Subgroups — #25002329

Order 11008.

854 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582662	C₂	1	2	yes	maximal
2	C₂×#14582662	C₂	1	2	yes	maximal
3	#19132825	C₂	1	2	yes	maximal
4	C₂²×#97×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₃	C₂²	1	4	yes	—
6	C₂×C₄₃⋊#95	—	2	4	no	—
7	#14582662	—	2	4	no	—
8	#14582662	C₄	1	8	yes	complemented
9	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×#97×C₄₃	—	2	4	no	—
13	C₂×#97×C₄₃	C₄	1	4	yes	complemented
14	C₂³×Dic₄₃	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	—	2	4	no	—
16	C₂×C₄⋊Dic₄₃	—	2	4	no	—
17	C₂×C₄⋊Dic₄₃	—	2	4	no	—
18	C₂×C₄⋊Dic₄₃	—	2	4	no	—
19	C₂×C₄⋊Dic₄₃	—	2	4	no	—
20	C₂×C₄⋊Dic₄₃	C₂×C₄	1	8	yes	—
21	C₄₃⋊#95	—	4	8	no	—
22	C₂³×C₄×C₄₃	D₄	1	2	yes	—
23	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₄₃	—	2	2	no	—
25	C₂²×C₈×C₄₃	—	2	2	no	—
26	C₂²×C₈×C₄₃	—	2	2	no	—
27	C₂²×C₈×C₄₃	—	2	2	no	—
28	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₄₃	D₄	1	4	yes	—
32	C₂×C₄²×C₄₃	—	2	2	no	—
33	C₂×C₄²×C₄₃	—	2	2	no	—
34	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
36	#97×C₄₃	—	2	8	no	—
37	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	2	4	no	—
41	C₂²×C₄×C₄₃	—	2	4	no	—
42	C₂²×C₄×C₄₃	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	#41	1	2	yes	—
48	C₂²×C₄×C₄₃	#41	1	2	yes	—
49	C₂²×C₄×C₄₃	#41	1	2	yes	—
50	C₂²×C₄×C₄₃	#41	1	2	yes	—
51	C₂²×Dic₄₃	—	2	8	no	—
52	C₂²×Dic₄₃	—	4	4	no	—
53	C₂²×Dic₄₃	—	4	4	no	—
54	C₂²×Dic₄₃	—	4	4	no	—
55	C₂²×Dic₄₃	—	4	4	no	—
56	C₂²×Dic₄₃	—	4	4	no	—
57	C₂²×Dic₄₃	—	4	4	no	—
58	C₂×C₈×C₄₃	—	2	2	no	—
59	C₂×C₈×C₄₃	—	2	2	no	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	—	2	8	no	—
63	C₂×C₈×C₄₃	—	2	8	no	—
64	C₂×C₈×C₄₃	C₄²	1	4	yes	—
65	C₂×C₈×C₄₃	C₂²⋊C₄	1	4	yes	—
66	C₄²×C₄₃	—	2	2	no	—
67	C₄²×C₄₃	—	2	2	no	—
68	C₄²×C₄₃	—	2	4	no	—
69	C₄⋊Dic₄₃	—	2	8	no	—
70	C₄⋊Dic₄₃	—	2	8	no	—
71	C₄⋊Dic₄₃	—	4	4	no	—
72	C₄⋊Dic₄₃	—	4	4	no	—
73	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₄₃	C₄×D₄	1	2	yes	—
75	C₂³×C₄₃	#121	1	2	yes	—
76	C₂³×C₄₃	#124	1	2	yes	—
77	C₂³×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
78	C₂³×C₄₃	#137	1	4	yes	—
79	C₂×C₄×C₄₃	—	2	4	no	—
80	C₂×C₄×C₄₃	—	2	8	no	—
81	C₂×C₄×C₄₃	—	4	4	no	—
82	C₂×C₄×C₄₃	—	4	2	no	—
83	C₂×C₄×C₄₃	—	4	2	no	—
84	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	#117	1	4	yes	—
87	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
88	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
92	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₄₃	#123	1	4	yes	—
95	C₂×C₄×C₄₃	#125	1	2	yes	—
96	C₂×C₄×C₄₃	#126	1	4	yes	—
97	C₂×Dic₄₃	—	4	8	no	—
