Subgroups — #19149000

Order 5504.

1258 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583641	C₂	1	4	yes	maximal, complemented
2	#14583615	C₂	1	2	yes	maximal, complemented
3	#14583614	C₂	1	4	yes	maximal, complemented
4	D₄×D₁₇₂	C₂	1	4	yes	maximal, complemented
5	#14583534	C₂	1	2	yes	maximal, complemented
6	#14583465	C₂	1	8	yes	maximal, complemented
7	#14583463	C₂	1	4	yes	maximal, complemented
8	#14583449	C₂	1	2	yes	maximal, complemented
9	C₄₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	C₂²	1	4	yes	complemented
11	C₂²×D₁₇₂	C₂²	1	4	yes	complemented
12	C₂×D₄×D₄₃	C₂²	1	8	yes	complemented
13	#13063325	C₂²	1	2	yes	complemented
14	#13063322	C₂²	1	2	yes	—
15	#13063321	C₂²	1	8	yes	complemented
16	#13063314	C₂²	1	8	yes	complemented
17	#13063313	C₂²	1	4	yes	complemented
18	#13063311	C₂²	1	16	yes	complemented
19	#13063300	C₂²	1	8	yes	complemented
20	#13063299	C₂²	1	8	yes	complemented
21	#13063299	C₂²	1	4	yes	complemented
22	#13063299	C₂²	1	4	yes	complemented
23	#13063296	C₂²	1	8	yes	complemented
24	#13063295	C₂²	1	16	yes	complemented
25	#13063293	C₂²	1	8	yes	complemented
26	#13063293	C₂²	1	8	yes	complemented
27	C₄×D₁₇₂	C₂²	1	4	yes	complemented
28	C₄×D₁₇₂	C₂²	1	4	yes	complemented
29	#13063284	C₂²	1	4	yes	complemented
30	#13063283	C₂²	1	8	yes	complemented
31	C₂×#41×C₄₃	C₂²	1	2	yes	complemented
32	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
33	#127×C₄₃	C₂²	1	1	yes	characteristic
34	#124×C₄₃	C₂²	1	2	yes	complemented
35	#121×C₄₃	C₂²	1	4	yes	complemented
36	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
37	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic, complemented
38	C₂³×D₄₃	C₂³	1	8	yes	—
39	#41×C₄₃	—	4	4	no	—
40	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
41	C₂×D₄×C₄₃	C₂³	1	4	yes	—
42	C₂×D₄×C₄₃	C₂³	1	4	yes	—
43	C₂×D₄×C₄₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
47	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
48	C₂×C₂²⋊D₄₃	C₂³	1	16	yes	—
49	D₄×D₄₃	—	4	16	no	—
50	#10621898	—	4	8	no	—
51	C₂×D₁₇₂	—	4	8	no	—
52	C₂×D₁₇₂	C₂³	1	4	yes	—
53	C₂×D₁₇₂	C₂³	1	4	yes	—
54	C₂×D₁₇₂	C₂³	1	4	yes	—
55	C₂×D₁₇₂	C₂³	1	8	yes	—
56	C₂×C₄×D₄₃	C₂³	1	8	yes	—
57	C₂×Dic₈₆	C₂³	1	4	yes	—
58	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
59	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
60	C₄²×C₄₃	C₂³	1	2	yes	—
61	C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
64	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
65	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
66	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
67	C₄⋊Dic₄₃	C₂³	1	4	yes	—
68	C₄⋊Dic₄₃	C₂³	1	4	yes	—
69	C₄×Dic₄₃	C₂³	1	8	yes	—
70	C₂³×C₄₃	C₂⁴	1	4	yes	—
71	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²×D₄₃	—	4	8	no	—
73	C₂²×D₄₃	—	4	16	no	—
74	C₂²×D₄₃	C₂⁴	1	8	yes	—
