Subgroups — #19137604

Order 5504.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582806	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582798	C₂	1	2	yes	maximal, complemented
5	#14582798	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582785	C₂	1	2	yes	maximal, complemented
7	#14582774	C₂	1	2	yes	maximal
8	C₄⋊C₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₄×Dic₄₃	C₂	1	2	yes	maximal
10	C₄×C₂²⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂²⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×Dic₄₃	C₂²	1	2	yes	—
13	Q₈×Dic₄₃	—	2	4	no	—
14	C₂×C₄⋊Dic₄₃	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
26	C₄₃⋊#95	C₂²	1	1	yes	characteristic
27	C₄₃⋊#95	C₂²	1	2	yes	—
28	C₄₃⋊#95	C₂²	1	2	yes	—
29	C₄₃⋊#95	C₂²	1	2	yes	—
30	C₄₃⋊#95	C₂²	1	1	yes	characteristic
31	C₄₃⋊#95	C₂²	1	2	yes	—
32	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
34	C₂²⋊Q₈×C₄₃	C₄	1	8	yes	complemented
35	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
36	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
38	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₄₃	—	2	2	no	—
40	C₂×Q₈×C₄₃	—	2	2	no	—
41	C₂×Q₈×C₄₃	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₄₃	—	2	2	no	—
43	C₂²×C₄×C₄₃	—	2	2	no	—
44	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×Dic₄₃	—	2	4	no	—
50	C₂²×Dic₄₃	—	2	4	no	—
51	C₂²×Dic₄₃	—	4	2	no	—
52	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
53	C₂²×Dic₄₃	C₂³	1	2	yes	—
54	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
55	C₂²×Dic₄₃	C₂³	1	2	yes	—
56	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	complemented
59	C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	complemented
60	C₂²⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
61	C₂²⋊Dic₄₃	—	2	2	no	—
62	C₂²⋊Dic₄₃	Q₈	1	4	yes	complemented
63	C₄⋊Dic₄₃	—	2	2	no	—
64	C₄⋊Dic₄₃	—	2	4	no	—
65	C₄⋊Dic₄₃	—	2	4	no	—
66	C₄⋊Dic₄₃	—	4	2	no	—
67	C₄⋊Dic₄₃	D₄	1	4	yes	—
68	C₄×Dic₄₃	—	2	2	no	—
69	C₄×Dic₄₃	—	2	2	no	—
70	C₄×Dic₄₃	—	2	4	no	—
71	C₄×Dic₄₃	—	4	2	no	—
72	C₂³×C₄₃	—	2	1	no	—
73	C₂³×C₄₃	—	2	1	no	—
74	C₂³×C₄₃	—	2	1	no	—
75	C₂³×C₄₃	—	2	1	no	—
76	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
77	C₂³×C₄₃	C₂×Q₈	1	2	yes	—
78	C₂³×C₄₃	#41	1	2	yes	—
79	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
80	Q₈×C₄₃	—	2	8	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	1	no	—
83	C₂×C₄×C₄₃	—	2	1	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	2	no	—
87	C₂×C₄×C₄₃	—	2	8	no	—
88	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₄₃	#41	1	2	yes	—
94	C₂×Dic₄₃	—	2	1	no	—
95	C₂×Dic₄₃	—	2	1	no	—
96	C₂×Dic₄₃	—	2	2	no	—
97	C₂×Dic₄₃	—	2	1	no	—
98	C₂×Dic₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	2	1	no	—
100	C₂×Dic₄₃	—	2	1	no	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	2	1	no	—
