Subgroups — C₄×#14582960 #25094741

Order 11008.

970 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582960	C₂	1	2	yes	maximal
2	#19134452	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	#19134361	C₂	1	2	yes	maximal
5	C₄×C₂²⋊(C₄₃⋊C₈)	C₂	1	2	yes	maximal
6	#19134347	C₂	1	2	yes	maximal, complemented
7	C₄×#13063236	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
10	#14582960	C₄	1	16	yes	direct product factor
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
12	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
13	C₂×#13063236	C₂²	1	2	yes	—
14	C₂×#13063236	C₂²	1	2	yes	—
15	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
19	#14582654	C₂²	1	4	yes	—
20	#14582654	C₂²	1	4	yes	—
21	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₄₃	C₄	1	4	yes	—
25	C₄×C₂²⋊C₄×C₄₃	C₄	1	4	yes	—
26	C₄₃×#376	C₂²	1	1	yes	characteristic
27	C₄₃×#376	C₂²	1	1	yes	characteristic
28	C₄³×C₄₃	C₂²	1	1	yes	characteristic
29	C₄³×C₄₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
31	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
32	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	16	yes	complemented
33	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	16	yes	—
34	#13063236	—	2	4	no	—
35	#13063236	C₂×C₄	1	8	yes	complemented
36	C₄×C₄₃⋊C₈	—	2	4	no	—
37	C₄×C₄₃⋊C₈	—	2	4	no	—
38	C₄×C₄₃⋊C₈	—	2	4	no	—
39	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	complemented
40	C₂³×C₄×C₄₃	C₂×C₄	1	2	yes	—
41	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	#117×C₄₃	—	2	4	no	—
43	#117×C₄₃	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₄₃	—	2	4	no	—
49	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	#95×C₄₃	C₂×C₄	1	4	yes	—
59	#95×C₄₃	C₂×C₄	1	4	yes	—
60	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	4	no	—
64	C₂²×C₄×C₄₃	—	2	8	no	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₄₃	—	2	4	no	—
75	C₄⋊C₄×C₄₃	C₄²	1	8	yes	—
76	C₂²⋊C₄×C₄₃	—	2	4	no	—
77	C₂²⋊C₄×C₄₃	C₄²	1	8	yes	—
78	C₄²×C₄₃	—	2	4	no	—
79	C₄²×C₄₃	—	2	8	no	—
80	C₄²×C₄₃	—	2	8	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
85	C₄²×C₄₃	#41	1	4	yes	—
86	C₄²×C₄₃	#41	1	4	yes	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
92	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
93	C₂³×C₄₃	—	2	1	no	—
94	C₂³×C₄₃	—	2	1	no	—
95	C₂³×C₄₃	—	2	4	no	—
96	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
97	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	—	2	1	no	—
99	C₂×C₄×C₄₃	—	2	1	no	—
100	C₂×C₄×C₄₃	—	2	4	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	8	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	4	8	no	—
109	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
110	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₄₃	#117	1	4	yes	—
113	C₂×C₄×C₄₃	#117	1	4	yes	—
114	C₂×C₄×C₄₃	#131	1	8	yes	—
115	C₂×C₄×C₄₃	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
118	C₄₃⋊C₈	—	2	8	no	—
119	C₄₃⋊C₈	—	2	8	no	—
120	C₄×#433	—	43	1	no	maximal, Sylow
121	C₂²×C₄₃	—	2	4	no	—
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	—	4	2	no	—
124	C₂²×C₄₃	—	4	2	no	—
125	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
126	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
127	C₂²×C₄₃	C₂×#117	1	1	yes	characteristic
128	C₂²×C₄₃	C₂×#131	1	4	yes	—
129	C₄×C₄₃	—	2	4	no	—
130	C₄×C₄₃	—	2	4	no	—
131	C₄×C₄₃	—	2	4	no	—
132	C₄×C₄₃	—	2	2	no	—
133	C₄×C₄₃	—	4	2	no	—
134	C₄×C₄₃	—	4	2	no	—
135	C₄×C₄₃	#433	1	8	yes	complemented
136	C₄×C₄₃	C₄×#41	1	4	yes	—
137	C₂×#433	—	43	2	no	—
138	C₄×#117	D₄₃	1	1	yes	characteristic
139	#1768	—	43	2	no	—
140	C₄×C₄⋊C₈	—	43	2	no	—
141	#1750	—	43	2	no	—
142	C₄×C₂²⋊C₈	—	43	2	no	—
143	#1746	—	43	2	no	—
144	C₄×#97	—	43	1	no	—
145	C₄²×C₈	—	43	1	no	—
146	C₂×C₄₃	—	4	2	no	—
147	C₂×C₄₃	C₄×#117	1	1	yes	characteristic
148	C₂×C₄₃	C₄×#131	1	2	yes	—
149	C₂×C₄₃	C₂×#433	1	2	yes	—
150	C₂×C₄₃	#2922	1	2	yes	—
151	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
152	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
153	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
154	#433	—	43	16	no	—
155	C₂×C₄⋊C₈	—	43	4	no	—
156	C₂×C₂²⋊C₈	—	43	4	no	—
