Subgroups — #19148469

Order 5504.

1232 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583635	C₂	1	4	yes	maximal, complemented
2	#14583569	C₂	1	4	yes	maximal, complemented
3	#14583543	C₂	1	8	yes	maximal, complemented
4	#14583503	C₂	1	4	yes	maximal, complemented
5	#14583466	C₂	1	2	yes	maximal, complemented
6	#14583456	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583447	C₂	1	4	yes	maximal, complemented
8	#14583374	C₂	1	2	yes	maximal, complemented
9	C₄×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#10621898	C₂²	1	1	yes	characteristic
12	C₂×#10621900	C₂²	1	8	yes	complemented
13	#13063325	C₂²	1	2	yes	complemented
14	#13063321	C₂²	1	4	yes	complemented
15	#13063317	C₂²	1	2	yes	complemented
16	#13063312	C₂²	1	4	yes	complemented
17	#13063312	C₂²	1	8	yes	complemented
18	#13063306	C₂²	1	16	yes	complemented
19	#13063304	C₂²	1	4	yes	—
20	#13063297	C₂²	1	16	yes	complemented
21	#13063296	C₂²	1	8	yes	complemented
22	#13063296	C₂²	1	8	yes	complemented
23	#13063294	C₂²	1	8	yes	—
24	C₄×Dic₈₆	C₂²	1	4	yes	complemented
25	C₄×Dic₈₆	C₂²	1	2	yes	complemented
26	C₄×D₁₇₂	C₂²	1	2	yes	complemented
27	D₄×Dic₄₃	C₂²	1	8	yes	complemented
28	D₄×Dic₄₃	C₂²	1	8	yes	complemented
29	D₄×Dic₄₃	C₂²	1	8	yes	complemented
30	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
31	#13063284	C₂²	1	1	yes	characteristic
32	#13063284	C₂²	1	2	yes	—
33	#13063282	C₂²	1	2	yes	complemented
34	C₂×C₂²⋊Dic₄₃	C₂²	1	8	yes	—
35	C₄²×D₄₃	C₂²	1	2	yes	complemented
36	C₂²×D₄×C₄₃	C₂²	1	2	yes	—
37	C₂²×D₄×C₄₃	C₂²	1	4	yes	—
38	#127×C₄₃	C₂²	1	4	yes	—
39	#127×C₄₃	C₂²	1	2	yes	—
40	#127×C₄₃	C₂²	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
42	C₂⁴×C₄₃	C₂³	1	4	yes	—
43	C₂×D₄×C₄₃	—	2	8	no	—
44	C₂×D₄×C₄₃	—	4	8	no	—
45	C₂×D₄×C₄₃	C₂³	1	4	yes	—
46	C₂×D₄×C₄₃	C₂³	1	4	yes	—
47	C₂×D₄×C₄₃	C₂³	1	8	yes	—
48	C₂×D₄×C₄₃	C₂³	1	8	yes	—
49	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
52	C₂×C₂²⋊D₄₃	C₂³	1	16	yes	—
53	C₂²×Dic₄₃	C₂³	1	16	yes	—
54	#10621900	—	4	16	no	—
55	#10621898	—	2	4	no	—
56	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₄₃	C₂³	1	2	yes	—
58	C₂×C₄×D₄₃	C₂³	1	4	yes	—
59	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
60	C₂×Dic₈₆	C₂³	1	4	yes	—
61	C₄²×C₄₃	C₂³	1	1	yes	characteristic
62	C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄²×C₄₃	C₂³	1	2	yes	—
64	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
65	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
66	C₂²⋊Dic₄₃	C₂³	1	16	yes	—
67	C₂²⋊Dic₄₃	C₂³	1	16	yes	—
68	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
69	C₄⋊Dic₄₃	C₂³	1	2	yes	—
70	C₄⋊Dic₄₃	C₂³	1	4	yes	—
71	C₄⋊Dic₄₃	C₂³	1	4	yes	—
72	Dic₄₃⋊C₄	C₂³	1	4	yes	—
73	Dic₄₃⋊C₄	C₂³	1	8	yes	—
74	C₄×Dic₄₃	C₂³	1	2	yes	—
75	C₄×Dic₄₃	C₂³	1	4	yes	—
76	C₄×Dic₄₃	C₂³	1	4	yes	—
77	C₂³×C₄₃	—	2	8	no	—
