Subgroups — C₂×C₄×#10621898 #19147775

Order 5504.

1728 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10621898	C₂	1	1	yes	maximal, characteristic
2	C₄×#10621898	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₈₆	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₄₃	C₂	1	4	yes	maximal, complemented
6	C₂×#13063284	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13063282	C₂	1	2	yes	maximal, complemented
8	C₂×C₄²×D₄₃	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄²×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10621898	C₄	1	16	yes	direct product factor
11	C₂×#10621898	C₂²	1	8	yes	—
12	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
13	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₄₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₄₃	C₂²	1	4	yes	—
16	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
17	C₄×Dic₈₆	C₂²	1	16	yes	complemented
18	C₄×D₁₇₂	C₂²	1	16	yes	complemented
19	C₄×C₂²⋊D₄₃	C₂²	1	32	yes	complemented
20	#13063284	C₂²	1	8	yes	complemented
21	#13063282	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	—
23	C₂×Dic₄₃⋊C₄	C₂²	1	4	yes	—
24	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
25	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	—
26	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₄₃	C₂²	1	4	yes	—
28	C₄²×D₄₃	C₂²	1	16	yes	complemented
29	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
30	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₄₃	C₂²	1	2	yes	—
33	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	C₂²	1	8	yes	complemented
35	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
36	C₂³×D₄₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₄₃	—	2	8	no	—
38	C₂²×C₄×C₄₃	C₂×C₄	1	8	yes	complemented
39	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
41	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
44	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₄₃	C₂×C₄	1	16	yes	complemented
46	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
47	C₂²×Dic₄₃	C₂³	1	4	yes	—
48	C₂²×Dic₄₃	C₂³	1	2	yes	—
49	#10621898	C₂×C₄	1	64	yes	direct product factor
50	C₂×D₁₇₂	C₂×C₄	1	8	yes	complemented
51	C₂×D₁₇₂	C₂³	1	4	yes	—
52	C₂×C₄×D₄₃	—	2	16	no	—
53	C₂×C₄×D₄₃	C₂×C₄	1	16	yes	complemented
54	C₂×C₄×D₄₃	C₂³	1	16	yes	—
55	C₂×C₄×D₄₃	C₂³	1	8	yes	—
56	C₂×Dic₈₆	C₂×C₄	1	8	yes	complemented
57	C₂×Dic₈₆	C₂³	1	4	yes	—
58	C₄²×C₄₃	C₂³	1	4	yes	—
59	C₄²×C₄₃	C₂³	1	4	yes	—
60	C₄²×C₄₃	C₂³	1	8	yes	—
61	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
62	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
63	C₄⋊Dic₄₃	C₂³	1	8	yes	—
64	Dic₄₃⋊C₄	C₂³	1	16	yes	—
65	C₄×Dic₄₃	C₂³	1	8	yes	—
66	C₄×Dic₄₃	C₂³	1	16	yes	—
67	C₂³×C₄₃	—	2	4	no	—
68	C₂³×C₄₃	C₂²×C₄	1	4	yes	—
69	C₂³×C₄₃	C₂⁴	1	2	yes	—
70	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
71	C₂²×D₄₃	—	2	8	no	—
72	C₂²×D₄₃	C₂²×C₄	1	8	yes	—
73	C₂²×D₄₃	C₂⁴	1	4	yes	—
74	C₂×C₄×C₄₃	—	2	4	no	—
75	C₂×C₄×C₄₃	—	2	16	no	—
