Subgroups — C₂×#26022526 #36709904

Order 22784.

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×#19152002	C₂	1	2	yes	maximal
2	#26022526	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151424	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151423	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#15668011	—	2	2	no	—
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	#19152002	—	2	4	no	—
10	#19152002	C₂²	1	8	yes	—
11	C₂×#15667946	C₂²	1	2	yes	—
12	C₂×#15667945	—	2	2	no	—
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151424	C₂²	1	4	yes	—
15	#19151423	C₂²	1	8	yes	—
16	C₁₆×Dic₈₉	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#107×C₈₉	C₂²	1	2	yes	—
19	#366×C₈₉	C₂²	1	4	yes	—
20	C₂²×Dic₁₇₈	—	4	2	no	—
21	C₂²×#10623494	D₄	1	1	yes	characteristic
22	#15668011	—	2	8	no	—
23	#15668011	—	4	4	no	—
24	C₂×C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×#13083553	—	4	2	no	—
26	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
27	C₂×#13083549	Q₈	1	4	yes	complemented
28	C₂×#13083549	C₂³	1	2	yes	—
29	#15667946	—	2	2	no	—
30	#15667946	C₂³	1	4	yes	—
31	#15667945	—	2	8	no	—
32	C₈×Dic₈₉	D₄	1	2	yes	—
33	C₈×Dic₈₉	C₂³	1	2	yes	—
34	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
38	#107×C₈₉	—	2	2	no	—
39	#107×C₈₉	C₂³	1	4	yes	—
40	C₂×Dic₁₇₈	—	4	8	no	—
41	C₂×Dic₁₇₈	—	8	8	no	—
42	C₂²×Dic₈₉	—	4	2	no	—
43	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
44	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
45	C₂×#10623494	Q₁₆	1	4	yes	—
46	C₂×#10623494	C₂×D₄	1	2	yes	—
47	C₄⋊Dic₈₉	—	2	4	no	—
48	C₄⋊Dic₈₉	—	4	2	no	—
49	#13083553	—	4	8	no	—
50	C₄×Dic₈₉	D₈	1	2	yes	—
51	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
52	#13083549	C₂×Q₈	1	4	yes	complemented
53	C₂²×C₄×C₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₈₉	—	2	4	no	—
60	C₄⋊C₄×C₈₉	—	4	2	no	—
61	C₁₆×C₈₉	C₂×Q₈	1	4	yes	complemented
62	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₈₉	—	4	4	no	—
64	C₂×C₄×C₈₉	—	8	4	no	—
65	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
66	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
68	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
69	C₂×Dic₈₉	—	4	4	no	—
70	C₂×Dic₈₉	—	8	4	no	—
71	C₂×Dic₈₉	D₁₆	1	4	yes	—
72	C₂×Dic₈₉	Q₃₂	1	4	yes	—
73	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
74	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
75	Dic₁₇₈	—	8	16	no	—
76	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
77	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
79	#10623494	C₂×Q₁₆	1	4	yes	—
80	C₂²×C₈₉	#500	1	2	yes	—
81	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
82	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
83	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
84	C₄×C₈₉	—	8	4	no	—
85	C₄×C₈₉	#500	1	2	yes	—
86	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
88	Dic₈₉	—	8	4	no	—
89	Dic₈₉	C₂×D₁₆	1	4	yes	—
90	Dic₈₉	C₂×Q₃₂	1	4	yes	—
91	C₂×#2252	—	89	1	no	maximal, Sylow
92	C₂×C₈₉	#2252	1	4	yes	complemented
93	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
94	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
95	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
96	C₂×#500	—	89	2	no	—
97	#2252	—	89	8	no	—
98	C₂×#366	—	89	1	no	—
99	C₂×#366	—	89	2	no	—
100	C₂×#366	D₈₉	1	1	yes	characteristic
101	C₂×C₄×C₁₆	—	89	1	no	—
