Subgroups — #19138293

Order 5504.

608 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063301	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₄×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#14582832	C₂	1	2	yes	maximal, complemented
4	#14582818	C₂	1	2	yes	maximal, complemented
5	#14582798	C₂	1	1	yes	maximal, characteristic, complemented
6	#14582797	C₂	1	1	yes	maximal, characteristic, complemented
7	#14582770	C₂	1	2	yes	maximal
8	#14582768	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582756	C₂	1	1	yes	maximal, characteristic
10	C₄×(C₂×D₄₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
13	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
14	#13063301	—	2	8	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	2	yes	—
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×Dic₄₃⋊C₄	C₂²	1	2	yes	—
22	C₄⋊C₄×D₄₃	—	2	4	no	—
23	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₄₃⋊#95	C₂²	1	2	yes	—
28	C₄₃⋊#95	C₂²	1	2	yes	—
29	C₄₃⋊#95	C₂²	1	1	yes	characteristic
30	C₄₃⋊#95	C₂²	1	2	yes	—
31	C₄₃⋊#95	C₂²	1	1	yes	characteristic
32	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
33	C₄⋊Q₈×C₄₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
38	C₂³×D₄₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₄₃	—	2	4	no	—
40	C₂×Q₈×C₄₃	—	2	4	no	—
41	C₂×Q₈×C₄₃	—	4	2	no	—
42	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₄₃	C₂³	1	2	yes	—
48	C₂²×Dic₄₃	C₂³	1	2	yes	—
49	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
50	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₄₃	—	2	2	no	—
53	C₂×C₄×D₄₃	—	2	2	no	—
54	C₂×C₄×D₄₃	—	2	4	no	—
55	C₂×C₄×D₄₃	—	2	4	no	—
56	C₂×C₄×D₄₃	—	4	2	no	—
57	C₂×C₄×D₄₃	Q₈	1	4	yes	complemented
58	C₄⋊C₄×C₄₃	—	2	4	no	—
59	C₄⋊C₄×C₄₃	—	2	2	no	—
60	C₄⋊C₄×C₄₃	D₄	1	4	yes	—
61	C₄²×C₄₃	—	2	2	no	—
62	(C₂×D₄₃)⋊C₄	—	2	4	no	—
63	(C₂×D₄₃)⋊C₄	D₄	1	4	yes	—
64	(C₂×D₄₃)⋊C₄	Q₈	1	4	yes	complemented
65	C₄⋊Dic₄₃	—	2	2	no	—
66	C₄⋊Dic₄₃	—	2	2	no	—
67	C₄⋊Dic₄₃	—	2	4	no	—
68	Dic₄₃⋊C₄	—	2	4	no	—
69	Dic₄₃⋊C₄	—	2	4	no	—
70	Dic₄₃⋊C₄	—	2	4	no	—
71	Dic₄₃⋊C₄	—	4	2	no	—
72	C₄×Dic₄₃	—	2	2	no	—
73	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
74	C₂²×D₄₃	—	2	1	no	—
75	C₂²×D₄₃	—	2	1	no	—
76	C₂²×D₄₃	—	2	1	no	—
77	C₂²×D₄₃	—	2	1	no	—
78	C₂²×D₄₃	C₂×D₄	1	2	yes	—
79	C₂²×D₄₃	C₂×Q₈	1	2	yes	—
80	C₂²×D₄₃	C₂×Q₈	1	2	yes	—
81	Q₈×C₄₃	—	4	8	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	1	no	—
84	C₂×C₄×C₄₃	—	2	1	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	1	no	—
87	C₂×C₄×C₄₃	—	2	1	no	—
88	C₂×C₄×C₄₃	—	2	2	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	2	no	—
91	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
93	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₄₃	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₄₃	#41	1	2	yes	—
