Subgroups — #24986161

Order 11008.

638 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582923	C₂	1	1	yes	maximal, characteristic
2	#19136993	C₂	1	1	yes	maximal, characteristic
3	#19132435	C₂	1	4	yes	maximal, complemented
4	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10621877	—	2	1	no	—
6	#14582923	—	2	2	no	—
7	#14582923	C₄	1	4	yes	complemented
8	C₂×C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	#14582767	—	2	1	no	—
11	#14582765	C₄	1	2	yes	—
12	#14582762	—	2	1	no	—
13	#14582759	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₄₃	—	2	1	no	—
16	#14582745	—	2	1	no	—
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#431	—	2	2	no	—
19	C₄₃×#431	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₄₃	—	2	1	no	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
23	C₂×#10621877	—	2	4	no	—
24	C₂×#10621877	—	4	4	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄×Dic₄₃	—	2	2	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
32	C₄₃⋊#95	—	2	2	no	—
33	C₄₃⋊#95	—	2	2	no	—
34	C₄₃⋊#95	—	2	2	no	—
35	C₄₃⋊#95	—	2	2	no	—
36	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₄₃	—	2	4	no	—
49	C₂²⋊C₈×C₄₃	—	2	4	no	—
50	C₄×C₈×C₄₃	—	2	2	no	—
51	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
52	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	—	2	4	no	—
54	C₂²×C₄×C₄₃	—	2	2	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	4	2	no	—
57	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	—	2	4	no	—
70	C₂×C₈×C₄₃	—	2	4	no	—
71	C₂×C₈×C₄₃	—	4	4	no	—
72	C₄²×C₄₃	C₄²	1	2	yes	—
73	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
76	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	—	4	4	no	—
83	C₄⋊Dic₄₃	—	2	8	no	—
84	C₄×Dic₄₃	—	4	2	no	—
85	C₄×Dic₄₃	—	4	2	no	—
86	#10621877	—	4	8	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	4	4	no	—
89	C₂³×C₄₃	—	2	2	no	—
90	C₂³×C₄₃	—	2	2	no	—
91	C₂³×C₄₃	—	2	2	no	—
92	C₂³×C₄₃	#95	1	2	yes	—
93	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₄₃	—	2	2	no	—
95	C₂×C₄×C₄₃	—	2	2	no	—
96	C₂×C₄×C₄₃	—	2	2	no	—
97	C₂×C₄×C₄₃	—	2	2	no	—
98	C₂×C₄×C₄₃	—	2	4	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	#95	1	2	yes	—
106	C₂×C₄×C₄₃	#95	1	2	yes	—
107	C₂×C₄×C₄₃	#95	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#131	1	4	yes	—
115	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₄₃	#137	1	2	yes	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₈×C₄₃	—	4	4	no	—
121	C₄₃⋊C₈	—	4	4	no	—
122	#7899	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	—	2	2	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	4	4	no	—
127	C₂²×C₄₃	C₂×#95	1	1	yes	characteristic
128	C₂²×C₄₃	#408	1	2	yes	—
129	C₂²×C₄₃	#417	1	2	yes	—
130	C₂²×C₄₃	#421	1	2	yes	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	—	2	1	no	—
133	C₄×C₄₃	—	2	2	no	—
134	C₄×C₄₃	—	2	1	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	#405	1	2	yes	—
137	C₄×C₄₃	#418	1	1	yes	characteristic
138	C₄×C₄₃	#419	1	1	yes	characteristic
139	C₄×C₄₃	#424	1	2	yes	—
140	C₄×C₄₃	#428	1	2	yes	—
141	Dic₄₃	—	8	4	no	—
142	C₂×#431	D₄₃	1	1	yes	characteristic
143	C₂×#423	—	43	1	no	—
144	#2283	—	43	1	no	—
145	#1279	—	43	4	no	—
146	C₂×C₄₃	—	4	2	no	—
147	C₂×C₄₃	#1495	1	4	yes	—
148	C₂×C₄₃	#1729	1	1	yes	characteristic
149	C₂×C₄₃	#1733	1	1	yes	characteristic
150	C₂×C₄₃	#1736	1	1	yes	characteristic
151	C₂²×M₁₆	—	86	1	no	—
152	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
153	#431	—	2	2	no	—
154	#431	Dic₄₃	1	4	yes	complemented
155	#423	—	43	4	no	—
