Subgroups — #19144638

Order 5504.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583232	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583215	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583196	C₂	1	2	yes	maximal, complemented
5	#14583111	C₂	1	2	yes	maximal, complemented
6	#14583049	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583032	C₂	1	1	yes	maximal, characteristic
8	C₄×#10621884	C₂	1	2	yes	maximal, complemented
9	C₄×#10621882	C₂	1	2	yes	maximal, complemented
10	#14582952	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₃	—	2	1	no	—
14	C₂×#10621884	C₂²	1	2	yes	—
15	C₂×#10621882	C₂²	1	2	yes	—
16	C₄×D₁₇₂	—	2	1	no	—
17	C₄×D₁₇₂	—	2	1	no	—
18	C₄×D₁₇₂	C₂²	1	2	yes	—
19	C₄×D₁₇₂	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
22	#13063258	C₂²	1	2	yes	complemented
23	#13063256	C₂²	1	2	yes	complemented
24	C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	complemented
25	#13063249	C₂²	1	2	yes	complemented
26	#13063246	C₂²	1	2	yes	complemented
27	#13063245	C₂²	1	2	yes	—
28	#13063245	C₂²	1	2	yes	—
29	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
31	C₄×C₄₃⋊C₈	C₂²	1	2	yes	complemented
32	#126×C₄₃	—	2	1	no	—
33	#125×C₄₃	C₂²	1	1	yes	characteristic, complemented
34	#124×C₄₃	C₂²	1	1	yes	characteristic, complemented
35	#123×C₄₃	—	2	1	no	—
36	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
37	#121×C₄₃	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₄₃	—	2	1	no	—
40	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	#117×C₄₃	—	2	1	no	—
42	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
43	C₂³×D₄₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₃	—	2	1	no	—
52	C₂×D₁₇₂	—	2	1	no	—
53	C₂×D₁₇₂	—	2	4	no	—
54	C₂×D₁₇₂	C₂³	1	2	yes	—
55	C₂×C₄×D₄₃	—	2	2	no	—
56	C₂×C₄×D₄₃	—	2	2	no	—
57	C₂×C₄×D₄₃	—	4	4	no	—
58	C₄⋊C₄×C₄₃	—	2	1	no	—
59	C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₄⋊C₄×C₄₃	—	2	1	no	—
61	C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₄₃	—	2	1	no	—
65	C₂²⋊C₄×C₄₃	—	2	1	no	—
66	C₂²⋊C₄×C₄₃	—	2	1	no	—
67	C₂²⋊C₄×C₄₃	—	2	1	no	—
68	C₂²⋊C₄×C₄₃	—	2	1	no	—
69	C₄²×C₄₃	—	2	1	no	—
70	C₄²×C₄₃	D₄	1	1	yes	characteristic
71	C₄²×C₄₃	D₄	1	1	yes	characteristic
72	C₄²×C₄₃	C₂³	1	1	yes	characteristic
73	C₄²×C₄₃	C₂³	1	1	yes	characteristic
74	#10621884	—	2	4	no	—
75	#10621882	—	2	4	no	—
76	(C₂×D₄₃)⋊C₄	—	2	2	no	—
77	(C₂×D₄₃)⋊C₄	—	2	2	no	—
78	C₄⋊Dic₄₃	—	2	1	no	—
79	C₄⋊Dic₄₃	C₂³	1	2	yes	—
80	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
81	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
82	C₂³×C₄₃	—	2	1	no	—
83	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₄₃	—	2	4	no	—
85	C₂²×D₄₃	—	2	2	no	—
86	C₂²×D₄₃	—	4	2	no	—
87	C₂²×D₄₃	—	4	2	no	—
88	Q₈×C₄₃	—	2	2	no	—
89	D₄×C₄₃	—	2	2	no	—
90	D₄×C₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	2	no	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	1	no	—
95	C₂×C₄×C₄₃	—	2	1	no	—
96	C₂×C₄×C₄₃	—	2	1	no	—
97	C₂×C₄×C₄₃	—	4	2	no	—
98	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂×Dic₄₃	—	2	2	no	—
105	C₂×Dic₄₃	—	4	2	no	—
106	D₁₇₂	—	2	4	no	—
107	D₁₇₂	—	2	2	no	—
108	D₁₇₂	#41	1	4	yes	—
109	C₄×D₄₃	—	4	4	no	—
110	C₄×D₄₃	—	4	4	no	—
111	C₄₃⋊C₈	—	2	2	no	—
112	C₄₃⋊C₈	—	2	2	no	—
113	C₂²×C₄₃	—	2	1	no	—
114	C₂²×C₄₃	—	2	1	no	—
115	C₂²×C₄₃	—	4	1	no	—
116	C₂²×C₄₃	—	4	1	no	—
117	C₂²×C₄₃	—	4	1	no	—
118	C₂²×C₄₃	#135	1	2	yes	—
119	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₄₃	—	2	4	no	—
121	C₂×D₄₃	—	4	8	no	—
122	C₂×D₄₃	—	4	2	no	—
123	C₂×D₄₃	—	4	2	no	—
124	C₄×C₄₃	—	2	1	no	—
125	C₄×C₄₃	—	2	1	no	—
126	C₄×C₄₃	—	2	1	no	—
127	C₄×C₄₃	—	2	1	no	—
128	C₄×C₄₃	—	4	1	no	—
129	C₄×C₄₃	—	4	1	no	—
130	C₄×C₄₃	—	4	1	no	—
131	C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
132	C₄×C₄₃	C₂×#41	1	2	yes	—
133	Dic₄₃	—	4	2	no	—
134	#3105	—	43	1	no	maximal, Sylow
135	C₂×C₄₃	—	2	2	no	—
136	C₂×C₄₃	—	8	1	no	—
137	C₂×C₄₃	#525	1	1	yes	characteristic
138	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
