Subgroups — #19147985

Order 5504.

1040 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583520	C₂	1	4	yes	maximal, complemented
2	#14583511	C₂	1	2	yes	maximal, complemented
3	#14583504	C₂	1	2	yes	maximal, complemented
4	#14583496	C₂	1	4	yes	maximal, complemented
5	#14583487	C₂	1	4	yes	maximal, complemented
6	#14583456	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583449	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583427	C₂	1	8	yes	maximal, complemented
9	#14583410	C₂	1	2	yes	maximal, complemented
10	C₄×#10621898	C₂	1	2	yes	maximal, complemented
11	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	2	yes	—
13	#13063325	C₂²	1	1	yes	characteristic, complemented
14	#13063325	C₂²	1	2	yes	complemented
15	#13063322	C₂²	1	1	yes	characteristic, complemented
16	#13063320	C₂²	1	8	yes	complemented
17	#13063318	C₂²	1	8	yes	—
18	#13063317	C₂²	1	2	yes	complemented
19	#13063315	C₂²	1	8	yes	complemented
20	#13063313	C₂²	1	2	yes	complemented
21	#13063311	C₂²	1	8	yes	—
22	#13063310	C₂²	1	8	yes	complemented
23	#13063308	C₂²	1	4	yes	—
24	#13063307	C₂²	1	8	yes	—
25	#13063304	C₂²	1	4	yes	—
26	#13063304	C₂²	1	4	yes	—
27	#13063303	C₂²	1	8	yes	complemented
28	#13063299	C₂²	1	4	yes	—
29	#13063299	C₂²	1	4	yes	—
30	#13063298	C₂²	1	8	yes	—
31	#13063294	C₂²	1	4	yes	—
32	C₄×Dic₈₆	C₂²	1	4	yes	complemented
33	C₄×Dic₈₆	C₂²	1	4	yes	complemented
34	C₄×D₁₇₂	C₂²	1	4	yes	complemented
35	C₄×D₁₇₂	C₂²	1	4	yes	complemented
36	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	—
37	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	—
38	#13063284	C₂²	1	2	yes	—
39	#13063282	C₂²	1	4	yes	complemented
40	C₄²×D₄₃	C₂²	1	4	yes	complemented
41	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
42	#117×C₄₃	C₂²	1	4	yes	—
43	#117×C₄₃	C₂²	1	2	yes	—
44	#117×C₄₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
46	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
49	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
59	#10621898	—	2	8	no	—
60	C₂×D₁₇₂	C₂³	1	2	yes	—
61	C₂×D₁₇₂	C₂³	1	4	yes	—
62	C₂×C₄×D₄₃	C₂³	1	4	yes	—
63	C₂×C₄×D₄₃	C₂³	1	8	yes	—
64	C₂×Dic₈₆	C₂³	1	2	yes	—
65	C₂×Dic₈₆	C₂³	1	4	yes	—
66	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
67	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
68	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
69	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
70	C₄²×C₄₃	C₂³	1	2	yes	—
71	C₄²×C₄₃	C₂³	1	2	yes	—
72	C₄²×C₄₃	C₂³	1	2	yes	—
73	C₄²×C₄₃	C₂³	1	1	yes	characteristic
74	C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
76	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
77	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
80	C₄⋊Dic₄₃	C₂³	1	4	yes	—
81	C₄⋊Dic₄₃	C₂³	1	4	yes	—
82	C₄⋊Dic₄₃	C₂³	1	4	yes	—
83	Dic₄₃⋊C₄	C₂³	1	8	yes	—
84	Dic₄₃⋊C₄	C₂³	1	8	yes	—
85	Dic₄₃⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₄₃	C₂³	1	4	yes	—
87	C₄×Dic₄₃	C₂³	1	8	yes	—
