Subgroups — #25105464

Order 11008.

662 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	#19145117	C₂	1	1	yes	maximal, characteristic
2	#19143530	C₂	1	1	yes	maximal, characteristic
3	#19143525	C₂	1	1	yes	maximal, characteristic
4	#19135576	C₂	1	1	yes	maximal, characteristic
5	#19135347	C₂	1	4	yes	maximal
6	#19134769	C₂	1	1	yes	maximal, characteristic
7	#19134473	C₂	1	4	yes	maximal
8	#19134282	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#1995	C₂	1	1	yes	maximal, characteristic
10	#14583456	C₂²	1	1	yes	characteristic
11	#14583309	C₄	1	4	yes	complemented
12	#14583305	C₄	1	4	yes	complemented
13	#14583030	—	2	2	no	—
14	#14583026	—	2	2	no	—
15	Q₈×C₄₃⋊C₈	—	2	2	no	—
16	#14583023	—	2	2	no	—
17	C₂×#13063260	C₂²	1	4	yes	—
18	#14582935	C₂²	1	4	yes	—
19	#14582927	C₂²	1	2	yes	—
20	#14582926	C₂²	1	2	yes	—
21	#14582923	C₂²	1	1	yes	characteristic
22	#14582857	C₂²	1	2	yes	—
23	#14582855	C₂²	1	2	yes	—
24	C₂×#13063236	C₂²	1	2	yes	—
25	C₂×#13063236	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
28	#14582659	C₂²	1	4	yes	—
29	C₄₃×#533	C₂²	1	1	yes	characteristic
30	C₄₃×#423	C₂²	1	1	yes	characteristic
31	C₄₃×#376	C₂²	1	1	yes	characteristic
32	C₄₃×#339	C₂²	1	4	yes	—
33	C₂×#10621877	C₂³	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
36	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
37	#13063325	C₂×C₄	1	2	yes	—
38	#13063317	C₂×C₄	1	2	yes	—
39	#13063304	—	2	2	no	—
40	C₄×Dic₈₆	—	2	2	no	—
41	#13063284	C₂³	1	2	yes	—
42	#13063260	—	2	8	no	—
43	#13063258	C₂×C₄	1	8	yes	—
44	#13063256	C₂×C₄	1	8	yes	—
45	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
47	Dic₄₃⋊C₈	—	2	4	no	—
48	#13063236	—	2	2	no	—
49	#13063236	—	2	4	no	—
50	#13063236	C₂×C₄	1	2	yes	—
51	#13063236	C₂×C₄	1	2	yes	—
52	#13063235	—	2	2	no	—
53	C₄×C₄₃⋊C₈	—	2	2	no	—
54	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
55	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
56	C₄×C₄₃⋊C₈	Q₈	1	8	yes	complemented
57	C₈×Dic₄₃	—	2	2	no	—
58	C₂×M₁₆×C₄₃	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
60	#125×C₄₃	C₂×C₄	1	2	yes	—
61	C₂²⋊Q₈×C₄₃	—	2	2	no	—
62	C₄×Q₈×C₄₃	—	2	2	no	—
63	#117×C₄₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	—	2	2	no	—
67	#95×C₄₃	—	2	2	no	—
68	C₂×Q₈×C₄₃	—	2	2	no	—
69	C₂²×C₄×C₄₃	—	2	2	no	—
70	C₂²×C₄×C₄₃	#41	1	2	yes	—
71	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₈₆	—	2	2	no	—
73	M₁₆×C₄₃	—	2	4	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₄⋊C₄×C₄₃	—	2	2	no	—
77	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
78	C₂²⋊C₄×C₄₃	—	2	2	no	—
79	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₃	C₂×D₄	1	2	yes	—
84	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
85	C₂²⋊Dic₄₃	—	2	2	no	—
86	C₄⋊Dic₄₃	—	2	2	no	—
87	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
88	Dic₄₃⋊C₄	—	2	4	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	#10621877	—	2	4	no	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
95	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
98	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
99	C₂×C₄₃⋊C₈	#41	1	4	yes	—
100	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
101	Q₈×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂²×Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
112	C₂×Dic₄₃	—	2	4	no	—
113	Dic₈₆	—	4	4	no	—
114	C₈×C₄₃	—	4	2	no	—
115	C₄₃⋊C₈	—	2	2	no	—
116	C₄₃⋊C₈	—	4	2	no	—
117	C₄₃⋊C₈	C₄×Q₈	1	8	yes	—
118	C₄₃⋊C₈	C₄⋊Q₈	1	2	yes	—
119	C₄₃⋊C₈	C₄⋊Q₈	1	2	yes	—
120	#21735	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	—	2	2	no	—
122	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₄₃	#533	1	2	yes	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	—	2	2	no	—
126	C₄×C₄₃	—	4	4	no	—
127	C₄×C₄₃	—	4	4	no	—
128	C₄×C₄₃	C₂×C₄×Q₈	1	2	yes	—
129	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₄₃	#532	1	1	yes	characteristic
131	Dic₄₃	—	4	4	no	—
132	#3162	—	43	1	no	—
133	#2960	—	43	1	no	—
134	#2960	—	43	1	no	—
135	#1995	—	43	1	no	—
136	#1995	D₄₃	1	1	yes	characteristic
137	#1946	—	43	4	no	—
