Subgroups — #19138467

Order 5504.

548 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063317	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#14582834	C₂	1	2	yes	maximal, complemented
4	#14582823	C₂	1	2	yes	maximal, complemented
5	#14582801	C₂	1	1	yes	maximal, characteristic
6	#14582801	C₂	1	1	yes	maximal, characteristic
7	#14582797	C₂	1	1	yes	maximal, characteristic, complemented
8	#14582797	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582770	C₂	1	2	yes	maximal, complemented
10	#14582758	C₂	1	2	yes	maximal, complemented
11	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
14	#13063317	—	2	4	no	—
15	C₄×D₁₇₂	—	2	4	no	—
16	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
27	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
28	C₄₃⋊#95	C₂²	1	2	yes	—
29	C₄₃⋊#95	C₂²	1	2	yes	—
30	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
35	#95×C₄₃	C₂²	1	1	yes	characteristic
36	#95×C₄₃	C₂²	1	1	yes	characteristic
37	C₂³×D₄₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₄₃	C₂³	1	2	yes	—
46	C₂²×Dic₄₃	C₂³	1	2	yes	—
47	C₂²×Dic₄₃	C₂³	1	2	yes	—
48	C₂×D₁₇₂	—	2	2	no	—
49	C₂×D₁₇₂	—	2	2	no	—
50	C₂×D₁₇₂	Q₈	1	4	yes	—
51	C₂×C₄×D₄₃	—	2	4	no	—
52	C₂×C₄×D₄₃	—	2	4	no	—
53	C₂×C₄×D₄₃	—	4	2	no	—
54	C₄⋊C₄×C₄₃	—	2	2	no	—
55	C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₄²×C₄₃	—	2	2	no	—
57	C₄²×C₄₃	—	2	2	no	—
58	C₄²×C₄₃	—	2	2	no	—
59	(C₂×D₄₃)⋊C₄	—	2	4	no	—
60	(C₂×D₄₃)⋊C₄	—	2	4	no	—
61	(C₂×D₄₃)⋊C₄	—	4	2	no	—
62	(C₂×D₄₃)⋊C₄	—	4	2	no	—
63	(C₂×D₄₃)⋊C₄	—	4	2	no	—
64	C₄⋊Dic₄₃	—	2	2	no	—
65	C₄⋊Dic₄₃	—	2	2	no	—
66	C₄⋊Dic₄₃	D₄	1	4	yes	—
67	Dic₄₃⋊C₄	—	2	4	no	—
68	Dic₄₃⋊C₄	—	2	4	no	—
69	Dic₄₃⋊C₄	—	4	2	no	—
70	Dic₄₃⋊C₄	—	4	2	no	—
71	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
72	C₂²×D₄₃	—	2	2	no	—
73	C₂²×D₄₃	—	2	2	no	—
74	C₂²×D₄₃	—	2	2	no	—
75	C₂²×D₄₃	—	2	2	no	—
76	C₂²×D₄₃	—	2	2	no	—
77	C₂²×D₄₃	—	2	2	no	—
78	C₂²×D₄₃	C₂×Q₈	1	4	yes	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	1	no	—
81	C₂×C₄×C₄₃	—	2	1	no	—
82	C₂×C₄×C₄₃	—	2	1	no	—
83	C₂×C₄×C₄₃	—	2	1	no	—
84	C₂×C₄×C₄₃	—	2	1	no	—
85	C₂×C₄×C₄₃	—	2	1	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	2	1	no	—
89	C₂×C₄×C₄₃	—	2	1	no	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	1	no	—
92	C₂×C₄×C₄₃	—	2	1	no	—
93	C₂×C₄×C₄₃	—	2	1	no	—
94	C₂×C₄×C₄₃	—	2	1	no	—
95	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₄₃	#41	1	2	yes	—
98	C₂×C₄×C₄₃	#41	1	2	yes	—
99	C₂×C₄×C₄₃	#41	1	2	yes	—
100	C₂×Dic₄₃	—	2	2	no	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	2	2	no	—
103	C₂×Dic₄₃	—	2	2	no	—
104	C₂×Dic₄₃	—	2	2	no	—
105	C₂×Dic₄₃	—	2	2	no	—
106	C₂×Dic₄₃	—	2	2	no	—
107	C₂×Dic₄₃	—	2	2	no	—
108	C₂×Dic₄₃	C₂×D₄	1	4	yes	—
109	D₁₇₂	—	2	8	no	—
110	C₄×D₄₃	—	4	8	no	—
111	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
114	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
115	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×D₄₃	—	2	4	no	—
119	C₂×D₄₃	—	2	4	no	—
120	C₂×D₄₃	—	2	4	no	—
121	C₂×D₄₃	—	4	2	no	—
122	C₂×D₄₃	—	4	2	no	—
123	C₂×D₄₃	—	4	2	no	—
124	C₂×D₄₃	—	4	2	no	—
125	C₄×C₄₃	—	2	2	no	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	4	1	no	—
128	C₄×C₄₃	—	4	1	no	—
129	C₄×C₄₃	—	4	1	no	—
130	C₄×C₄₃	—	4	1	no	—
131	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	—
132	Dic₄₃	—	2	4	no	—
133	Dic₄₃	—	4	2	no	—
134	Dic₄₃	—	4	2	no	—
135	#2454	—	43	1	no	maximal, Sylow
136	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄₃	#529	1	1	yes	characteristic
138	C₂×C₄₃	#540	1	1	yes	characteristic
139	C₂×C₄₃	#546	1	1	yes	characteristic
140	C₂×C₄₃	#550	1	1	yes	characteristic
141	C₂×C₄₃	#554	1	1	yes	characteristic