75	Q₈×C₄₃	—	4	2	no	—
76	D₄×C₄₃	—	2	8	no	—
77	D₄×C₄₃	—	4	4	no	—
78	D₄×C₄₃	—	4	2	no	—
79	C₂×C₄×C₄₃	—	2	8	no	—
80	C₂×C₄×C₄₃	—	2	4	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
83	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
85	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
87	C₂²⋊D₄₃	—	4	8	no	—
88	C₂²⋊D₄₃	—	4	16	no	—
89	C₂×Dic₄₃	C₂⁴	1	8	yes	—
90	D₁₇₂	—	2	8	no	—
91	D₁₇₂	—	2	8	no	—
92	D₁₇₂	—	4	8	no	—
93	D₁₇₂	—	4	4	no	—
94	C₄×D₄₃	—	2	16	no	—
95	Dic₈₆	—	4	4	no	—
96	C₂²×C₄₃	—	2	8	no	—
97	C₂²×C₄₃	—	2	4	no	—
98	C₂²×C₄₃	—	2	2	no	—
99	C₂²×C₄₃	—	2	1	no	—
100	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
101	C₂×D₄₃	—	2	8	no	—
102	C₂×D₄₃	—	2	8	no	—
103	C₂×D₄₃	—	2	8	no	—
104	C₂×D₄₃	—	8	16	no	—
105	C₄×C₄₃	—	2	2	no	—
106	C₄×C₄₃	—	2	4	no	—
107	C₄×C₄₃	—	2	1	no	—
108	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
109	C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
110	Dic₄₃	—	2	8	no	—
111	#3549	—	43	1	no	maximal, Sylow
112	C₂×C₄₃	—	4	4	no	—
113	C₂×C₄₃	—	4	1	no	—
114	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
115	C₂×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
116	D₄₃	—	4	8	no	—
117	H(3,𝔽₄)	—	43	4	no	—
118	#555	—	43	2	no	—
119	#555	—	43	4	no	—
120	D₄²	—	43	2	no	—
121	D₄²	—	43	4	no	—
122	#535	—	43	4	no	—
123	#535	—	43	8	no	—
124	#532	—	43	2	no	—
125	#532	D₄₃	1	1	yes	characteristic, complemented
126	C₄₃	#3549	1	1	yes	characteristic, complemented, Sylow
127	C₂×#41	—	43	4	no	—
128	C₂×#41	C₂×D₄₃	1	2	yes	complemented
129	C₂²×D₄	—	43	8	no	—
130	C₂²×D₄	—	43	4	no	—
131	C₄⋊Q₈	—	43	2	no	—
132	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
133	#127	—	43	8	no	—
134	#127	—	43	2	no	—
135	#127	C₂×D₄₃	1	1	yes	characteristic
136	#124	—	43	8	no	—
137	#124	—	43	16	no	—
138	#124	—	43	4	no	—
139	#124	C₂×D₄₃	1	2	yes	complemented
140	#121	—	43	8	no	—
141	#121	—	43	4	no	—
142	#121	—	43	4	no	—
143	#121	—	43	8	no	—
144	#121	—	43	8	no	—
145	#121	C₂×D₄₃	1	4	yes	complemented
146	#120	—	43	8	no	—
147	#120	—	43	8	no	—
148	#120	—	43	16	no	—
149	C₄×D₄	—	43	4	no	—
150	C₄×D₄	—	43	4	no	—
151	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
152	#117	—	43	4	no	—
153	#117	—	43	8	no	—
154	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic, complemented
155	C₂⁴	—	43	8	no	—
156	#41	—	4	4	no	—
157	#41	—	172	8	no	—
158	C₂×Q₈	—	43	4	no	—
159	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
160	C₂×D₄	—	43	16	no	—
161	C₂×D₄	—	43	8	no	—
162	C₂×D₄	—	43	4	no	—
163	C₂×D₄	—	43	4	no	—
164	C₂×D₄	—	43	4	no	—
165	C₂×D₄	—	43	8	no	—
166	C₂×D₄	—	172	8	no	—
167	C₂×D₄	—	172	16	no	—
168	C₂×D₄	C₂²×D₄₃	1	4	yes	—
169	C₂×D₄	C₂²×D₄₃	1	2	yes	—
170	C₂×D₄	C₂²×D₄₃	1	4	yes	—
171	C₂²×C₄	—	43	8	no	—
172	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
174	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
