Subgroups — #25100033

Order 11008.

914 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582948	C₂	1	1	yes	maximal, characteristic
2	#19143185	C₂	1	1	yes	maximal, characteristic
3	#19143158	C₂	1	2	yes	maximal
4	#19135348	C₂	1	4	yes	maximal, complemented
5	#19135343	C₂	1	4	yes	maximal, complemented
6	C₂×#14582659	C₂	1	2	yes	maximal
7	C₂×C₄×M₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10621877	C₂²	1	2	yes	—
9	C₂×#13063284	C₂²	1	2	yes	—
10	#14582948	—	2	4	no	—
11	C₂×#13063261	C₂²	1	4	yes	—
12	#14582941	C₄	1	8	yes	complemented
13	#14582937	C₂²	1	4	yes	—
14	#14582927	—	2	4	no	—
15	#14582925	C₂²	1	8	yes	—
16	#14582922	—	2	4	no	—
17	#14582659	C₂²	1	4	yes	—
18	#14582659	C₂²	1	4	yes	—
19	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₄₃	—	2	4	no	—
23	C₂×#97×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	—	2	4	no	—
27	C₂×#10621877	—	2	4	no	—
28	C₂×#10621877	C₂³	1	4	yes	—
29	C₂²×C₄₃⋊C₈	—	2	2	no	—
30	#13063284	—	2	2	no	—
31	#13063284	—	2	2	no	—
32	#13063284	C₂×C₄	1	4	yes	—
33	#13063284	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
36	C₂×C₄×Dic₄₃	—	2	2	no	—
37	#13063261	—	2	8	no	—
38	#13063259	C₂×C₄	1	8	yes	complemented
39	#13063257	C₂×C₄	1	8	yes	complemented
40	Dic₄₃⋊C₈	—	2	16	no	—
41	C₂³×C₄×C₄₃	—	2	1	no	—
42	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₄₃	—	2	2	no	—
44	C₂×M₁₆×C₄₃	—	2	2	no	—
45	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×M₁₆×C₄₃	D₄	1	4	yes	—
47	C₂×M₁₆×C₄₃	D₄	1	4	yes	—
48	C₂×M₁₆×C₄₃	Q₈	1	4	yes	—
49	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	—	2	1	no	—
53	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	#97×C₄₃	—	2	4	no	—
57	C₄×C₈×C₄₃	—	2	4	no	—
58	C₂⁴×C₄₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	—	4	4	no	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₄₃	#41	1	2	yes	—
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	2	4	no	—
72	M₁₆×C₄₃	—	2	8	no	—
73	M₁₆×C₄₃	C₄⋊C₄	1	16	yes	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
80	C₄²×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	—	2	1	no	—
83	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
85	C₂²⋊Dic₄₃	—	2	4	no	—
86	C₂²⋊Dic₄₃	—	2	4	no	—
87	C₄⋊Dic₄₃	—	2	4	no	—
88	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
89	C₄×Dic₄₃	—	2	4	no	—
90	C₄×Dic₄₃	—	2	4	no	—
91	#10621877	—	2	4	no	—
92	#10621877	—	2	4	no	—
93	#10621877	—	4	4	no	—
94	C₂×C₄₃⋊C₈	—	2	4	no	—
95	C₂×C₄₃⋊C₈	—	4	4	no	—
96	C₂³×C₄₃	—	4	2	no	—
97	C₂³×C₄₃	C₄×D₄	1	2	yes	—
98	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₄₃	#125	1	2	yes	—
100	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	4	8	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₄₃	#121	1	4	yes	—
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	#125	1	2	yes	—
112	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
113	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₄₃	—	2	8	no	—
120	C₂×Dic₄₃	—	8	4	no	—
121	C₈×C₄₃	—	2	4	no	—
122	C₈×C₄₃	C₂×C₄⋊C₄	1	8	yes	—
123	C₄₃⋊C₈	—	4	4	no	—
124	#21165	—	43	1	no	maximal, Sylow
125	C₂²×C₄₃	—	2	4	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	—	4	2	no	—
128	C₂²×C₄₃	—	4	2	no	—
129	C₂²×C₄₃	#389	1	4	yes	—
130	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂²×C₄₃	C₂×#125	1	1	yes	characteristic
133	C₄×C₄₃	—	2	4	no	—
134	C₄×C₄₃	—	4	2	no	—
135	C₄×C₄₃	#389	1	4	yes	—
136	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
138	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
139	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	Dic₄₃	—	4	8	no	—
141	C₂×#430	—	43	1	no	—
142	#2911	—	43	1	no	—
143	#2905	—	43	2	no	—
144	C₂×C₄×M₁₆	D₄₃	1	1	yes	characteristic
145	#1946	—	43	4	no	—
146	#1945	—	43	4	no	—
147	C₂×#339	—	43	2	no	—
148	C₂×C₄₃	—	2	1	no	—
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	—	2	4	no	—
151	C₂×C₄₃	C₂×#389	1	1	yes	characteristic
152	C₂×C₄₃	#2956	1	2	yes	—
153	C₂²×M₁₆	—	43	2	no	—
154	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
155	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
156	C₂×#117	—	43	2	no	—
157	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
158	#430	—	86	4	no	—