139	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	D₄₃	—	4	4	no	—
141	#529	D₄₃	1	1	yes	characteristic, complemented
142	C₂×C₄×D₄	—	43	1	no	—
143	#484	—	43	1	no	—
144	#481	—	43	1	no	—
145	#478	—	43	2	no	—
146	#463	—	43	2	no	—
147	#450	—	43	1	no	—
148	#447	—	43	1	no	—
149	C₄×SD₁₆	—	43	2	no	—
150	C₄×D₈	—	43	2	no	—
151	#431	—	43	1	no	—
152	C₄₃	#3105	1	1	yes	characteristic, complemented, Sylow
153	C₂²×D₄	—	43	1	no	—
154	C₂³×C₄	—	86	1	no	—
155	C₂×SD₁₆	—	43	2	no	—
156	C₂×D₈	—	43	2	no	—
157	#126	—	2	1	no	—
158	#125	C₂×D₄₃	1	1	yes	characteristic, complemented
159	#124	C₂×D₄₃	1	1	yes	characteristic, complemented
160	#123	—	2	1	no	—
161	C₂²⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
162	#121	C₂×D₄₃	1	1	yes	characteristic
163	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	43	2	no	—
166	C₄×D₄	—	43	2	no	—
167	C₄×D₄	—	86	1	no	—
168	C₄×D₄	—	86	1	no	—
169	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic, complemented
170	#117	—	2	1	no	—
171	C₂×C₄⋊C₄	—	43	1	no	—
172	C₂×C₂²⋊C₄	—	86	1	no	—
173	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
174	#107	—	43	2	no	—
175	#106	—	43	2	no	—
176	C₄⋊C₈	—	43	2	no	—
177	#103	—	43	2	no	—
178	#102	—	43	2	no	—
179	#102	—	43	2	no	—
180	#102	—	43	2	no	—
181	C₂²⋊C₈	—	43	1	no	—
182	C₂²⋊C₈	—	43	1	no	—
183	C₄×C₈	—	43	2	no	—
184	C₂⁴	—	86	1	no	—
185	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	43	2	no	—
188	C₂×D₄	—	86	4	no	—
189	C₂×D₄	—	86	1	no	—
190	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	86	1	no	—
193	C₂²×C₄	—	86	2	no	—
194	C₂²×C₄	—	86	2	no	—
195	C₂²×C₄	—	172	4	no	—
196	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
199	SD₁₆	—	86	4	no	—
200	D₈	—	86	4	no	—
201	C₂×C₈	—	43	2	no	—
202	C₂×C₈	—	43	2	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	43	2	no	—
208	C₄⋊C₄	—	86	1	no	—
209	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	86	2	no	—
217	C₂²⋊C₄	—	86	2	no	—
218	C₄²	—	2	1	no	—
219	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
220	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
221	C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	C₂³	—	2	1	no	—
224	C₂³	—	86	4	no	—
225	C₂³	—	86	2	no	—
226	C₂³	—	172	2	no	—
227	C₂³	—	172	2	no	—
228	C₂³	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	43	4	no	—
233	D₄	—	86	4	no	—
234	D₄	—	86	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	86	2	no	—
243	C₂×C₄	—	172	4	no	—
244	C₂×C₄	—	172	2	no	—
245	C₂×C₄	—	172	4	no	—
246	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
252	C₈	—	86	2	no	—
253	C₈	—	86	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	86	4	no	—
260	C₂²	—	172	8	no	—
261	C₂²	—	172	2	no	—
262	C₂²	—	172	2	no	—
263	C₂²	#13063345	1	2	yes	—
264	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	172	2	no	—
273	C₄	#13063350	1	2	yes	—
274	C₄	#41×D₄₃	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	8	1	no	—
277	C₂	—	172	4	no	—
278	C₂	#14583406	1	1	yes	characteristic
279	C₂	C₂×#13063345	1	1	yes	characteristic
280	C₂	C₂×#13063350	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#529	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#529	D₄₃
hypocenter	C₄₃	#3105
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×D₁₇₂	C₂	semidirect
2. #14583232	C₂	semidirect
3. #14583215	C₂	semidirect
4. #14583196	C₂	semidirect
5. #14583111	C₂	semidirect
6. #14583049	C₂	semidirect
8. C₄×#10621884	C₂	semidirect
9. C₄×#10621882	C₂	semidirect
10. #14582952	C₂	semidirect
11. C₄₃×#529	C₂	semidirect
22. #13063258	C₂²	semidirect
23. #13063256	C₂²	semidirect
24. C₄⋊(C₄₃⋊C₈)	C₂²	semidirect
25. #13063249	C₂²	semidirect
26. #13063246	C₂²	semidirect
31. C₄×C₄₃⋊C₈	C₂²	semidirect
33. #125×C₄₃	C₂²	semidirect
34. #124×C₄₃	C₂²	semidirect
38. C₄×Q₈×C₄₃	C₂²	semidirect
40. C₄×D₄×C₄₃	C₂²	semidirect
141. #529	D₄₃	semidirect
152. C₄₃	#3105	semidirect
158. #125	C₂×D₄₃	semidirect
159. #124	C₂×D₄₃	semidirect
163. C₄×Q₈	C₂×D₄₃	semidirect
169. C₄×D₄	C₂×D₄₃	semidirect