Subgroups — C₂×#24909737 #35559880

Order 21248.

558 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128105	C₂	1	2	yes	maximal
2	#24909737	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19127527	C₂	1	1	yes	maximal, characteristic
4	C₂×#19127526	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#366×C₈₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14581791	—	2	2	no	—
8	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
9	#19128105	—	2	4	no	—
10	#19128105	C₂²	1	8	yes	—
11	C₂×#14581726	C₂²	1	2	yes	—
12	C₂×#14581725	—	2	2	no	—
13	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
14	#19127527	C₂²	1	4	yes	—
15	#19127526	C₂²	1	8	yes	—
16	C₁₆×Dic₈₃	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×#107×C₈₃	C₂²	1	2	yes	—
19	#366×C₈₃	C₂²	1	4	yes	—
20	C₂²×Dic₁₆₆	—	4	2	no	—
21	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
22	#14581791	—	2	8	no	—
23	#14581791	—	4	4	no	—
24	C₂×C₄⋊Dic₈₃	—	2	2	no	—
25	C₂×Dic₈₃⋊C₄	—	4	2	no	—
26	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₁₆	Q₈	1	4	yes	complemented
28	C₂×C₈₃⋊C₁₆	C₂³	1	2	yes	—
29	#14581726	—	2	2	no	—
30	#14581726	C₂³	1	4	yes	—
31	#14581725	—	2	8	no	—
32	C₈×Dic₈₃	D₄	1	2	yes	—
33	C₈×Dic₈₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
36	C₂×C₁₆×C₈₃	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
38	#107×C₈₃	—	2	2	no	—
39	#107×C₈₃	C₂³	1	4	yes	—
40	C₂×Dic₁₆₆	—	4	8	no	—
41	C₂×Dic₁₆₆	—	8	8	no	—
42	C₂²×Dic₈₃	—	4	2	no	—
43	C₂²×Dic₈₃	D₈	1	1	yes	characteristic
44	C₂²×Dic₈₃	D₈	1	1	yes	characteristic
45	C₂×C₈₃⋊C₈	Q₁₆	1	4	yes	—
46	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
47	C₄⋊Dic₈₃	—	2	4	no	—
48	C₄⋊Dic₈₃	—	4	2	no	—
49	Dic₈₃⋊C₄	—	4	8	no	—
50	C₄×Dic₈₃	D₈	1	2	yes	—
51	C₄×Dic₈₃	C₂×D₄	1	2	yes	—
52	C₈₃⋊C₁₆	C₂×Q₈	1	4	yes	complemented
53	C₂²×C₄×C₈₃	—	4	2	no	—
54	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₈₃	—	2	4	no	—
60	C₄⋊C₄×C₈₃	—	4	2	no	—
61	C₁₆×C₈₃	C₂×Q₈	1	4	yes	complemented
62	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₈₃	—	4	4	no	—
64	C₂×C₄×C₈₃	—	8	4	no	—
65	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
69	C₂×Dic₈₃	—	4	4	no	—
70	C₂×Dic₈₃	—	8	4	no	—
71	C₂×Dic₈₃	D₁₆	1	4	yes	—
72	C₂×Dic₈₃	Q₃₂	1	4	yes	—
73	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
74	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
75	Dic₁₆₆	—	8	16	no	—
76	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
77	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
79	C₈₃⋊C₈	C₂×Q₁₆	1	4	yes	—
80	C₂²×C₈₃	#500	1	2	yes	—
81	C₂²×C₈₃	C₂×D₁₆	1	2	yes	—
82	C₂²×C₈₃	C₂×Q₃₂	1	2	yes	—
83	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
84	C₄×C₈₃	—	8	4	no	—
85	C₄×C₈₃	#500	1	2	yes	—
86	C₄×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
88	Dic₈₃	—	8	4	no	—
89	Dic₈₃	C₂×D₁₆	1	4	yes	—
90	Dic₈₃	C₂×Q₃₂	1	4	yes	—
91	C₂×#2252	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	#2252	1	4	yes	complemented
93	C₂×C₈₃	C₂×#500	1	1	yes	characteristic
94	C₂×C₈₃	C₂²×D₁₆	1	1	yes	characteristic
95	C₂×C₈₃	C₂²×Q₃₂	1	1	yes	characteristic
96	C₂×#500	—	83	2	no	—
97	#2252	—	83	8	no	—
98	C₂×#366	—	83	1	no	—
99	C₂×#366	—	83	2	no	—
100	C₂×#366	D₈₃	1	1	yes	characteristic
101	C₂×C₄×C₁₆	—	83	1	no	—
102	C₈₃	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	166	2	no	—
