Subgroups — C₂×#14583523 #19147976

Order 5504.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583523	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063320	C₂	1	2	yes	maximal, complemented
3	C₂×#13063314	C₂	1	2	yes	maximal, complemented
4	C₂×#13063313	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13063310	C₂	1	2	yes	maximal, complemented
6	C₂×#13063305	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063301	C₂	1	2	yes	maximal, complemented
8	C₂×#13063300	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063290	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063287	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#13063282	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
14	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
15	C₂²×D₁₇₂	C₂²	1	2	yes	—
16	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
18	#13063320	C₂²	1	16	yes	complemented
19	#13063314	C₂²	1	16	yes	complemented
20	#13063313	C₂²	1	8	yes	complemented
21	#13063310	C₂²	1	16	yes	complemented
22	#13063305	C₂²	1	8	yes	complemented
23	#13063301	C₂²	1	16	yes	complemented
24	#13063300	C₂²	1	8	yes	—
25	#13063290	C₂²	1	8	yes	complemented
26	#13063287	C₂²	1	8	yes	complemented
27	#13063282	C₂²	1	8	yes	complemented
28	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
29	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
32	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
35	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
37	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
38	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
39	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
41	C₂²×Q₈×C₄₃	C₂²	1	2	yes	—
42	C₄⋊Q₈×C₄₃	C₂²	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
47	C₂³×D₄₃	C₂³	1	1	yes	characteristic
48	C₂³×D₄₃	C₂³	1	2	yes	—
49	C₂×Q₈×C₄₃	—	4	4	no	—
50	C₂×Q₈×C₄₃	C₂³	1	8	yes	—
51	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
57	C₂²×Dic₄₃	C₂³	1	2	yes	—
58	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₁₇₂	—	2	4	no	—
61	C₂×D₁₇₂	—	4	4	no	—
62	C₂×D₁₇₂	C₂³	1	4	yes	—
63	C₂×D₁₇₂	C₂³	1	8	yes	—
64	C₂×C₄×D₄₃	—	2	4	no	—
65	C₂×C₄×D₄₃	—	4	4	no	—
66	C₂×C₄×D₄₃	C₂³	1	4	yes	—
67	C₂×C₄×D₄₃	C₂³	1	8	yes	—
68	C₂×Dic₈₆	—	2	4	no	—
69	C₂×Dic₈₆	C₂³	1	4	yes	—
70	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
71	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
72	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
73	C₄²×C₄₃	C₂³	1	4	yes	—
74	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
75	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
76	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
80	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
81	C₄⋊Dic₄₃	C₂³	1	8	yes	—
82	Dic₄₃⋊C₄	C₂³	1	4	yes	—
83	Dic₄₃⋊C₄	C₂³	1	4	yes	—
84	Dic₄₃⋊C₄	C₂³	1	8	yes	—
85	C₄×Dic₄₃	C₂³	1	4	yes	—
86	C₄×Dic₄₃	C₂³	1	8	yes	—
87	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×D₄₃	—	2	2	no	—
89	C₂²×D₄₃	—	2	4	no	—
90	C₂²×D₄₃	—	2	4	no	—
91	C₂²×D₄₃	—	2	2	no	—
92	C₂²×D₄₃	—	2	2	no	—
93	C₂²×D₄₃	—	2	4	no	—
94	C₂²×D₄₃	C₂⁴	1	2	yes	—
95	C₂²×D₄₃	C₂⁴	1	4	yes	—
96	Q₈×C₄₃	—	4	8	no	—
97	C₂×C₄×C₄₃	—	2	4	no	—