76	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₄₃	C₂²×C₄	1	16	yes	complemented
79	C₂×C₄×C₄₃	#41	1	8	yes	complemented
80	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
81	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
82	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
84	C₂²⋊D₄₃	C₂²×C₄	1	32	yes	complemented
85	C₂×Dic₄₃	—	2	8	no	—
86	C₂×Dic₄₃	C₂²×C₄	1	8	yes	—
87	C₂×Dic₄₃	C₂⁴	1	8	yes	—
88	C₂×Dic₄₃	C₂⁴	1	4	yes	—
89	D₁₇₂	C₂²×C₄	1	16	yes	complemented
90	C₄×D₄₃	—	2	32	no	—
91	C₄×D₄₃	C₂²×C₄	1	32	yes	complemented
92	Dic₈₆	C₂²×C₄	1	16	yes	complemented
93	C₂²×C₄₃	—	2	8	no	—
94	C₂²×C₄₃	—	2	4	no	—
95	C₂²×C₄₃	C₂³×C₄	1	4	yes	—
96	C₂²×C₄₃	C₂³×C₄	1	2	yes	—
97	C₂²×C₄₃	C₂×#41	1	4	yes	—
98	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₄₃	—	2	16	no	—
100	C₂×D₄₃	—	2	8	no	—
101	C₂×D₄₃	C₂³×C₄	1	8	yes	—
102	C₄×C₄₃	—	2	4	no	—
103	C₄×C₄₃	C₂³×C₄	1	4	yes	—
104	C₄×C₄₃	C₂³×C₄	1	4	yes	—
105	C₄×C₄₃	C₂×#41	1	8	yes	complemented
106	Dic₄₃	—	2	8	no	—
107	Dic₄₃	C₂³×C₄	1	8	yes	—
108	C₂×C₄×#41	—	43	1	no	maximal, Sylow
109	C₂×C₄₃	—	2	4	no	—
110	C₂×C₄₃	C₄×#41	1	4	yes	complemented
111	C₂×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
112	C₂×C₄₃	C₂²×#41	1	2	yes	—
113	D₄₃	—	2	8	no	—
114	C₂²×#41	—	43	1	no	—
115	C₄×#41	—	43	16	no	—
116	C₂×C₄×Q₈	—	43	2	no	—
117	C₂×C₄×D₄	—	43	4	no	—
118	C₂×C₄×D₄	—	43	2	no	—
119	C₂×#117	—	43	1	no	—
120	C₂×#117	—	43	2	no	—
121	C₂²×C₄²	—	43	2	no	—
122	C₂²×C₄²	D₄₃	1	1	yes	characteristic, complemented
123	C₄₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	43	8	no	—
125	C₂×#41	—	43	16	no	—
126	C₂²×Q₈	—	43	1	no	—
127	C₂²×D₄	—	43	2	no	—
128	C₂²×D₄	—	43	1	no	—
129	C₂³×C₄	—	43	2	no	—
130	C₂³×C₄	—	43	4	no	—
131	C₂³×C₄	C₂×D₄₃	1	2	yes	—
132	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic
133	C₄×Q₈	—	43	16	no	—
134	C₄×D₄	—	43	32	no	—
135	C₄×D₄	—	43	16	no	—
136	#117	—	43	8	no	—
137	#117	—	43	16	no	—
138	C₂×C₄⋊C₄	—	43	4	no	—
139	C₂×C₄⋊C₄	—	43	2	no	—
140	C₂×C₂²⋊C₄	—	43	2	no	—
141	C₂×C₂²⋊C₄	—	43	4	no	—
142	C₂×C₄²	—	43	2	no	—
143	C₂×C₄²	—	43	4	no	—
144	C₂×C₄²	—	43	16	no	—
145	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
146	C₂×C₄²	C₂×D₄₃	1	2	yes	—
147	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×D₄₃	1	8	yes	complemented
149	C₂⁴	—	43	2	no	—
150	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
151	#41	—	43	64	no	—
152	C₂×Q₈	—	43	8	no	—
153	C₂×Q₈	—	43	4	no	—
154	C₂×D₄	—	43	16	no	—
155	C₂×D₄	—	43	8	no	—
156	C₂×D₄	—	43	8	no	—
157	C₂×D₄	—	43	4	no	—
158	C₂²×C₄	—	2	8	no	—
159	C₂²×C₄	—	43	2	no	—
160	C₂²×C₄	—	43	16	no	—
161	C₂²×C₄	—	43	8	no	—
162	C₂²×C₄	—	43	4	no	—
163	C₂²×C₄	—	43	16	no	—
164	C₂²×C₄	—	86	16	no	—
165	C₂²×C₄	C₄×D₄₃	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
168	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
171	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
172	C₄⋊C₄	—	43	16	no	—
173	C₄⋊C₄	—	43	8	no	—
174	C₂²⋊C₄	—	43	8	no	—
175	C₂²⋊C₄	—	43	16	no	—
176	C₄²	—	43	8	no	—
