Subgroups — #25097116

Order 11008.

906 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582956	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582952	C₂	1	1	yes	maximal, characteristic
3	#19134749	C₂	1	2	yes	maximal, complemented
4	#19134715	C₂	1	2	yes	maximal, complemented
5	#19134575	C₂	1	4	yes	maximal
6	#19134452	C₂	1	4	yes	maximal, complemented
7	C₂×C₄₃×#376	C₂	1	1	yes	maximal, characteristic
8	#14582956	—	2	4	no	—
9	#14582952	—	2	4	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
13	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13063236	C₂²	1	2	yes	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
17	#14582654	C₂²	1	8	yes	—
18	C₂²×C₄²×C₄₃	C₂²	1	2	yes	—
19	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
20	C₄₃×#376	C₄	1	2	yes	—
21	C₄₃×#376	C₄	1	2	yes	—
22	C₄₃×#376	C₂²	1	1	yes	characteristic
23	C₄₃×#376	C₂²	1	2	yes	—
24	C₄₃×#376	C₂²	1	1	yes	characteristic
25	C₂×#95×C₄₃	C₄	1	2	yes	—
26	C₂×#95×C₄₃	C₄	1	2	yes	—
27	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
28	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
29	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
31	C₄⋊(C₄₃⋊C₈)	—	2	16	no	—
32	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
33	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
34	#13063236	—	2	4	no	—
35	C₄×C₄₃⋊C₈	—	2	4	no	—
36	C₂³×C₄×C₄₃	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	—	2	8	no	—
41	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	4	yes	—
45	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
50	#95×C₄₃	C₂×C₄	1	2	yes	—
51	#95×C₄₃	C₂×C₄	1	2	yes	—
52	#95×C₄₃	C₂×C₄	1	4	yes	—
53	#95×C₄₃	C₂×C₄	1	4	yes	—
54	#95×C₄₃	C₂×C₄	1	2	yes	—
55	#95×C₄₃	C₂×C₄	1	2	yes	—
56	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	—	2	4	no	—
58	C₂²×C₄×C₄₃	—	2	8	no	—
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	—	2	8	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	—	4	4	no	—
63	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
75	C₄²×C₄₃	—	2	8	no	—
76	C₄²×C₄₃	—	2	4	no	—
77	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
78	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
79	C₄²×C₄₃	C₂×D₄	1	2	yes	—
80	C₄²×C₄₃	C₂×D₄	1	2	yes	—
81	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
83	C₄²×C₄₃	#41	1	8	yes	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	2	8	no	—
87	C₂×C₄₃⋊C₈	—	2	8	no	—
88	C₂³×C₄₃	—	2	2	no	—
89	C₂³×C₄₃	—	2	2	no	—
90	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₄₃	#131	1	4	yes	—
92	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	2	4	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	2	8	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	4	4	no	—
107	C₂×C₄×C₄₃	—	4	16	no	—
108	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₄₃	#117	1	8	yes	—
114	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×#41	1	4	yes	—
117	C₄₃⋊C₈	—	4	4	no	—
118	C₄₃⋊C₈	—	4	4	no	—
119	#20814	—	43	1	no	maximal, Sylow
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	—	2	8	no	—
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	#424	1	4	yes	—
124	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₄₃	C₂×#117	1	2	yes	—
126	C₂²×C₄₃	C₂×#131	1	2	yes	—
127	C₂²×C₄₃	#568	1	2	yes	—
128	C₄×C₄₃	—	2	2	no	—
129	C₄×C₄₃	—	2	2	no	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	—	2	8	no	—
133	C₄×C₄₃	—	2	2	no	—
134	C₄×C₄₃	—	4	4	no	—
135	C₄×C₄₃	#532	1	2	yes	—
136	C₄×C₄₃	#533	1	2	yes	—
137	C₂×#432	—	43	1	no	—
138	C₂×#431	—	43	1	no	—
139	C₂×#376	D₄₃	1	1	yes	characteristic
140	#1843	—	43	2	no	—
141	#1832	—	43	2	no	—
142	#1797	—	43	4	no	—
143	#1768	—	43	4	no	—
144	C₂×C₄₃	—	2	4	no	—
145	C₂×C₄₃	#2285	1	1	yes	characteristic