104	C₂²×C₁₆	—	83	1	no	—
105	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
106	#500	—	83	8	no	—
107	#500	—	166	4	no	—
108	C₂×#107	—	83	2	no	—
109	C₂×#107	—	166	2	no	—
110	C₂×#107	C₂×D₈₃	1	2	yes	—
111	C₂×C₄×C₈	—	83	1	no	—
112	#366	—	83	4	no	—
113	#366	—	83	8	no	—
114	#366	C₂×D₈₃	1	4	yes	—
115	C₄×C₁₆	—	83	4	no	—
116	C₂²×Q₈	—	332	2	no	—
117	C₂²×C₈	—	83	1	no	—
118	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
119	C₄⋊Q₈	—	332	4	no	—
120	C₄⋊Q₈	—	166	8	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	332	2	no	—
123	C₂×C₄⋊C₄	—	166	2	no	—
124	C₂×C₄²	—	83	1	no	—
125	C₂×C₁₆	—	83	4	no	—
126	C₂×C₁₆	—	83	2	no	—
127	C₂×C₁₆	Dic₁₆₆	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
129	#107	—	2	2	no	—
130	#107	—	83	4	no	—
131	#107	—	166	8	no	—
132	#107	—	166	2	no	—
133	#107	C₂²×D₈₃	1	4	yes	—
134	C₄×C₈	—	83	2	no	—
135	C₄×C₈	—	83	2	no	—
136	C₂×Q₈	—	332	8	no	—
137	C₂×Q₈	—	664	8	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	332	2	no	—
140	C₂²×C₄	—	83	1	no	—
141	C₂²×C₄	—	83	1	no	—
142	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
143	C₂×C₈	—	83	4	no	—
144	C₂×C₈	—	83	2	no	—
145	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₈₃	1	2	yes	—
147	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
148	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	332	8	no	—
152	C₄⋊C₄	—	332	2	no	—
153	C₄⋊C₄	—	166	4	no	—
154	C₄²	—	83	2	no	—
155	C₄²	—	83	2	no	—
156	C₁₆	—	83	4	no	—
157	C₁₆	C₂×Dic₁₆₆	1	4	yes	complemented
158	C₂³	D₈×D₈₃	1	1	yes	characteristic
159	Q₈	—	664	16	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	332	4	no	—
163	C₂×C₄	—	83	4	no	—
164	C₂×C₄	—	83	4	no	—
165	C₂×C₄	—	83	2	no	—
166	C₂×C₄	—	83	2	no	—
167	C₂×C₄	—	664	4	no	—
168	C₂×C₄	#14581791	1	2	yes	—
169	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
170	C₂×C₄	Q₁₆×D₈₃	1	2	yes	—
171	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
172	C₈	—	83	4	no	—
173	C₈	#14581791	1	2	yes	—
174	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₆₆	1	1	yes	characteristic
176	C₂²	#19128105	1	2	yes	—
177	C₂²	D₁₆×D₈₃	1	2	yes	—
178	C₂²	Q₃₂×D₈₃	1	2	yes	—
179	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
180	C₄	—	8	4	no	—
181	C₄	—	83	4	no	—
182	C₄	—	83	4	no	—
183	C₄	—	664	4	no	—
184	C₄	#19128105	1	2	yes	—
185	C₄	C₂×#14581791	1	1	yes	characteristic
186	C₄	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
187	C₂	#24909737	1	4	yes	direct product factor
188	C₂	C₂×#19128105	1	1	yes	characteristic
189	C₂	C₂×D₁₆×D₈₃	1	1	yes	characteristic
190	C₂	C₂×Q₃₂×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#366×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
hypercenter	C₂×#366	D₈₃
hypocenter	C₈₃	C₂×#2252
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #24909737	C₂	direct (N × H)
27. C₂×C₈₃⋊C₁₆	Q₈	semidirect
36. C₂×C₁₆×C₈₃	Q₈	semidirect
52. C₈₃⋊C₁₆	C₂×Q₈	semidirect
61. C₁₆×C₈₃	C₂×Q₈	semidirect
92. C₂×C₈₃	#2252	semidirect
102. C₈₃	C₂×#2252	semidirect
127. C₂×C₁₆	Dic₁₆₆	semidirect
157. C₁₆	C₂×Dic₁₆₆	semidirect
187. C₂	#24909737	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19128105	2
#24909737	2
C₂×#19127527	2
C₂×#19127526	2
C₂×C₁₆×Dic₈₃	2
C₂×#366×C₈₃	2
C₂×#2252	83

Subgroups — C2×#24909737 #35559880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909737 #35559880