Subgroups — C₂×#19128501 #24922210

Order 10624.

1484 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13050590	C₂	1	2	yes	maximal, complemented
2	#19128501	C₂	1	16	yes	maximal, direct product factor
3	C₂×#14581825	C₂	1	4	yes	maximal, complemented
4	C₂×#14581820	C₂	1	4	yes	maximal, complemented
5	C₂×#14581814	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13050590	—	2	2	no	—
8	C₂×#13050590	—	2	2	no	—
9	C₂×#13050590	—	2	8	no	—
10	C₂×#13050590	C₂²	1	4	yes	—
11	C₂×#13050590	C₂²	1	4	yes	complemented
12	C₂×#13050590	C₂²	1	16	yes	complemented
13	C₂²×Dic₁₆₆	C₂²	1	2	yes	complemented
14	C₂²×D₃₃₂	C₂²	1	2	yes	complemented
15	C₂²×C₂²⋊D₈₃	—	2	2	no	—
16	C₂²×C₄×D₈₃	—	2	2	no	—
17	#14581825	C₂²	1	32	yes	complemented
18	#14581820	C₂²	1	32	yes	complemented
19	#14581814	C₂²	1	32	yes	complemented
20	C₂×Dic₃₃₂	C₂²	1	4	yes	complemented
21	C₂×#13050579	C₂²	1	8	yes	complemented
22	C₂×D₆₆₄	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₈₃	C₂²	1	4	yes	complemented
25	C₂×M₁₆×C₈₃	C₂²	1	8	yes	complemented
26	C₂²×C₈×C₈₃	C₂²	1	2	yes	complemented
27	C₂³×D₈₃	—	2	2	no	—
28	#13050590	—	2	8	no	—
29	#13050590	—	2	8	no	—
30	#13050590	—	2	16	no	—
31	#13050590	—	2	16	no	—
32	#13050590	C₂³	1	16	yes	—
33	#13050590	C₂³	1	16	yes	complemented
34	C₂×Dic₁₆₆	—	2	4	no	—
35	C₂×Dic₁₆₆	—	2	2	no	—
36	C₂×Dic₁₆₆	C₂³	1	4	yes	—
37	C₂×Dic₁₆₆	C₂³	1	8	yes	complemented
38	C₂×D₃₃₂	—	2	4	no	—
39	C₂×D₃₃₂	—	2	2	no	—
40	C₂×D₃₃₂	C₂³	1	4	yes	—
41	C₂×D₃₃₂	C₂³	1	8	yes	complemented
42	C₂×C₂²⋊D₈₃	—	2	4	no	—
43	C₂×C₂²⋊D₈₃	—	2	4	no	—
44	C₂×C₂²⋊D₈₃	—	2	16	no	—
45	C₂²×Dic₈₃	—	2	2	no	—
46	C₂×C₄×D₈₃	—	2	4	no	—
47	C₂×C₄×D₈₃	—	2	16	no	—
48	C₂×C₄×D₈₃	—	2	4	no	—
49	Dic₃₃₂	C₂³	1	16	yes	complemented
50	#13050579	C₂³	1	32	yes	complemented
51	D₆₆₄	C₂³	1	16	yes	complemented
52	C₂⁴×C₈₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	D₄	1	2	yes	—
54	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₃	D₄	1	4	yes	—
56	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₈₃	C₂³	1	4	yes	—
59	M₁₆×C₈₃	C₂³	1	16	yes	complemented
60	C₂×C₈×C₈₃	C₂³	1	4	yes	—
61	C₂×C₈×C₈₃	C₂³	1	8	yes	complemented
62	C₂³×C₈₃	—	4	2	no	—
63	C₂³×C₈₃	C₂×D₄	1	2	yes	—
64	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₈₃	C₂×D₄	1	4	yes	—
66	C₂²×D₈₃	—	2	4	no	—
67	C₂²×D₈₃	—	2	8	no	—
68	C₂²×D₈₃	—	4	8	no	—
69	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₈₃	C₂×D₄	1	8	yes	—
71	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₈₃	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
78	C₂²⋊D₈₃	—	2	16	no	—
79	C₂²⋊D₈₃	—	2	16	no	—
80	C₂×Dic₈₃	—	2	4	no	—
81	C₂×Dic₈₃	—	2	8	no	—
82	D₃₃₂	—	2	4	no	—
83	D₃₃₂	—	2	8	no	—
84	D₃₃₂	C₂⁴	1	8	yes	—
85	C₄×D₈₃	—	2	16	no	—
