Subgroups — #19147330

Order 5504.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583618	C₂	1	1	yes	maximal, characteristic
3	#14583258	C₂	1	1	yes	maximal, characteristic
4	#14583257	C₂	1	1	yes	maximal, characteristic
5	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583187	C₂	1	1	yes	maximal, characteristic
7	#14583185	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583119	C₂	1	1	yes	maximal, characteristic
10	#14583107	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583028	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583006	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583004	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#464	C₂	1	1	yes	maximal, characteristic
16	C₂×#10621885	C₂²	1	1	yes	characteristic
17	#13063325	—	2	1	no	—
18	#13063325	C₂²	1	1	yes	characteristic
19	#13063324	—	2	1	no	—
20	#13063324	—	2	1	no	—
21	#13063324	C₂²	1	1	yes	characteristic
22	#13063324	C₂²	1	1	yes	characteristic
23	#13063323	C₂²	1	1	yes	characteristic
24	#13063318	—	2	1	no	—
25	#13063318	—	2	1	no	—
26	#13063312	C₂²	1	1	yes	characteristic
27	#13063302	—	2	1	no	—
28	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
29	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
30	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
31	C₄×Dic₈₆	—	2	1	no	—
32	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
33	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	#13063281	—	2	1	no	—
36	#13063257	C₂²	1	1	yes	characteristic
37	#13063257	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	#13063250	C₂²	1	1	yes	characteristic
42	#13063250	C₂²	1	1	yes	characteristic
43	#13063249	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063236	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	#124×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₄₃	—	2	1	no	—
63	C₂×Dic₈₆	—	2	1	no	—
64	C₂×Dic₈₆	—	2	1	no	—
65	C₂×Dic₈₆	—	2	1	no	—
66	C₂×Dic₈₆	D₄	1	1	yes	characteristic
67	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
68	Q₁₆×C₄₃	—	2	2	no	—
69	SD₁₆×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₄₃	—	2	1	no	—
73	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₄₃	—	2	1	no	—
75	C₂²⋊C₄×C₄₃	—	2	1	no	—
76	C₄²×C₄₃	—	2	1	no	—
77	C₄²×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₄₃	—	2	1	no	—
79	C₂²⋊Dic₄₃	—	2	1	no	—
80	C₂²⋊Dic₄₃	—	2	1	no	—
81	#10621885	—	2	2	no	—
82	C₄⋊Dic₄₃	—	2	1	no	—
83	C₄⋊Dic₄₃	—	2	1	no	—
84	C₄⋊Dic₄₃	—	2	1	no	—
85	C₄⋊Dic₄₃	—	2	1	no	—
86	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
87	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
92	Dic₄₃⋊C₄	—	2	1	no	—
93	Dic₄₃⋊C₄	—	2	1	no	—
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
96	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂³×C₄₃	—	2	1	no	—
106	Q₈×C₄₃	—	2	1	no	—
107	Q₈×C₄₃	—	2	1	no	—
108	Q₈×C₄₃	—	2	1	no	—
109	Q₈×C₄₃	#41	1	2	yes	—
110	D₄×C₄₃	—	2	1	no	—
111	D₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
127	Dic₈₆	—	2	2	no	—
128	Dic₈₆	—	2	1	no	—
129	Dic₈₆	—	2	1	no	—
130	Dic₈₆	—	8	1	no	—
131	Dic₈₆	—	8	1	no	—
132	Dic₈₆	—	8	1	no	—
133	C₈×C₄₃	—	2	1	no	—
134	C₈×C₄₃	—	2	1	no	—
135	C₄₃⋊C₈	—	2	1	no	—
136	C₄₃⋊C₈	—	2	1	no	—
137	C₂²×C₄₃	—	4	1	no	—
138	C₂²×C₄₃	—	4	1	no	—
139	C₂²×C₄₃	—	4	1	no	—
140	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₄₃	—	2	2	no	—
142	C₄×C₄₃	—	2	1	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	4	1	no	—
145	C₄×C₄₃	—	4	1	no	—
146	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₄₃	—	2	1	no	—
149	Dic₄₃	—	2	1	no	—
150	Dic₄₃	—	2	1	no	—
151	Dic₄₃	—	4	1	no	—
152	Dic₄₃	—	4	1	no	—
153	Dic₄₃	—	4	1	no	—
154	Dic₄₃	—	4	1	no	—
155	Dic₄₃	#137	1	2	yes	—
156	#3369	—	43	1	no	maximal, Sylow
157	C₂×C₄₃	—	8	1	no	—
158	C₂×C₄₃	#548	1	1	yes	characteristic
159	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
160	C₂×C₄₃	#577	1	1	yes	characteristic
161	Q₈²	—	43	1	no	—
162	#556	—	43	1	no	—
163	#489	—	43	1	no	—
164	#489	—	43	1	no	—
