Subgroups — #25105213

Order 11008.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145023	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583020	C₂	1	4	yes	maximal, complemented
3	#19143306	C₂	1	2	yes	maximal, complemented
4	#19135486	C₂	1	4	yes	maximal, complemented
5	#19135422	C₂	1	2	yes	maximal, complemented
6	C₄×#13063235	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063347	C₂²	1	4	yes	—
9	C₂×#13063333	C₂²	1	4	yes	—
10	#14583449	C₂²	1	2	yes	—
11	#14583283	C₄	1	4	yes	complemented
12	#14583279	C₄	1	1	yes	characteristic, complemented
13	#14583276	C₄	1	1	yes	characteristic, complemented
14	#14583272	C₄	1	2	yes	complemented
15	#14583020	—	2	16	no	—
16	#14582976	—	2	4	no	—
17	#14582964	—	2	4	no	—
18	C₂×#13063251	C₂²	1	8	yes	—
19	#14582867	C₂²	1	4	yes	—
20	C₂×#13063235	C₂²	1	2	yes	—
21	C₂×#13063235	C₂²	1	2	yes	—
22	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₄₃	C₂²	1	4	yes	—
25	C₄₃×#423	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#10621898	C₂³	1	4	yes	—
28	#13063347	—	2	8	no	—
29	#13063347	—	2	8	no	—
30	C₂×Dic₁₇₂	C₂×C₄	1	4	yes	—
31	C₂×#10621873	C₂×C₄	1	8	yes	—
32	C₂×D₃₄₄	C₂×C₄	1	4	yes	—
33	#13063333	—	2	16	no	—
34	C₂×#10621872	—	2	4	no	—
35	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
36	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
37	C₂×C₈×D₄₃	—	2	4	no	—
38	#13063325	C₂×C₄	1	2	yes	—
39	#13063322	C₂×C₄	1	2	yes	—
40	#13063313	C₂×C₄	1	4	yes	—
41	#13063299	—	2	4	no	—
42	C₄×D₁₇₂	—	2	4	no	—
43	C₂×C₄×Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
45	#13063251	—	2	16	no	—
46	(C₂×D₄₃)⋊C₈	—	2	8	no	—
47	#13063235	—	2	4	no	—
48	#13063235	D₄	1	8	yes	complemented
49	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
50	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
51	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#108×C₄₃	—	2	8	no	—
54	C₄⋊C₈×C₄₃	—	2	4	no	—
55	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	#41	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₃	—	2	4	no	—
61	C₂²×Dic₄₃	—	2	2	no	—
62	#10621898	—	2	8	no	—
63	#10621898	—	2	8	no	—
64	C₂×D₁₇₂	—	2	4	no	—
65	C₂×D₁₇₂	C₂²×C₄	1	4	yes	—
66	C₂×C₄×D₄₃	—	2	4	no	—
67	C₂×Dic₈₆	C₂²×C₄	1	4	yes	—
68	M₁₆×C₄₃	—	2	8	no	—
69	C₂×C₈×C₄₃	—	2	4	no	—
70	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
73	C₂×C₈×C₄₃	#41	1	4	yes	—
74	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₃	C₂×D₄	1	2	yes	—
77	(C₂×D₄₃)⋊C₄	—	2	8	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄×Dic₄₃	—	2	2	no	—
80	C₄×Dic₄₃	—	4	4	no	—
81	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
82	C₂×C₄₃⋊C₈	—	2	2	no	—
83	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
84	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
85	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
86	Dic₁₇₂	—	2	8	no	—
87	D₃₄₄	—	2	8	no	—
88	#10621873	—	2	16	no	—
89	#10621872	—	4	8	no	—
90	C₈×D₄₃	—	4	8	no	—
91	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
92	C₂²×D₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	2	4	no	—
94	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
100	C₂²⋊D₄₃	—	4	8	no	—
101	C₂×Dic₄₃	—	2	4	no	—
102	C₂×Dic₄₃	—	2	4	no	—
103	C₂×Dic₄₃	—	4	4	no	—
104	D₁₇₂	—	2	8	no	—
105	D₁₇₂	—	4	8	no	—
106	C₄×D₄₃	—	4	4	no	—
107	C₄×D₄₃	—	4	4	no	—
108	Dic₈₆	—	2	8	no	—
109	C₈×C₄₃	—	4	4	no	—
110	C₈×C₄₃	C₄×D₄	1	4	yes	—
111	C₈×C₄₃	C₄×D₄	1	4	yes	—
112	C₄₃⋊C₈	—	2	2	no	—
113	C₄₃⋊C₈	#127	1	4	yes	—
114	#21706	—	43	1	no	maximal, Sylow
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂²×C₄₃	#532	1	2	yes	—
118	C₂×D₄₃	—	4	4	no	—
119	C₂×D₄₃	—	4	8	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	#444	1	4	yes	complemented
122	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
123	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
124	C₄×C₄₃	#532	1	1	yes	characteristic
125	Dic₄₃	—	4	4	no	—
126	Dic₄₃	—	4	4	no	—
127	#3147	—	43	1	no	—
128	C₂×#444	—	43	4	no	—
129	#2931	—	43	2	no	—
130	#1975	—	43	4	no	—
131	#1962	—	43	2	no	—
132	#1848	D₄₃	1	1	yes	characteristic, complemented
133	C₄×#97	—	43	1	no	—
134	C₂×C₄₃	—	2	2	no	—
135	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
136	C₂×C₄₃	C₂×#444	1	2	yes	—
137	D₄₃	—	8	4	no	—
138	C₂×#135	—	43	4	no	—
139	C₂×#131	—	43	4	no	—
140	#532	—	43	2	no	—