97	C₂×Dic₄₃	—	2	1	no	—
98	C₂×Dic₄₃	—	2	1	no	—
99	C₂×Dic₄₃	—	2	1	no	—
100	C₂×Dic₄₃	—	2	1	no	—
101	C₂×Dic₄₃	—	2	1	no	—
102	C₂×Dic₄₃	—	2	1	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	—	2	2	no	—
109	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
110	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
111	C₂×Dic₄₃	C₂×Q₈	1	2	yes	—
112	C₄×D₄₃	—	2	8	no	—
113	C₄×D₄₃	—	4	8	no	—
114	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂²×C₄₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×D₄₃	—	2	2	no	—
122	C₂×D₄₃	—	2	2	no	—
123	C₂×D₄₃	—	2	2	no	—
124	C₂×D₄₃	—	2	2	no	—
125	C₂×D₄₃	—	2	2	no	—
126	C₂×D₄₃	—	2	2	no	—
127	C₂×D₄₃	C₄⋊Q₈	1	4	yes	complemented
128	C₄×C₄₃	—	2	4	no	—
129	C₄×C₄₃	—	2	2	no	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	4	2	no	—
132	C₄×C₄₃	C₂²⋊Q₈	1	4	yes	complemented
133	Dic₄₃	—	2	2	no	—
134	Dic₄₃	—	2	2	no	—
135	Dic₄₃	—	2	2	no	—
136	Dic₄₃	—	4	2	no	—
137	Dic₄₃	—	4	2	no	—
138	#2440	—	43	1	no	maximal, Sylow
139	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₂×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂×C₄₃	#544	1	1	yes	characteristic
142	C₂×C₄₃	D₄²	1	1	yes	characteristic
143	C₂×C₄₃	#547	1	1	yes	characteristic
144	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
145	C₂×C₄₃	#554	1	1	yes	characteristic
146	D₄₃	—	2	4	no	—
147	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
148	C₂×C₂²⋊Q₈	—	43	2	no	—
149	C₂²×C₄⋊C₄	—	43	1	no	—
150	#398	—	43	2	no	—
151	#395	—	43	2	no	—
152	#389	—	43	1	no	—
153	#389	—	43	1	no	—
154	#385	—	43	2	no	—
155	#384	—	43	1	no	—
156	#380	—	43	1	no	—
157	C₄×C₂²⋊C₄	—	43	1	no	—
158	C₄₃	#2440	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
160	C₂³×C₄	—	43	1	no	—
161	C₂³×C₄	—	43	2	no	—
162	C₄⋊Q₈	—	2	4	no	—
163	C₂²⋊Q₈	—	86	8	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	—	43	2	no	—
169	C₂×C₄⋊C₄	—	43	1	no	—
170	C₂×C₄⋊C₄	—	43	2	no	—
171	C₂×C₄⋊C₄	—	86	4	no	—
172	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
173	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	—	43	1	no	—
176	C₂×C₂²⋊C₄	—	43	1	no	—
177	C₂×C₂²⋊C₄	—	43	2	no	—
178	C₂×C₄²	—	43	1	no	—
179	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
180	#95	—	43	1	no	—
181	#95	—	43	1	no	—
182	#95	—	43	2	no	—
183	#95	—	43	2	no	—
184	#95	—	43	2	no	—
185	C₂⁴	—	43	1	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂²×C₄	—	43	1	no	—
190	C₂²×C₄	—	43	1	no	—
191	C₂²×C₄	—	43	1	no	—
192	C₂²×C₄	—	43	4	no	—
193	C₂²×C₄	—	43	2	no	—
194	C₂²×C₄	—	43	2	no	—
195	C₂²×C₄	—	86	4	no	—
196	C₂²×C₄	—	86	2	no	—
197	C₂²×C₄	—	86	2	no	—
198	C₂²×C₄	—	86	4	no	—
199	C₂²×C₄	—	172	2	no	—
200	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
