Subgroups — #25251975

Order 11008.

590 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583301	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136840	C₂	1	2	yes	maximal, complemented
3	#19136836	C₂	1	2	yes	maximal, complemented
4	#19136826	C₂	1	4	yes	maximal, complemented
5	#19136307	C₂	1	4	yes	maximal, complemented
6	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063317	—	2	1	no	—
9	C₂×#13063265	C₂²	1	2	yes	—
10	#14583301	—	2	1	no	—
11	#14583301	—	2	1	no	—
12	#14583301	C₂²	1	2	yes	—
13	#14583301	C₂²	1	2	yes	—
14	C₂×#13063259	C₂²	1	2	yes	—
15	C₂×#13063257	—	2	1	no	—
16	#14582703	C₂²	1	8	yes	—
17	#14582700	C₂²	1	8	yes	—
18	#14582675	C₂²	1	2	yes	—
19	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
20	C₂×#135×C₄₃	—	2	1	no	—
21	C₂×#135×C₄₃	C₂²	1	2	yes	—
22	C₄₃×#532	—	2	1	no	—
23	C₄₃×#495	—	2	1	no	—
24	C₄₃×#495	C₂²	1	2	yes	—
25	C₄₃×#494	C₂²	1	1	yes	characteristic
26	C₄₃×#493	C₂²	1	1	yes	characteristic
27	C₄₃×#492	—	2	1	no	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	#13063317	—	2	2	no	—
30	#13063317	—	2	2	no	—
31	#13063317	—	4	2	no	—
32	C₂×C₄⋊Dic₄₃	—	2	2	no	—
33	C₂×Dic₄₃⋊C₄	—	4	2	no	—
34	#13063265	—	2	4	no	—
35	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
36	#13063259	—	2	2	no	—
37	#13063259	C₂³	1	4	yes	—
38	#13063257	—	2	2	no	—
39	#13063257	—	2	2	no	—
40	C₂×#41×C₄₃	—	2	2	no	—
41	#135×C₄₃	—	2	8	no	—
42	#135×C₄₃	—	4	4	no	—
43	C₂×Q₁₆×C₄₃	—	2	1	no	—
44	C₂×Q₁₆×C₄₃	C₂³	1	2	yes	—
45	C₂×SD₁₆×C₄₃	—	2	2	no	—
46	C₂×SD₁₆×C₄₃	—	2	2	no	—
47	C₂×D₈×C₄₃	—	2	1	no	—
48	C₂×D₈×C₄₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈×C₄₃	—	2	1	no	—
52	#127×C₄₃	—	2	1	no	—
53	#124×C₄₃	—	2	2	no	—
54	#121×C₄₃	—	4	2	no	—
55	C₄×D₄×C₄₃	—	4	2	no	—
56	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	#41×C₄₃	—	4	4	no	—
62	#41×C₄₃	—	4	4	no	—
63	C₂×Q₈×C₄₃	—	2	2	no	—
64	C₂×D₄×C₄₃	—	2	2	no	—
65	C₂×D₄×C₄₃	—	4	2	no	—
66	C₂×D₄×C₄₃	—	4	2	no	—
67	C₂²×C₄×C₄₃	—	4	2	no	—
68	C₂²×C₄×C₄₃	D₈	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	4	2	no	—
71	Q₁₆×C₄₃	—	2	4	no	—
72	Q₁₆×C₄₃	—	4	2	no	—
73	SD₁₆×C₄₃	—	4	4	no	—
74	SD₁₆×C₄₃	—	4	4	no	—
75	D₈×C₄₃	—	2	4	no	—
76	D₈×C₄₃	—	4	2	no	—
77	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₄₃	#41	1	4	yes	—
82	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊C₄×C₄₃	—	4	4	no	—
85	C₄²×C₄₃	D₈	1	2	yes	—
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₄⋊Dic₄₃	—	2	4	no	—
89	C₄⋊Dic₄₃	—	4	2	no	—
90	Dic₄₃⋊C₄	—	4	4	no	—
91	Dic₄₃⋊C₄	—	4	4	no	—
92	C₄₃⋊C₁₆	—	2	4	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₄₃	—	4	2	no	—
96	Q₈×C₄₃	—	4	2	no	—
97	D₄×C₄₃	—	4	2	no	—
98	D₄×C₄₃	—	4	2	no	—
99	D₄×C₄₃	—	8	4	no	—
100	D₄×C₄₃	—	8	4	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	—	8	4	no	—
104	C₂×C₄×C₄₃	#124	1	2	yes	—
105	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
109	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×Dic₄₃	—	4	4	no	—
111	C₂×Dic₄₃	—	8	4	no	—
112	C₈×C₄₃	#124	1	2	yes	—
113	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
114	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
115	C₈×C₄₃	C₂×#41	1	2	yes	—
116	#33708	—	43	1	no	maximal, Sylow
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	—	8	2	no	—
119	C₂²×C₄₃	—	8	4	no	—
120	C₂²×C₄₃	#486	1	2	yes	—
121	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₄₃	—	2	2	no	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	—	8	2	no	—
125	C₄×C₄₃	#486	1	2	yes	—
126	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
127	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
128	Dic₄₃	—	8	4	no	—
129	C₂×#499	—	43	1	no	—
130	#3147	D₄₃	1	1	yes	characteristic
131	#2245	—	43	2	no	—
132	#2244	—	43	2	no	—
133	#2242	—	43	4	no	—
134	#2139	—	43	4	no	—
135	C₄×M₃₂	—	43	1	no	—
136	C₂×C₄₃	—	2	2	no	—
137	C₂×C₄₃	—	16	2	no	—