88	C₂³×C₄₃	—	2	2	no	—
89	C₂³×C₄₃	—	4	2	no	—
90	C₂³×C₄₃	C₂⁴	1	2	yes	—
91	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂²×D₄₃	C₂⁴	1	4	yes	—
93	C₂×C₄×C₄₃	—	2	2	no	—
94	C₂×C₄×C₄₃	—	2	4	no	—
95	C₂×C₄×C₄₃	—	2	4	no	—
96	C₂×C₄×C₄₃	—	4	2	no	—
97	C₂×C₄×C₄₃	—	4	2	no	—
98	C₂×C₄×C₄₃	#41	1	4	yes	—
99	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
100	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
102	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
103	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₂²⋊D₄₃	—	2	8	no	—
106	C₂²⋊D₄₃	—	4	8	no	—
107	C₂×Dic₄₃	C₂⁴	1	4	yes	—
108	C₂×Dic₄₃	C₂⁴	1	8	yes	—
109	D₁₇₂	—	2	4	no	—
110	D₁₇₂	—	4	4	no	—
111	C₄×D₄₃	—	2	8	no	—
112	C₄×D₄₃	—	4	8	no	—
113	Dic₈₆	—	2	4	no	—
114	Dic₈₆	—	4	4	no	—
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	2	4	no	—
118	C₂²×C₄₃	—	4	4	no	—
119	C₂²×C₄₃	C₂×#41	1	2	yes	—
120	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
121	C₂×D₄₃	—	2	4	no	—
122	C₂×D₄₃	—	2	8	no	—
123	C₄×C₄₃	—	2	2	no	—
124	C₄×C₄₃	—	2	2	no	—
125	C₄×C₄₃	—	2	4	no	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	C₂×#41	1	2	yes	—
128	C₄×C₄₃	C₂×#41	1	2	yes	—
129	Dic₄₃	—	2	4	no	—
130	Dic₄₃	—	2	8	no	—
131	#3455	—	43	1	no	maximal, Sylow
132	C₂×C₄₃	—	2	2	no	—
133	C₂×C₄₃	—	4	2	no	—
134	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
135	C₂×C₄₃	#585	1	2	yes	—
136	D₄₃	—	4	4	no	—
137	#542	—	43	4	no	—
138	#541	—	43	2	no	—
139	#540	—	43	2	no	—
140	#539	—	43	4	no	—
141	#538	—	43	4	no	—
142	#533	—	43	1	no	—
143	#532	—	43	1	no	—
144	#529	—	43	8	no	—
145	#525	—	43	2	no	—
146	C₄×#41	—	43	2	no	—
147	C₂×#117	D₄₃	1	1	yes	characteristic, complemented
148	C₄₃	#3455	1	1	yes	characteristic, complemented, Sylow
149	C₂×#41	—	43	2	no	—
150	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
151	C₄⋊Q₈	—	43	1	no	—
152	C₄⋊Q₈	—	43	2	no	—
153	#127	—	43	1	no	—
154	#126	—	43	8	no	—
155	#126	—	43	8	no	—
156	#125	—	43	8	no	—
157	#125	—	43	2	no	—
158	#124	—	43	8	no	—
159	#124	—	43	2	no	—
160	#123	—	43	4	no	—
161	#123	—	43	8	no	—
162	#123	—	43	8	no	—
163	C₂²⋊Q₈	—	43	4	no	—
164	C₂²⋊Q₈	—	43	4	no	—
165	C₂²⋊Q₈	—	43	8	no	—
166	#121	—	43	4	no	—
167	#121	—	43	4	no	—
168	#121	—	43	8	no	—
169	#120	—	43	4	no	—
170	C₄×Q₈	—	43	4	no	—
171	C₄×Q₈	—	43	4	no	—
172	C₄×D₄	—	43	8	no	—
173	C₄×D₄	—	43	8	no	—
174	C₄×D₄	—	43	4	no	—
175	C₄×D₄	—	43	4	no	—
176	#117	—	43	2	no	—
177	#117	—	43	4	no	—
178	#117	C₂×D₄₃	1	4	yes	—
179	#117	C₂×D₄₃	1	2	yes	—
180	#117	C₂×D₄₃	1	2	yes	—
181	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
182	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
183	C₂×C₄²	—	43	4	no	—
184	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
185	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
186	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
187	#41	—	86	8	no	—
188	C₂×Q₈	—	43	2	no	—
189	C₂×Q₈	—	43	4	no	—
190	C₂×D₄	—	43	4	no	—
191	C₂×D₄	—	43	8	no	—
192	C₂×D₄	—	43	2	no	—