78	C₂³×C₄₃	—	4	8	no	—
79	C₂³×C₄₃	C₂⁴	1	8	yes	—
80	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₂²×D₄₃	C₂⁴	1	2	yes	—
82	D₄×C₄₃	—	2	8	no	—
83	D₄×C₄₃	—	2	8	no	—
84	D₄×C₄₃	—	2	16	no	—
85	D₄×C₄₃	—	4	8	no	—
86	C₂×C₄×C₄₃	—	2	4	no	—
87	C₂×C₄×C₄₃	#41	1	4	yes	complemented
88	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
89	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
92	C₂²⋊D₄₃	—	2	4	no	—
93	C₂²⋊D₄₃	—	4	16	no	—
94	C₂×Dic₄₃	—	4	16	no	—
95	C₂×Dic₄₃	C₂⁴	1	2	yes	—
96	C₂×Dic₄₃	C₂⁴	1	4	yes	—
97	C₂×Dic₄₃	C₂⁴	1	8	yes	—
98	D₁₇₂	—	2	2	no	—
99	C₄×D₄₃	—	2	4	no	—
100	C₄×D₄₃	—	2	8	no	—
101	Dic₈₆	—	2	2	no	—
102	Dic₈₆	—	2	8	no	—
103	C₂²×C₄₃	—	2	2	no	—
104	C₂²×C₄₃	—	2	8	no	—
105	C₂²×C₄₃	—	2	16	no	—
106	C₂²×C₄₃	—	8	8	no	—
107	C₂²×C₄₃	C₂×#41	1	2	yes	—
108	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
109	C₂×D₄₃	—	2	2	no	—
110	C₂×D₄₃	—	2	4	no	—
111	C₄×C₄₃	—	2	2	no	—
112	C₄×C₄₃	C₂×#41	1	2	yes	—
113	C₄×C₄₃	C₂×#41	1	2	yes	—
114	C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
115	Dic₄₃	—	2	2	no	—
116	Dic₄₃	—	2	8	no	—
117	Dic₄₃	—	2	4	no	—
118	#3508	—	43	1	no	maximal, Sylow
119	C₂×C₄₃	—	2	2	no	—
120	C₂×C₄₃	—	4	8	no	—
121	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
122	C₂×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
123	D₄₃	—	4	2	no	—
124	#559	—	43	4	no	—
125	#550	—	43	4	no	—
126	#546	—	43	8	no	—
127	#540	—	43	4	no	—
128	#535	—	43	2	no	—
129	#533	—	43	1	no	—
130	#532	—	43	4	no	—
131	C₂×#127	D₄₃	1	1	yes	characteristic, complemented
132	#518	—	43	2	no	—
133	C₄×#41	—	43	1	no	—
134	C₄₃	#3508	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	43	8	no	—
136	C₂×#41	—	43	1	no	—
137	C₂²×D₄	C₂×D₄₃	1	2	yes	—
138	C₂²×D₄	C₂×D₄₃	1	4	yes	—
139	C₄⋊Q₈	—	43	2	no	—
140	#127	—	43	4	no	—
141	#127	C₂×D₄₃	1	2	yes	—
142	#127	C₂×D₄₃	1	2	yes	—
143	#127	C₂×D₄₃	1	4	yes	—
144	#125	—	43	2	no	—
145	#124	—	43	8	no	—
146	#124	—	43	4	no	—
147	#123	—	43	16	no	—
148	C₂²⋊Q₈	—	43	4	no	—
149	#121	—	43	8	no	—
150	#121	—	43	8	no	—
151	#121	—	43	16	no	—
152	#120	—	43	8	no	—
153	C₄×Q₈	—	43	2	no	—
154	C₄×Q₈	—	43	4	no	—
155	C₄×D₄	—	43	8	no	—
156	C₄×D₄	—	43	8	no	—
157	C₄×D₄	—	43	8	no	—
158	C₄×D₄	—	43	4	no	—
159	C₄×D₄	—	43	2	no	—
160	#117	—	43	1	no	—
161	#117	—	43	2	no	—
162	#117	—	43	2	no	—
163	C₂×C₂²⋊C₄	—	43	8	no	—
164	C₂×C₄²	—	43	2	no	—
165	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
166	C₂⁴	C₂²×D₄₃	1	4	yes	—
167	#41	—	86	4	no	—
168	#41	—	172	16	no	—
169	C₂×Q₈	—	43	1	no	—
170	C₂×Q₈	—	43	4	no	—
171	C₂×D₄	—	2	8	no	—
172	C₂×D₄	—	4	8	no	—
173	C₂×D₄	—	43	2	no	—
174	C₂×D₄	—	43	16	no	—
175	C₂×D₄	—	43	1	no	—
176	C₂×D₄	C₂²×D₄₃	1	4	yes	—
177	C₂×D₄	C₂²×D₄₃	1	4	yes	—
178	C₂×D₄	C₂²×D₄₃	1	8	yes	—
179	C₂×D₄	C₂²×D₄₃	1	8	yes	—
180	C₂²×C₄	—	43	16	no	—
181	C₂²×C₄	—	43	2	no	—
182	C₂²×C₄	—	43	4	no	—