103	C₂×Dic₄₃	—	2	1	no	—
104	C₂×Dic₄₃	—	2	1	no	—
105	C₂×Dic₄₃	—	2	2	no	—
106	C₂×Dic₄₃	—	2	1	no	—
107	C₂×Dic₄₃	—	2	2	no	—
108	C₂×Dic₄₃	—	2	1	no	—
109	C₂×Dic₄₃	—	4	8	no	—
110	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
111	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
112	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	—	2	2	no	—
115	C₂²×C₄₃	—	2	2	no	—
116	C₂²×C₄₃	—	2	2	no	—
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	—	2	2	no	—
119	C₂²×C₄₃	C₄×Q₈	1	4	yes	complemented
120	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
121	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
124	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
125	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	—	2	4	no	—
129	C₄×C₄₃	—	2	4	no	—
130	C₄×C₄₃	C₄×D₄	1	4	yes	—
131	Dic₄₃	—	2	4	no	—
132	Dic₄₃	—	2	2	no	—
133	Dic₄₃	—	2	2	no	—
134	Dic₄₃	—	4	1	no	—
135	Dic₄₃	—	4	2	no	—
136	Dic₄₃	—	4	1	no	—
137	#2360	—	43	1	no	maximal, Sylow
138	C₂×C₄₃	—	2	4	no	—
139	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
141	C₂×C₄₃	#520	1	1	yes	characteristic
142	C₂×C₄₃	#546	1	1	yes	characteristic
143	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
144	C₂×C₄₃	#552	1	1	yes	characteristic
145	C₂×C₄₃	#554	1	1	yes	characteristic
146	C₂×C₂²⋊Q₈	D₄₃	1	1	yes	characteristic
147	C₂×C₄×Q₈	—	43	1	no	—
148	C₂²×C₄⋊C₄	—	43	1	no	—
149	#391	—	43	1	no	—
150	#389	—	43	1	no	—
151	#389	—	43	2	no	—
152	#387	—	43	2	no	—
153	#385	—	43	2	no	—
154	C₄×C₄⋊C₄	—	43	1	no	—
155	C₄×C₂²⋊C₄	—	43	1	no	—
156	C₄×C₂²⋊C₄	—	43	2	no	—
157	C₄₃	#2360	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
159	C₂³×C₄	—	43	2	no	—
160	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
161	C₂²⋊Q₈	Dic₄₃	1	8	yes	complemented
162	C₄×Q₈	—	86	4	no	—
163	C₂×C₄⋊C₄	—	43	1	no	—
164	C₂×C₄⋊C₄	—	43	2	no	—
165	C₂×C₄⋊C₄	—	43	1	no	—
166	C₂×C₄⋊C₄	—	43	2	no	—
167	C₂×C₄⋊C₄	—	43	2	no	—
168	C₂×C₄⋊C₄	—	86	4	no	—
169	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
171	C₂×C₂²⋊C₄	—	43	1	no	—
172	C₂×C₂²⋊C₄	—	43	1	no	—
173	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
174	C₂×C₄²	—	43	2	no	—
175	C₂×C₄²	—	43	1	no	—
176	C₂×C₄²	—	43	1	no	—
177	C₂×C₄²	—	43	2	no	—
178	#95	—	43	1	no	—
179	#95	—	43	2	no	—
180	#95	—	43	1	no	—
181	#95	—	43	2	no	—
182	#95	—	43	2	no	—
183	#95	—	43	2	no	—
184	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₂×Dic₄₃	1	4	yes	complemented
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	43	1	no	—
191	C₂²×C₄	—	43	1	no	—
192	C₂²×C₄	—	43	2	no	—
193	C₂²×C₄	—	43	1	no	—
194	C₂²×C₄	—	43	2	no	—
195	C₂²×C₄	—	43	1	no	—
196	C₂²×C₄	—	86	4	no	—
197	C₂²×C₄	—	86	4	no	—
198	C₂²×C₄	—	172	2	no	—
199	C₂²×C₄	C₂×Dic₄₃	1	4	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	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
