Subgroups — #25105022

Order 11008.

1422 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583276	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582981	C₂	1	4	yes	maximal, complemented
3	#19137157	C₂	1	2	yes	maximal, complemented
4	#19135459	C₂	1	4	yes	maximal, complemented
5	#19135400	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₄₄	C₂²	1	4	yes	—
9	C₂×#13063322	C₂²	1	2	yes	—
10	C₂×#13063287	—	2	4	no	—
11	C₂×#13063287	C₂²	1	4	yes	—
12	#14583276	C₄	1	8	yes	complemented
13	#14582981	—	2	16	no	—
14	C₂×#13063245	C₂²	1	8	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
17	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
18	#14582800	—	2	4	no	—
19	#14582800	C₂²	1	4	yes	—
20	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×#107×C₄₃	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#384	C₂²	1	2	yes	—
24	C₂²×D₁₇₂	—	2	4	no	—
25	C₂²×D₁₇₂	C₂³	1	4	yes	—
26	C₂²×C₄×D₄₃	—	2	4	no	—
27	C₂×D₃₄₄	—	2	8	no	—
28	C₂×D₃₄₄	—	2	8	no	—
29	C₂×D₃₄₄	C₂×C₄	1	16	yes	complemented
30	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
31	#13063322	—	2	4	no	—
32	#13063322	C₂×C₄	1	8	yes	complemented
33	#13063287	—	2	16	no	—
34	#13063287	—	4	16	no	—
35	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	8	no	—
37	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
38	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
39	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
41	#13063245	—	2	16	no	—
42	C₄×C₄₃⋊C₈	—	2	2	no	—
43	C₈×Dic₄₃	—	2	4	no	—
44	C₈×Dic₄₃	D₄	1	8	yes	complemented
45	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
49	#107×C₄₃	—	2	8	no	—
50	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	complemented
51	C₂³×D₄₃	—	2	4	no	—
52	C₂²×C₄×C₄₃	—	2	4	no	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
55	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
57	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂×D₁₇₂	—	2	16	no	—
59	C₂×D₁₇₂	—	2	4	no	—
60	C₂×D₁₇₂	—	2	8	no	—
61	C₂×D₁₇₂	—	2	8	no	—
62	C₂×D₁₇₂	—	4	8	no	—
63	C₂×D₁₇₂	—	4	8	no	—
64	C₂×D₁₇₂	C₂²×C₄	1	8	yes	—
65	C₂×C₄×D₄₃	—	4	8	no	—
66	C₂×C₄×D₄₃	—	4	16	no	—
67	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₄₃	#41	1	4	yes	—
70	C₄⋊C₄×C₄₃	—	2	8	no	—
71	C₄⋊C₄×C₄₃	—	4	8	no	—
72	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₄₃	#41	1	2	yes	—
74	(C₂×D₄₃)⋊C₄	—	4	8	no	—
75	(C₂×D₄₃)⋊C₄	—	4	16	no	—
76	C₄×Dic₄₃	—	2	4	no	—
77	C₄×Dic₄₃	—	2	2	no	—
78	C₄×Dic₄₃	D₈	1	8	yes	complemented
79	C₄×Dic₄₃	C₂×D₄	1	4	yes	—
80	C₂×C₄₃⋊C₈	—	2	2	no	—
81	C₂×C₄₃⋊C₈	—	2	2	no	—
82	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
83	D₃₄₄	—	2	32	no	—
84	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₄₃	—	2	8	no	—
86	C₂²×D₄₃	—	4	4	no	—
87	C₂²×D₄₃	—	4	4	no	—
88	C₂²×D₄₃	—	4	8	no	—
89	C₂²×D₄₃	—	4	8	no	—
90	C₂×C₄×C₄₃	—	4	4	no	—
91	C₂×C₄×C₄₃	—	4	8	no	—
92	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₄₃	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₄₃	#135	1	4	yes	—