175	C₄⋊C₄	—	43	4	no	—
176	C₄⋊C₄	—	43	4	no	—
177	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
178	C₂²⋊C₄	—	43	8	no	—
179	C₂²⋊C₄	—	43	16	no	—
180	C₂²⋊C₄	—	43	8	no	—
181	C₂²⋊C₄	—	43	8	no	—
182	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
183	C₄²	—	43	8	no	—
184	C₄²	C₂²×D₄₃	1	1	yes	characteristic
185	C₄²	C₂²×D₄₃	1	2	yes	—
186	C₄²	C₂²×D₄₃	1	1	yes	characteristic
187	C₂³	—	43	8	no	—
188	C₂³	—	172	8	no	—
189	C₂³	—	172	16	no	—
190	C₂³	C₂³×D₄₃	1	4	yes	—
191	C₂³	C₂³×D₄₃	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	172	4	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	2	no	—
196	D₄	—	4	4	no	—
197	D₄	—	86	8	no	—
198	D₄	—	86	8	no	—
199	D₄	—	172	8	no	—
200	D₄	—	172	4	no	—
201	D₄	—	172	16	no	—
202	D₄	—	172	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	43	8	no	—
207	C₂×C₄	—	86	16	no	—
208	C₂×C₄	C₂³×D₄₃	1	4	yes	—
209	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
210	C₂×C₄	C₂³×D₄₃	1	2	yes	—
211	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
212	C₂×C₄	C₂³×D₄₃	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	86	8	no	—
218	C₂²	—	86	8	no	—
219	C₂²	—	86	8	no	—
220	C₂²	—	344	16	no	—
221	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	86	8	no	—
226	C₄	#13063370	1	2	yes	—
227	C₄	#13063371	1	4	yes	—
228	C₂	—	4	4	no	—
229	C₂	—	4	1	no	—
230	C₂	—	172	8	no	—
231	C₂	C₂×#13063370	1	1	yes	characteristic
232	C₂	C₂×#13063371	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#532	C₂
hypercenter	#532	D₄₃
hypocenter	C₄₃	#3549
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #14583641	C₂	semidirect
2. #14583615	C₂	semidirect
3. #14583614	C₂	semidirect
4. D₄×D₁₇₂	C₂	semidirect
5. #14583534	C₂	semidirect
6. #14583465	C₂	semidirect
7. #14583463	C₂	semidirect
8. #14583449	C₂	semidirect
9. C₄₃×#532	C₂	semidirect
10. C₂×#10621898	C₂²	semidirect
11. C₂²×D₁₇₂	C₂²	semidirect
12. C₂×D₄×D₄₃	C₂²	semidirect
13. #13063325	C₂²	semidirect
15. #13063321	C₂²	semidirect
16. #13063314	C₂²	semidirect
17. #13063313	C₂²	semidirect
18. #13063311	C₂²	semidirect
19. #13063300	C₂²	semidirect
20. #13063299	C₂²	semidirect
21. #13063299	C₂²	semidirect
22. #13063299	C₂²	semidirect
23. #13063296	C₂²	semidirect
24. #13063295	C₂²	semidirect
25. #13063293	C₂²	semidirect
26. #13063293	C₂²	semidirect
27. C₄×D₁₇₂	C₂²	semidirect
28. C₄×D₁₇₂	C₂²	semidirect
29. #13063284	C₂²	semidirect
30. #13063283	C₂²	semidirect
31. C₂×#41×C₄₃	C₂²	semidirect
32. C₄⋊Q₈×C₄₃	C₂²	semidirect
34. #124×C₄₃	C₂²	semidirect
35. #121×C₄₃	C₂²	semidirect
36. C₄×D₄×C₄₃	C₂²	semidirect
37. C₂×C₄²×C₄₃	C₂²	semidirect
125. #532	D₄₃	semidirect
126. C₄₃	#3549	semidirect
128. C₂×#41	C₂×D₄₃	semidirect
132. C₄⋊Q₈	C₂×D₄₃	semidirect
139. #124	C₂×D₄₃	semidirect
145. #121	C₂×D₄₃	semidirect
151. C₄×D₄	C₂×D₄₃	semidirect
154. C₂×C₄²	C₂×D₄₃	semidirect