183	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
184	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
185	C₄⋊C₄	—	43	2	no	—
186	C₄⋊C₄	—	43	4	no	—
187	C₄⋊C₄	—	43	4	no	—
188	C₄⋊C₄	—	43	4	no	—
189	C₄⋊C₄	—	43	8	no	—
190	C₂²⋊C₄	—	43	2	no	—
191	C₂²⋊C₄	—	43	4	no	—
192	C₂²⋊C₄	—	43	16	no	—
193	C₂²⋊C₄	—	43	16	no	—
194	C₂²⋊C₄	—	43	4	no	—
195	C₄²	—	43	2	no	—
196	C₄²	—	43	4	no	—
197	C₄²	—	43	4	no	—
198	C₄²	C₂²×D₄₃	1	1	yes	characteristic
199	C₄²	C₂²×D₄₃	1	1	yes	characteristic
200	C₄²	C₂²×D₄₃	1	2	yes	—
201	C₂³	—	2	8	no	—
202	C₂³	—	4	8	no	—
203	C₂³	—	43	2	no	—
204	C₂³	C₂³×D₄₃	1	8	yes	—
205	C₂³	C₂³×D₄₃	1	1	yes	characteristic
206	Q₈	—	86	2	no	—
207	Q₈	—	86	8	no	—
208	D₄	—	2	8	no	—
209	D₄	—	2	8	no	—
210	D₄	—	2	16	no	—
211	D₄	—	4	8	no	—
212	D₄	—	86	4	no	—
213	D₄	—	86	2	no	—
214	D₄	—	172	16	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	43	2	no	—
217	C₂×C₄	—	43	4	no	—
218	C₂×C₄	—	43	8	no	—
219	C₂×C₄	—	86	4	no	—
220	C₂×C₄	—	86	8	no	—
221	C₂×C₄	—	172	16	no	—
222	C₂×C₄	#10621900	1	4	yes	complemented
223	C₂×C₄	C₂³×D₄₃	1	2	yes	—
224	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
225	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
226	C₂×C₄	C₂³×D₄₃	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	8	no	—
229	C₂²	—	2	16	no	—
230	C₂²	—	8	8	no	—
231	C₂²	—	86	2	no	—
232	C₂²	—	86	4	no	—
233	C₂²	C₂×#10621900	1	2	yes	—
234	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	86	2	no	—
237	C₄	—	86	4	no	—
238	C₄	—	86	8	no	—
239	C₄	C₂×#10621900	1	2	yes	—
240	C₄	C₂×#10621900	1	2	yes	—
241	C₄	#13063370	1	4	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	8	no	—
244	C₂	—	172	2	no	—
245	C₂	C₂²×#10621900	1	1	yes	characteristic
246	C₂	C₂×#13063370	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂×#127×C₄₃	C₂
hypercenter	C₂×#127	D₄₃
hypocenter	C₄₃	#3508
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #14583635	C₂	semidirect
2. #14583569	C₂	semidirect
3. #14583543	C₂	semidirect
4. #14583503	C₂	semidirect
5. #14583466	C₂	semidirect
6. #14583456	C₂	semidirect
7. #14583447	C₂	semidirect
8. #14583374	C₂	semidirect
9. C₄×#10621898	C₂	semidirect
10. C₂×#127×C₄₃	C₂	semidirect
12. C₂×#10621900	C₂²	semidirect
13. #13063325	C₂²	semidirect
14. #13063321	C₂²	semidirect
15. #13063317	C₂²	semidirect
16. #13063312	C₂²	semidirect
17. #13063312	C₂²	semidirect
18. #13063306	C₂²	semidirect
20. #13063297	C₂²	semidirect
21. #13063296	C₂²	semidirect
22. #13063296	C₂²	semidirect
24. C₄×Dic₈₆	C₂²	semidirect
25. C₄×Dic₈₆	C₂²	semidirect
26. C₄×D₁₇₂	C₂²	semidirect
27. D₄×Dic₄₃	C₂²	semidirect
28. D₄×Dic₄₃	C₂²	semidirect
29. D₄×Dic₄₃	C₂²	semidirect
33. #13063282	C₂²	semidirect
35. C₄²×D₄₃	C₂²	semidirect
87. C₂×C₄×C₄₃	#41	semidirect
131. C₂×#127	D₄₃	semidirect
134. C₄₃	#3508	semidirect
222. C₂×C₄	#10621900	semidirect