98	C₂×C₄×C₄₃	—	2	4	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	#41	1	4	yes	—
102	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
103	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
104	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
105	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
106	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
107	C₂×Dic₄₃	—	2	4	no	—
108	C₂×Dic₄₃	—	2	2	no	—
109	C₂×Dic₄₃	—	2	2	no	—
110	C₂×Dic₄₃	—	2	2	no	—
111	C₂×Dic₄₃	C₂⁴	1	2	yes	—
112	C₂×Dic₄₃	C₂⁴	1	2	yes	—
113	C₂×Dic₄₃	C₂⁴	1	4	yes	—
114	C₂×Dic₄₃	C₂⁴	1	2	yes	—
115	D₁₇₂	—	2	8	no	—
116	D₁₇₂	—	4	8	no	—
117	C₄×D₄₃	—	2	8	no	—
118	C₄×D₄₃	—	4	8	no	—
119	Dic₈₆	—	2	8	no	—
120	C₂²×C₄₃	C₂×#41	1	2	yes	—
121	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
122	C₂²×C₄₃	2₋¹⁺⁴	1	2	yes	—
123	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
124	C₂×D₄₃	—	2	2	no	—
125	C₂×D₄₃	—	2	4	no	—
126	C₂×D₄₃	—	2	4	no	—
127	C₂×D₄₃	—	2	2	no	—
128	C₂×D₄₃	—	2	2	no	—
129	C₂×D₄₃	—	2	4	no	—
130	C₂×D₄₃	—	4	4	no	—
131	C₂×D₄₃	—	4	8	no	—
132	C₄×C₄₃	—	2	4	no	—
133	C₄×C₄₃	—	2	4	no	—
134	C₄×C₄₃	—	2	2	no	—
135	C₄×C₄₃	—	2	2	no	—
136	C₄×C₄₃	C₂×#41	1	4	yes	—
137	Dic₄₃	—	2	4	no	—
138	Dic₄₃	—	2	2	no	—
139	Dic₄₃	—	2	2	no	—
140	Dic₄₃	—	2	2	no	—
141	C₂×#542	—	43	1	no	maximal, Sylow
142	C₂×C₄₃	#542	1	4	yes	complemented
143	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
144	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
146	D₄₃	—	4	2	no	—
147	D₄₃	—	4	4	no	—
148	#542	—	43	16	no	—
149	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
150	C₂×#126	—	43	2	no	—
151	C₂×#124	—	43	2	no	—
152	C₂×#124	—	43	1	no	—
153	C₂×#123	—	43	2	no	—
154	C₂×C₂²⋊Q₈	—	43	2	no	—
155	C₂×C₂²⋊Q₈	—	43	1	no	—
156	C₂×#121	—	43	1	no	—
157	C₂×C₄×Q₈	—	43	1	no	—
158	C₂×C₄×D₄	—	43	1	no	—
159	C₂×#117	—	43	1	no	—
160	C₄₃	C₂×#542	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	43	1	no	—
162	C₂²×Q₈	C₂×D₄₃	1	2	yes	—
163	C₂²×D₄	—	43	1	no	—
164	C₂²×D₄	—	43	2	no	—
165	C₂³×C₄	—	43	1	no	—
166	C₂³×C₄	—	43	2	no	—
167	C₄⋊Q₈	C₂×D₄₃	1	8	yes	complemented
168	#126	—	43	16	no	—
169	#124	—	43	16	no	—
170	#124	—	43	8	no	—
171	#123	—	43	16	no	—
172	C₂²⋊Q₈	—	43	16	no	—
173	C₂²⋊Q₈	—	43	8	no	—
174	#121	—	43	8	no	—
175	C₄×Q₈	—	43	8	no	—
176	C₄×D₄	—	43	8	no	—
177	#117	—	43	8	no	—
178	C₂×C₄⋊C₄	—	43	1	no	—
179	C₂×C₄⋊C₄	—	43	1	no	—
180	C₂×C₄⋊C₄	—	43	2	no	—
181	C₂×C₄⋊C₄	—	43	2	no	—
182	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
183	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
185	C₂×C₂²⋊C₄	—	43	1	no	—
186	C₂×C₂²⋊C₄	—	43	1	no	—
187	C₂×C₂²⋊C₄	—	43	2	no	—
188	C₂×C₂²⋊C₄	—	43	2	no	—
189	C₂×C₂²⋊C₄	—	43	2	no	—
190	C₂×C₂²⋊C₄	—	43	2	no	—
191	C₂×C₂²⋊C₄	—	43	2	no	—
192	C₂×C₄²	—	43	1	no	—
193	C₂×C₄²	—	43	2	no	—
194	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
195	C₂⁴	—	43	1	no	—
196	C₂⁴	—	43	2	no	—
197	C₂×Q₈	—	4	4	no	—
198	C₂×Q₈	—	43	4	no	—
199	C₂×Q₈	—	86	4	no	—
200	C₂×Q₈	C₂²×D₄₃	1	8	yes	—
201	C₂×D₄	—	43	4	no	—
202	C₂×D₄	—	43	8	no	—
203	C₂×D₄	—	86	4	no	—
204	C₂×D₄	—	172	4	no	—
205	C₂²×C₄	—	43	1	no	—
206	C₂²×C₄	—	43	2	no	—
207	C₂²×C₄	—	43	1	no	—
208	C₂²×C₄	—	43	1	no	—
209	C₂²×C₄	—	43	4	no	—
210	C₂²×C₄	—	43	8	no	—
