Subgroups — #19148416

Order 5504.

988 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583627	C₂	1	4	yes	maximal, complemented
2	#14583609	C₂	1	4	yes	maximal, complemented
3	#14583603	C₂	1	8	yes	maximal, complemented
4	#14583527	C₂	1	2	yes	maximal, complemented
5	#14583526	C₂	1	4	yes	maximal, complemented
6	#14583455	C₂	1	2	yes	maximal, complemented
7	C₂×#13063317	C₂	1	1	yes	maximal, characteristic, complemented
8	#125×D₄₃	C₂	1	2	yes	maximal, complemented
9	#14583379	C₂	1	2	yes	maximal
10	C₄×#10621898	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	#13063325	C₂²	1	4	yes	—
14	#13063324	C₂²	1	4	yes	—
15	#13063323	C₂²	1	8	yes	—
16	#13063317	C₂²	1	4	yes	—
17	#13063317	C₂²	1	4	yes	—
18	#13063317	C₂²	1	2	yes	—
19	#13063317	C₂²	1	2	yes	—
20	#13063316	C₂²	1	4	yes	—
21	#13063315	C₂²	1	8	yes	—
22	#13063315	C₂²	1	8	yes	—
23	#13063303	C₂²	1	8	yes	—
24	#13063303	C₂²	1	8	yes	—
25	#13063302	C₂²	1	8	yes	—
26	#13063301	C₂²	1	8	yes	—
27	C₄×Dic₈₆	C₂²	1	2	yes	—
28	C₄×Dic₈₆	C₂²	1	4	yes	—
29	#13063290	C₂²	1	8	yes	—
30	C₄×D₁₇₂	C₂²	1	2	yes	—
31	C₄×D₁₇₂	C₂²	1	4	yes	—
32	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
33	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	—
34	#13063284	C₂²	1	1	yes	characteristic
35	#13063283	C₂²	1	4	yes	—
36	#13063282	C₂²	1	2	yes	—
37	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
38	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
39	C₄⋊C₄×D₄₃	C₂²	1	8	yes	—
40	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
41	C₄²×D₄₃	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
43	#125×C₄₃	C₂²	1	1	yes	characteristic
44	#125×C₄₃	C₂²	1	1	yes	characteristic
45	C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	—
46	C₄×Q₈×C₄₃	C₂²	1	4	yes	—
47	#117×C₄₃	C₂²	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
49	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
53	C₂²×Dic₄₃	C₂³	1	4	yes	—
54	#10621898	Q₈	1	8	yes	complemented
55	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₄₃	C₂³	1	4	yes	—
57	C₂×C₄×D₄₃	C₂³	1	8	yes	—
58	C₂×C₄×D₄₃	C₂³	1	2	yes	—
59	C₂×Dic₈₆	C₂³	1	8	yes	—
60	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
64	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
65	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
66	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
67	C₄²×C₄₃	C₂³	1	2	yes	—
68	C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₄²×C₄₃	C₂³	1	4	yes	—
71	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
72	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
74	C₄⋊Dic₄₃	C₂³	1	4	yes	—
75	C₄⋊Dic₄₃	C₂³	1	8	yes	—
76	C₄⋊Dic₄₃	C₂³	1	4	yes	—
77	C₄⋊Dic₄₃	C₂³	1	2	yes	—
78	Dic₄₃⋊C₄	C₂³	1	8	yes	—
79	Dic₄₃⋊C₄	C₂³	1	8	yes	—
