Subgroups — C₂×#14583634 #19148183

Order 5504.

1240 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583634	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13063322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063314	C₂	1	2	yes	maximal, complemented
4	C₂×#13063313	C₂	1	2	yes	maximal, complemented
5	C₂×#13063310	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
8	C₂×#13063287	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₂²×D₁₇₂	C₂²	1	2	yes	—
12	C₂²×D₁₇₂	C₂²	1	2	yes	—
13	C₂²×D₁₇₂	C₂²	1	2	yes	—
14	C₂²×C₄×D₄₃	C₂²	1	4	yes	—
15	#13063322	C₂²	1	8	yes	—
16	#13063314	C₂²	1	16	yes	complemented
17	#13063313	C₂²	1	16	yes	complemented
18	#13063310	C₂²	1	32	yes	complemented
19	C₄×Dic₈₆	C₂²	1	8	yes	complemented
20	C₄×D₁₇₂	C₂²	1	16	yes	complemented
21	#13063287	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
25	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
26	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
27	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
28	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₄₃	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
34	C₂³×D₄₃	C₂³	1	4	yes	—
35	C₂×Q₈×C₄₃	—	2	4	no	—
36	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₄₃	C₂³	1	2	yes	—
43	C₂²×Dic₄₃	C₂³	1	2	yes	—
44	C₂×D₁₇₂	—	2	8	no	—
45	C₂×D₁₇₂	—	2	8	no	—
46	C₂×D₁₇₂	—	4	4	no	—
47	C₂×D₁₇₂	C₂³	1	8	yes	—
48	C₂×D₁₇₂	C₂³	1	8	yes	—
49	C₂×D₁₇₂	C₂³	1	8	yes	—
50	C₂×C₄×D₄₃	—	4	8	no	—
51	C₂×C₄×D₄₃	C₂³	1	16	yes	—
52	C₂×Dic₈₆	—	2	4	no	—
53	C₂×Dic₈₆	C₂³	1	4	yes	—
54	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
55	C₄⋊C₄×C₄₃	C₂³	1	8	yes	—
56	C₄²×C₄₃	C₂³	1	8	yes	—
57	C₄²×C₄₃	C₂³	1	4	yes	—
58	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
59	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
60	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
61	C₄⋊Dic₄₃	C₂³	1	4	yes	—
62	Dic₄₃⋊C₄	C₂³	1	8	yes	—
63	C₄×Dic₄₃	C₂³	1	8	yes	—
64	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
65	C₂²×D₄₃	—	2	8	no	—
66	C₂²×D₄₃	—	2	8	no	—
67	C₂²×D₄₃	—	2	8	no	—
68	C₂²×D₄₃	C₂⁴	1	8	yes	—
69	Q₈×C₄₃	—	2	8	no	—
70	C₂×C₄×C₄₃	—	2	4	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	—	2	2	no	—
73	C₂×C₄×C₄₃	#41	1	4	yes	—
74	C₂×C₄×C₄₃	#41	1	4	yes	—
75	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
76	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
77	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
78	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
79	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
80	C₂×Dic₄₃	—	2	4	no	—
81	C₂×Dic₄₃	—	2	4	no	—
82	C₂×Dic₄₃	C₂⁴	1	4	yes	—
83	C₂×Dic₄₃	C₂⁴	1	4	yes	—
84	D₁₇₂	—	2	16	no	—
85	D₁₇₂	—	2	16	no	—
86	D₁₇₂	—	4	8	no	—
87	C₄×D₄₃	—	4	16	no	—
88	Dic₈₆	—	2	8	no	—
89	C₂²×C₄₃	C₂×#41	1	2	yes	—
90	C₂²×C₄₃	C₂×#41	1	2	yes	—
91	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
92	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
93	C₂×D₄₃	—	2	8	no	—
94	C₂×D₄₃	—	2	8	no	—
95	C₂×D₄₃	—	2	8	no	—
96	C₂×D₄₃	—	4	16	no	—
97	C₄×C₄₃	—	2	4	no	—
98	C₄×C₄₃	—	2	4	no	—
99	C₄×C₄₃	—	2	2	no	—
100	C₄×C₄₃	C₂×#41	1	4	yes	—
101	C₄×C₄₃	C₂×#41	1	4	yes	—
102	Dic₄₃	—	2	4	no	—
103	Dic₄₃	—	2	4	no	—
104	C₂×#559	—	43	1	no	maximal, Sylow
105	C₂×C₄₃	#559	1	4	yes	complemented
106	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
107	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
108	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
109	D₄₃	—	4	8	no	—
110	#559	—	43	16	no	—
111	C₂×#127	—	43	1	no	—
112	C₂×#124	—	43	2	no	—
113	C₂×#124	—	43	2	no	—
114	C₂×#123	—	43	4	no	—
