Subgroups — #19137277

Order 5504.

556 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×#13063304	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582805	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582797	C₂	1	1	yes	maximal, characteristic, complemented
5	#14582794	C₂	1	2	yes	maximal, complemented
6	#14582793	C₂	1	2	yes	maximal
7	#14582787	C₂	1	2	yes	maximal
8	#14582780	C₂	1	2	yes	maximal, complemented
9	#14582759	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#117×C₄₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
13	#13063304	C₄	1	8	yes	complemented
14	C₄×Dic₈₆	—	2	4	no	—
15	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
25	C₄₃⋊#95	C₂²	1	2	yes	—
26	C₄₃⋊#95	C₂²	1	2	yes	—
27	C₄₃⋊#95	C₂²	1	2	yes	—
28	C₄₃⋊#95	C₂²	1	2	yes	—
29	C₄₃⋊#95	C₂²	1	2	yes	—
30	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
31	#117×C₄₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
38	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	2	2	no	—
41	C₂²×C₄×C₄₃	C₂×C₄	1	4	yes	—
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×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₄₃	C₂³	1	2	yes	—
49	C₂²×Dic₄₃	C₂³	1	2	yes	—
50	C₂²×Dic₄₃	C₂³	1	2	yes	—
51	C₂²×Dic₄₃	C₂³	1	2	yes	—
52	C₂×Dic₈₆	—	2	2	no	—
53	C₂×Dic₈₆	—	2	2	no	—
54	C₂×Dic₈₆	C₂×C₄	1	4	yes	—
55	C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₄⋊C₄×C₄₃	—	2	2	no	—
57	C₂²⋊C₄×C₄₃	—	2	2	no	—
58	C₂²⋊C₄×C₄₃	—	2	2	no	—
59	C₄²×C₄₃	—	2	2	no	—
60	C₄²×C₄₃	D₄	1	4	yes	—
61	C₂²⋊Dic₄₃	—	4	2	no	—
62	C₂²⋊Dic₄₃	C₂×C₄	1	8	yes	—
63	C₄⋊Dic₄₃	—	2	2	no	—
64	C₄⋊Dic₄₃	—	2	2	no	—
65	C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
66	Dic₄₃⋊C₄	—	2	4	no	—
67	Dic₄₃⋊C₄	—	4	2	no	—
68	Dic₄₃⋊C₄	C₂×C₄	1	8	yes	—
69	C₄×Dic₄₃	—	2	4	no	—
70	C₄×Dic₄₃	—	4	2	no	—
71	C₂³×C₄₃	—	2	1	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	—	2	1	no	—
77	C₂³×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
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	8	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	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₄₃	#41	1	2	yes	—
98	C₂×Dic₄₃	—	2	2	no	—
99	C₂×Dic₄₃	—	2	2	no	—
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	C₂²×C₄	1	4	yes	—
109	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
110	Dic₈₆	—	2	8	no	—
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	—	2	2	no	—
113	C₂²×C₄₃	—	2	2	no	—
114	C₂²×C₄₃	—	4	1	no	—
115	C₂²×C₄₃	—	4	1	no	—
116	C₂²×C₄₃	—	4	1	no	—
117	C₂²×C₄₃	—	4	1	no	—
118	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
119	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₄₃	—	2	2	no	—
126	C₄×C₄₃	—	2	2	no	—
127	C₄×C₄₃	—	2	2	no	—
128	C₄×C₄₃	—	4	1	no	—
129	C₄×C₄₃	—	4	1	no	—
130	C₄×C₄₃	C₄×D₄	1	4	yes	—
131	Dic₄₃	—	2	4	no	—
132	Dic₄₃	—	2	4	no	—
133	Dic₄₃	—	4	2	no	—
134	Dic₄₃	—	4	2	no	—
135	#2323	—	43	1	no	maximal, Sylow
136	C₂×C₄₃	—	4	2	no	—
137	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₄₃	#519	1	1	yes	characteristic
139	C₂×C₄₃	#520	1	1	yes	characteristic
140	C₂×C₄₃	#536	1	1	yes	characteristic
141	C₂×C₄₃	#537	1	1	yes	characteristic
142	C₂×C₄₃	#541	1	1	yes	characteristic
143	C₂×C₄₃	#542	1	1	yes	characteristic
144	C₂×C₂²⋊Q₈	—	43	1	no	—
145	C₂×C₄×Q₈	—	43	1	no	—
146	C₂×#117	D₄₃	1	1	yes	characteristic