86	C₄×D₈₃	—	2	16	no	—
87	Dic₁₆₆	—	2	4	no	—
88	Dic₁₆₆	—	2	8	no	—
89	Dic₁₆₆	C₂⁴	1	8	yes	—
90	C₈×C₈₃	C₂⁴	1	8	yes	—
91	C₂²×C₈₃	—	2	4	no	—
92	C₂²×C₈₃	—	2	2	no	—
93	C₂²×C₈₃	—	4	4	no	—
94	C₂²×C₈₃	C₂²×D₄	1	4	yes	—
95	C₂²×C₈₃	C₂²×D₄	1	2	yes	—
96	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₈₃	—	2	8	no	—
98	C₂×D₈₃	—	4	16	no	—
99	C₂×D₈₃	—	4	8	no	—
100	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
101	C₄×C₈₃	C₂²×D₄	1	4	yes	—
102	C₄×C₈₃	C₂²×D₄	1	2	yes	—
103	C₄×C₈₃	C₂⁵	1	1	yes	characteristic
104	Dic₈₃	—	2	8	no	—
105	C₂×C₈₃	—	2	4	no	—
106	C₂×C₈₃	—	2	2	no	—
107	C₂×C₈₃	#575	1	2	yes	complemented
108	C₂×C₈₃	C₂³×D₄	1	1	yes	characteristic
109	D₈₃	—	4	8	no	—
110	C₂×#575	—	83	1	no	maximal, Sylow
111	C₈₃	C₂×#575	1	1	yes	characteristic, complemented, Sylow
112	C₂²×#41	—	83	2	no	—
113	#575	—	83	16	no	—
114	C₂×#137	—	83	4	no	—
115	C₂×AGL(1,ℤ/8)	—	83	4	no	—
116	C₂×#135	—	83	4	no	—
117	C₂²×M₁₆	D₈₃	1	1	yes	characteristic, complemented
118	C₂×#41	—	83	4	no	—
119	C₂×#41	—	83	4	no	—
120	C₂×#41	—	83	16	no	—
121	C₂×#41	—	166	2	no	—
122	C₂×#41	—	166	8	no	—
123	C₂×#41	—	166	2	no	—
124	C₂²×Q₈	—	83	2	no	—
125	C₂²×D₄	—	83	2	no	—
126	C₂²×D₄	—	166	2	no	—
127	C₂³×C₄	—	166	2	no	—
128	C₂³×C₄	C₂×D₈₃	1	1	yes	characteristic
129	#137	—	83	32	no	—
130	AGL(1,ℤ/8)	—	83	32	no	—
131	#135	—	83	32	no	—
132	C₂×Q₁₆	—	83	4	no	—
133	C₂×SD₁₆	—	83	8	no	—
134	C₂×D₈	—	83	4	no	—
135	C₂×M₁₆	C₂×D₈₃	1	4	yes	complemented
136	C₂×M₁₆	C₂×D₈₃	1	8	yes	complemented
137	C₂²×C₈	C₂×D₈₃	1	2	yes	complemented
138	C₂⁴	—	166	2	no	—
139	C₂⁴	D₃₃₂	1	1	yes	characteristic
140	#41	—	83	16	no	—
141	#41	—	83	16	no	—
142	#41	—	166	16	no	—
143	#41	—	166	8	no	—
144	#41	—	166	8	no	—
145	#41	—	166	16	no	—
146	C₂×Q₈	—	83	4	no	—
147	C₂×Q₈	—	83	8	no	—
148	C₂×Q₈	—	166	2	no	—
149	C₂×Q₈	—	166	4	no	—
150	C₂×D₄	—	83	4	no	—
151	C₂×D₄	—	83	8	no	—
152	C₂×D₄	—	166	4	no	—
153	C₂×D₄	—	166	4	no	—
154	C₂×D₄	—	166	16	no	—
155	C₂×D₄	—	166	2	no	—
156	C₂×D₄	—	166	4	no	—
157	C₂²×C₄	—	166	2	no	—
158	C₂²×C₄	—	166	4	no	—
159	C₂²×C₄	—	166	16	no	—
160	C₂²×C₄	—	166	4	no	—
161	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
162	C₂²×C₄	D₃₃₂	1	4	yes	—
163	C₂²×C₄	D₃₃₂	1	2	yes	—
164	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
165	C₂²×C₄	C₂²×D₈₃	1	4	yes	—
166	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
167	Q₁₆	—	83	16	no	—
168	SD₁₆	—	83	32	no	—
169	D₈	—	83	16	no	—
170	M₁₆	C₂²×D₈₃	1	16	yes	complemented
171	C₂×C₈	C₂²×D₈₃	1	4	yes	—
172	C₂×C₈	C₂²×D₈₃	1	8	yes	complemented
173	C₂³	—	4	2	no	—
174	C₂³	—	332	8	no	—
175	C₂³	—	166	8	no	—
176	C₂³	—	166	4	no	—
177	C₂³	C₂×D₃₃₂	1	2	yes	—
178	C₂³	C₂×D₃₃₂	1	1	yes	characteristic
179	C₂³	C₂×D₃₃₂	1	4	yes	—
180	Q₈	—	83	8	no	—