138	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
139	C₂×C₄₃	#3414	1	2	yes	—
140	C₂×#135	—	2	1	no	—
141	C₂×#135	C₂×D₄₃	1	2	yes	—
142	#532	—	2	1	no	—
143	C₂×#125	—	86	1	no	—
144	C₂×M₃₂	—	43	2	no	—
145	#499	—	43	2	no	—
146	#499	—	43	2	no	—
147	#499	—	86	1	no	—
148	#499	—	86	1	no	—
149	#495	—	2	1	no	—
150	#495	C₂×D₄₃	1	2	yes	—
151	#494	C₂×D₄₃	1	1	yes	characteristic
152	#493	C₂×D₄₃	1	1	yes	characteristic
153	#492	—	2	1	no	—
154	C₂×#107	—	43	2	no	—
155	C₂×#106	—	86	1	no	—
156	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
157	#358	—	43	8	no	—
158	#357	—	43	8	no	—
159	#346	—	43	2	no	—
160	C₄×C₁₆	—	43	2	no	—
161	C₄₃	#33708	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	2	2	no	—
163	#135	—	2	8	no	—
164	#135	—	4	4	no	—
165	C₂×Q₁₆	—	2	1	no	—
166	C₂×Q₁₆	C₂²×D₄₃	1	2	yes	—
167	C₂×SD₁₆	—	2	2	no	—
168	C₂×SD₁₆	—	2	2	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	C₂²×D₄₃	1	2	yes	—
171	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	2	1	no	—
174	#127	—	2	1	no	—
175	#125	—	86	2	no	—
176	#125	—	86	2	no	—
177	#125	—	172	2	no	—
178	#124	—	2	2	no	—
179	#121	—	4	2	no	—
180	C₄×D₄	—	4	2	no	—
181	C₂×C₄⋊C₄	—	86	2	no	—
182	C₂×C₄⋊C₄	—	172	2	no	—
183	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
184	M₃₂	—	86	4	no	—
185	C₂×C₁₆	—	43	4	no	—
186	#107	—	43	4	no	—
187	#107	—	86	2	no	—
188	#106	—	86	2	no	—
189	#106	—	86	2	no	—
190	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
191	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
192	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
194	#41	—	4	4	no	—
195	#41	—	4	4	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	172	2	no	—
202	C₂²×C₄	#10621882	1	2	yes	—
203	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	4	2	no	—
206	SD₁₆	—	4	4	no	—
207	SD₁₆	—	4	4	no	—
208	D₈	—	2	4	no	—
209	D₈	—	4	2	no	—
210	C₂×C₈	#10621883	1	4	yes	—
211	C₂×C₈	#10621900	1	4	yes	—
212	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
213	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
214	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
215	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	86	4	no	—
218	C₄⋊C₄	—	172	2	no	—
219	C₄⋊C₄	—	172	4	no	—
220	C₄⋊C₄	—	172	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	#10621882	1	2	yes	—
223	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
225	C₁₆	—	86	4	no	—
226	C₂³	—	4	2	no	—
227	C₂³	C₂×#10621882	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	—	344	4	no	—
238	C₂×C₄	—	172	4	no	—
239	C₂×C₄	#13063312	1	2	yes	—
240	C₂×C₄	C₂×#10621882	1	2	yes	—
241	C₂×C₄	C₂×#10621882	1	2	yes	—
242	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
243	C₂×C₄	C₂×#10621883	1	2	yes	—
244	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
245	C₈	#13063312	1	2	yes	—
246	C₈	C₂×#10621883	1	2	yes	—
247	C₈	C₂×#10621883	1	2	yes	—
248	C₈	C₂×#10621900	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	#14583245	1	2	yes	—
253	C₂²	C₂²×#10621882	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	344	4	no	—
258	C₄	#14583245	1	2	yes	—
259	C₄	C₂×#13063312	1	1	yes	characteristic
260	C₄	C₂²×#10621883	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	16	2	no	—
263	C₂	C₂×#14583245	1	1	yes	characteristic
264	C₂	#19147729	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3147	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	#3147	D₄₃
hypocenter	C₄₃	#33708
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#14583301	C₂	semidirect
2. #19136840	C₂	semidirect
3. #19136836	C₂	semidirect
4. #19136826	C₂	semidirect
5. #19136307	C₂	semidirect
6. C₄×#13063265	C₂	semidirect
161. C₄₃	#33708	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583301	2
#19136840	2
#19136836	2
#19136826	2
#19136307	2
C₄×#13063265	2
C₄₃×#3147	2
#33708	43