147	#391	—	43	1	no	—
148	#389	—	43	2	no	—
149	#389	—	43	1	no	—
150	#388	—	43	2	no	—
151	#388	—	43	2	no	—
152	#387	—	43	2	no	—
153	#380	—	43	1	no	—
154	C₄×C₄⋊C₄	—	43	1	no	—
155	C₄₃	#2323	1	1	yes	characteristic, complemented, Sylow
156	C₂²×Q₈	—	43	1	no	—
157	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
158	C₂²⋊Q₈	—	43	8	no	—
159	C₄×Q₈	—	86	4	no	—
160	#117	—	2	4	no	—
161	C₂×C₄⋊C₄	—	43	2	no	—
162	C₂×C₄⋊C₄	—	43	1	no	—
163	C₂×C₄⋊C₄	—	43	2	no	—
164	C₂×C₄⋊C₄	—	43	1	no	—
165	C₂×C₄⋊C₄	—	43	1	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	C₂×D₄₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂×D₄₃	1	1	yes	characteristic
173	C₂×C₄²	—	43	2	no	—
174	C₂×C₄²	—	43	2	no	—
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	—	43	2	no	—
180	#95	—	43	2	no	—
181	#95	—	43	2	no	—
182	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×Q₈	—	43	4	no	—
184	C₂×Q₈	—	86	2	no	—
185	C₂×Q₈	—	86	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	43	2	no	—
189	C₂²×C₄	—	43	2	no	—
190	C₂²×C₄	—	43	2	no	—
191	C₂²×C₄	—	43	2	no	—
192	C₂²×C₄	C₄×D₄₃	1	4	yes	—
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	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	43	8	no	—
202	C₄⋊C₄	—	43	4	no	—
203	C₄⋊C₄	—	86	2	no	—
204	C₄⋊C₄	—	86	4	no	—
205	C₄⋊C₄	—	86	2	no	—
206	C₄⋊C₄	—	172	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	43	8	no	—
210	C₂²⋊C₄	—	172	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	86	4	no	—
213	C₄²	—	172	2	no	—
214	C₄²	C₂²⋊D₄₃	1	4	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	C₂×C₄×D₄₃	1	2	yes	—
222	C₂³	C₂³×D₄₃	1	1	yes	characteristic
223	Q₈	—	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	8	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	—	43	4	no	—
238	C₂×C₄	—	43	4	no	—
239	C₂×C₄	—	86	2	no	—
240	C₂×C₄	—	86	2	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	C₂×C₄×D₄₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
251	C₂×C₄	#10621898	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
253	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
254	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
263	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂²	C₂×#10621898	1	1	yes	characteristic
265	C₂²	#13063371	1	1	yes	characteristic
266	C₂²	#13063372	1	1	yes	characteristic
267	C₂²	#13063372	1	1	yes	characteristic
268	C₂²	#13063372	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	86	4	no	—
275	C₄	—	86	4	no	—
276	C₄	—	172	2	no	—
277	C₄	—	172	2	no	—
278	C₄	C₄×C₂²⋊D₄₃	1	4	yes	—
279	C₂	—	4	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₄₃	1	1	yes	characteristic
281	C₂	#14583376	1	1	yes	characteristic
282	C₂	#14583379	1	1	yes	characteristic
283	C₂	#14583470	1	1	yes	characteristic
284	C₂	#14583477	1	1	yes	characteristic
285	C₂	#14583511	1	1	yes	characteristic
286	C₂	#14583520	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂³×C₄
Fitting	C₂×#117×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×#117	D₄₃
hypocenter	C₄₃	#2323
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₈₆	C₂	semidirect
3. #14582805	C₂	semidirect
4. #14582797	C₂	semidirect
5. #14582794	C₂	semidirect
8. #14582780	C₂	semidirect
10. C₄×C₄⋊Dic₄₃	C₂	semidirect
13. #13063304	C₄	semidirect
155. C₄₃	#2323	semidirect