80	Dic₄₃⋊C₄	C₂³	1	8	yes	—
81	Dic₄₃⋊C₄	C₂³	1	8	yes	—
82	Dic₄₃⋊C₄	C₂³	1	4	yes	—
83	Dic₄₃⋊C₄	C₂³	1	8	yes	—
84	C₄×Dic₄₃	C₂³	1	2	yes	—
85	C₄×Dic₄₃	C₂³	1	4	yes	—
86	C₄×Dic₄₃	C₂³	1	8	yes	—
87	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×D₄₃	C₂⁴	1	2	yes	—
89	Q₈×C₄₃	—	4	4	no	—
90	C₂×C₄×C₄₃	—	4	2	no	—
91	C₂×C₄×C₄₃	C₂×Q₈	1	4	yes	—
92	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
93	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
94	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
95	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
96	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₄₃	C₂×Q₈	1	8	yes	—
99	C₂×Dic₄₃	—	4	4	no	—
100	C₂×Dic₄₃	C₂⁴	1	4	yes	—
101	C₂×Dic₄₃	C₂⁴	1	8	yes	—
102	C₂×Dic₄₃	C₂⁴	1	2	yes	—
103	D₁₇₂	C₂×Q₈	1	4	yes	—
104	C₄×D₄₃	—	4	4	no	—
105	C₄×D₄₃	—	4	8	no	—
106	C₄×D₄₃	C₂×Q₈	1	8	yes	—
107	Dic₈₆	—	4	8	no	—
108	Dic₈₆	C₂×Q₈	1	4	yes	—
109	C₂²×C₄₃	—	2	2	no	—
110	C₂²×C₄₃	C₂²×Q₈	1	2	yes	—
111	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
112	C₂×D₄₃	—	2	4	no	—
113	C₂×D₄₃	C₂²×Q₈	1	4	yes	—
114	C₄×C₄₃	—	2	4	no	—
115	C₄×C₄₃	—	2	2	no	—
116	C₄×C₄₃	—	2	4	no	—
117	C₄×C₄₃	—	2	2	no	—
118	C₄×C₄₃	C₂²×Q₈	1	2	yes	—
119	C₄×C₄₃	C₂²×Q₈	1	2	yes	—
120	Dic₄₃	—	2	4	no	—
121	Dic₄₃	—	2	8	no	—
122	Dic₄₃	C₂²×Q₈	1	4	yes	—
123	#3504	—	43	1	no	maximal, Sylow
124	C₂×C₄₃	—	2	2	no	—
125	C₂×C₄₃	C₂³×Q₈	1	1	yes	characteristic
126	C₂×C₄₃	#585	1	2	yes	—
127	D₄₃	—	2	4	no	—
128	#557	—	43	4	no	—
129	#554	—	43	8	no	—
130	#554	—	43	4	no	—
131	#544	—	43	4	no	—
132	#544	—	43	2	no	—
133	#533	—	43	2	no	—
134	#533	D₄₃	1	1	yes	characteristic, complemented
135	C₂×#125	—	43	2	no	—
136	C₂×#125	—	43	1	no	—
137	#520	—	43	2	no	—
138	C₄×#41	—	43	1	no	—
139	C₄₃	#3504	1	1	yes	characteristic, complemented, Sylow
140	C₂×#41	—	43	1	no	—
141	C₄⋊Q₈	—	43	8	no	—
142	C₄⋊Q₈	—	43	4	no	—
143	C₄⋊Q₈	—	43	4	no	—
144	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
145	#125	—	43	4	no	—
146	#125	—	43	8	no	—
147	#125	—	43	8	no	—
148	#125	—	43	2	no	—
149	#125	—	43	2	no	—
150	#125	—	43	4	no	—
151	#125	—	43	4	no	—
152	#125	C₂×D₄₃	1	1	yes	characteristic
153	#125	C₂×D₄₃	1	1	yes	characteristic
154	C₂²⋊Q₈	—	43	8	no	—
155	C₂²⋊Q₈	—	43	8	no	—
156	C₂²⋊Q₈	—	43	8	no	—
157	C₂²⋊Q₈	—	43	8	no	—
158	C₂²⋊Q₈	C₂×D₄₃	1	4	yes	—
159	C₄×Q₈	—	43	8	no	—
160	C₄×Q₈	—	43	2	no	—
161	C₄×Q₈	—	43	4	no	—
162	C₄×Q₈	C₂×D₄₃	1	4	yes	—
163	C₄×D₄	—	43	4	no	—
164	C₄×D₄	—	43	8	no	—
165	C₄×D₄	—	43	2	no	—
166	C₄×D₄	—	43	4	no	—
167	#117	—	43	1	no	—
168	#117	—	43	4	no	—
169	#117	—	43	2	no	—
170	#117	C₂×D₄₃	1	2	yes	—
171	C₂×C₄⋊C₄	—	43	2	no	—
172	C₂×C₄⋊C₄	—	43	4	no	—
173	C₂×C₄⋊C₄	—	43	4	no	—
174	C₂×C₄⋊C₄	—	43	8	no	—
175	C₂×C₄²	—	43	2	no	—