157	C₂×#97	—	43	2	no	—
158	C₂×#97	—	43	2	no	—
159	C₂×C₄×C₈	—	43	2	no	—
160	C₂×C₄×C₈	—	43	2	no	—
161	C₂×C₄×C₈	—	43	2	no	—
162	C₂×C₄×C₈	—	43	2	no	—
163	C₄×C₄⋊C₄	Dic₄₃	1	4	yes	—
164	C₄×C₂²⋊C₄	Dic₄₃	1	4	yes	—
165	#376	C₂×D₄₃	1	1	yes	characteristic
166	#376	C₂×D₄₃	1	1	yes	characteristic
167	C₄³	C₂×D₄₃	1	1	yes	characteristic
168	C₄³	C₂×D₄₃	1	1	yes	characteristic
169	#336	—	43	4	no	—
170	#336	—	43	4	no	—
171	C₄₃	C₄×#433	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₄	C₂×Dic₄₃	1	2	yes	—
173	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
174	C₂²×C₈	—	43	4	no	—
175	C₂²×C₈	—	43	4	no	—
176	#117	—	2	4	no	—
177	#117	C₄×D₄₃	1	8	yes	—
178	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
180	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
181	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	2	yes	—
182	C₂×C₄²	—	2	4	no	—
183	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
184	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
185	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
186	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
188	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
189	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
192	C₄⋊C₈	—	43	16	no	—
193	C₂²⋊C₈	—	43	16	no	—
194	#97	—	43	8	no	—
195	#97	—	86	4	no	—
196	C₄×C₈	—	43	8	no	—
197	C₄×C₈	—	86	4	no	—
198	C₄×C₈	—	86	4	no	—
199	C₄×C₈	—	86	4	no	—
200	#95	C₂×Dic₄₃	1	4	yes	—
201	#95	C₂×Dic₄₃	1	4	yes	—
202	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	C₂×C₄×D₄₃	1	4	yes	—
208	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
210	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
211	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
215	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
216	C₂×C₈	—	43	8	no	—
217	C₂×C₈	—	43	8	no	—
218	C₂×C₈	—	86	4	no	—
219	C₂×C₈	—	86	4	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	—	86	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	C₄×Dic₄₃	1	8	yes	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	C₄×Dic₄₃	1	8	yes	—
226	C₄²	—	2	4	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	8	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	C₂×C₄×D₄₃	1	4	yes	—
232	C₄²	C₂×C₄×D₄₃	1	4	yes	—
233	C₄²	#10621898	1	4	yes	—
234	C₄²	#10621898	1	4	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
239	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	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	8	no	—
251	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
252	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
254	C₂×C₄	#13063284	1	4	yes	—
255	C₂×C₄	#13063284	1	4	yes	—
256	C₂×C₄	#13063332	1	8	yes	—
257	C₂×C₄	C₂²×C₄×D₄₃	1	2	yes	—
258	C₂×C₄	C₂×#10621898	1	2	yes	—
259	C₂×C₄	C₂×#10621898	1	2	yes	—
260	C₈	—	86	8	no	—
261	C₈	—	86	8	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
267	C₂²	C₂×#13063284	1	1	yes	characteristic
268	C₂²	C₂×#13063284	1	1	yes	characteristic
269	C₂²	C₂×#13063332	1	4	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	4	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	#14582960	1	8	yes	direct product factor
277	C₄	C₄×#10621898	1	4	yes	—
278	C₂	—	4	2	no	—
279	C₂	C₄×#13063284	1	1	yes	characteristic
280	C₂	C₄×#13063332	1	2	yes	—
281	C₂	C₂×#14582960	1	2	yes	—
282	C₂	#19143239	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₄×#117	D₄₃
hypocenter	C₄₃	C₄×#433
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #19134452	C₂	semidirect
3. C₄×C₄⋊(C₄₃⋊C₈)	C₂	semidirect
6. #19134347	C₂	semidirect
7. C₄×#13063236	C₂	semidirect
8. C₄²×C₄₃⋊C₈	C₂	semidirect
10. #14582960	C₄	direct (N × H)
32. C₄⋊(C₄₃⋊C₈)	C₂×C₄	semidirect
35. #13063236	C₂×C₄	semidirect
39. C₄×C₄₃⋊C₈	C₂×C₄	semidirect
135. C₄×C₄₃	#433	semidirect
171. C₄₃	C₄×#433	semidirect
276. C₄	#14582960	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#14582960	2
#19134452	2
C₄×C₄⋊(C₄₃⋊C₈)	2
#19134361	2
C₄×C₂²⋊(C₄₃⋊C₈)	2
#19134347	2
C₄×#13063236	2
C₄²×C₄₃⋊C₈	2
C₄×#117×C₄₃	2
C₄×#433	43

Subgroups — C4×#14582960 #25094741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#14582960 #25094741