204	C₄⋊C₄	—	43	4	no	—
205	C₄⋊C₄	—	86	4	no	—
206	C₄⋊C₄	—	86	4	no	—
207	C₄⋊C₄	—	86	2	no	—
208	C₄⋊C₄	—	172	2	no	—
209	C₄⋊C₄	C₂×Dic₄₃	1	4	yes	complemented
210	C₄⋊C₄	C₂×Dic₄₃	1	8	yes	complemented
211	C₂²⋊C₄	—	43	4	no	—
212	C₂²⋊C₄	—	86	2	no	—
213	C₂²⋊C₄	C₂×Dic₄₃	1	8	yes	—
214	C₄²	—	86	4	no	—
215	C₄²	—	86	2	no	—
216	C₄²	—	86	2	no	—
217	C₄²	—	172	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	Q₈×D₄₃	1	2	yes	—
223	C₂³	#10621902	1	2	yes	—
224	C₂³	C₂²×Dic₄₃	1	2	yes	—
225	C₂³	C₂³×D₄₃	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	43	4	no	—
235	C₂×C₄	—	43	2	no	—
236	C₂×C₄	—	43	2	no	—
237	C₂×C₄	—	86	1	no	—
238	C₂×C₄	—	86	1	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	86	1	no	—
241	C₂×C₄	—	86	2	no	—
242	C₂×C₄	—	86	1	no	—
243	C₂×C₄	—	86	1	no	—
244	C₂×C₄	—	86	2	no	—
245	C₂×C₄	—	86	1	no	—
246	C₂×C₄	—	86	1	no	—
247	C₂×C₄	—	86	1	no	—
248	C₂×C₄	—	86	2	no	—
249	C₂×C₄	—	86	1	no	—
250	C₂×C₄	—	86	2	no	—
251	C₂×C₄	—	86	1	no	—
252	C₂×C₄	—	172	8	no	—
253	C₂×C₄	D₄×D₄₃	1	2	yes	—
254	C₂×C₄	#10621900	1	2	yes	—
255	C₂×C₄	C₂²×Dic₄₃	1	4	yes	—
256	C₂×C₄	C₂²×Dic₄₃	1	4	yes	—
257	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
258	C₂×C₄	C₂²×Dic₄₃	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	Q₈×Dic₄₃	1	4	yes	complemented
266	C₂²	C₂³×Dic₄₃	1	1	yes	characteristic
267	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
268	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
269	C₂²	C₂×#10621900	1	1	yes	characteristic
270	C₂²	C₂×#10621902	1	1	yes	characteristic
271	C₂²	#13063371	1	1	yes	characteristic
272	C₂²	#13063372	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	86	2	no	—
277	C₄	—	86	2	no	—
278	C₄	—	86	4	no	—
279	C₄	—	172	1	no	—
280	C₄	—	172	2	no	—
281	C₄	—	172	1	no	—
282	C₄	D₄×Dic₄₃	1	4	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₂×D₄×Dic₄₃	1	1	yes	characteristic
285	C₂	C₂×Q₈×Dic₄₃	1	1	yes	characteristic
286	C₂	#14583382	1	1	yes	characteristic
287	C₂	#14583540	1	1	yes	characteristic
288	C₂	#14583562	1	1	yes	characteristic
289	C₂	#14583591	1	1	yes	characteristic
290	C₂	#14583607	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×C₂²⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₄₃
hypocenter	C₄₃	#2360
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₄₃	C₂	semidirect
3. #14582806	C₂	semidirect
4. #14582798	C₂	semidirect
5. #14582798	C₂	semidirect
6. #14582785	C₂	semidirect
8. C₄⋊C₄×Dic₄₃	C₂	semidirect
34. C₂²⋊Q₈×C₄₃	C₄	semidirect
41. C₂×Q₈×C₄₃	C₂×C₄	semidirect
58. C₄⋊C₄×C₄₃	C₂×C₄	semidirect
59. C₄⋊C₄×C₄₃	C₂×C₄	semidirect
62. C₂²⋊Dic₄₃	Q₈	semidirect
119. C₂²×C₄₃	C₄×Q₈	semidirect
157. C₄₃	#2360	semidirect
161. C₂²⋊Q₈	Dic₄₃	semidirect
187. C₂×Q₈	C₂×Dic₄₃	semidirect
209. C₄⋊C₄	C₂×Dic₄₃	semidirect
210. C₄⋊C₄	C₂×Dic₄₃	semidirect
265. C₂²	Q₈×Dic₄₃	semidirect