95	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
98	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
99	C₂×Dic₄₃	—	2	2	no	—
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	#127	1	2	yes	—
103	C₂×Dic₄₃	#127	1	2	yes	—
104	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
105	C₂×Dic₄₃	C₂×D₈	1	4	yes	—
106	D₁₇₂	—	2	16	no	—
107	D₁₇₂	—	4	8	no	—
108	D₁₇₂	—	4	32	no	—
109	C₄×D₄₃	—	8	16	no	—
110	C₈×C₄₃	C₄×D₄	1	8	yes	complemented
111	C₄₃⋊C₈	—	2	4	no	—
112	C₄×#493	—	43	1	no	maximal, Sylow
113	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
115	C₂²×C₄₃	#532	1	1	yes	characteristic
116	C₂²×C₄₃	C₂²×D₈	1	2	yes	—
117	C₂²×C₄₃	C₂×#135	1	2	yes	—
118	C₂×D₄₃	—	4	8	no	—
119	C₂×D₄₃	—	4	16	no	—
120	C₂×D₄₃	—	8	4	no	—
121	C₂×D₄₃	—	8	4	no	—
122	C₂×D₄₃	—	8	8	no	—
123	C₄×C₄₃	—	8	4	no	—
124	C₄×C₄₃	C₄×D₈	1	8	yes	complemented
125	C₄×C₄₃	C₂×C₄×D₄	1	2	yes	—
126	C₄×C₄₃	#532	1	2	yes	—
127	Dic₄₃	—	2	2	no	—
128	Dic₄₃	—	2	4	no	—
129	Dic₄₃	#493	1	4	yes	complemented
130	C₂×#493	—	43	1	no	—
131	C₂×C₄×D₈	—	43	4	no	—
132	C₄×#127	—	43	2	no	—
133	#1970	—	43	4	no	—
134	#1957	—	43	2	no	—
135	#1845	D₄₃	1	1	yes	characteristic, complemented
136	C₄²×C₈	—	43	1	no	—
137	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
138	C₂×C₄₃	C₂×C₄×D₈	1	2	yes	—
139	C₂×C₄₃	C₂×#493	1	1	yes	characteristic
140	C₂×C₄₃	#3147	1	1	yes	characteristic
141	C₂×C₄₃	#3169	1	2	yes	—
142	D₄₃	—	8	8	no	—
143	C₂²×D₈	—	43	4	no	—
144	C₂×#127	—	43	2	no	—
145	C₂×C₄×D₄	—	43	4	no	—
146	C₂×C₄×D₄	—	86	4	no	—
147	#493	—	43	8	no	—
148	C₄×D₈	—	86	16	no	—
149	C₂×#107	C₂×D₄₃	1	4	yes	—
150	C₂×#102	—	43	8	no	—
151	C₂×C₄×C₈	—	43	2	no	—
152	C₂×C₄×C₈	—	43	2	no	—
153	C₂×C₄×C₈	—	43	1	no	—
154	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
155	#390	—	43	4	no	—
156	#390	—	86	4	no	—
157	#384	C₂×D₄₃	1	2	yes	—
158	C₄³	—	43	1	no	—
159	C₄₃	C₄×#493	1	1	yes	characteristic, complemented, Sylow
160	C₂²×D₄	—	43	4	no	—
161	C₂²×D₄	—	86	4	no	—
162	C₂³×C₄	—	86	4	no	—
163	C₂×D₈	—	43	16	no	—
164	C₂×D₈	—	86	8	no	—
165	C₂×D₈	—	86	8	no	—
166	C₂²×C₈	—	43	2	no	—
167	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
168	#127	—	43	8	no	—
169	#127	—	86	4	no	—
170	C₄×D₄	—	86	16	no	—
171	C₄×D₄	—	172	16	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
174	C₂×C₂²⋊C₄	—	86	4	no	—
175	C₂×C₂²⋊C₄	—	86	8	no	—
176	C₂×C₄²	—	43	2	no	—
177	C₂×C₄²	—	43	1	no	—
178	C₂×C₄²	—	43	2	no	—
179	C₂×C₄²	—	43	1	no	—
180	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
181	#107	—	2	8	no	—
182	#102	—	86	16	no	—
183	C₄×C₈	—	43	8	no	—
184	C₄×C₈	—	86	4	no	—
185	C₄×C₈	—	86	2	no	—
186	C₄×C₈	C₄×D₄₃	1	4	yes	complemented
187	C₂⁴	—	86	4	no	—
188	C₂×D₄	—	43	8	no	—
189	C₂×D₄	—	86	16	no	—
190	C₂×D₄	—	86	8	no	—
191	C₂×D₄	—	86	8	no	—
192	C₂×D₄	—	86	4	no	—
193	C₂×D₄	—	172	8	no	—
194	C₂×D₄	—	172	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	43	1	no	—
197	C₂²×C₄	—	43	2	no	—
198	C₂²×C₄	—	43	1	no	—
199	C₂²×C₄	—	172	8	no	—
200	C₂²×C₄	—	172	16	no	—
