Subgroups — #25099875

Order 11008.

1038 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582926	C₂	1	2	yes	maximal
2	C₂×#14582923	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#19135314	C₂	1	8	yes	maximal, complemented
5	#19134292	C₂	1	2	yes	maximal
6	C₄₃×#2285	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10621877	C₂²	1	2	yes	—
8	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
10	#14582926	—	2	8	no	—
11	#14582923	—	2	4	no	—
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	8	yes	—
14	C₂×C₄×C₄₃⋊C₈	—	2	4	no	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	4	yes	—
16	#14582654	C₂²	1	8	yes	—
17	C₂×#117×C₄₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#384	C₄	1	2	yes	—
20	C₄₃×#382	C₄	1	2	yes	—
21	C₄₃×#380	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
23	C₂×#10621877	—	2	4	no	—
24	C₂×#10621877	—	2	4	no	—
25	C₂×#10621877	—	2	4	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
31	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
32	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
35	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	16	no	—
37	C₄×C₄₃⋊C₈	—	2	8	no	—
38	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	#117×C₄₃	—	2	4	no	—
41	#117×C₄₃	—	2	4	no	—
42	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
44	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₃	D₄	1	4	yes	—
48	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
49	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
50	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
51	#95×C₄₃	C₂×C₄	1	8	yes	—
52	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	—	2	8	no	—
54	C₂²×C₄×C₄₃	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	8	yes	—
56	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₄₃	#41	1	2	yes	—
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
66	C₄⋊C₄×C₄₃	—	2	8	no	—
67	C₄⋊C₄×C₄₃	—	2	16	no	—
68	C₂²⋊C₄×C₄₃	—	2	8	no	—
69	C₄²×C₄₃	—	2	4	no	—
70	C₄²×C₄₃	—	2	4	no	—
71	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
72	C₄²×C₄₃	C₄⋊C₄	1	4	yes	—
73	C₄²×C₄₃	C₂×D₄	1	4	yes	—
74	C₄²×C₄₃	C₂×Q₈	1	4	yes	—
75	#10621877	—	4	8	no	—
76	C₂×C₄₃⋊C₈	—	2	8	no	—
77	C₂×C₄₃⋊C₈	—	2	8	no	—
78	C₂×C₄₃⋊C₈	—	2	2	no	—
79	C₂×C₄₃⋊C₈	—	2	2	no	—
80	C₂×C₄₃⋊C₈	—	2	2	no	—
81	C₂×C₄₃⋊C₈	—	2	2	no	—
82	C₂×C₄₃⋊C₈	—	2	4	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
86	C₂×C₄₃⋊C₈	C₂×Q₈	1	4	yes	—
87	C₂³×C₄₃	—	2	4	no	—
88	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₂³×C₄₃	C₄×D₄	1	2	yes	—
90	C₂³×C₄₃	C₄×Q₈	1	2	yes	—
91	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	2	16	no	—
94	C₂×C₄×C₄₃	—	2	8	no	—
95	C₂×C₄×C₄₃	—	2	16	no	—
96	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₄₃	#121	1	4	yes	—
102	C₂×C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₄₃	#125	1	4	yes	—
104	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₄₃	#131	1	8	yes	—
106	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
112	C₄₃⋊C₈	—	2	4	no	—
113	C₄₃⋊C₈	—	2	4	no	—
114	C₄₃⋊C₈	—	4	4	no	—
115	#21143	—	43	1	no	maximal, Sylow
116	C₂²×C₄₃	—	2	4	no	—
117	C₂²×C₄₃	—	2	4	no	—
118	C₂²×C₄₃	—	2	4	no	—
119	C₂²×C₄₃	#389	1	4	yes	—
120	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂²×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₂²×C₄₃	C₂×#131	1	4	yes	—
124	C₄×C₄₃	—	2	4	no	—
125	C₄×C₄₃	—	4	8	no	—
126	C₄×C₄₃	#389	1	4	yes	—
127	C₄×C₄₃	#424	1	8	yes	complemented
128	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
129	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₄×C₄₃	C₂×#125	1	1	yes	characteristic
131	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×#424	—	43	2	no	—
133	C₂×#423	—	43	1	no	—
134	C₂²×C₄×C₈	—	43	1	no	—
135	#2285	D₄₃	1	1	yes	characteristic
136	#1939	—	43	8	no	—
137	#1729	—	43	2	no	—
138	C₂×C₄₃	—	2	4	no	—
139	C₂×C₄₃	C₂×#389	1	1	yes	characteristic
140	C₂×C₄₃	C₂×#424	1	2	yes	—
141	C₂×C₄₃	#2931	1	2	yes	—
142	C₂×C₄₃	#2960	1	2	yes	—
143	C₂²×M₁₆	—	43	2	no	—
144	C₂³×C₈	—	43	1	no	—
145	C₂³×C₈	—	43	1	no	—
146	C₂×#117	C₂×D₄₃	1	2	yes	—
147	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
148	#424	—	86	8	no	—
149	#423	—	86	4	no	—