141	#495	—	43	4	no	—
142	#494	—	43	1	no	—
143	#493	—	43	1	no	—
144	#492	—	43	2	no	—
145	#444	—	86	16	no	—
146	#436	—	86	4	no	—
147	C₈×D₄	—	86	4	no	—
148	C₂×#108	C₂×D₄₃	1	4	yes	—
149	#423	C₂×D₄₃	1	2	yes	—
150	C₂×#104	—	43	8	no	—
151	#408	—	43	4	no	—
152	C₂×#97	—	43	2	no	—
153	C₂×#97	—	43	2	no	—
154	C₂×C₄×C₈	—	43	1	no	—
155	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
156	C₄³	—	43	1	no	—
157	C₄₃	#21706	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	43	4	no	—
159	#135	—	86	8	no	—
160	#135	—	86	8	no	—
161	C₂×Q₁₆	—	43	4	no	—
162	C₂×SD₁₆	—	43	8	no	—
163	C₂×D₈	—	43	4	no	—
164	#131	—	86	16	no	—
165	C₂×M₁₆	—	86	4	no	—
166	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
167	C₂²×C₈	—	43	1	no	—
168	C₂²×C₈	—	43	1	no	—
169	C₂²×C₈	—	86	4	no	—
170	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
171	C₄⋊Q₈	—	43	2	no	—
172	#127	—	43	2	no	—
173	#124	—	43	4	no	—
174	#121	—	86	4	no	—
175	C₄×D₄	—	86	4	no	—
176	C₂×C₄²	—	43	2	no	—
177	C₂×C₄²	—	86	2	no	—
178	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
179	#108	—	2	8	no	—
180	C₄⋊C₈	—	2	4	no	—
181	#104	—	86	16	no	—
182	C₂²⋊C₈	—	86	8	no	—
183	#97	—	43	8	no	—
184	#97	—	86	4	no	—
185	C₄×C₈	—	43	4	no	—
186	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
187	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
188	C₄×C₈	C₄×D₄₃	1	2	yes	—
189	#41	—	86	8	no	—
190	#41	—	86	8	no	—
191	C₂×Q₈	—	43	4	no	—
192	C₂×D₄	—	43	4	no	—
193	C₂×D₄	—	86	4	no	—
194	C₂×D₄	—	86	4	no	—
195	C₂²×C₄	—	86	2	no	—
196	C₂²×C₄	—	86	4	no	—
197	C₂²×C₄	#10621902	1	2	yes	—
198	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
199	Q₁₆	—	86	8	no	—
200	SD₁₆	—	86	16	no	—
201	D₈	—	86	8	no	—
202	M₁₆	—	2	8	no	—
203	M₁₆	—	172	8	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	43	4	no	—
206	C₂×C₈	—	43	2	no	—
207	C₂×C₈	—	43	2	no	—
208	C₂×C₈	—	86	2	no	—
209	C₂×C₈	—	172	8	no	—
210	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
212	C₂×C₈	D₄×D₄₃	1	4	yes	—
213	C₂×C₈	#10621902	1	4	yes	—
214	C₄⋊C₄	—	86	4	no	—
215	C₂²⋊C₄	—	86	8	no	—
216	C₄²	—	43	4	no	—
217	C₄²	—	86	2	no	—
218	C₄²	—	172	4	no	—
219	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
220	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
221	C₄²	D₄×D₄₃	1	2	yes	—
222	C₂³	—	86	4	no	—
223	C₂³	C₂×#10621902	1	1	yes	characteristic
224	Q₈	—	86	8	no	—
225	D₄	—	86	8	no	—
226	D₄	—	172	8	no	—
227	D₄	—	172	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	86	4	no	—
230	C₂×C₄	—	86	4	no	—
231	C₂×C₄	—	172	4	no	—
232	C₂×C₄	—	172	4	no	—
233	C₂×C₄	—	172	4	no	—
234	C₂×C₄	#13063287	1	2	yes	—
235	C₂×C₄	#13063287	1	2	yes	—
236	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
238	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
239	C₂×C₄	C₂×#10621902	1	2	yes	—
240	C₈	—	4	4	no	—
241	C₈	—	43	4	no	—
242	C₈	—	86	2	no	—
243	C₈	#13063287	1	4	yes	—
244	C₈	#13063287	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	172	8	no	—
247	C₂²	—	172	4	no	—
248	C₂²	C₂×#13063287	1	1	yes	characteristic
249	C₂²	#14583450	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	172	4	no	—
252	C₄	—	172	4	no	—
253	C₄	#14583020	1	4	yes	complemented
254	C₄	C₂×#13063287	1	2	yes	—
255	C₄	#127×D₄₃	1	1	yes	characteristic
256	C₄	#14583450	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	344	4	no	—
259	C₂	#19137157	1	1	yes	characteristic
260	C₂	C₂×#14583020	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1848	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	#19145023	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1848	D₄₃
hypocenter	C₄₃	#21706
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. C₂×#14583020	C₂	semidirect
3. #19143306	C₂	semidirect
4. #19135486	C₂	semidirect
5. #19135422	C₂	semidirect
6. C₄×#13063235	C₂	semidirect
7. C₄₃×#1848	C₂	semidirect
11. #14583283	C₄	semidirect
12. #14583279	C₄	semidirect
13. #14583276	C₄	semidirect
14. #14583272	C₄	semidirect
48. #13063235	D₄	semidirect
121. C₄×C₄₃	#444	semidirect
132. #1848	D₄₃	semidirect
157. C₄₃	#21706	semidirect
253. C₄	#14583020	semidirect

Maximal subgroups

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

subgroup	index
#19145023	2
C₂×#14583020	2
#19143306	2
#19135486	2
#19135422	2
C₄×#13063235	2
C₄₃×#1848	2
#21706	43