Subgroups — #25255201

Order 11008.

2390 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137204	C₂	1	8	yes	maximal, complemented
2	#19137198	C₂	1	4	yes	maximal, complemented
3	#19137193	C₂	1	4	yes	maximal
4	#19137093	C₂	1	8	yes	maximal, complemented
5	#19137002	C₂	1	2	yes	maximal, complemented
6	#19136978	C₂	1	2	yes	maximal
7	C₄×#13063284	C₂	1	2	yes	maximal, complemented
8	C₂²×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	C₂²	1	4	yes	—
10	#14582792	C₂²	1	32	yes	—
11	#14582785	C₂²	1	32	yes	complemented
12	#14582769	C₂²	1	4	yes	complemented
13	#14582765	C₂²	1	4	yes	complemented
14	#14582762	C₂²	1	4	yes	—
15	#14582759	C₂²	1	8	yes	—
16	#14582759	C₂²	1	8	yes	—
17	#14582755	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₄₃	C₂²	1	8	yes	complemented
19	C₄×C₄⋊Dic₄₃	C₂²	1	8	yes	complemented
20	C₄×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
21	C₄×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
22	#14582745	C₂²	1	4	yes	complemented
23	C₄²×Dic₄₃	C₂²	1	4	yes	complemented
24	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₃	C₄	1	16	yes	complemented
26	C₂×#117×C₄₃	C₂²	1	4	yes	—
27	C₂×#117×C₄₃	C₂²	1	4	yes	—
28	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
30	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
32	#13063284	—	2	16	no	—
33	C₂×C₄⋊Dic₄₃	C₂³	1	8	yes	—
34	C₂×C₄⋊Dic₄₃	C₂³	1	16	yes	—
35	C₂×C₂²⋊Dic₄₃	C₂³	1	8	yes	—
36	C₂×C₂²⋊Dic₄₃	C₂³	1	16	yes	—
37	C₂×C₄×Dic₄₃	C₂³	1	8	yes	—
38	C₂×C₄×Dic₄₃	C₂³	1	16	yes	—
39	C₄₃⋊#95	C₂³	1	16	yes	—
40	C₄₃⋊#95	C₂³	1	16	yes	—
41	C₄₃⋊#95	C₂³	1	16	yes	—
42	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₄₃	—	2	8	no	—
44	C₂³×C₄×C₄₃	C₂×C₄	1	8	yes	—
45	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
47	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₄₃	C₂³	1	2	yes	—
50	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	#117×C₄₃	C₂×C₄	1	32	yes	—
52	#117×C₄₃	C₂×C₄	1	32	yes	—
53	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	16	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
55	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	16	yes	—
56	C₂×C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
57	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
58	C₂×C₄²×C₄₃	C₂×C₄	1	8	yes	—
59	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
62	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
63	C₂⁴×C₄₃	—	2	4	no	—
64	C₂⁴×C₄₃	—	4	4	no	—
65	C₂⁴×C₄₃	C₂²×C₄	1	4	yes	—
66	C₂⁴×C₄₃	C₂⁴	1	2	yes	—
67	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	—	2	32	no	—
69	C₂²×C₄×C₄₃	—	2	8	no	—
70	C₂²×C₄×C₄₃	—	2	8	no	—
71	C₂²×C₄×C₄₃	—	2	4	no	—
72	C₂²×C₄×C₄₃	—	4	4	no	—
73	C₂²×C₄×C₄₃	—	4	4	no	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
76	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₄₃	#41	1	8	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₄₃	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₄₃	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₄₃	C₂⁴	1	16	yes	—
87	C₄⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
88	C₂²⋊C₄×C₄₃	C₂²×C₄	1	32	yes	—
89	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
90	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
91	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
92	C₄²×C₄₃	C₂²×C₄	1	8	yes	—
93	C₂²⋊Dic₄₃	—	2	16	no	—
94	C₂²⋊Dic₄₃	—	4	16	no	—
95	C₄⋊Dic₄₃	—	2	16	no	—
96	C₄⋊Dic₄₃	—	4	16	no	—
97	C₄×Dic₄₃	—	2	16	no	—