193	C₂×D₄	—	43	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	43	4	no	—
196	C₂²×C₄	—	43	8	no	—
197	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
199	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
200	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
203	C₄⋊C₄	—	43	4	no	—
204	C₄⋊C₄	—	43	8	no	—
205	C₄⋊C₄	—	43	4	no	—
206	C₄⋊C₄	—	43	8	no	—
207	C₄⋊C₄	—	43	8	no	—
208	C₄⋊C₄	—	43	4	no	—
209	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
210	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
211	C₂²⋊C₄	—	43	4	no	—
212	C₂²⋊C₄	—	43	8	no	—
213	C₂²⋊C₄	—	43	8	no	—
214	C₂²⋊C₄	—	43	8	no	—
215	C₂²⋊C₄	—	43	8	no	—
216	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
217	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
218	C₄²	—	43	8	no	—
219	C₄²	—	43	4	no	—
220	C₄²	C₂²×D₄₃	1	2	yes	—
221	C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	C₄²	C₂²×D₄₃	1	1	yes	characteristic
223	C₄²	C₂²×D₄₃	1	2	yes	—
224	C₄²	C₂²×D₄₃	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	43	4	no	—
228	C₂³	C₂³×D₄₃	1	2	yes	—
229	C₂³	C₂³×D₄₃	1	1	yes	characteristic
230	Q₈	—	86	4	no	—
231	Q₈	—	172	4	no	—
232	D₄	—	86	8	no	—
233	D₄	—	86	4	no	—
234	D₄	—	172	4	no	—
235	D₄	—	172	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	43	4	no	—
242	C₂×C₄	—	43	8	no	—
243	C₂×C₄	—	86	8	no	—
244	C₂×C₄	—	172	8	no	—
245	C₂×C₄	#10621898	1	4	yes	—
246	C₂×C₄	C₂³×D₄₃	1	4	yes	—
247	C₂×C₄	C₂³×D₄₃	1	2	yes	—
248	C₂×C₄	C₂³×D₄₃	1	2	yes	—
249	C₂×C₄	C₂³×D₄₃	1	2	yes	—
250	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
251	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	86	4	no	—
257	C₂²	—	86	8	no	—
258	C₂²	C₂×#10621898	1	2	yes	—
259	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	86	4	no	—
265	C₄	—	86	8	no	—
266	C₄	C₂×#10621898	1	2	yes	—
267	C₄	C₂×#10621898	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	—	172	4	no	—
271	C₂	C₂²×#10621898	1	1	yes	characteristic
272	C₂	#14583735	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×#117×C₄₃	C₂
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#3455
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #14583520	C₂	semidirect
2. #14583511	C₂	semidirect
3. #14583504	C₂	semidirect
4. #14583496	C₂	semidirect
5. #14583487	C₂	semidirect
6. #14583456	C₂	semidirect
7. #14583449	C₂	semidirect
8. #14583427	C₂	semidirect
9. #14583410	C₂	semidirect
10. C₄×#10621898	C₂	semidirect
11. C₂×#117×C₄₃	C₂	semidirect
13. #13063325	C₂²	semidirect
14. #13063325	C₂²	semidirect
15. #13063322	C₂²	semidirect
16. #13063320	C₂²	semidirect
18. #13063317	C₂²	semidirect
19. #13063315	C₂²	semidirect
20. #13063313	C₂²	semidirect
22. #13063310	C₂²	semidirect
27. #13063303	C₂²	semidirect
32. C₄×Dic₈₆	C₂²	semidirect
33. C₄×Dic₈₆	C₂²	semidirect
34. C₄×D₁₇₂	C₂²	semidirect
35. C₄×D₁₇₂	C₂²	semidirect
39. #13063282	C₂²	semidirect
40. C₄²×D₄₃	C₂²	semidirect
147. C₂×#117	D₄₃	semidirect
148. C₄₃	#3455	semidirect