138	#1849	—	43	1	no	—
139	#1772	—	43	4	no	—
140	#1726	—	43	1	no	—
141	C₂×C₄₃	—	2	2	no	—
142	C₂×C₄₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₄₃	#2955	1	2	yes	—
144	#533	—	43	1	no	—
145	#533	C₂×D₄₃	1	1	yes	characteristic
146	#501	—	43	4	no	—
147	#501	—	43	4	no	—
148	#446	—	86	2	no	—
149	#446	—	86	2	no	—
150	C₈×Q₈	—	86	2	no	—
151	C₈×Q₈	—	86	2	no	—
152	C₂×#108	—	43	4	no	—
153	#427	—	43	4	no	—
154	#424	—	43	2	no	—
155	#424	—	43	2	no	—
156	#423	—	43	1	no	—
157	#423	C₂×D₄₃	1	1	yes	characteristic
158	#405	—	43	2	no	—
159	#405	—	43	2	no	—
160	C₂×#97	—	43	1	no	—
161	C₂×#97	—	43	2	no	—
162	C₂×C₄×C₈	—	43	2	no	—
163	C₂×C₄×C₈	—	43	1	no	—
164	#376	C₂×D₄₃	1	1	yes	characteristic
165	#339	—	43	4	no	—
166	#339	C₂×D₄₃	1	4	yes	—
167	C₄₃	#21735	1	1	yes	characteristic, complemented, Sylow
168	C₂×M₁₆	—	43	2	no	—
169	C₂×M₁₆	C₂²×D₄₃	1	2	yes	—
170	C₂²×C₈	—	43	1	no	—
171	C₂²×C₈	—	43	2	no	—
172	C₂²×C₈	—	43	1	no	—
173	C₄⋊Q₈	—	43	2	no	—
174	C₄⋊Q₈	C₄×D₄₃	1	2	yes	—
175	#125	—	43	2	no	—
176	#125	C₄×D₄₃	1	2	yes	—
177	C₂²⋊Q₈	—	2	2	no	—
178	C₂²⋊Q₈	—	86	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	86	2	no	—
181	#117	—	43	2	no	—
182	#117	C₂²×D₄₃	1	2	yes	—
183	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
184	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	#108	—	86	8	no	—
186	#107	—	43	8	no	—
187	#106	—	43	8	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	86	4	no	—
190	C₄⋊C₈	—	86	4	no	—
191	C₄⋊C₈	—	86	2	no	—
192	#97	—	43	2	no	—
193	#97	—	43	2	no	—
194	#97	—	86	4	no	—
195	#97	—	86	2	no	—
196	#97	—	86	2	no	—
197	C₄×C₈	—	43	2	no	—
198	C₄×C₈	—	43	2	no	—
199	C₄×C₈	—	43	8	no	—
200	C₄×C₈	—	86	2	no	—
201	C₄×C₈	—	86	2	no	—
202	#95	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	86	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	#10621900	1	2	yes	—
207	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	86	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	43	4	no	—
212	C₂×C₈	—	43	1	no	—
213	C₂×C₈	—	43	1	no	—
214	C₂×C₈	—	43	4	no	—
215	C₂×C₈	—	43	4	no	—
216	C₂×C₈	—	43	4	no	—
217	C₂×C₈	—	43	2	no	—
218	C₂×C₈	—	86	2	no	—
219	C₂×C₈	—	86	2	no	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	43	4	no	—
223	C₄⋊C₄	—	86	4	no	—
224	C₄⋊C₄	—	86	2	no	—
225	C₄⋊C₄	C₂×C₄×D₄₃	1	4	yes	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	86	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	86	2	no	—
231	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
232	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
233	C₄²	D₄×D₄₃	1	2	yes	—
234	C₄²	Q₈×D₄₃	1	4	yes	—
235	C₂³	C₂×#10621900	1	1	yes	characteristic
236	Q₈	—	4	4	no	—
237	Q₈	—	172	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	86	4	no	—
243	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
244	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
245	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂×Q₈×D₄₃	1	2	yes	—
248	C₂×C₄	#41×D₄₃	1	2	yes	—
249	C₈	—	4	2	no	—
250	C₈	—	43	8	no	—
251	C₈	—	43	2	no	—
252	C₈	—	43	2	no	—
253	C₈	—	86	2	no	—
254	C₈	—	172	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
257	C₂²	#14583451	1	2	yes	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	4	no	—
262	C₄	—	172	4	no	—
263	C₄	C₄×Q₈×D₄₃	1	2	yes	—
264	C₄	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
265	C₄	#14583448	1	1	yes	characteristic
266	C₂	—	2	2	no	—
267	C₂	#19137164	1	1	yes	characteristic
268	C₂	#19143494	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1995	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1995	D₄₃
hypocenter	C₄₃	#21735
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

Complements

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

N	H ≅ G/N	type
11. #14583309	C₄	semidirect
12. #14583305	C₄	semidirect
56. C₄×C₄₃⋊C₈	Q₈	semidirect
167. C₄₃	#21735	semidirect

Maximal subgroups

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

subgroup	index
#19145117	2
#19143530	2
#19143525	2
#19135576	2
#19135347	2
#19134769	2
#19134473	2
#19134282	2
C₄₃×#1995	2
#21735	43