98	C₄×Dic₄₃	—	4	16	no	—
99	C₂³×C₄₃	—	2	4	no	—
100	C₂³×C₄₃	—	2	8	no	—
101	C₂³×C₄₃	—	2	4	no	—
102	C₂³×C₄₃	—	2	8	no	—
103	C₂³×C₄₃	—	4	8	no	—
104	C₂³×C₄₃	—	4	16	no	—
105	C₂³×C₄₃	C₂³×C₄	1	4	yes	—
106	C₂³×C₄₃	C₂³×C₄	1	2	yes	—
107	C₂³×C₄₃	C₂×#41	1	4	yes	—
108	C₂³×C₄₃	C₂⁵	1	1	yes	characteristic
109	C₂×C₄×C₄₃	—	2	8	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	8	no	—
112	C₂×C₄×C₄₃	—	2	8	no	—
113	C₂×C₄×C₄₃	—	2	16	no	—
114	C₂×C₄×C₄₃	—	2	16	no	—
115	C₂×C₄×C₄₃	—	4	8	no	—
116	C₂×C₄×C₄₃	—	4	8	no	—
117	C₂×C₄×C₄₃	#117	1	16	yes	—
118	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
119	C₂×C₄×C₄₃	C₂³×C₄	1	8	yes	—
120	C₂×C₄×C₄₃	C₂³×C₄	1	4	yes	—
121	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
123	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
124	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
125	C₂×Dic₄₃	—	2	16	no	—
126	C₂×Dic₄₃	—	2	32	no	—
127	#34048	—	43	1	no	maximal, Sylow
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	2	8	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	8	no	—
132	C₂²×C₄₃	—	4	16	no	—
133	C₂²×C₄₃	—	4	8	no	—
134	C₂²×C₄₃	C₂×#117	1	4	yes	—
135	C₂²×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
136	C₂²×C₄₃	C₂²×#41	1	2	yes	—
137	C₂²×C₄₃	#585	1	4	yes	—
138	C₄×C₄₃	—	2	4	no	—
139	C₄×C₄₃	—	2	8	no	—
140	C₄×C₄₃	—	2	8	no	—
141	C₄×C₄₃	C₂×#117	1	4	yes	—
142	C₄×C₄₃	C₂×#117	1	4	yes	—
143	Dic₄₃	—	4	16	no	—
144	C₂²×#117	D₄₃	1	1	yes	characteristic
145	#2314	—	43	8	no	—
146	#2313	—	43	4	no	—
147	#2312	—	43	4	no	—
148	#2298	—	43	8	no	—
149	#2285	—	43	2	no	—
150	#2280	—	43	2	no	—
151	C₄×#117	—	43	2	no	—
152	C₂×C₄₃	—	2	4	no	—
153	C₂×C₄₃	—	4	4	no	—
154	C₂×C₄₃	C₂²×#117	1	1	yes	characteristic
155	C₂×C₄₃	#3428	1	2	yes	—
156	C₂×C₄₃	#3455	1	4	yes	—
157	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
158	C₂×#117	—	43	4	no	—
159	C₂×#117	Dic₄₃	1	16	yes	complemented
160	C₂×#117	C₂×D₄₃	1	4	yes	—
161	C₂×#117	C₂×D₄₃	1	4	yes	—
162	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
163	C₂²×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
164	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
165	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
166	#388	—	43	32	no	—
167	#387	—	43	32	no	—
168	#384	—	43	4	no	—
169	#382	—	43	4	no	—
170	#381	—	43	4	no	—
171	#380	—	43	8	no	—
172	#380	—	43	8	no	—
173	#379	—	43	4	no	—
174	C₄×C₄⋊C₄	—	43	8	no	—
175	C₄×C₄⋊C₄	—	43	8	no	—
176	C₄×C₂²⋊C₄	—	43	8	no	—
177	C₄×C₂²⋊C₄	—	43	8	no	—
178	#376	—	43	4	no	—
179	C₄³	—	43	4	no	—
180	C₄₃	#34048	1	1	yes	characteristic, complemented, Sylow
181	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
182	C₂³×C₄	—	2	8	no	—
183	C₂³×C₄	C₂×Dic₄₃	1	8	yes	—
184	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
186	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
187	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
188	C₂³×C₄	C₂²×D₄₃	1	2	yes	—
189	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
190	#117	—	86	16	no	—
191	#117	C₂×Dic₄₃	1	32	yes	—
192	#117	C₂×Dic₄₃	1	32	yes	—
193	C₂×C₄⋊C₄	—	43	8	no	—
194	C₂×C₄⋊C₄	—	43	16	no	—
195	C₂×C₄⋊C₄	C₂×Dic₄₃	1	16	yes	—
196	C₂×C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
197	C₂×C₂²⋊C₄	—	43	8	no	—
198	C₂×C₂²⋊C₄	—	43	16	no	—
199	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	16	yes	—
200	C₂×C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
201	C₂×C₄²	—	43	8	no	—
202	C₂×C₄²	—	43	16	no	—
203	C₂×C₄²	C₂×Dic₄₃	1	8	yes	—
