Subgroups — #25013069

Order 11008.

420 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063265	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582720	C₂	1	2	yes	maximal, complemented
3	#19136206	C₂	1	1	yes	maximal, characteristic
4	#19132677	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063265	—	2	4	no	—
7	C₂×#13063265	—	2	1	no	—
8	C₂×#13063265	C₂²	1	1	yes	characteristic
9	C₂×#13063265	C₂²	1	1	yes	characteristic
10	C₂×#13063265	C₂²	1	2	yes	—
11	C₂²×C₄₃⋊C₁₆	—	2	1	no	—
12	#14582720	—	2	4	no	—
13	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
14	C₂²⋊(C₄₃⋊C₁₆)	—	2	1	no	—
15	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₄₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
18	C₄×C₂²⋊C₄×C₄₃	C₄	1	1	yes	characteristic
19	C₄₃×#336	C₄	1	2	yes	—
20	#13063265	—	2	4	no	—
21	#13063265	—	2	2	no	—
22	#13063265	—	2	2	no	—
23	#13063265	—	2	2	no	—
24	#13063265	D₄	1	4	yes	—
25	C₂×C₄₃⋊C₁₆	—	2	4	no	—
26	C₂×C₄₃⋊C₁₆	—	2	2	no	—
27	C₂×C₄₃⋊C₁₆	—	2	1	no	—
28	C₂×C₄₃⋊C₁₆	—	2	1	no	—
29	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₄₃	—	2	2	no	—
31	C₂²×C₈×C₄₃	—	2	2	no	—
32	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
40	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
41	C₂²⋊C₈×C₄₃	C₈	1	2	yes	—
42	C₂²⋊C₈×C₄₃	C₈	1	2	yes	—
43	C₄×C₈×C₄₃	—	2	2	no	—
44	C₄×C₈×C₄₃	—	2	2	no	—
45	#95×C₄₃	—	2	1	no	—
46	C₂⁴×C₄₃	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	—	2	2	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	—	2	2	no	—
50	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₄₃	—	2	2	no	—
57	C₂×C₈×C₄₃	—	2	2	no	—
58	C₂×C₈×C₄₃	—	2	2	no	—
59	C₂×C₈×C₄₃	—	2	1	no	—
60	C₂×C₈×C₄₃	—	2	1	no	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	—	2	2	no	—
63	C₂×C₈×C₄₃	—	2	1	no	—
64	C₂×C₈×C₄₃	—	2	1	no	—
65	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₄₃	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₄₃	#41	1	2	yes	—
72	C₂²⋊C₄×C₄₃	—	2	4	no	—
73	C₄²×C₄₃	—	2	2	no	—
74	C₄²×C₄₃	—	2	2	no	—
75	C₄₃⋊C₁₆	—	2	4	no	—
76	C₄₃⋊C₁₆	—	4	1	no	—
77	C₄₃⋊C₁₆	—	4	1	no	—
78	C₂³×C₄₃	—	2	2	no	—
79	C₂³×C₄₃	—	2	2	no	—
80	C₂³×C₄₃	—	4	1	no	—
81	C₂³×C₄₃	C₄⋊C₈	1	1	yes	characteristic
82	C₂³×C₄₃	C₄⋊C₈	1	1	yes	characteristic
83	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	1	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	2	1	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	—	4	2	no	—
94	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₄₃	#131	1	2	yes	—
100	C₈×C₄₃	—	2	1	no	—
101	C₈×C₄₃	—	2	1	no	—
102	C₈×C₄₃	—	2	4	no	—
103	C₈×C₄₃	#120	1	1	yes	characteristic
104	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
105	C₈×C₄₃	#123	1	1	yes	characteristic
106	#12914	—	43	1	no	maximal, Sylow
107	C₂²×C₄₃	—	2	1	no	—
108	C₂²×C₄₃	—	2	1	no	—
109	C₂²×C₄₃	—	2	1	no	—
110	C₂²×C₄₃	—	2	1	no	—
111	C₂²×C₄₃	—	2	1	no	—
112	C₂²×C₄₃	—	2	1	no	—
113	C₂²×C₄₃	—	2	1	no	—
114	C₂²×C₄₃	—	4	4	no	—
115	C₂²×C₄₃	#364	1	2	yes	complemented
116	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₄₃	#435	1	2	yes	—
118	C₄×C₄₃	—	2	1	no	—
119	C₄×C₄₃	—	2	1	no	—
120	C₄×C₄₃	—	4	4	no	—
121	C₄×C₄₃	#385	1	1	yes	characteristic
122	C₄×C₄₃	#424	1	1	yes	characteristic
123	C₄×C₄₃	C₈×D₄	1	2	yes	—
124	C₂²×M₃₂	—	43	1	no	—
125	C₂×#364	—	43	2	no	—
126	#2112	—	43	1	no	—
127	C₈×C₂²⋊C₄	D₄₃	1	1	yes	characteristic
128	#1376	—	43	2	no	—
129	C₂×C₄₃	—	2	2	no	—
130	C₂×C₄₃	—	2	2	no	—
131	C₂×C₄₃	—	4	1	no	—
