Subgroups — C₄×#13063345 #19143365

Order 5504.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063345	C₂	1	1	yes	maximal, characteristic
2	C₄×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#10621885	C₂	1	2	yes	maximal, complemented
4	C₄×#10621884	C₂	1	2	yes	maximal, complemented
5	C₄×#10621883	C₂	1	2	yes	maximal, complemented
6	C₄×#10621882	C₂	1	2	yes	maximal, complemented
7	#14582935	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582895	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582893	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	#13063345	C₄	1	8	yes	direct product factor
14	C₂×#10621885	C₂²	1	1	yes	characteristic
15	C₂×#10621883	C₂²	1	1	yes	characteristic
16	C₂×#10621884	C₂²	1	1	yes	characteristic
17	C₂×#10621882	C₂²	1	1	yes	characteristic
18	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₈₆	C₂²	1	2	yes	—
21	C₄×D₁₇₂	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₄₃	—	2	2	no	—
23	#13063284	C₂²	1	1	yes	characteristic
24	#13063282	—	2	1	no	—
25	C₄²×D₄₃	—	2	1	no	—
26	#13063258	C₂²	1	2	yes	—
27	#13063256	C₂²	1	2	yes	—
28	#13063249	C₂²	1	2	yes	—
29	#13063248	C₂²	1	2	yes	—
30	#13063246	C₂²	1	2	yes	—
31	#13063245	C₂²	1	2	yes	—
32	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
33	C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
36	C₄×D₄×C₄₃	—	2	2	no	—
37	C₄×D₄×C₄₃	C₂²	1	2	yes	—
38	#117×C₄₃	—	2	1	no	—
39	#117×C₄₃	C₂²	1	1	yes	characteristic
40	C₂×C₄²×C₄₃	—	2	1	no	—
41	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
42	#41×C₄₃	—	2	2	no	—
43	#41×C₄₃	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₄₃	—	2	1	no	—
46	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	—	2	1	no	—
48	C₂²×C₄×C₄₃	—	2	2	no	—
49	C₂²×C₄×C₄₃	D₄	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₃	—	2	1	no	—
52	#10621898	—	2	2	no	—
53	#10621898	C₂×C₄	1	4	yes	complemented
54	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₄₃	—	2	1	no	—
56	C₂×C₄×D₄₃	—	2	2	no	—
57	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₄₃	—	2	2	no	—
59	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
60	C₂²⋊C₄×C₄₃	—	2	2	no	—
61	C₂²⋊C₄×C₄₃	—	2	1	no	—
62	C₄²×C₄₃	—	2	1	no	—
63	C₄²×C₄₃	—	2	2	no	—
64	C₄²×C₄₃	D₄	1	1	yes	characteristic
65	C₄²×C₄₃	D₄	1	1	yes	characteristic
66	C₄²×C₄₃	C₂³	1	2	yes	—
67	C₂²⋊Dic₄₃	—	2	1	no	—
68	#10621885	C₂×C₄	1	4	yes	complemented
69	#10621884	C₂×C₄	1	4	yes	complemented
70	#10621883	C₂×C₄	1	4	yes	complemented
71	#10621882	C₂×C₄	1	4	yes	complemented
72	(C₂×D₄₃)⋊C₄	—	2	2	no	—
73	C₄⋊Dic₄₃	C₂³	1	2	yes	—
74	Dic₄₃⋊C₄	—	2	2	no	—
75	C₄×Dic₄₃	—	2	1	no	—
76	C₄×Dic₄₃	—	2	2	no	—
77	C₂×C₄₃⋊C₈	—	2	2	no	—
78	C₂×C₄₃⋊C₈	C₂×C₄	1	4	yes	complemented
79	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
80	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂³×C₄₃	—	2	1	no	—
83	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	1	no	—
85	Q₈×C₄₃	—	2	1	no	—
86	Q₈×C₄₃	C₂²×C₄	1	2	yes	—
87	D₄×C₄₃	—	2	4	no	—
88	D₄×C₄₃	—	2	1	no	—
89	D₄×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	4	no	—
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	4	4	no	—
95	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₄₃	#41	1	2	yes	—
100	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₄₃	—	2	4	no	—
102	C₂×Dic₄₃	—	2	1	no	—
103	C₂×Dic₄₃	—	2	2	no	—
104	D₁₇₂	—	2	1	no	—
105	D₁₇₂	C₂²×C₄	1	2	yes	—
106	C₄×D₄₃	—	2	4	no	—
107	C₄×D₄₃	—	4	4	no	—
108	Dic₈₆	—	2	1	no	—
109	Dic₈₆	C₂²×C₄	1	2	yes	—
110	C₄₃⋊C₈	—	2	2	no	—
111	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
112	C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	—	2	2	no	—
115	C₂²×C₄₃	—	4	2	no	—
116	C₂²×C₄₃	C₄×D₄	1	2	yes	—
117	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₄₃	—	2	2	no	—
119	C₂×D₄₃	—	4	2	no	—
120	C₄×C₄₃	—	2	2	no	—
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	—	4	2	no	—
123	C₄×C₄₃	C₄×D₄	1	2	yes	—
124	C₄×C₄₃	#135	1	4	yes	complemented
125	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
127	Dic₄₃	—	2	2	no	—
128	Dic₄₃	—	4	2	no	—
129	C₄×#135	—	43	1	no	maximal, Sylow
130	C₂×C₄₃	—	2	2	no	—
131	C₂×C₄₃	—	4	2	no	—
132	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₄₃	C₂×#135	1	2	yes	—
134	D₄₃	—	4	2	no	—
135	C₂×#135	—	43	1	no	—
136	C₄×#41	—	43	1	no	—
137	C₄×#41	D₄₃	1	1	yes	characteristic, complemented