204	C₂×C₄²	C₂×Dic₄₃	1	8	yes	—
205	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
206	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
207	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
208	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
209	#95	—	43	16	no	—
210	#95	—	43	16	no	—
211	#95	—	43	16	no	—
212	C₂⁴	—	2	4	no	—
213	C₂⁴	—	4	4	no	—
214	C₂⁴	C₂²×Dic₄₃	1	4	yes	—
215	C₂⁴	C₂³×D₄₃	1	2	yes	—
216	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	32	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	43	16	no	—
224	C₂²×C₄	#10621898	1	8	yes	—
225	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
227	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
228	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
229	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
230	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
231	C₂²×C₄	C₂³×D₄₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
233	C₂²×C₄	C₂³×D₄₃	1	4	yes	—
234	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	C₄⋊C₄	—	86	16	no	—
237	C₄⋊C₄	—	172	16	no	—
238	C₄⋊C₄	C₂²×Dic₄₃	1	32	yes	—
239	C₂²⋊C₄	—	86	16	no	—
240	C₂²⋊C₄	—	172	16	no	—
241	C₂²⋊C₄	C₂²×Dic₄₃	1	32	yes	—
242	C₄²	—	86	16	no	—
243	C₄²	—	172	16	no	—
244	C₄²	C₂²×Dic₄₃	1	8	yes	—
245	C₄²	C₂²×Dic₄₃	1	8	yes	—
246	C₄²	C₂²×Dic₄₃	1	8	yes	—
247	C₄²	C₂²×Dic₄₃	1	8	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	8	no	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	8	no	—
252	C₂³	—	4	8	no	—
253	C₂³	—	4	16	no	—
254	C₂³	C₂³×Dic₄₃	1	4	yes	—
255	C₂³	C₂³×Dic₄₃	1	2	yes	—
256	C₂³	C₂×#10621898	1	4	yes	—
257	C₂³	C₂⁴×D₄₃	1	1	yes	characteristic
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	86	16	no	—
267	C₂×C₄	—	86	32	no	—
268	C₂×C₄	#13063284	1	16	yes	—
269	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
270	C₂×C₄	C₂³×Dic₄₃	1	8	yes	—
271	C₂×C₄	C₂³×Dic₄₃	1	4	yes	—
272	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
273	C₂×C₄	C₂³×Dic₄₃	1	2	yes	—
274	C₂×C₄	C₂×#10621898	1	4	yes	—
275	C₂×C₄	C₂×#10621898	1	4	yes	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	16	no	—
281	C₂²	—	4	8	no	—
282	C₂²	C₂×#13063284	1	4	yes	—
283	C₂²	C₂⁴×Dic₄₃	1	1	yes	characteristic
284	C₂²	C₂²×#10621898	1	2	yes	—
285	C₂²	#14583735	1	4	yes	—
286	C₄	—	2	8	no	—
287	C₄	—	2	8	no	—
288	C₄	—	2	4	no	—
289	C₄	—	172	16	no	—
290	C₄	C₂×#13063284	1	4	yes	—
291	C₄	C₂×#13063284	1	4	yes	—
292	C₂	—	2	4	no	—
293	C₂	—	4	4	no	—
294	C₂	C₂²×#13063284	1	1	yes	characteristic
295	C₂	#19147798	1	2	yes	—
296	C₂	#19147985	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₄₃
Fitting	C₂²×#117×C₄₃	C₂
socle, cosocle	C₂³×C₄₃	C₂⁵
prime radical	C₂⁵×C₄₃	C₂³
hypercenter	C₂²×#117	D₄₃
hypocenter	C₄₃	#34048
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. #19137204	C₂	semidirect
2. #19137198	C₂	semidirect
4. #19137093	C₂	semidirect
5. #19137002	C₂	semidirect
7. C₄×#13063284	C₂	semidirect
11. #14582785	C₂²	semidirect
12. #14582769	C₂²	semidirect
13. #14582765	C₂²	semidirect
18. C₄×C₄⋊Dic₄₃	C₂²	semidirect
19. C₄×C₄⋊Dic₄₃	C₂²	semidirect
22. #14582745	C₂²	semidirect
23. C₄²×Dic₄₃	C₂²	semidirect
25. C₂×#117×C₄₃	C₄	semidirect
159. C₂×#117	Dic₄₃	semidirect
180. C₄₃	#34048	semidirect

Maximal subgroups

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

subgroup	index
#19137204	2
#19137198	2
#19137193	2
#19137093	2
#19137002	2
#19136978	2
C₄×#13063284	2
C₂²×#117×C₄₃	2
#34048	43