211	C₂²×C₄	—	86	4	no	—
212	C₂²×C₄	—	172	4	no	—
213	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
214	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
217	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊C₄	—	43	4	no	—
219	C₄⋊C₄	—	43	4	no	—
220	C₄⋊C₄	—	43	8	no	—
221	C₄⋊C₄	—	43	8	no	—
222	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
223	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
224	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
225	C₂²⋊C₄	—	43	4	no	—
226	C₂²⋊C₄	—	43	4	no	—
227	C₂²⋊C₄	—	43	8	no	—
228	C₂²⋊C₄	—	43	8	no	—
229	C₂²⋊C₄	—	43	8	no	—
230	C₂²⋊C₄	—	43	8	no	—
231	C₂²⋊C₄	—	43	8	no	—
232	C₄²	—	43	4	no	—
233	C₄²	—	43	8	no	—
234	C₄²	C₂²×D₄₃	1	4	yes	—
235	C₂³	—	43	2	no	—
236	C₂³	—	43	4	no	—
237	C₂³	—	86	2	no	—
238	C₂³	—	86	4	no	—
239	C₂³	—	86	4	no	—
240	C₂³	—	86	2	no	—
241	C₂³	—	86	2	no	—
242	C₂³	—	86	4	no	—
243	C₂³	C₂³×D₄₃	1	1	yes	characteristic
244	Q₈	—	4	8	no	—
245	Q₈	—	86	8	no	—
246	D₄	—	86	8	no	—
247	D₄	—	172	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	43	2	no	—
253	C₂×C₄	—	43	4	no	—
254	C₂×C₄	—	43	2	no	—
255	C₂×C₄	—	43	2	no	—
256	C₂×C₄	—	86	2	no	—
257	C₂×C₄	—	86	4	no	—
258	C₂×C₄	—	86	2	no	—
259	C₂×C₄	—	86	2	no	—
260	C₂×C₄	—	86	8	no	—
261	C₂×C₄	—	172	8	no	—
262	C₂×C₄	#10621902	1	4	yes	—
263	C₂×C₄	C₂³×D₄₃	1	4	yes	—
264	C₂×C₄	C₂³×D₄₃	1	2	yes	—
265	C₂×C₄	C₂³×D₄₃	1	2	yes	—
266	C₂×C₄	C₂³×D₄₃	1	4	yes	—
267	C₂×C₄	C₂³×D₄₃	1	2	yes	—
268	C₂²	—	86	2	no	—
269	C₂²	—	86	4	no	—
270	C₂²	—	86	4	no	—
271	C₂²	—	86	2	no	—
272	C₂²	—	86	2	no	—
273	C₂²	—	86	4	no	—
274	C₂²	—	172	4	no	—
275	C₂²	—	172	8	no	—
276	C₂²	C₂×#10621902	1	2	yes	—
277	C₂²	#13063370	1	2	yes	—
278	C₂²	#13063373	1	2	yes	—
279	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	86	2	no	—
285	C₄	—	86	4	no	—
286	C₄	—	86	2	no	—
287	C₄	—	86	2	no	—
288	C₄	C₂×#10621902	1	4	yes	—
289	C₂	—	172	2	no	—
290	C₂	—	172	4	no	—
291	C₂	#14583523	1	4	yes	direct product factor
292	C₂	C₂²×#10621902	1	1	yes	characteristic
293	C₂	C₂×#13063370	1	1	yes	characteristic
294	C₂	C₂×#13063373	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×C₄⋊Q₈×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₄₃
hypocenter	C₄₃	C₂×#542
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #14583523	C₂	direct (N × H)
2. C₂×#13063320	C₂	semidirect
3. C₂×#13063314	C₂	semidirect
4. C₂×#13063313	C₂	semidirect
5. C₂×#13063310	C₂	semidirect
6. C₂×#13063305	C₂	semidirect
7. C₂×#13063301	C₂	semidirect
9. C₂×#13063290	C₂	semidirect
10. C₂×#13063287	C₂	semidirect
11. C₂×#13063282	C₂	semidirect
12. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
18. #13063320	C₂²	semidirect
19. #13063314	C₂²	semidirect
20. #13063313	C₂²	semidirect
21. #13063310	C₂²	semidirect
22. #13063305	C₂²	semidirect
23. #13063301	C₂²	semidirect
25. #13063290	C₂²	semidirect
26. #13063287	C₂²	semidirect
27. #13063282	C₂²	semidirect
42. C₄⋊Q₈×C₄₃	C₂²	semidirect
142. C₂×C₄₃	#542	semidirect
149. C₂×C₄⋊Q₈	D₄₃	semidirect
160. C₄₃	C₂×#542	semidirect
167. C₄⋊Q₈	C₂×D₄₃	semidirect
291. C₂	#14583523	direct (N × H)

Subgroups — C2×#14583523 #19147976

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583523 #19147976