165	#487	—	43	1	no	—
166	#476	—	43	1	no	—
167	#475	—	43	1	no	—
168	#475	—	43	1	no	—
169	#464	—	43	1	no	—
170	#464	D₄₃	1	1	yes	characteristic
171	#462	—	43	1	no	—
172	#446	—	43	1	no	—
173	#441	—	43	1	no	—
174	#441	—	43	1	no	—
175	C₄×SD₁₆	—	43	1	no	—
176	C₄₃	#3369	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	—	43	1	no	—
178	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
179	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	43	1	no	—
181	C₄⋊Q₈	—	43	1	no	—
182	C₄⋊Q₈	—	43	1	no	—
183	C₄⋊Q₈	—	43	1	no	—
184	C₄⋊Q₈	—	86	1	no	—
185	C₄⋊Q₈	—	86	1	no	—
186	C₄⋊Q₈	—	86	1	no	—
187	#126	—	86	1	no	—
188	#126	—	86	1	no	—
189	#124	—	43	1	no	—
190	#124	C₂×D₄₃	1	1	yes	characteristic
191	C₂²⋊Q₈	—	86	1	no	—
192	C₄×Q₈	—	43	1	no	—
193	C₄×Q₈	—	43	1	no	—
194	C₄×Q₈	—	43	1	no	—
195	C₄×Q₈	—	43	1	no	—
196	C₄×Q₈	—	43	1	no	—
197	C₄×Q₈	—	86	1	no	—
198	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
199	C₄×D₄	—	43	1	no	—
200	#117	—	86	1	no	—
201	#106	—	43	1	no	—
202	#106	—	43	1	no	—
203	#106	—	43	1	no	—
204	C₄⋊C₈	—	43	1	no	—
205	C₄⋊C₈	—	43	1	no	—
206	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
207	#103	—	43	1	no	—
208	#103	—	43	1	no	—
209	#103	—	43	1	no	—
210	#103	—	43	1	no	—
211	#103	—	43	1	no	—
212	#103	—	43	1	no	—
213	#103	C₂×D₄₃	1	1	yes	characteristic
214	#102	—	43	1	no	—
215	#102	—	43	1	no	—
216	#102	C₂×D₄₃	1	1	yes	characteristic
217	#97	—	43	1	no	—
218	#97	—	43	1	no	—
219	C₄×C₈	—	43	1	no	—
220	C₂×Q₈	—	43	1	no	—
221	C₂×Q₈	—	43	1	no	—
222	C₂×Q₈	—	86	1	no	—
223	C₂×Q₈	—	86	1	no	—
224	C₂×Q₈	—	86	1	no	—
225	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
227	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	—	86	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	86	2	no	—
231	SD₁₆	—	2	2	no	—
232	C₂×C₈	—	43	1	no	—
233	C₂×C₈	—	43	1	no	—
234	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	43	1	no	—
238	C₄⋊C₄	—	43	1	no	—
239	C₄⋊C₄	—	43	1	no	—
240	C₄⋊C₄	—	43	1	no	—
241	C₄⋊C₄	—	43	1	no	—
242	C₄⋊C₄	—	43	1	no	—
243	C₄⋊C₄	—	86	1	no	—
244	C₄⋊C₄	—	86	1	no	—
245	C₄⋊C₄	—	86	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	1	no	—
248	C₄⋊C₄	—	86	1	no	—
249	C₄⋊C₄	—	86	1	no	—
250	C₄⋊C₄	—	86	1	no	—
251	C₄⋊C₄	—	86	1	no	—
252	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	86	1	no	—
256	C₂²⋊C₄	—	86	1	no	—
257	C₂²⋊C₄	—	86	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	43	1	no	—
260	C₄²	—	43	1	no	—
261	C₄²	—	86	1	no	—
262	C₄²	—	86	1	no	—
263	C₄²	—	86	1	no	—
264	C₄²	—	86	1	no	—
265	C₄²	C₂²×D₄₃	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	86	2	no	—
271	Q₈	—	86	1	no	—
272	Q₈	—	86	1	no	—
273	Q₈	—	344	1	no	—
274	Q₈	—	344	1	no	—
275	Q₈	—	344	1	no	—
276	Q₈	#10621900	1	2	yes	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	43	1	no	—
283	C₂×C₄	—	43	1	no	—
284	C₂×C₄	—	43	1	no	—
285	C₂×C₄	—	43	1	no	—
286	C₂×C₄	—	86	1	no	—
287	C₂×C₄	—	86	1	no	—
288	C₂×C₄	—	86	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	172	2	no	—
291	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
292	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	86	1	no	—
297	C₈	—	86	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	43	2	no	—
308	C₄	—	86	1	no	—
309	C₄	—	86	1	no	—
310	C₄	—	86	1	no	—
311	C₄	—	172	1	no	—
312	C₄	—	172	1	no	—
313	C₄	—	172	1	no	—
314	C₄	—	172	1	no	—
315	C₄	C₂×#10621900	1	1	yes	characteristic
316	C₄	#13063372	1	1	yes	characteristic
317	C₂	—	8	1	no	—
318	C₂	#14583555	1	1	yes	characteristic
319	C₂	#137×D₄₃	1	1	yes	characteristic
320	C₂	#14583706	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#464	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#464	D₄₃
hypocenter	C₄₃	#3369
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₈₆	C₂	semidirect
5. #14583248	C₂	semidirect
7. #14583185	C₂	semidirect
8. #14583182	C₂	semidirect
10. #14583107	C₂	semidirect
11. #14583028	C₂	semidirect
12. #14583006	C₂	semidirect
13. #14583004	C₂	semidirect
176. C₄₃	#3369	semidirect