#14582857	C₂²	1	8	yes	—
26	#14582854	C₂²	1	8	yes	—
27	M₁₆×Dic₄₃	C₂²	1	8	yes	—
28	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
30	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
31	C₄₃×#405	C₂²	1	8	yes	—
32	C₂×#97×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
34	C₂×#10621877	—	2	8	no	—
35	C₂×#10621877	C₂³	1	4	yes	—
36	C₂×#10621877	C₂³	1	4	yes	—
37	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
38	#13063325	C₂×C₄	1	8	yes	—
39	#13063324	C₂×C₄	1	16	yes	—
40	#13063317	C₂×C₄	1	8	yes	—
41	#13063316	C₂×C₄	1	16	yes	—
42	#13063304	C₂×C₄	1	16	yes	—
43	#13063302	C₂×C₄	1	32	yes	—
44	C₄×Dic₈₆	C₂³	1	16	yes	—
45	#13063284	C₂×C₄	1	4	yes	—
46	#13063284	C₂×C₄	1	4	yes	—
47	#13063284	C₂³	1	4	yes	—
48	#13063284	C₂³	1	4	yes	—
49	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
52	C₄⋊(C₄₃⋊C₈)	C₂³	1	16	yes	—
53	Dic₄₃⋊C₈	C₂³	1	16	yes	—
54	Dic₄₃⋊C₈	C₂³	1	16	yes	—
55	#13063235	C₂³	1	16	yes	—
56	C₈×Dic₄₃	C₂³	1	16	yes	—
57	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₄₃	Q₈	1	8	yes	complemented
59	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₄₃	—	2	4	no	—
61	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
62	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
65	#117×C₄₃	C₂×C₄	1	4	yes	—
66	#117×C₄₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
70	#97×C₄₃	C₂³	1	8	yes	—
71	C₄×C₈×C₄₃	C₂³	1	8	yes	—
72	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₄₃	#41	1	4	yes	—
78	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
80	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
81	C₂×Dic₈₆	C₂²×C₄	1	16	yes	—
82	M₁₆×C₄₃	C₂×Q₈	1	16	yes	—
83	C₂×C₈×C₄₃	—	2	8	no	—
84	C₂×C₈×C₄₃	—	2	2	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
88	C₂×C₈×C₄₃	C₂⁴	1	2	yes	—
89	C₂×C₈×C₄₃	C₂⁴	1	4	yes	—
90	C₄⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
91	C₂²⋊C₄×C₄₃	C₂²×C₄	1	8	yes	—
92	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
93	C₄²×C₄₃	C₂⁴	1	4	yes	—
94	C₂²⋊Dic₄₃	C₂²×C₄	1	8	yes	—
95	C₂²⋊Dic₄₃	C₂²×C₄	1	8	yes	—
96	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
97	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
98	C₄⋊Dic₄₃	C₂²×C₄	1	16	yes	—
99	Dic₄₃⋊C₄	C₂²×C₄	1	32	yes	—
100	C₄×Dic₄₃	C₂²×C₄	1	8	yes	—
101	C₄×Dic₄₃	C₂⁴	1	8	yes	—
102	#10621877	—	2	8	no	—
103	#10621877	—	2	8	no	—
104	C₂×C₄₃⋊C₈	—	2	8	no	—
105	C₂×C₄₃⋊C₈	C₂⁴	1	8	yes	—
106	C₂³×C₄₃	—	4	2	no	—
107	C₂³×C₄₃	C₄×Q₈	1	4	yes	—
108	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
110	C₂×C₄×C₄₃	—	4	4	no	—
111	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
112	C₂×C₄×C₄₃	C₄×Q₈	1	8	yes	—
113	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₄₃	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
119	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