176	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
177	#41	—	43	8	no	—
178	C₂×Q₈	—	43	1	no	—
179	C₂×Q₈	—	43	8	no	—
180	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
181	C₂×D₄	—	43	2	no	—
182	C₂×D₄	—	43	1	no	—
183	C₂²×C₄	—	43	2	no	—
184	C₂²×C₄	—	43	4	no	—
185	C₂²×C₄	—	43	4	no	—
186	C₂²×C₄	—	43	8	no	—
187	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
188	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
189	C₄⋊C₄	—	43	4	no	—
190	C₄⋊C₄	—	43	8	no	—
191	C₄⋊C₄	—	43	8	no	—
192	C₄⋊C₄	—	43	4	no	—
193	C₄⋊C₄	—	43	8	no	—
194	C₄⋊C₄	—	43	8	no	—
195	C₄⋊C₄	—	43	8	no	—
196	C₄⋊C₄	—	43	4	no	—
197	C₄⋊C₄	—	43	2	no	—
198	C₄⋊C₄	—	43	8	no	—
199	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
200	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
201	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
202	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
203	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
204	C₂²⋊C₄	—	43	2	no	—
205	C₂²⋊C₄	—	43	4	no	—
206	C₂²⋊C₄	—	43	8	no	—
207	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
208	C₄²	—	43	2	no	—
209	C₄²	—	43	4	no	—
210	C₄²	—	43	8	no	—
211	C₄²	C₂²×D₄₃	1	1	yes	characteristic
212	C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	C₄²	C₂²×D₄₃	1	2	yes	—
214	C₄²	C₂²×D₄₃	1	4	yes	—
215	C₂³	—	43	2	no	—
216	C₂³	C₂³×D₄₃	1	1	yes	characteristic
217	Q₈	—	4	4	no	—
218	Q₈	—	43	4	no	—
219	Q₈	—	172	8	no	—
220	D₄	—	43	8	no	—
221	D₄	—	43	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	43	2	no	—
224	C₂×C₄	—	43	8	no	—
225	C₂×C₄	—	43	4	no	—
226	C₂×C₄	—	43	8	no	—
227	C₂×C₄	—	172	4	no	—
228	C₂×C₄	—	172	8	no	—
229	C₂×C₄	—	172	4	no	—
230	C₂×C₄	Q₈×D₄₃	1	4	yes	—
231	C₂×C₄	C₂³×D₄₃	1	2	yes	—
232	C₂×C₄	C₂³×D₄₃	1	4	yes	—
233	C₂×C₄	C₂³×D₄₃	1	2	yes	—
234	C₂×C₄	C₂³×D₄₃	1	4	yes	—
235	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	43	4	no	—
239	C₂²	—	86	4	no	—
240	C₂²	C₂×Q₈×D₄₃	1	2	yes	—
241	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	4	no	—
245	C₄	—	2	4	no	—
246	C₄	—	43	4	no	—
247	C₄	—	86	8	no	—
248	C₄	—	86	4	no	—
249	C₄	C₂×Q₈×D₄₃	1	2	yes	—
250	C₄	C₂×Q₈×D₄₃	1	2	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	86	4	no	—
253	C₂	C₂²×Q₈×D₄₃	1	1	yes	characteristic
254	C₂	#14583735	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#533	C₂
hypercenter	#533	D₄₃
hypocenter	C₄₃	#3504
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #14583627	C₂	semidirect
2. #14583609	C₂	semidirect
3. #14583603	C₂	semidirect
4. #14583527	C₂	semidirect
5. #14583526	C₂	semidirect
6. #14583455	C₂	semidirect
7. C₂×#13063317	C₂	semidirect
8. #125×D₄₃	C₂	semidirect
11. C₄₃×#533	C₂	semidirect
54. #10621898	Q₈	semidirect
134. #533	D₄₃	semidirect
139. C₄₃	#3504	semidirect