132	C₂×C₄₃	#1850	1	1	yes	characteristic
133	C₂×C₄₃	C₂×#364	1	1	yes	characteristic
134	C₂×C₄₃	#2153	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
136	C₂×M₃₂	—	43	2	no	—
137	C₂×M₃₂	—	43	1	no	—
138	C₂×M₃₂	—	43	1	no	—
139	C₂×M₃₂	—	86	4	no	—
140	C₂×M₃₂	—	86	1	no	—
141	C₂²×C₁₆	—	86	1	no	—
142	C₂×C₂²⋊C₈	Dic₄₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
144	C₄×C₂²⋊C₄	Dic₄₃	1	1	yes	characteristic
145	#364	—	86	4	no	—
146	C₂²⋊C₁₆	—	86	1	no	—
147	C₂²⋊C₁₆	—	86	1	no	—
148	#336	Dic₄₃	1	2	yes	—
149	C₄₃	#12914	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
154	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
155	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
156	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
158	C₂²×C₈	C₂×Dic₄₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	—	2	1	no	—
161	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
162	M₃₂	—	43	4	no	—
163	M₃₂	—	86	4	no	—
164	M₃₂	—	86	2	no	—
165	M₃₂	—	86	2	no	—
166	M₃₂	—	86	2	no	—
167	C₂×C₁₆	—	86	4	no	—
168	C₂×C₁₆	—	86	2	no	—
169	C₂×C₁₆	—	86	1	no	—
170	C₂×C₁₆	—	86	1	no	—
171	C₂²⋊C₈	C₄₃⋊C₈	1	2	yes	—
172	C₂²⋊C₈	C₄₃⋊C₈	1	2	yes	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	2	2	no	—
175	#95	—	2	1	no	—
176	C₂⁴	#10621877	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	#10621877	1	1	yes	characteristic
183	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	C₂×C₄₃⋊C₈	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
197	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
198	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
199	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
200	C₂×C₈	D₄×D₄₃	1	2	yes	—
201	C₂×C₈	#10621900	1	2	yes	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₁₆	—	86	4	no	—
206	C₁₆	—	172	1	no	—
207	C₁₆	—	172	1	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
212	C₂³	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
213	C₂³	C₂×#10621877	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
225	C₂×C₄	C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
227	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
228	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
229	C₂×C₄	#13063332	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	#13063295	1	1	yes	characteristic
234	C₈	#13063302	1	2	yes	—
235	C₈	#13063309	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	4	4	no	—
244	C₂²	#14582720	1	2	yes	complemented
245	C₂²	C₂×C₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
246	C₂²	#14582971	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	4	no	—
250	C₄	#14582774	1	1	yes	characteristic
251	C₄	#14582926	1	1	yes	characteristic
252	C₄	D₄×C₄₃⋊C₈	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	—	4	1	no	—
256	C₂	#19134775	1	1	yes	characteristic
257	C₂	C₂×#14582720	1	1	yes	characteristic
258	C₂	#19136361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₈×C₂²⋊C₄×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₄₃	M₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	C₈×C₂²⋊C₄	D₄₃
hypocenter	C₄₃	#12914
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#14582720	C₂	semidirect
4. #19132677	C₂	semidirect
115. C₂²×C₄₃	#364	semidirect
149. C₄₃	#12914	semidirect
244. C₂²	#14582720	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063265	2
C₂×#14582720	2
#19136206	2
#19132677	2
C₈×C₂²⋊C₄×C₄₃	2
#12914	43