138	C₄×Q₁₆	—	43	2	no	—
139	C₄×SD₁₆	—	43	2	no	—
140	C₄×SD₁₆	—	43	2	no	—
141	C₄×D₈	—	43	2	no	—
142	#427	—	43	1	no	—
143	#416	—	43	1	no	—
144	#416	—	43	1	no	—
145	C₂×C₄×C₈	—	43	1	no	—
146	C₄₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	43	1	no	—
148	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
149	#135	—	43	8	no	—
150	C₂×Q₁₆	—	43	1	no	—
151	C₂×SD₁₆	—	43	1	no	—
152	C₂×SD₁₆	—	43	1	no	—
153	C₂×D₈	—	43	1	no	—
154	C₂²×C₈	—	43	1	no	—
155	C₂²×C₈	—	43	1	no	—
156	C₄×Q₈	—	43	2	no	—
157	C₄×Q₈	C₂×D₄₃	1	2	yes	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	43	2	no	—
160	C₄×D₄	—	86	2	no	—
161	C₄×D₄	C₂×D₄₃	1	2	yes	—
162	#117	—	2	1	no	—
163	#117	—	43	1	no	—
164	#117	—	86	1	no	—
165	#117	C₂×D₄₃	1	1	yes	characteristic
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	—	86	1	no	—
168	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
169	#107	—	43	2	no	—
170	#106	—	43	2	no	—
171	#103	—	43	2	no	—
172	#103	—	43	2	no	—
173	#102	—	43	2	no	—
174	#102	—	43	2	no	—
175	C₄×C₈	—	43	2	no	—
176	C₄×C₈	—	43	2	no	—
177	#41	—	2	2	no	—
178	#41	—	43	4	no	—
179	#41	—	86	2	no	—
180	#41	C₄×D₄₃	1	4	yes	complemented
181	C₂×Q₈	—	43	1	no	—
182	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	43	1	no	—
185	C₂×D₄	—	86	1	no	—
186	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	86	1	no	—
190	C₂²×C₄	—	86	2	no	—
191	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
192	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
193	Q₁₆	—	43	4	no	—
194	SD₁₆	—	43	4	no	—
195	SD₁₆	—	43	4	no	—
196	D₈	—	43	4	no	—
197	C₂×C₈	—	43	4	no	—
198	C₂×C₈	—	43	1	no	—
199	C₂×C₈	—	43	1	no	—
200	C₂×C₈	—	43	2	no	—
201	C₂×C₈	—	86	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	43	2	no	—
204	C₄⋊C₄	—	86	2	no	—
205	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	86	1	no	—
209	C₂²⋊C₄	—	86	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	86	1	no	—
213	C₄²	—	86	2	no	—
214	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
215	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
216	C₄²	C₂²×D₄₃	1	2	yes	—
217	C₂³	—	2	1	no	—
218	C₂³	—	86	1	no	—
219	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	43	2	no	—
222	Q₈	—	86	1	no	—
223	Q₈	C₂×C₄×D₄₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	43	2	no	—
227	D₄	—	86	4	no	—
228	D₄	—	86	1	no	—
229	D₄	C₂×C₄×D₄₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	86	1	no	—
236	C₂×C₄	—	86	4	no	—
237	C₂×C₄	—	86	2	no	—
238	C₂×C₄	—	172	4	no	—
239	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
240	C₂×C₄	#10621898	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
244	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
245	C₈	—	43	2	no	—
246	C₈	—	43	2	no	—
247	C₈	—	86	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	86	2	no	—
252	C₂²	—	172	2	no	—
253	C₂²	C₄×C₂²⋊D₄₃	1	2	yes	—
254	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	4	2	no	—
258	C₄	—	86	2	no	—
259	C₄	—	172	2	no	—
260	C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
261	C₄	#13063345	1	4	yes	direct product factor
262	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
263	C₄	C₂×#10621898	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	172	2	no	—
267	C₂	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
268	C₂	C₂×#13063345	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×#41×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×#41	D₄₃
hypocenter	C₄₃	C₄×#135
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#10621898	C₂	semidirect
3. C₄×#10621885	C₂	semidirect
4. C₄×#10621884	C₂	semidirect
5. C₄×#10621883	C₂	semidirect
6. C₄×#10621882	C₂	semidirect
7. #14582935	C₂	semidirect
8. #14582895	C₂	semidirect
9. #14582893	C₂	semidirect
10. C₂×C₄×C₄₃⋊C₈	C₂	semidirect
11. C₄×#41×C₄₃	C₂	semidirect
13. #13063345	C₄	direct (N × H)
43. #41×C₄₃	C₂×C₄	semidirect
53. #10621898	C₂×C₄	semidirect
68. #10621885	C₂×C₄	semidirect
69. #10621884	C₂×C₄	semidirect
70. #10621883	C₂×C₄	semidirect
71. #10621882	C₂×C₄	semidirect
78. C₂×C₄₃⋊C₈	C₂×C₄	semidirect
124. C₄×C₄₃	#135	semidirect
137. C₄×#41	D₄₃	semidirect
146. C₄₃	C₄×#135	semidirect
180. #41	C₄×D₄₃	semidirect
261. C₄	#13063345	direct (N × H)

Subgroups — C4×#13063345 #19143365

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13063345 #19143365