150	C₂×C₄⋊C₈	—	43	4	no	—
151	C₂×C₄⋊C₈	—	43	8	no	—
152	C₂×C₄×C₈	—	43	4	no	—
153	C₂×C₄×C₈	—	86	4	no	—
154	#384	Dic₄₃	1	2	yes	—
155	#382	Dic₄₃	1	2	yes	—
156	#380	Dic₄₃	1	4	yes	—
157	C₄×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
158	#336	—	43	8	no	—
159	C₄₃	#21143	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
161	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
162	C₂×M₁₆	—	86	4	no	—
163	C₂×M₁₆	—	86	4	no	—
164	C₂×M₁₆	—	86	4	no	—
165	C₂²×C₈	—	43	4	no	—
166	C₂²×C₈	—	43	4	no	—
167	C₂²×C₈	—	43	2	no	—
168	C₂²×C₈	—	43	2	no	—
169	C₂²×C₈	—	43	4	no	—
170	C₂²×C₈	—	86	2	no	—
171	C₂²×C₈	—	86	2	no	—
172	C₂²×C₈	—	86	2	no	—
173	C₂²×C₈	—	86	2	no	—
174	#117	—	2	4	no	—
175	#117	—	2	4	no	—
176	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
177	C₂×C₄⋊C₄	C₂×Dic₄₃	1	8	yes	—
178	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
179	C₂×C₄²	Dic₈₆	1	4	yes	—
180	C₂×C₄²	D₁₇₂	1	4	yes	—
181	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
184	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
185	C₄⋊C₈	—	86	16	no	—
186	C₄⋊C₈	—	86	8	no	—
187	C₄×C₈	—	86	8	no	—
188	#95	C₂×Dic₄₃	1	8	yes	—
189	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	8	no	—
191	C₂²×C₄	C₄⋊Dic₄₃	1	8	yes	—
192	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
193	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
194	C₂²×C₄	D₄×D₄₃	1	2	yes	—
195	C₂²×C₄	#10621900	1	2	yes	—
196	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
197	C₂²×C₄	#10621902	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₄₃	1	8	yes	—
199	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
203	M₁₆	—	172	8	no	—
204	C₂×C₈	—	43	4	no	—
205	C₂×C₈	—	43	4	no	—
206	C₂×C₈	—	86	8	no	—
207	C₂×C₈	—	86	8	no	—
208	C₂×C₈	—	86	2	no	—
209	C₂×C₈	—	86	2	no	—
210	C₂×C₈	—	86	2	no	—
211	C₂×C₈	—	86	2	no	—
212	C₂×C₈	—	86	4	no	—
213	C₂×C₈	—	86	4	no	—
214	C₂×C₈	—	86	4	no	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	16	no	—
217	C₂²⋊C₄	—	2	8	no	—
218	C₄²	—	2	4	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₄⋊Dic₄₃	1	4	yes	—
221	C₄²	C₄⋊Dic₄₃	1	4	yes	—
222	C₄²	C₂×Dic₈₆	1	4	yes	—
223	C₄²	C₂×D₁₇₂	1	4	yes	—
224	C₂³	—	2	4	no	—
225	C₂³	C₂×C₄⋊Dic₄₃	1	2	yes	—
226	C₂³	D₄×Dic₄₃	1	2	yes	—
227	C₂³	Q₈×Dic₄₃	1	2	yes	—
228	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	16	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊Dic₄₃	1	2	yes	—
235	C₂×C₄	C₂×C₄⋊Dic₄₃	1	4	yes	—
236	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
237	C₂×C₄	Q₈×Dic₄₃	1	2	yes	—
238	C₂×C₄	#13063300	1	4	yes	—
239	C₂×C₄	#13063305	1	4	yes	—
240	C₂×C₄	#13063316	1	4	yes	—
241	C₂×C₄	#13063324	1	4	yes	—
242	C₂×C₄	#13063332	1	8	yes	—
243	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
245	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
246	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
247	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
248	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
249	C₈	—	86	4	no	—
250	C₈	—	86	4	no	—
251	C₈	—	172	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	#14582798	1	4	yes	—
256	C₂²	C₂²×C₄⋊Dic₄₃	1	1	yes	characteristic
257	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
258	C₂²	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
259	C₂²	C₂×#13063332	1	4	yes	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	#14582798	1	4	yes	—
263	C₄	#14582926	1	8	yes	complemented
264	C₄	C₂×#13063300	1	1	yes	characteristic
265	C₄	C₂×#13063305	1	1	yes	characteristic
266	C₄	C₂×#13063316	1	1	yes	characteristic
267	C₄	C₂×#13063324	1	1	yes	characteristic
268	C₂	—	2	4	no	—
269	C₂	C₂×#14582798	1	1	yes	characteristic
270	C₂	C₂×#14582926	1	2	yes	—
271	C₂	#19143307	1	2	yes	—
272	C₂	#19143530	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#2285	C₂
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2285	D₄₃
hypocenter	C₄₃	#21143
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
4. #19135314	C₂	semidirect
127. C₄×C₄₃	#424	semidirect
159. C₄₃	#21143	semidirect
263. C₄	#14582926	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582926	2
C₂×#14582923	2
C₂²×C₄×C₄₃⋊C₈	2
#19135314	2
#19134292	2
C₄₃×#2285	2
#21143	43