177	C₄²	—	43	16	no	—
178	C₄²	C₂²×D₄₃	1	4	yes	—
179	C₄²	C₂²×D₄₃	1	4	yes	—
180	C₄²	C₂²×D₄₃	1	8	yes	—
181	C₂³	—	2	4	no	—
182	C₂³	—	43	8	no	—
183	C₂³	—	43	4	no	—
184	C₂³	—	86	8	no	—
185	C₂³	C₂×C₄×D₄₃	1	4	yes	—
186	C₂³	C₂³×D₄₃	1	2	yes	—
187	C₂³	C₂³×D₄₃	1	1	yes	characteristic
188	Q₈	—	43	16	no	—
189	D₄	—	43	32	no	—
190	D₄	—	43	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	43	8	no	—
194	C₂×C₄	—	43	4	no	—
195	C₂×C₄	—	43	32	no	—
196	C₂×C₄	—	43	8	no	—
197	C₂×C₄	—	86	8	no	—
198	C₂×C₄	—	86	32	no	—
199	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
200	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
201	C₂×C₄	C₂×C₄×D₄₃	1	16	yes	complemented
202	C₂×C₄	#10621898	1	8	yes	direct product factor
203	C₂×C₄	C₂³×D₄₃	1	4	yes	—
204	C₂×C₄	C₂³×D₄₃	1	2	yes	—
205	C₂×C₄	C₂³×D₄₃	1	2	yes	—
206	C₂×C₄	C₂³×D₄₃	1	4	yes	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	43	8	no	—
210	C₂²	—	86	16	no	—
211	C₂²	—	86	8	no	—
212	C₂²	C₂²×C₄×D₄₃	1	4	yes	—
213	C₂²	C₂²×C₄×D₄₃	1	2	yes	—
214	C₂²	C₂×#10621898	1	4	yes	—
215	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	43	8	no	—
218	C₄	—	86	8	no	—
219	C₄	C₂²×C₄×D₄₃	1	4	yes	—
220	C₄	C₂²×C₄×D₄₃	1	4	yes	—
221	C₄	C₂×#10621898	1	8	yes	direct product factor
222	C₂	—	2	4	no	—
223	C₂	—	86	8	no	—
224	C₂	C₄×#10621898	1	4	yes	direct product factor
225	C₂	C₂³×C₄×D₄₃	1	1	yes	characteristic
226	C₂	C₂²×#10621898	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₄₃
commutator	C₂×C₄₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂²×C₄²×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂²×C₄²	D₄₃
hypocenter	C₄₃	C₂×C₄×#41
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#10621898	C₂	direct (N × H)
3. C₂×C₄×Dic₈₆	C₂	semidirect
4. C₂×C₄×D₁₇₂	C₂	semidirect
5. C₂×C₄×C₂²⋊D₄₃	C₂	semidirect
6. C₂×#13063284	C₂	semidirect
7. C₂×#13063282	C₂	semidirect
8. C₂×C₄²×D₄₃	C₂	semidirect
9. C₂²×C₄²×C₄₃	C₂	semidirect
10. C₂×#10621898	C₄	direct (N × H)
17. C₄×Dic₈₆	C₂²	semidirect
18. C₄×D₁₇₂	C₂²	semidirect
19. C₄×C₂²⋊D₄₃	C₂²	semidirect
20. #13063284	C₂²	semidirect
21. #13063282	C₂²	semidirect
28. C₄²×D₄₃	C₂²	semidirect
34. C₂×C₄²×C₄₃	C₂²	semidirect
38. C₂²×C₄×C₄₃	C₂×C₄	semidirect
45. C₂×C₂²⋊D₄₃	C₂×C₄	semidirect
49. #10621898	C₂×C₄	direct (N × H)
50. C₂×D₁₇₂	C₂×C₄	semidirect
53. C₂×C₄×D₄₃	C₂×C₄	semidirect
56. C₂×Dic₈₆	C₂×C₄	semidirect
78. C₂×C₄×C₄₃	C₂²×C₄	semidirect
79. C₂×C₄×C₄₃	#41	semidirect
84. C₂²⋊D₄₃	C₂²×C₄	semidirect
89. D₁₇₂	C₂²×C₄	semidirect
91. C₄×D₄₃	C₂²×C₄	semidirect
92. Dic₈₆	C₂²×C₄	semidirect
105. C₄×C₄₃	C₂×#41	semidirect
110. C₂×C₄₃	C₄×#41	semidirect
122. C₂²×C₄²	D₄₃	semidirect
123. C₄₃	C₂×C₄×#41	semidirect
148. C₂×C₄²	C₂×D₄₃	semidirect
165. C₂²×C₄	C₄×D₄₃	semidirect
201. C₂×C₄	C₂×C₄×D₄₃	semidirect
202. C₂×C₄	#10621898	direct (N × H)
221. C₄	C₂×#10621898	direct (N × H)
224. C₂	C₄×#10621898	direct (N × H)

Subgroups — C2×C4×#10621898 #19147775

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×#10621898 #19147775