159	C₂×#108	—	43	4	no	—
160	#428	—	43	8	no	—
161	#427	—	43	4	no	—
162	#424	—	43	8	no	—
163	#424	—	86	4	no	—
164	#423	—	86	4	no	—
165	C₄×M₁₆	—	2	4	no	—
166	C₂×#97	C₂×D₄₃	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
168	#339	—	43	4	no	—
169	#339	—	43	4	no	—
170	C₄₃	#21165	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	2	1	no	—
172	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	43	4	no	—
176	C₂×M₁₆	—	86	2	no	—
177	C₂×M₁₆	—	86	4	no	—
178	C₂×M₁₆	—	86	4	no	—
179	C₂×M₁₆	Dic₈₆	1	4	yes	—
180	C₂×M₁₆	C₄×D₄₃	1	4	yes	—
181	C₂×M₁₆	D₁₇₂	1	4	yes	—
182	C₂×M₁₆	C₂²⋊D₄₃	1	4	yes	—
183	C₂²×C₈	—	86	2	no	—
184	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
186	#117	—	43	4	no	—
187	#117	—	43	4	no	—
188	#117	—	86	2	no	—
189	#117	—	86	2	no	—
190	C₂×C₄⋊C₄	—	43	4	no	—
191	C₂×C₂²⋊C₄	—	86	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	86	2	no	—
195	C₂×C₄²	Dic₈₆	1	4	yes	—
196	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
198	#108	—	86	8	no	—
199	#107	—	43	8	no	—
200	#106	—	43	8	no	—
201	C₄⋊C₈	—	86	16	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	C₂⁴	#10621898	1	1	yes	characteristic
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	86	4	no	—
210	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
211	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
213	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
214	C₂²×C₄	#10621898	1	1	yes	characteristic
215	C₂²×C₄	#10621898	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
217	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
218	M₁₆	—	2	8	no	—
219	M₁₆	—	86	4	no	—
220	M₁₆	—	86	4	no	—
221	M₁₆	—	172	4	no	—
222	M₁₆	Dic₄₃⋊C₄	1	16	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	86	4	no	—
226	C₂×C₈	—	172	4	no	—
227	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
228	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
229	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
230	C₂×C₈	C₂×C₂²⋊D₄₃	1	4	yes	—
231	C₄⋊C₄	—	43	8	no	—
232	C₄⋊C₄	—	86	4	no	—
233	C₂²⋊C₄	—	86	4	no	—
234	C₂²⋊C₄	—	86	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	86	4	no	—
239	C₄²	—	86	4	no	—
240	C₄²	C₂×Dic₈₆	1	2	yes	—
241	C₄²	C₂×Dic₈₆	1	2	yes	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×D₁₇₂	1	2	yes	—
244	C₂³	#13063304	1	2	yes	—
245	C₂³	#13063317	1	2	yes	—
246	C₂³	C₂×#10621898	1	1	yes	characteristic
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	8	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	86	8	no	—
253	C₂×C₄	—	344	4	no	—
254	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
255	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
256	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
257	C₂×C₄	#13063299	1	4	yes	—
258	C₂×C₄	#13063304	1	2	yes	—
259	C₂×C₄	#13063317	1	2	yes	—
260	C₂×C₄	#13063325	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
265	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
266	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	172	4	no	—
269	C₈	C₂×Dic₄₃⋊C₄	1	8	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	#14582797	1	4	yes	—
275	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
276	C₂²	C₂×#13063304	1	1	yes	characteristic
277	C₂²	C₂×#13063317	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	4	2	no	—
280	C₄	—	172	8	no	—
281	C₄	#14582797	1	4	yes	—
282	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
283	C₄	C₂×C₄×Dic₈₆	1	1	yes	characteristic
284	C₄	C₂×#13063299	1	1	yes	characteristic
285	C₄	C₂×#13063325	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	4	no	—
288	C₂	—	2	1	no	—
289	C₂	C₂×#14582797	1	1	yes	characteristic
290	C₂	#19143506	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄×M₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₄₃
hypocenter	C₄₃	#21165
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #19135348	C₂	semidirect
5. #19135343	C₂	semidirect
12. #14582941	C₄	semidirect
38. #13063259	C₂×C₄	semidirect
39. #13063257	C₂×C₄	semidirect
170. C₄₃	#21165	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582948	2
#19143185	2
#19143158	2
#19135348	2
#19135343	2
C₂×#14582659	2
C₂×C₄×M₁₆×C₄₃	2
#21165	43