156	#423	—	86	2	no	—
157	C₂×C₄⋊C₈	—	2	1	no	—
158	C₂×C₄⋊C₈	—	43	2	no	—
159	C₂×C₄⋊C₈	—	86	1	no	—
160	C₂×C₂²⋊C₈	—	2	1	no	—
161	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
162	#383	—	86	1	no	—
163	#382	—	43	2	no	—
164	#381	—	86	1	no	—
165	#380	—	43	2	no	—
166	C₄×C₄⋊C₄	—	43	2	no	—
167	C₄×C₂²⋊C₄	—	86	1	no	—
168	#376	—	86	1	no	—
169	C₄₃	#7899	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
171	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
172	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
173	C₂×M₁₆	—	86	4	no	—
174	C₂×M₁₆	—	172	4	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	86	2	no	—
177	C₂×C₄⋊C₄	—	43	4	no	—
178	C₂×C₂²⋊C₄	—	86	2	no	—
179	C₂×C₄²	—	86	2	no	—
180	C₂×C₄²	Dic₈₆	1	2	yes	—
181	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₄₃	1	2	yes	—
183	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
187	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	43	4	no	—
190	C₄⋊C₈	—	86	4	no	—
191	C₄⋊C₈	—	86	2	no	—
192	C₂²⋊C₈	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
195	#95	—	86	2	no	—
196	#95	—	86	2	no	—
197	#95	—	86	2	no	—
198	#95	—	86	2	no	—
199	C₂⁴	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	86	4	no	—
205	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
206	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
207	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
208	C₂²×C₄	C₄⋊Dic₄₃	1	2	yes	—
209	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
212	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
213	C₂²×C₄	C₂²⋊Dic₄₃	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
217	M₁₆	—	172	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	—	172	4	no	—
222	C₄⋊C₄	—	86	8	no	—
223	C₂²⋊C₄	—	172	4	no	—
224	C₄²	—	172	2	no	—
225	C₄²	—	172	2	no	—
226	C₄²	C₄×Dic₄₃	1	2	yes	—
227	C₄²	Dic₄₃⋊C₄	1	2	yes	—
228	C₄²	Dic₄₃⋊C₄	1	2	yes	—
229	C₄²	C₄⋊Dic₄₃	1	2	yes	—
230	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
231	C₄²	C₂²⋊Dic₄₃	1	2	yes	—
232	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
234	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
235	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	C₄₃⋊#95	1	2	yes	—
240	C₂³	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	172	4	no	—
253	C₂×C₄	—	172	4	no	—
254	C₂×C₄	—	172	4	no	—
255	C₂×C₄	C₄₃⋊#95	1	2	yes	—
256	C₂×C₄	C₄₃⋊#95	1	2	yes	—
257	C₂×C₄	C₄₃⋊#95	1	2	yes	—
258	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
264	C₂×C₄	#13063331	1	4	yes	—
265	C₂×C₄	#13063353	1	2	yes	—
266	C₂×C₄	#13063358	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	172	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂×C₄₃⋊#95	1	1	yes	characteristic
274	C₂²	#14582865	1	2	yes	—
275	C₂²	#14582901	1	2	yes	—
276	C₂²	#14582917	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	344	4	no	—
283	C₄	#14582854	1	2	yes	—
284	C₄	#14582905	1	1	yes	characteristic
285	C₄	#14582909	1	1	yes	characteristic
286	C₄	#14582925	1	2	yes	—
287	C₄	#14582941	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	#19133061	1	4	yes	—
290	C₂	#19134291	1	1	yes	characteristic
291	C₂	#19134305	1	1	yes	characteristic
292	C₂	#19134315	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#7899
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #19132435	C₂	semidirect
7. #14582923	C₄	semidirect
19. C₄₃×#431	C₄	semidirect
154. #431	Dic₄₃	semidirect
169. C₄₃	#7899	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582923	2
#19136993	2
#19132435	2
C₂×C₄₃×#431	2
#7899	43