120	C₂×C₄×C₄₃	C₂⁵	1	1	yes	characteristic
121	C₂×Dic₄₃	—	4	8	no	—
122	C₂×Dic₄₃	C₂³×C₄	1	16	yes	—
123	Dic₈₆	—	4	16	no	—
124	C₈×C₄₃	—	2	2	no	—
125	C₈×C₄₃	—	2	2	no	—
126	C₈×C₄₃	C₂²×Q₈	1	8	yes	—
127	C₄₃⋊C₈	—	2	8	no	—
128	#60193	—	43	1	no	maximal, Sylow
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	4	4	no	—
132	C₂²×C₄₃	C₂×C₄×Q₈	1	2	yes	—
133	C₂²×C₄₃	C₂×C₄×Q₈	1	4	yes	—
134	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
135	C₄×C₄₃	—	2	8	no	—
136	C₄×C₄₃	C₂×C₄×Q₈	1	4	yes	—
137	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
138	C₄×C₄₃	C₂³×Q₈	1	1	yes	characteristic
139	C₄×C₄₃	C₂²×#41	1	1	yes	characteristic
140	Dic₄₃	—	2	16	no	—
141	#3458	—	43	1	no	—
142	#2962	—	43	16	no	—
143	#2961	—	43	4	no	—
144	#2960	—	43	4	no	—
145	#2905	—	43	1	no	—
146	C₂×#424	—	43	2	no	—
147	#2873	—	43	2	no	—
148	C₂×#405	D₄₃	1	1	yes	characteristic
149	C₂×C₄₃	—	2	4	no	—
150	C₂×C₄₃	—	2	2	no	—
151	C₂×C₄₃	C₂²×C₄×Q₈	1	1	yes	characteristic
152	C₂×C₄₃	#3573	1	2	yes	—
153	C₂²×M₁₆	—	43	2	no	—
154	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
155	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
156	#544	—	43	8	no	—
157	#544	—	43	4	no	—
158	#543	—	43	4	no	—
159	#533	—	43	8	no	—
160	#533	—	43	4	no	—
161	C₂×#117	—	43	2	no	—
162	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
163	#446	—	43	32	no	—
164	#446	—	43	16	no	—
165	C₈×Q₈	—	43	16	no	—
166	#424	—	43	4	no	—
167	#424	—	43	4	no	—
168	#424	—	43	8	no	—
169	#424	—	43	8	no	—
170	#423	—	43	4	no	—
171	C₂×C₄⋊C₈	—	43	8	no	—
172	#405	—	43	8	no	—
173	#405	—	43	8	no	—
174	#405	C₂×D₄₃	1	8	yes	—
175	C₄×M₁₆	—	43	8	no	—
176	C₂×#97	C₂×D₄₃	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
178	C₄₃	#60193	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
180	C₂×M₁₆	—	43	4	no	—
181	C₂×M₁₆	—	43	4	no	—
182	C₂×M₁₆	—	86	8	no	—
183	C₂×M₁₆	Dic₈₆	1	8	yes	complemented
184	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	43	4	no	—
187	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
190	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
191	C₄⋊Q₈	—	43	16	no	—
192	C₄⋊Q₈	—	43	8	no	—
193	#125	—	43	16	no	—
194	#125	—	43	8	no	—
195	C₂²⋊Q₈	—	43	16	no	—
196	C₂²⋊Q₈	—	43	32	no	—
197	C₄×Q₈	—	43	16	no	—
198	#117	—	43	4	no	—
199	#117	—	43	4	no	—
200	#117	—	43	4	no	—
201	#117	—	43	4	no	—
202	#117	C₄×D₄₃	1	4	yes	—
203	#117	C₂²×D₄₃	1	4	yes	—
204	C₂×C₄⋊C₄	—	43	4	no	—
205	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
206	C₂×C₂²⋊C₄	—	43	4	no	—
207	C₂×C₂²⋊C₄	C₄×D₄₃	1	2	yes	—
208	C₂×C₄²	—	43	4	no	—
209	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
210	C₄⋊C₈	—	43	16	no	—
211	C₄⋊C₈	—	43	16	no	—
212	C₄⋊C₈	—	43	16	no	—
213	#97	—	43	16	no	—
214	#97	C₂²×D₄₃	1	8	yes	—
215	C₄×C₈	—	43	16	no	—
216	C₄×C₈	C₂²×D₄₃	1	8	yes	—