102	C₈₉	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	178	2	no	—
104	C₂²×C₁₆	—	89	1	no	—
105	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
106	#500	—	89	8	no	—
107	#500	—	178	4	no	—
108	C₂×#107	—	89	2	no	—
109	C₂×#107	—	178	2	no	—
110	C₂×#107	C₂×D₈₉	1	2	yes	—
111	C₂×C₄×C₈	—	89	1	no	—
112	#366	—	89	4	no	—
113	#366	—	89	8	no	—
114	#366	C₂×D₈₉	1	4	yes	—
115	C₄×C₁₆	—	89	4	no	—
116	C₂²×Q₈	—	356	2	no	—
117	C₂²×C₈	—	89	1	no	—
118	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
119	C₄⋊Q₈	—	356	4	no	—
120	C₄⋊Q₈	—	178	8	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	356	2	no	—
123	C₂×C₄⋊C₄	—	178	2	no	—
124	C₂×C₄²	—	89	1	no	—
125	C₂×C₁₆	—	89	4	no	—
126	C₂×C₁₆	—	89	2	no	—
127	C₂×C₁₆	Dic₁₇₈	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
129	#107	—	2	2	no	—
130	#107	—	89	4	no	—
131	#107	—	178	8	no	—
132	#107	—	178	2	no	—
133	#107	C₂²×D₈₉	1	4	yes	—
134	C₄×C₈	—	89	2	no	—
135	C₄×C₈	—	89	2	no	—
136	C₂×Q₈	—	356	8	no	—
137	C₂×Q₈	—	712	8	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	89	1	no	—
140	C₂²×C₄	—	89	1	no	—
141	C₂²×C₄	—	356	2	no	—
142	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
143	C₂×C₈	—	89	4	no	—
144	C₂×C₈	—	89	2	no	—
145	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₈₉	1	2	yes	—
147	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
148	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	356	8	no	—
152	C₄⋊C₄	—	356	2	no	—
153	C₄⋊C₄	—	178	4	no	—
154	C₄²	—	89	2	no	—
155	C₄²	—	89	2	no	—
156	C₁₆	—	89	4	no	—
157	C₁₆	C₂×Dic₁₇₈	1	4	yes	complemented
158	C₂³	D₈×D₈₉	1	1	yes	characteristic
159	Q₈	—	712	16	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	89	4	no	—
163	C₂×C₄	—	89	4	no	—
164	C₂×C₄	—	89	2	no	—
165	C₂×C₄	—	89	2	no	—
166	C₂×C₄	—	356	4	no	—
167	C₂×C₄	—	712	4	no	—
168	C₂×C₄	#15668011	1	2	yes	—
169	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
170	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
171	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
172	C₈	—	89	4	no	—
173	C₈	#15668011	1	2	yes	—
174	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
175	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
176	C₂²	#19152002	1	2	yes	—
177	C₂²	D₁₆×D₈₉	1	2	yes	—
178	C₂²	Q₃₂×D₈₉	1	2	yes	—
179	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
180	C₄	—	8	4	no	—
181	C₄	—	89	4	no	—
182	C₄	—	89	4	no	—
183	C₄	—	712	4	no	—
184	C₄	#19152002	1	2	yes	—
185	C₄	C₂×#15668011	1	1	yes	characteristic
186	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
187	C₂	#26022526	1	4	yes	direct product factor
188	C₂	C₂×#19152002	1	1	yes	characteristic
189	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
190	C₂	C₂×Q₃₂×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	C₂×#2252
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
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. #26022526	C₂	direct (N × H)
27. C₂×#13083549	Q₈	semidirect
36. C₂×C₁₆×C₈₉	Q₈	semidirect
52. #13083549	C₂×Q₈	semidirect
61. C₁₆×C₈₉	C₂×Q₈	semidirect
92. C₂×C₈₉	#2252	semidirect
102. C₈₉	C₂×#2252	semidirect
127. C₂×C₁₆	Dic₁₇₈	semidirect
157. C₁₆	C₂×Dic₁₇₈	semidirect
187. C₂	#26022526	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×#19152002	2
#26022526	2
C₂×#19151424	2
C₂×#19151423	2
C₂×C₁₆×Dic₈₉	2
C₂×#366×C₈₉	2
C₂×#2252	89

Subgroups — C2×#26022526 #36709904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022526 #36709904