115	C₂×C₄×Q₈	—	43	1	no	—
116	C₂×C₄×Q₈	D₄₃	1	1	yes	characteristic, complemented
117	C₂×C₄×D₄	—	43	2	no	—
118	C₂×C₄×D₄	—	43	2	no	—
119	C₄₃	C₂×#559	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	43	1	no	—
121	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
122	C₂²×D₄	—	43	2	no	—
123	C₂²×D₄	—	43	2	no	—
124	C₂²×D₄	—	43	2	no	—
125	C₂³×C₄	—	43	4	no	—
126	#127	—	43	8	no	—
127	#124	—	43	16	no	—
128	#124	—	43	16	no	—
129	#123	—	43	32	no	—
130	C₄×Q₈	—	43	8	no	—
131	C₄×Q₈	C₂×D₄₃	1	8	yes	complemented
132	C₄×D₄	—	43	16	no	—
133	C₄×D₄	—	43	16	no	—
134	C₂×C₄⋊C₄	—	43	1	no	—
135	C₂×C₄⋊C₄	—	43	2	no	—
136	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
138	C₂×C₂²⋊C₄	—	43	4	no	—
139	C₂×C₂²⋊C₄	—	43	4	no	—
140	C₂×C₂²⋊C₄	—	43	4	no	—
141	C₂×C₄²	—	43	2	no	—
142	C₂×C₄²	C₂×D₄₃	1	2	yes	—
143	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
144	C₂⁴	—	43	4	no	—
145	C₂×Q₈	—	2	4	no	—
146	C₂×Q₈	—	43	4	no	—
147	C₂×Q₈	—	86	4	no	—
148	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
149	C₂×D₄	—	43	8	no	—
150	C₂×D₄	—	43	8	no	—
151	C₂×D₄	—	43	8	no	—
152	C₂×D₄	—	86	8	no	—
153	C₂×D₄	—	86	8	no	—
154	C₂×D₄	—	172	4	no	—
155	C₂²×C₄	—	43	2	no	—
156	C₂²×C₄	—	43	2	no	—
157	C₂²×C₄	—	43	16	no	—
158	C₂²×C₄	—	172	8	no	—
159	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
161	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
164	C₄⋊C₄	—	43	4	no	—
165	C₄⋊C₄	—	43	8	no	—
166	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
167	C₄⋊C₄	C₂²×D₄₃	1	8	yes	—
168	C₂²⋊C₄	—	43	16	no	—
169	C₂²⋊C₄	—	43	16	no	—
170	C₂²⋊C₄	—	43	16	no	—
171	C₄²	—	43	8	no	—
172	C₄²	C₂²×D₄₃	1	8	yes	—
173	C₄²	C₂²×D₄₃	1	4	yes	—
174	C₂³	—	43	8	no	—
175	C₂³	—	86	8	no	—
176	C₂³	—	86	8	no	—
177	C₂³	—	86	8	no	—
178	C₂³	C₂³×D₄₃	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	86	8	no	—
181	D₄	—	86	16	no	—
182	D₄	—	86	16	no	—
183	D₄	—	172	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	43	4	no	—
188	C₂×C₄	—	43	4	no	—
189	C₂×C₄	—	86	4	no	—
190	C₂×C₄	—	86	4	no	—
191	C₂×C₄	—	172	16	no	—
192	C₂×C₄	#10621898	1	4	yes	—
193	C₂×C₄	#10621902	1	4	yes	—
194	C₂×C₄	C₂³×D₄₃	1	4	yes	—
195	C₂×C₄	C₂³×D₄₃	1	4	yes	—
196	C₂×C₄	C₂³×D₄₃	1	2	yes	—
197	C₂×C₄	C₂³×D₄₃	1	2	yes	—
198	C₂×C₄	C₂³×D₄₃	1	2	yes	—
199	C₂²	—	86	8	no	—
200	C₂²	—	86	8	no	—
201	C₂²	—	86	8	no	—
202	C₂²	—	172	16	no	—
203	C₂²	C₂×#10621898	1	2	yes	—
204	C₂²	C₂×#10621902	1	2	yes	—
205	C₂²	#13063371	1	2	yes	—
206	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	86	4	no	—
211	C₄	—	86	4	no	—
212	C₄	C₂×#10621898	1	4	yes	—
213	C₄	C₂×#10621902	1	4	yes	—
214	C₂	—	172	8	no	—
215	C₂	#14583634	1	4	yes	direct product factor
216	C₂	C₂²×#10621898	1	1	yes	characteristic
217	C₂	C₂²×#10621902	1	1	yes	characteristic
218	C₂	C₂×#13063371	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×#559
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. #14583634	C₂	direct (N × H)
3. C₂×#13063314	C₂	semidirect
4. C₂×#13063313	C₂	semidirect
5. C₂×#13063310	C₂	semidirect
6. C₂×C₄×Dic₈₆	C₂	semidirect
7. C₂×C₄×D₁₇₂	C₂	semidirect
8. C₂×#13063287	C₂	semidirect
9. C₂×C₄×Q₈×C₄₃	C₂	semidirect
16. #13063314	C₂²	semidirect
17. #13063313	C₂²	semidirect
18. #13063310	C₂²	semidirect
19. C₄×Dic₈₆	C₂²	semidirect
20. C₄×D₁₇₂	C₂²	semidirect
21. #13063287	C₂²	semidirect
29. C₄×Q₈×C₄₃	C₂²	semidirect
105. C₂×C₄₃	#559	semidirect
116. C₂×C₄×Q₈	D₄₃	semidirect
119. C₄₃	C₂×#559	semidirect
131. C₄×Q₈	C₂×D₄₃	semidirect
215. C₂	#14583634	direct (N × H)

Subgroups — C2×#14583634 #19148183

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#14583634 #19148183