217	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂×Q₈	—	43	16	no	—
219	C₂²×C₄	—	43	8	no	—
220	C₂²×C₄	C₂×Dic₈₆	1	4	yes	—
221	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
222	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
223	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
224	C₂²×C₄	#10621898	1	4	yes	—
225	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
226	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
227	M₁₆	—	86	8	no	—
228	M₁₆	—	86	8	no	—
229	M₁₆	C₂×Dic₈₆	1	16	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	43	8	no	—
234	C₂×C₈	—	86	8	no	—
235	C₂×C₈	C₂×Dic₈₆	1	8	yes	—
236	C₂×C₈	C₂³×D₄₃	1	2	yes	—
237	C₂×C₈	C₂³×D₄₃	1	2	yes	—
238	C₂×C₈	C₂³×D₄₃	1	4	yes	—
239	C₄⋊C₄	—	43	8	no	—
240	C₄⋊C₄	—	43	8	no	—
241	C₄⋊C₄	—	43	32	no	—
242	C₄⋊C₄	—	43	16	no	—
243	C₄⋊C₄	C₂×C₄×D₄₃	1	8	yes	—
244	C₂²⋊C₄	—	43	8	no	—
245	C₂²⋊C₄	—	43	8	no	—
246	C₂²⋊C₄	C₂×C₄×D₄₃	1	8	yes	—
247	C₄²	—	43	8	no	—
248	C₄²	—	43	8	no	—
249	C₄²	C₂×C₄×D₄₃	1	4	yes	—
250	C₄²	C₂³×D₄₃	1	4	yes	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄×Dic₈₆	1	4	yes	—
253	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
254	C₂³	C₂²×C₄×D₄₃	1	2	yes	—
255	Q₈	—	172	16	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	43	16	no	—
258	C₂×C₄	—	172	8	no	—
259	C₂×C₄	C₄×Dic₈₆	1	8	yes	—
260	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₄₃	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₈₆	1	2	yes	—
265	C₂×C₄	C₂²×Dic₈₆	1	4	yes	—
266	C₂×C₄	C₂×#10621898	1	4	yes	—
267	C₂×C₄	C₂×#10621898	1	2	yes	—
268	C₂×C₄	C₂⁴×D₄₃	1	1	yes	characteristic
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	—	86	8	no	—
272	C₈	C₂²×Dic₈₆	1	8	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂×C₄×Dic₈₆	1	2	yes	—
277	C₂²	C₂×C₄×Dic₈₆	1	4	yes	—
278	C₂²	C₂³×C₄×D₄₃	1	1	yes	characteristic
279	C₄	—	2	8	no	—
280	C₄	—	86	16	no	—
281	C₄	C₂×C₄×Dic₈₆	1	4	yes	—
282	C₄	C₂×C₄×Dic₈₆	1	2	yes	—
283	C₄	C₂³×Dic₈₆	1	1	yes	characteristic
284	C₄	C₂²×#10621898	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	—	2	4	no	—
287	C₂	C₂²×C₄×Dic₈₆	1	1	yes	characteristic
288	C₂	#19149306	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁴×C₄
Fitting	C₂×C₄₃×#405	C₂
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂×C₄×C₄₃	C₂⁵
hypercenter	C₂×#405	D₄₃
hypocenter	C₄₃	#60193
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #19143539	C₂	semidirect
3. #19143535	C₂	semidirect
4. #19143529	C₂	semidirect
16. #14583029	C₂²	semidirect
17. #14583027	C₂²	semidirect
18. C₈×Dic₈₆	C₂²	semidirect
58. C₂×M₁₆×C₄₃	Q₈	semidirect
178. C₄₃	#60193	semidirect
183. C₂×M₁₆	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19148000	2
#19143539	2
#19143535	2
#19143529	2
#19143159	2
C₂×#14582925	2
#19143013	2
C₂×C₄₃×#405	2
#60193	43