204	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	86	4	no	—
208	C₄⋊C₄	—	86	4	no	—
209	C₄⋊C₄	—	86	4	no	—
210	C₄⋊C₄	—	86	2	no	—
211	C₄⋊C₄	—	86	4	no	—
212	C₄⋊C₄	—	86	2	no	—
213	C₄⋊C₄	—	172	2	no	—
214	C₄⋊C₄	C₂²⋊D₄₃	1	4	yes	—
215	C₂²⋊C₄	—	43	4	no	—
216	C₂²⋊C₄	—	43	4	no	—
217	C₂²⋊C₄	—	86	4	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	86	2	no	—
220	C₂³	—	43	2	no	—
221	C₂³	—	43	2	no	—
222	C₂³	—	43	2	no	—
223	C₂³	—	86	1	no	—
224	C₂³	—	86	1	no	—
225	C₂³	—	86	1	no	—
226	C₂³	—	86	1	no	—
227	C₂³	C₂³×D₄₃	1	1	yes	characteristic
228	Q₈	—	4	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	43	2	no	—
239	C₂×C₄	—	43	2	no	—
240	C₂×C₄	—	43	2	no	—
241	C₂×C₄	—	86	1	no	—
242	C₂×C₄	—	86	1	no	—
243	C₂×C₄	—	86	1	no	—
244	C₂×C₄	—	86	1	no	—
245	C₂×C₄	—	86	1	no	—
246	C₂×C₄	—	86	1	no	—
247	C₂×C₄	—	86	8	no	—
248	C₂×C₄	—	86	2	no	—
249	C₂×C₄	—	86	2	no	—
250	C₂×C₄	—	86	2	no	—
251	C₂×C₄	—	86	2	no	—
252	C₂×C₄	—	86	2	no	—
253	C₂×C₄	—	86	2	no	—
254	C₂×C₄	—	172	8	no	—
255	C₂×C₄	D₄×D₄₃	1	2	yes	—
256	C₂×C₄	Q₈×D₄₃	1	2	yes	—
257	C₂×C₄	Q₈×D₄₃	1	2	yes	—
258	C₂×C₄	#10621902	1	2	yes	—
259	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
260	C₂×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
261	C₂²	—	43	4	no	—
262	C₂²	—	86	2	no	—
263	C₂²	—	86	2	no	—
264	C₂²	—	86	2	no	—
265	C₂²	—	86	2	no	—
266	C₂²	—	86	2	no	—
267	C₂²	—	86	2	no	—
268	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
269	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
270	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
271	C₂²	C₂×Q₈×D₄₃	1	1	yes	characteristic
272	C₂²	C₂×#10621902	1	1	yes	characteristic
273	C₂²	#13063370	1	1	yes	characteristic
274	C₂²	#13063373	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	2	no	—
279	C₄	—	86	2	no	—
280	C₄	—	86	2	no	—
281	C₄	—	86	2	no	—
282	C₄	—	172	2	no	—
283	C₄	—	172	2	no	—
284	C₄	#13063301	1	4	yes	complemented
285	C₂	—	86	4	no	—
286	C₂	C₂×#13063301	1	1	yes	characteristic
287	C₂	C₄⋊Q₈×D₄₃	1	1	yes	characteristic
288	C₂	#14583526	1	1	yes	characteristic
289	C₂	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
290	C₂	#14583552	1	1	yes	characteristic
291	C₂	Q₈×C₂²⋊D₄₃	1	1	yes	characteristic
292	C₂	#14583610	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#2440
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #14582832	C₂	semidirect
4. #14582818	C₂	semidirect
5. #14582798	C₂	semidirect
6. #14582797	C₂	semidirect
8. #14582768	C₂	semidirect
11. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
57. C₂×C₄×D₄₃	Q₈	semidirect
64. (C₂×D₄₃)⋊C₄	Q₈	semidirect
127. C₂×D₄₃	C₄⋊Q₈	semidirect
132. C₄×C₄₃	C₂²⋊Q₈	semidirect
147. C₂×C₄⋊Q₈	D₄₃	semidirect
158. C₄₃	#2440	semidirect
284. C₄	#13063301	semidirect