142	C₂×C₄₃	#554	1	1	yes	characteristic
143	D₄₃	—	4	4	no	—
144	C₂×#125	—	43	1	no	—
145	C₂×C₄×D₄	—	43	1	no	—
146	#399	—	43	2	no	—
147	#396	—	43	2	no	—
148	#390	—	43	1	no	—
149	#390	—	43	1	no	—
150	#389	—	43	1	no	—
151	#389	—	43	1	no	—
152	#385	—	43	2	no	—
153	#380	—	43	2	no	—
154	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
155	C₄₃	#2454	1	1	yes	characteristic, complemented, Sylow
156	C₂²×D₄	—	43	1	no	—
157	C₂³×C₄	—	43	2	no	—
158	#125	—	86	4	no	—
159	C₄×D₄	—	86	4	no	—
160	C₂×C₄⋊C₄	—	43	1	no	—
161	C₂×C₄⋊C₄	—	43	2	no	—
162	C₂×C₄⋊C₄	—	43	2	no	—
163	C₂×C₄⋊C₄	—	43	1	no	—
164	C₂×C₄⋊C₄	—	43	1	no	—
165	C₂×C₄⋊C₄	—	43	2	no	—
166	C₂×C₄⋊C₄	—	43	2	no	—
167	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
168	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	43	2	no	—
170	C₂×C₂²⋊C₄	—	43	2	no	—
171	C₂×C₂²⋊C₄	—	43	2	no	—
172	C₂×C₂²⋊C₄	—	43	2	no	—
173	C₂×C₂²⋊C₄	—	43	2	no	—
174	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
176	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
177	#95	—	43	2	no	—
178	#95	—	43	2	no	—
179	#95	C₂×D₄₃	1	1	yes	characteristic
180	#95	C₂×D₄₃	1	1	yes	characteristic
181	C₂⁴	—	43	2	no	—
182	C₂×D₄	—	43	4	no	—
183	C₂×D₄	—	86	2	no	—
184	C₂×D₄	—	86	2	no	—
185	C₂²×C₄	—	43	2	no	—
186	C₂²×C₄	—	43	2	no	—
187	C₂²×C₄	—	43	2	no	—
188	C₂²×C₄	—	86	4	no	—
189	C₂²×C₄	—	86	4	no	—
190	C₂²×C₄	—	172	2	no	—
191	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	43	4	no	—
201	C₄⋊C₄	—	86	4	no	—
202	C₄⋊C₄	—	86	4	no	—
203	C₄⋊C₄	—	86	2	no	—
204	C₄⋊C₄	—	86	2	no	—
205	C₄⋊C₄	—	172	2	no	—
206	C₄⋊C₄	—	172	2	no	—
207	C₂²⋊C₄	—	86	4	no	—
208	C₂²⋊C₄	—	86	4	no	—
209	C₂²⋊C₄	—	172	2	no	—
210	C₂²⋊C₄	—	172	2	no	—
211	C₂²⋊C₄	—	172	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₂³	—	43	4	no	—
216	C₂³	—	86	2	no	—
217	C₂³	—	86	2	no	—
218	C₂³	—	86	2	no	—
219	C₂³	—	86	2	no	—
220	C₂³	—	86	2	no	—
221	C₂³	—	86	2	no	—
222	C₂³	C₂³×D₄₃	1	1	yes	characteristic
223	D₄	—	86	8	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	43	4	no	—
241	C₂×C₄	—	86	2	no	—
242	C₂×C₄	—	86	2	no	—
243	C₂×C₄	—	86	2	no	—
244	C₂×C₄	—	86	2	no	—
245	C₂×C₄	—	86	2	no	—
246	C₂×C₄	—	86	2	no	—
247	C₂×C₄	—	86	2	no	—
248	C₂×C₄	—	86	2	no	—
249	C₂×C₄	—	172	8	no	—
250	C₂×C₄	#10621898	1	2	yes	—
251	C₂×C₄	#10621898	1	2	yes	—
252	C₂×C₄	#10621898	1	2	yes	—
253	C₂×C₄	D₄×D₄₃	1	2	yes	—
254	C₂×C₄	Q₈×D₄₃	1	2	yes	—
255	C₂²	—	86	4	no	—
256	C₂²	—	86	4	no	—
257	C₂²	—	86	4	no	—
258	C₂²	—	172	2	no	—
259	C₂²	—	172	2	no	—
260	C₂²	—	172	2	no	—
261	C₂²	—	172	2	no	—
262	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
263	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
264	C₂²	C₂×#10621898	1	1	yes	characteristic
265	C₂²	C₂×#10621898	1	1	yes	characteristic
266	C₂²	C₂×#10621898	1	1	yes	characteristic
267	C₂²	#13063371	1	1	yes	characteristic
268	C₂²	#13063371	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	86	4	no	—
276	C₄	—	172	2	no	—
277	C₄	—	172	2	no	—
278	C₄	#13063303	1	4	yes	—
279	C₂	—	172	4	no	—
280	C₂	C₂×#13063303	1	1	yes	characteristic
281	C₂	#14583433	1	1	yes	characteristic
282	C₂	#14583504	1	1	yes	characteristic
283	C₂	#14583545	1	1	yes	characteristic
284	C₂	#14583570	1	1	yes	characteristic
285	C₂	#14583609	1	1	yes	characteristic
286	C₂	#14583609	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#2454
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13063317	C₂	semidirect
3. #14582834	C₂	semidirect
4. #14582823	C₂	semidirect
7. #14582797	C₂	semidirect
8. #14582797	C₂	semidirect
9. #14582770	C₂	semidirect
10. #14582758	C₂	semidirect
11. C₄×C₄⋊C₄×C₄₃	C₂	semidirect
154. C₄×C₄⋊C₄	D₄₃	semidirect
155. C₄₃	#2454	semidirect