201	C₂²×C₄	D₄×D₄₃	1	2	yes	—
202	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
203	D₈	—	86	32	no	—
204	C₂×C₈	—	43	4	no	—
205	C₂×C₈	—	86	2	no	—
206	C₂×C₈	—	86	2	no	—
207	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
208	C₂×C₈	D₄×D₄₃	1	4	yes	—
209	C₂×C₈	#10621902	1	4	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	4	8	no	—
212	C₂²⋊C₄	—	172	8	no	—
213	C₂²⋊C₄	—	172	16	no	—
214	C₄²	—	43	8	no	—
215	C₄²	—	43	4	no	—
216	C₄²	—	86	4	no	—
217	C₄²	—	86	2	no	—
218	C₄²	C₂×C₄×D₄₃	1	2	yes	—
219	C₄²	#10621902	1	2	yes	—
220	C₂³	—	86	8	no	—
221	C₂³	—	172	8	no	—
222	C₂³	—	172	8	no	—
223	C₂³	—	172	4	no	—
224	C₂³	—	172	4	no	—
225	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
226	D₄	—	86	16	no	—
227	D₄	—	172	8	no	—
228	D₄	—	172	32	no	—
229	C₂×C₄	—	4	8	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	43	4	no	—
232	C₂×C₄	—	43	2	no	—
233	C₂×C₄	—	43	4	no	—
234	C₂×C₄	—	43	2	no	—
235	C₂×C₄	—	86	2	no	—
236	C₂×C₄	—	86	2	no	—
237	C₂×C₄	—	86	2	no	—
238	C₂×C₄	—	344	16	no	—
239	C₂×C₄	#13063287	1	4	yes	—
240	C₂×C₄	D₈×D₄₃	1	4	yes	—
241	C₂×C₄	#13063348	1	4	yes	—
242	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₄₃	1	2	yes	—
244	C₂×C₄	C₂×#10621902	1	2	yes	—
245	C₂×C₄	C₂×#10621902	1	1	yes	characteristic
246	C₈	—	86	4	no	—
247	C₈	#13063287	1	8	yes	complemented
248	C₂²	—	344	4	no	—
249	C₂²	—	344	4	no	—
250	C₂²	—	344	8	no	—
251	C₂²	—	172	8	no	—
252	C₂²	—	172	16	no	—
253	C₂²	C₂×#13063287	1	1	yes	characteristic
254	C₂²	#127×D₄₃	1	1	yes	characteristic
255	C₂²	#14583450	1	1	yes	characteristic
256	C₂²	C₂×D₈×D₄₃	1	2	yes	—
257	C₂²	C₂×#13063348	1	2	yes	—
258	C₄	—	8	4	no	—
259	C₄	—	43	4	no	—
260	C₄	—	86	2	no	—
261	C₄	—	86	4	no	—
262	C₄	#14582981	1	8	yes	complemented
263	C₄	C₂×#13063287	1	2	yes	—
264	C₄	#14583450	1	2	yes	—
265	C₂	—	344	8	no	—
266	C₂	#19137157	1	1	yes	characteristic
267	C₂	C₂×#14582981	1	2	yes	—
268	C₂	#493×D₄₃	1	1	yes	characteristic
269	C₂	#19145024	1	1	yes	characteristic
270	C₂	#19145175	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1845	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂×#14583276	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1845	D₄₃
hypocenter	C₄₃	C₄×#493
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
2. C₂×#14582981	C₂	semidirect
3. #19137157	C₂	semidirect
4. #19135459	C₂	semidirect
5. #19135400	C₂	semidirect
6. C₄×C₈×Dic₄₃	C₂	semidirect
7. C₄₃×#1845	C₂	semidirect
12. #14583276	C₄	semidirect
29. C₂×D₃₄₄	C₂×C₄	semidirect
32. #13063322	C₂×C₄	semidirect
44. C₈×Dic₄₃	D₄	semidirect
50. C₄×C₈×C₄₃	C₂×C₄	semidirect
78. C₄×Dic₄₃	D₈	semidirect
110. C₈×C₄₃	C₄×D₄	semidirect
124. C₄×C₄₃	C₄×D₈	semidirect
129. Dic₄₃	#493	semidirect
135. #1845	D₄₃	semidirect
159. C₄₃	C₄×#493	semidirect
186. C₄×C₈	C₄×D₄₃	semidirect
247. C₈	#13063287	semidirect
262. C₄	#14582981	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583276	2
C₂×#14582981	2
#19137157	2
#19135459	2
#19135400	2
C₄×C₈×Dic₄₃	2
C₄₃×#1845	2
C₄×#493	43