Subgroups — #25101299

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137072	C₂	1	1	yes	maximal, characteristic
2	#19135531	C₂	1	2	yes	maximal
3	#19135313	C₂	1	2	yes	maximal
4	#19135152	C₂	1	4	yes	maximal
5	#19134453	C₂	1	2	yes	maximal
6	#19134448	C₂	1	1	yes	maximal, characteristic
7	#19134346	C₂	1	2	yes	maximal
8	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₈₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×Dic₄₃⋊C₈	C₂²	1	4	yes	—
12	C₂×#13063236	C₂²	1	2	yes	—
13	#14582805	C₄	1	2	yes	—
14	#14582797	C₄	1	2	yes	—
15	#14582783	C₄	1	4	yes	—
16	C₄×Dic₄₃⋊C₄	C₂²	1	2	yes	—
17	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	#14582745	C₂²	1	2	yes	—
19	#14582654	C₂²	1	2	yes	—
20	#14582653	C₂²	1	4	yes	—
21	#14582653	C₂²	1	4	yes	—
22	#14582653	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₄³×C₄₃	C₂²	1	1	yes	characteristic
28	C₂²×Dic₈₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
30	C₄×Dic₈₆	—	2	4	no	—
31	C₂×C₄⋊Dic₄₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
33	C₂×Dic₄₃⋊C₄	C₂×C₄	1	4	yes	—
34	C₂×Dic₄₃⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
36	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
37	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
38	Dic₄₃⋊C₈	—	2	8	no	—
39	#13063236	—	2	4	no	—
40	C₄₃⋊#95	C₂×C₄	1	4	yes	—
41	C₄₃⋊#95	C₂×C₄	1	4	yes	—
42	C₄₃⋊#95	C₂×C₄	1	4	yes	—
43	C₄₃⋊#95	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₄₃	—	2	4	no	—
52	C₄×C₈×C₄₃	—	2	4	no	—
53	C₄×C₈×C₄₃	—	2	2	no	—
54	C₄×C₈×C₄₃	Q₈	1	4	yes	complemented
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
61	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
62	C₂²×Dic₄₃	C₂²×C₄	1	2	yes	—
63	C₂×Dic₈₆	—	2	2	no	—
64	C₂×Dic₈₆	—	2	4	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	—	2	2	no	—
69	C₂×C₈×C₄₃	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₄₃	#41	1	4	yes	—
71	C₄²×C₄₃	—	2	4	no	—
72	C₄²×C₄₃	—	2	4	no	—
73	C₄²×C₄₃	—	2	2	no	—
74	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
75	C₄²×C₄₃	#41	1	4	yes	—
76	C₄²×C₄₃	#41	1	2	yes	—
77	C₄⋊Dic₄₃	—	2	2	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	Dic₄₃⋊C₄	—	2	4	no	—
80	Dic₄₃⋊C₄	—	2	8	no	—
81	C₄×Dic₄₃	—	2	4	no	—
82	C₄×Dic₄₃	—	2	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₃	—	2	2	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	#117	1	2	yes	—
93	C₂×C₄×C₄₃	#117	1	2	yes	—
94	C₂×C₄×C₄₃	C₄×Q₈	1	4	yes	—
95	C₂×C₄×C₄₃	#131	1	4	yes	—
96	C₂×C₄×C₄₃	#131	1	4	yes	—
97	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
100	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	2	2	no	—
103	C₂×Dic₄₃	—	2	4	no	—
104	C₂×Dic₄₃	—	2	4	no	—
105	C₂×Dic₄₃	—	2	4	no	—
106	C₂×Dic₄₃	—	2	4	no	—
107	C₂×Dic₄₃	—	2	4	no	—
108	Dic₈₆	—	4	4	no	—
109	C₈×C₄₃	—	2	4	no	—
110	C₈×C₄₃	—	2	4	no	—
111	C₄₃⋊C₈	—	4	4	no	—
112	#21341	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₂²×C₄₃	C₄×#41	1	1	yes	characteristic
116	C₂²×C₄₃	C₂×#131	1	2	yes	—
117	C₂²×C₄₃	C₂×#131	1	2	yes	—
118	C₄×C₄₃	—	2	2	no	—
119	C₄×C₄₃	—	2	4	no	—
120	C₄×C₄₃	—	2	4	no	—
121	C₄×C₄₃	#433	1	4	yes	—
122	C₄×C₄₃	#529	1	2	yes	—
123	C₄×C₄₃	#533	1	2	yes	—
124	Dic₄₃	—	4	2	no	—
125	Dic₄₃	—	4	2	no	—
126	Dic₄₃	—	4	4	no	—
127	#2295	—	43	1	no	—
128	#1984	—	43	2	no	—
129	#1939	—	43	2	no	—
130	#1916	—	43	4	no	—
131	#1768	—	43	2	no	—
132	#1767	—	43	1	no	—
133	#1746	—	43	2	no	—
134	C₄²×C₈	D₄₃	1	1	yes	characteristic
135	C₂×C₄₃	#2295	1	1	yes	characteristic
136	C₂×C₄₃	C₂×#433	1	1	yes	characteristic
137	C₂×C₄₃	#2922	1	1	yes	characteristic
138	C₂×C₄₃	#2960	1	2	yes	—
139	C₂×C₄₃	#2966	1	2	yes	—
140	C₂×C₄×Q₈	—	43	1	no	—