98	C₂×Dic₄₃	—	4	8	no	—
99	C₂×Dic₄₃	—	4	8	no	—
100	C₂×Dic₄₃	—	8	4	no	—
101	C₂×Dic₄₃	—	8	4	no	—
102	C₂×Dic₄₃	—	8	4	no	—
103	C₂×Dic₄₃	—	8	4	no	—
104	C₈×C₄₃	—	2	4	no	—
105	C₈×C₄₃	—	2	4	no	—
106	#10821	—	43	1	no	maximal, Sylow
107	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄₃	#380	1	1	yes	characteristic
109	C₂²×C₄₃	#382	1	1	yes	characteristic
110	C₂²×C₄₃	#388	1	4	yes	—
111	C₂²×C₄₃	#428	1	4	yes	—
112	C₂²×C₄₃	#440	1	4	yes	—
113	C₂²×C₄₃	#441	1	2	yes	—
114	C₂²×C₄₃	#442	1	2	yes	—
115	C₂²×C₄₃	#468	1	4	yes	—
116	C₂²×C₄₃	#469	1	2	yes	—
117	C₂²×C₄₃	#470	1	2	yes	—
118	C₂²×C₄₃	#489	1	4	yes	—
119	C₂²×C₄₃	#490	1	2	yes	—
120	C₂²×C₄₃	#491	1	2	yes	—
121	C₄×C₄₃	—	4	4	no	—
122	C₄×C₄₃	#376	1	1	yes	characteristic
123	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₄₃	#381	1	1	yes	characteristic
125	C₄×C₄₃	#383	1	1	yes	characteristic
126	C₄×C₄₃	#387	1	4	yes	—
127	Dic₄₃	—	8	8	no	—
128	C₂²×#97	D₄₃	1	1	yes	characteristic
129	C₂×#340	—	43	2	no	—
130	C₂×#340	—	43	2	no	—
131	C₄⋊#95	—	43	2	no	—
132	C₂×C₄₃	#1434	1	1	yes	characteristic
133	C₂×C₄₃	#1776	1	2	yes	—
134	C₂×C₄₃	#1809	1	2	yes	—
135	C₂×C₄₃	#1836	1	2	yes	—
136	C₂×C₄₃	#1935	1	4	yes	—
137	C₂×C₄₃	#1936	1	4	yes	—
138	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
139	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	—	43	4	no	—
141	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
142	C₂×#97	—	2	4	no	—
143	C₂×#97	Dic₄₃	1	4	yes	complemented
144	C₂×#95	—	86	4	no	—
145	#340	—	43	8	no	—
146	#340	—	86	4	no	—
147	C₄₃	#10821	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	—	86	4	no	—
149	C₂³×C₄	D₁₇₂	1	2	yes	—
150	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	C₄×D₄₃	1	4	yes	—
156	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
157	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
158	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
159	C₂×C₄⋊C₄	—	43	8	no	—
160	C₂×C₄⋊C₄	—	86	4	no	—
161	C₂×C₄⋊C₄	—	86	4	no	—
162	C₂×C₄⋊C₄	—	86	4	no	—
163	C₂×C₄⋊C₄	—	86	4	no	—
164	C₂×C₄⋊C₄	—	86	4	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	2	no	—
167	C₂×C₄²	Dic₈₆	1	2	yes	—
168	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
169	#97	—	2	8	no	—
170	#95	—	172	8	no	—
171	C₂⁴	C₂×D₁₇₂	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	86	8	no	—
177	C₂²×C₄	—	172	4	no	—
178	C₂²×C₄	—	172	4	no	—
179	C₂²×C₄	—	172	4	no	—
180	C₂²×C₄	—	172	4	no	—
181	C₂²×C₄	—	172	4	no	—
182	C₂²×C₄	—	172	4	no	—
183	C₂²×C₄	C₄⋊Dic₄₃	1	4	yes	—
184	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
186	C₂²×C₄	#10621898	1	2	yes	—