146	C₂×C₄₃	C₂×#424	1	1	yes	characteristic
147	C₂×C₄₃	#2905	1	1	yes	characteristic
148	C₂×C₄₃	#2924	1	4	yes	—
149	C₂²×C₄²	C₂×D₄₃	1	2	yes	—
150	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
151	#432	—	86	4	no	—
152	#431	—	86	4	no	—
153	C₂×C₄⋊C₈	—	43	8	no	—
154	C₂×C₄⋊C₈	—	43	2	no	—
155	C₂×C₄⋊C₈	—	43	2	no	—
156	C₂×C₂²⋊C₈	—	43	2	no	—
157	C₂×C₂²⋊C₈	—	43	2	no	—
158	C₂×#97	—	43	2	no	—
159	C₂×C₄×C₈	—	43	2	no	—
160	#376	Dic₄₃	1	2	yes	—
161	#376	Dic₄₃	1	2	yes	—
162	#376	C₂×D₄₃	1	1	yes	characteristic
163	#376	C₂×D₄₃	1	1	yes	characteristic
164	#376	C₂×D₄₃	1	2	yes	—
165	C₂×#95	Dic₄₃	1	2	yes	—
166	C₂×#95	Dic₄₃	1	2	yes	—
167	#336	—	43	8	no	—
168	C₄₃	#20814	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₄	C₂×Dic₄₃	1	4	yes	—
170	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
171	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
172	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
173	C₂²×C₈	—	43	4	no	—
174	C₂²×C₈	—	43	4	no	—
175	C₂×C₄²	—	2	8	no	—
176	C₂×C₄²	Dic₈₆	1	4	yes	—
177	C₂×C₄²	D₁₇₂	1	4	yes	—
178	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
179	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
182	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
185	C₄⋊C₈	—	86	4	no	—
186	C₄⋊C₈	—	86	4	no	—
187	C₄⋊C₈	—	86	16	no	—
188	C₂²⋊C₈	—	86	4	no	—
189	C₂²⋊C₈	—	86	4	no	—
190	#97	—	86	4	no	—
191	C₄×C₈	—	86	4	no	—
192	#95	C₂×Dic₄₃	1	2	yes	—
193	#95	C₂×Dic₄₃	1	2	yes	—
194	#95	C₂×Dic₄₃	1	4	yes	—
195	#95	C₂×Dic₄₃	1	4	yes	—
196	#95	C₂×Dic₄₃	1	2	yes	—
197	#95	C₂×Dic₄₃	1	2	yes	—
198	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
206	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
207	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
208	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
210	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
216	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
217	C₂×C₈	—	86	4	no	—
218	C₂×C₈	—	86	4	no	—
219	C₂×C₈	—	86	8	no	—
220	C₂×C₈	—	86	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	4	no	—
223	C₄²	C₄⋊Dic₄₃	1	4	yes	—
224	C₄²	C₄⋊Dic₄₃	1	4	yes	—
225	C₄²	C₂×Dic₈₆	1	2	yes	—
226	C₄²	C₂×Dic₈₆	1	2	yes	—
227	C₄²	C₂×D₁₇₂	1	2	yes	—
228	C₄²	C₂×D₁₇₂	1	2	yes	—
229	C₄²	#10621898	1	8	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
233	C₂³	#13063332	1	4	yes	—
234	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
251	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
255	C₂×C₄	#13063284	1	8	yes	—
256	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
257	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
258	C₂×C₄	C₂×#10621898	1	4	yes	—
259	C₈	—	172	4	no	—
260	C₈	—	172	4	no	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	#14582926	1	4	yes	—
265	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
266	C₂²	C₂×#13063284	1	2	yes	—
267	C₂²	C₂×#13063332	1	2	yes	—
268	C₂²	#14583666	1	2	yes	—
269	C₄	—	2	8	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	#14583449	1	2	yes	—
277	C₄	#14583456	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	#19137002	1	1	yes	characteristic
280	C₂	C₂×#14582926	1	1	yes	characteristic
281	C₂	#19143159	1	1	yes	characteristic
282	C₂	#19143253	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₂×C₄₃×#376	C₂
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#376	D₄₃
hypocenter	C₄₃	#20814
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
3. #19134749	C₂	semidirect
4. #19134715	C₂	semidirect
6. #19134452	C₂	semidirect
168. C₄₃	#20814	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582956	2
C₂×#14582952	2
#19134749	2
#19134715	2
#19134575	2
#19134452	2
C₂×C₄₃×#376	2
#20814	43