141	C₂×C₄⋊C₈	—	43	4	no	—
142	C₂×C₄⋊C₈	—	43	2	no	—
143	C₂×#97	—	43	2	no	—
144	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
145	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
148	#391	—	43	2	no	—
149	#389	—	43	2	no	—
150	#387	—	43	4	no	—
151	C₄×C₄⋊C₄	—	43	2	no	—
152	C₄×C₄⋊C₄	—	43	1	no	—
153	#376	—	43	2	no	—
154	C₄³	C₂×D₄₃	1	1	yes	characteristic
155	#336	—	43	4	no	—
156	#336	—	43	4	no	—
157	#336	—	43	4	no	—
158	#336	—	43	2	no	—
159	C₄₃	#21341	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	43	1	no	—
161	C₂²×C₈	—	43	4	no	—
162	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
163	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
164	C₄×Q₈	—	86	4	no	—
165	C₂×C₄⋊C₄	—	43	2	no	—
166	C₂×C₄⋊C₄	—	43	1	no	—
167	C₂×C₄⋊C₄	—	43	4	no	—
168	C₂×C₄⋊C₄	—	43	2	no	—
169	C₂×C₄²	—	43	2	no	—
170	C₂×C₄²	—	43	2	no	—
171	C₂×C₄²	C₄×D₄₃	1	2	yes	—
172	C₂×C₄²	C₄×D₄₃	1	2	yes	—
173	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
176	C₄⋊C₈	—	86	8	no	—
177	C₄⋊C₈	—	86	4	no	—
178	#97	—	86	4	no	—
179	C₄×C₈	—	2	4	no	—
180	C₄×C₈	—	2	4	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	Dic₈₆	1	4	yes	complemented
183	#95	—	43	4	no	—
184	#95	—	43	4	no	—
185	#95	—	43	2	no	—
186	#95	—	43	4	no	—
187	C₂×Q₈	—	86	2	no	—
188	C₂×Q₈	—	86	4	no	—
189	C₂²×C₄	—	43	4	no	—
190	C₂²×C₄	—	43	2	no	—
191	C₂²×C₄	—	43	2	no	—
192	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
194	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
196	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	86	4	no	—
202	C₂×C₈	—	86	4	no	—
203	C₂×C₈	—	86	4	no	—
204	C₂×C₈	C₂×Dic₈₆	1	4	yes	—
205	C₂×C₈	#10621898	1	4	yes	—
206	C₄⋊C₄	—	86	4	no	—
207	C₄⋊C₄	—	86	2	no	—
208	C₄⋊C₄	—	86	8	no	—
209	C₄⋊C₄	—	86	4	no	—
210	C₄²	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	86	4	no	—
214	C₄²	—	86	4	no	—
215	C₄²	C₂×Dic₈₆	1	2	yes	—
216	C₄²	#10621898	1	4	yes	—
217	C₄²	#10621898	1	2	yes	—
218	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
219	Q₈	—	172	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	86	2	no	—
226	C₂×C₄	—	86	2	no	—
227	C₂×C₄	—	86	4	no	—
228	C₂×C₄	—	86	4	no	—
229	C₂×C₄	—	86	4	no	—
230	C₂×C₄	—	86	4	no	—
231	C₂×C₄	—	86	4	no	—
232	C₂×C₄	#13063282	1	2	yes	—
233	C₂×C₄	#13063282	1	2	yes	—
234	C₂×C₄	C₄×Dic₈₆	1	4	yes	—
235	C₂×C₄	#13063331	1	4	yes	—
236	C₂×C₄	#13063331	1	4	yes	—
237	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
238	C₂×C₄	C₂×#10621898	1	1	yes	characteristic
239	C₂×C₄	C₂×#10621898	1	2	yes	—
240	C₂×C₄	C₂×#10621898	1	2	yes	—
241	C₈	—	2	4	no	—
242	C₈	—	2	4	no	—
243	C₈	—	172	4	no	—
244	C₂²	C₂×#13063282	1	1	yes	characteristic
245	C₂²	C₂×C₄×Dic₈₆	1	1	yes	characteristic
246	C₂²	C₄×#10621898	1	1	yes	characteristic
247	C₂²	C₂×#13063331	1	2	yes	—
248	C₂²	C₂×#13063331	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	2	4	no	—
251	C₄	—	2	4	no	—
252	C₄	—	172	2	no	—
253	C₄	—	172	2	no	—
254	C₄	—	172	4	no	—
255	C₄	#14582958	1	4	yes	—
256	C₄	#14583433	1	2	yes	—
257	C₄	#14583456	1	2	yes	—
258	C₂	#19137072	1	1	yes	characteristic
259	C₂	C₂×#14582958	1	1	yes	characteristic
260	C₂	#19143237	1	1	yes	characteristic
261	C₂	#19143529	1	2	yes	—
262	C₂	#19143570	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂³×C₄
Fitting	C₄²×C₈×C₄₃	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#21341
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
54. C₄×C₈×C₄₃	Q₈	semidirect
159. C₄₃	#21341	semidirect
182. C₄×C₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19137072	2
#19135531	2
#19135313	2
#19135152	2
#19134453	2
#19134448	2
#19134346	2
C₄²×C₈×C₄₃	2
#21341	43