187	C₂²×C₄	#10621898	1	2	yes	—
188	C₂²×C₄	#10621898	1	2	yes	—
189	C₂²×C₄	#10621898	1	2	yes	—
190	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
199	C₂×C₈	C₂²⋊Dic₄₃	1	4	yes	—
200	C₄⋊C₄	—	86	8	no	—
201	C₄⋊C₄	—	86	8	no	—
202	C₄⋊C₄	—	172	4	no	—
203	C₄⋊C₄	—	172	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₂³	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
208	C₂³	C₄×D₁₇₂	1	2	yes	—
209	C₂³	#13063299	1	2	yes	—
210	C₂³	#13063313	1	2	yes	—
211	C₂³	#13063353	1	4	yes	—
212	C₂³	#13063358	1	4	yes	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	344	4	no	—
219	C₂×C₄	—	344	4	no	—
220	C₂×C₄	—	344	4	no	—
221	C₂×C₄	—	344	4	no	—
222	C₂×C₄	—	172	8	no	—
223	C₂×C₄	—	172	8	no	—
224	C₂×C₄	—	172	8	no	—
225	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
227	C₂×C₄	#13063282	1	4	yes	—
228	C₂×C₄	#13063284	1	1	yes	characteristic
229	C₂×C₄	#13063284	1	1	yes	characteristic
230	C₂×C₄	#13063284	1	1	yes	characteristic
231	C₂×C₄	#13063284	1	1	yes	characteristic
232	C₂×C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
233	C₂×C₄	C₄×Dic₈₆	1	2	yes	—
234	C₂×C₄	#13063304	1	2	yes	—
235	C₂×C₄	#13063308	1	4	yes	—
236	C₂×C₄	#13063317	1	2	yes	—
237	C₂×C₄	#13063319	1	4	yes	—
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₂²	C₄×C₄⋊Dic₄₃	1	1	yes	characteristic
241	C₂²	#14582759	1	1	yes	characteristic
242	C₂²	#14582765	1	1	yes	characteristic
243	C₂²	#14582790	1	4	yes	—
244	C₂²	#14582941	1	4	yes	—
245	C₂²	#14583001	1	4	yes	—
246	C₂²	#14583007	1	2	yes	—
247	C₂²	#14583012	1	2	yes	—
248	C₂²	#14583145	1	4	yes	—
249	C₂²	#14583152	1	2	yes	—
250	C₂²	#14583157	1	2	yes	—
251	C₂²	#14583259	1	4	yes	—
252	C₂²	#14583264	1	2	yes	—
253	C₂²	#14583268	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	—	344	8	no	—
256	C₄	#14582745	1	1	yes	characteristic
257	C₄	C₄×C₂²⋊Dic₄₃	1	1	yes	characteristic
258	C₄	#14582762	1	1	yes	characteristic
259	C₄	#14582767	1	1	yes	characteristic
260	C₄	#14582783	1	4	yes	—
261	C₂	#19132822	1	1	yes	characteristic
262	C₂	#19134492	1	2	yes	—
263	C₂	#19134628	1	2	yes	—
264	C₂	#19134729	1	2	yes	—
265	C₂	#19135285	1	4	yes	—
266	C₂	#19135292	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂²×#97×C₄₃	C₂
socle, prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂²×#97	D₄₃
hypocenter	C₄₃	#10821
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
8. #14582662	C₄	semidirect
13. C₂×#97×C₄₃	C₄	semidirect
143. C₂×#97	Dic₄₃	semidirect
147. C₄₃	#10821	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582662	2
#19132825	2
C₂²×#97×C₄₃	2
#10821	43