Subgroups — C₂×#14583648 #19148724

Order 5504.

1024 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583648	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063323	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063320	C₂	1	2	yes	maximal, complemented
4	C₂×#13063319	C₂	1	2	yes	maximal, complemented
5	C₂×#13063315	C₂	1	2	yes	maximal, complemented
6	C₂×#13063314	C₂	1	1	yes	maximal, characteristic
7	C₂×#13063310	C₂	1	2	yes	maximal
8	C₂×#13063303	C₂	1	2	yes	maximal, complemented
9	C₂×#13063301	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
12	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
14	#13063323	C₂²	1	8	yes	complemented
15	#13063320	C₂²	1	16	yes	complemented
16	#13063319	C₂²	1	16	yes	complemented
17	#13063315	C₂²	1	16	yes	complemented
18	#13063314	C₂²	1	8	yes	—
19	#13063310	C₂²	1	16	yes	—
20	#13063303	C₂²	1	16	yes	complemented
21	#13063301	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
23	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
25	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
26	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
27	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
28	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
29	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
30	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
31	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
32	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
33	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄⋊Q₈×C₄₃	C₂²	1	8	yes	complemented
38	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
39	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
41	C₂³×D₄₃	C₂³	1	2	yes	—
42	C₂×Q₈×C₄₃	—	4	2	no	—
43	C₂×Q₈×C₄₃	—	4	2	no	—
44	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
45	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
46	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
51	C₂²×Dic₄₃	C₂³	1	2	yes	—
52	C₂²×Dic₄₃	C₂³	1	2	yes	—
53	C₂²×Dic₄₃	C₂³	1	2	yes	—
54	C₂×D₁₇₂	—	4	2	no	—
55	C₂×D₁₇₂	C₂³	1	4	yes	—
56	C₂×C₄×D₄₃	—	4	4	no	—
57	C₂×C₄×D₄₃	C₂³	1	8	yes	—
58	C₂×Dic₈₆	—	4	2	no	—
59	C₂×Dic₈₆	C₂³	1	4	yes	—
60	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
61	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
62	C₄²×C₄₃	C₂³	1	4	yes	—
63	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
64	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
65	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
66	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
67	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
68	C₄⋊Dic₄₃	C₂³	1	8	yes	—
69	Dic₄₃⋊C₄	C₂³	1	8	yes	—
70	Dic₄₃⋊C₄	C₂³	1	8	yes	—
71	Dic₄₃⋊C₄	C₂³	1	8	yes	—
72	Dic₄₃⋊C₄	C₂³	1	8	yes	—
73	Dic₄₃⋊C₄	C₂³	1	8	yes	—
74	C₄×Dic₄₃	C₂³	1	4	yes	—
75	C₄×Dic₄₃	C₂³	1	4	yes	—
76	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
77	C₂²×D₄₃	—	2	4	no	—
78	C₂²×D₄₃	—	2	8	no	—
79	C₂²×D₄₃	C₂⁴	1	4	yes	—
80	Q₈×C₄₃	—	4	4	no	—
81	Q₈×C₄₃	—	4	4	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	2	4	no	—
84	C₂×C₄×C₄₃	—	2	2	no	—
85	C₂×C₄×C₄₃	—	2	2	no	—
86	C₂×C₄×C₄₃	—	2	4	no	—
87	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
88	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
89	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
90	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
92	C₂×Dic₄₃	—	2	4	no	—
93	C₂×Dic₄₃	—	2	4	no	—
94	C₂×Dic₄₃	—	2	4	no	—
95	C₂×Dic₄₃	C₂⁴	1	4	yes	—
96	C₂×Dic₄₃	C₂⁴	1	4	yes	—
97	C₂×Dic₄₃	C₂⁴	1	4	yes	—
98	D₁₇₂	—	4	4	no	—
99	C₄×D₄₃	—	4	8	no	—
100	Dic₈₆	—	4	4	no	—
101	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₄₃	2₋¹⁺⁴	1	4	yes	—
103	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
104	C₂×D₄₃	—	2	4	no	—
105	C₂×D₄₃	—	2	8	no	—
106	C₂×D₄₃	—	4	8	no	—
107	C₄×C₄₃	—	2	2	no	—
108	C₄×C₄₃	—	2	4	no	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	2	2	no	—
111	C₄×C₄₃	—	2	4	no	—
112	Dic₄₃	—	2	4	no	—
113	Dic₄₃	—	2	4	no	—
114	Dic₄₃	—	2	4	no	—
115	C₂×#563	—	43	1	no	maximal, Sylow