181	Q₈	—	166	8	no	—
182	Q₈	—	166	4	no	—
183	D₄	—	83	8	no	—
184	D₄	—	166	16	no	—
185	D₄	—	166	16	no	—
186	D₄	—	166	8	no	—
187	D₄	—	166	4	no	—
188	C₂×C₄	—	166	8	no	—
189	C₂×C₄	—	166	4	no	—
190	C₂×C₄	—	166	16	no	—
191	C₂×C₄	—	166	16	no	—
192	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
193	C₂×C₄	C₂×D₃₃₂	1	8	yes	—
194	C₂×C₄	C₂×D₃₃₂	1	4	yes	—
195	C₂×C₄	C₂×D₃₃₂	1	2	yes	—
196	C₂×C₄	C₂×D₃₃₂	1	4	yes	—
197	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
198	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
199	C₂×C₄	C₂³×D₈₃	1	2	yes	—
200	C₂×C₄	C₂³×D₈₃	1	4	yes	—
201	C₈	C₂³×D₈₃	1	8	yes	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	332	16	no	—
206	C₂²	—	332	8	no	—
207	C₂²	—	166	8	no	—
208	C₂²	C₂²×D₃₃₂	1	4	yes	—
209	C₂²	C₂²×D₃₃₂	1	2	yes	—
210	C₂²	C₂²×D₃₃₂	1	1	yes	characteristic
211	C₄	—	166	8	no	—
212	C₄	C₂²×D₃₃₂	1	2	yes	—
213	C₄	C₂²×D₃₃₂	1	4	yes	—
214	C₄	C₂²×D₃₃₂	1	1	yes	characteristic
215	C₄	C₂⁴×D₈₃	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	—	2	2	no	—
218	C₂	—	332	8	no	—
219	C₂	#19128501	1	2	yes	direct product factor
220	C₂	C₂³×D₃₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₃₂
commutator, cosocle	C₄×C₈₃	C₂⁵
Frattini	C₄	C₂⁴×D₈₃
Fitting	C₂²×M₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	C₂²×M₁₆	D₈₃
hypocenter	C₈₃	C₂×#575
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13050590	C₂	semidirect
2. #19128501	C₂	direct (N × H)
3. C₂×#14581825	C₂	semidirect
4. C₂×#14581820	C₂	semidirect
5. C₂×#14581814	C₂	semidirect
6. C₂²×M₁₆×C₈₃	C₂	semidirect
11. C₂×#13050590	C₂²	semidirect
12. C₂×#13050590	C₂²	semidirect
13. C₂²×Dic₁₆₆	C₂²	semidirect
14. C₂²×D₃₃₂	C₂²	semidirect
17. #14581825	C₂²	semidirect
18. #14581820	C₂²	semidirect
19. #14581814	C₂²	semidirect
20. C₂×Dic₃₃₂	C₂²	semidirect
21. C₂×#13050579	C₂²	semidirect
22. C₂×D₆₆₄	C₂²	semidirect
24. C₂×M₁₆×C₈₃	C₂²	semidirect
25. C₂×M₁₆×C₈₃	C₂²	semidirect
26. C₂²×C₈×C₈₃	C₂²	semidirect
33. #13050590	C₂³	semidirect
37. C₂×Dic₁₆₆	C₂³	semidirect
41. C₂×D₃₃₂	C₂³	semidirect
49. Dic₃₃₂	C₂³	semidirect
50. #13050579	C₂³	semidirect
51. D₆₆₄	C₂³	semidirect
59. M₁₆×C₈₃	C₂³	semidirect
61. C₂×C₈×C₈₃	C₂³	semidirect
107. C₂×C₈₃	#575	semidirect
111. C₈₃	C₂×#575	semidirect
117. C₂²×M₁₆	D₈₃	semidirect
135. C₂×M₁₆	C₂×D₈₃	semidirect
136. C₂×M₁₆	C₂×D₈₃	semidirect
137. C₂²×C₈	C₂×D₈₃	semidirect
170. M₁₆	C₂²×D₈₃	semidirect
172. C₂×C₈	C₂²×D₈₃	semidirect
219. C₂	#19128501	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13050590	2
#19128501	2
C₂×#14581825	2
C₂×#14581820	2
C₂×#14581814	2
C₂²×M₁₆×C₈₃	2
C₂×#575	83

Subgroups — C2×#19128501 #24922210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19128501 #24922210