116	C₂×C₄₃	#563	1	4	yes	complemented
117	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄₃	C₂×2₋¹⁺⁴	1	2	yes	—
119	D₄₃	—	4	4	no	—
120	#563	—	43	16	no	—
121	C₂×C₄⋊Q₈	—	43	1	no	—
122	C₂×C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
123	C₂×#126	—	43	2	no	—
124	C₂×#126	—	43	2	no	—
125	C₂×#125	—	43	2	no	—
126	C₂×#124	—	43	1	no	—
127	C₂×#123	—	43	2	no	—
128	C₂×C₂²⋊Q₈	—	43	2	no	—
129	C₂×C₂²⋊Q₈	—	43	2	no	—
130	C₄₃	C₂×#563	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₈	—	43	1	no	—
132	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
133	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
134	C₂²×D₄	—	43	1	no	—
135	C₂³×C₄	—	43	2	no	—
136	C₄⋊Q₈	—	43	8	no	—
137	C₄⋊Q₈	C₂×D₄₃	1	8	yes	complemented
138	#126	—	43	16	no	—
139	#126	—	43	16	no	—
140	#125	—	43	16	no	—
141	#124	—	43	8	no	—
142	#123	—	43	16	no	—
143	C₂²⋊Q₈	—	43	16	no	—
144	C₂²⋊Q₈	—	43	16	no	—
145	C₂×C₄⋊C₄	—	43	2	no	—
146	C₂×C₄⋊C₄	—	43	2	no	—
147	C₂×C₄⋊C₄	—	43	2	no	—
148	C₂×C₄⋊C₄	—	43	2	no	—
149	C₂×C₄⋊C₄	—	43	2	no	—
150	C₂×C₄⋊C₄	—	43	2	no	—
151	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
153	C₂×C₂²⋊C₄	—	43	2	no	—
154	C₂×C₂²⋊C₄	—	43	2	no	—
155	C₂×C₂²⋊C₄	—	43	2	no	—
156	C₂×C₂²⋊C₄	—	43	2	no	—
157	C₂×C₂²⋊C₄	—	43	2	no	—
158	C₂×C₄²	—	43	1	no	—
159	C₂×C₄²	—	43	1	no	—
160	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
161	C₂⁴	—	43	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	43	4	no	—
165	C₂×Q₈	—	172	2	no	—
166	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
167	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
168	C₂×D₄	—	43	4	no	—
169	C₂×D₄	—	172	2	no	—
170	C₂²×C₄	—	43	2	no	—
171	C₂²×C₄	—	43	2	no	—
172	C₂²×C₄	—	43	2	no	—
173	C₂²×C₄	—	43	8	no	—
174	C₂²×C₄	—	172	4	no	—
175	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
178	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
180	C₄⋊C₄	—	43	8	no	—
181	C₄⋊C₄	—	43	8	no	—
182	C₄⋊C₄	—	43	8	no	—
183	C₄⋊C₄	—	43	8	no	—
184	C₄⋊C₄	—	43	8	no	—
185	C₄⋊C₄	—	43	8	no	—
186	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
187	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
188	C₂²⋊C₄	—	43	8	no	—
189	C₂²⋊C₄	—	43	8	no	—
190	C₂²⋊C₄	—	43	8	no	—
191	C₂²⋊C₄	—	43	8	no	—
192	C₂²⋊C₄	—	43	8	no	—
193	C₄²	—	43	4	no	—
194	C₄²	—	43	4	no	—
195	C₄²	C₂²×D₄₃	1	4	yes	—
196	C₂³	—	43	4	no	—
197	C₂³	—	86	4	no	—
198	C₂³	—	86	8	no	—
199	C₂³	C₂³×D₄₃	1	1	yes	characteristic
200	Q₈	—	4	4	no	—
201	Q₈	—	4	4	no	—
202	Q₈	—	172	4	no	—
203	D₄	—	172	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	43	4	no	—
210	C₂×C₄	—	43	4	no	—
211	C₂×C₄	—	43	4	no	—
212	C₂×C₄	—	86	4	no	—
213	C₂×C₄	—	86	4	no	—
214	C₂×C₄	—	86	4	no	—
215	C₂×C₄	—	172	8	no	—
216	C₂×C₄	C₂³×D₄₃	1	2	yes	—
217	C₂×C₄	C₂³×D₄₃	1	2	yes	—
218	C₂×C₄	C₂³×D₄₃	1	4	yes	—
219	C₂×C₄	C₂³×D₄₃	1	2	yes	—
220	C₂×C₄	C₂³×D₄₃	1	4	yes	—
221	C₂²	—	86	4	no	—
222	C₂²	—	86	8	no	—
223	C₂²	—	172	8	no	—
224	C₂²	#13063370	1	2	yes	—
225	C₂²	#13063373	1	4	yes	—
226	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	86	4	no	—
233	C₄	—	86	4	no	—
234	C₄	—	86	4	no	—
235	C₂	—	172	4	no	—
236	C₂	#14583648	1	4	yes	direct product factor
237	C₂	C₂×#13063370	1	1	yes	characteristic
238	C₂	C₂×#13063373	1	2	yes	—

	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×#563
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. #14583648	C₂	direct (N × H)
2. C₂×#13063323	C₂	semidirect
3. C₂×#13063320	C₂	semidirect
4. C₂×#13063319	C₂	semidirect
5. C₂×#13063315	C₂	semidirect
8. C₂×#13063303	C₂	semidirect
9. C₂×#13063301	C₂	semidirect
10. C₂×C₄⋊Q₈×C₄₃	C₂	semidirect
14. #13063323	C₂²	semidirect
15. #13063320	C₂²	semidirect
16. #13063319	C₂²	semidirect
17. #13063315	C₂²	semidirect
20. #13063303	C₂²	semidirect
21. #13063301	C₂²	semidirect
37. C₄⋊Q₈×C₄₃	C₂²	semidirect
116. C₂×C₄₃	#563	semidirect
122. C₂×C₄⋊Q₈	D₄₃	semidirect
130. C₄₃	C₂×#563	semidirect
137. C₄⋊Q₈	C₂×D₄₃	semidirect
236. C₂	#14583648	direct (N × H)

Subgroups — C2×#14583648 #19148724

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583648 #19148724