Finite Group Databasebeta

← C2×#26012344 #36368931

Subgroups — C2×#26012344 #36368931

Order 22144.

336 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149736C211yesmaximal, characteristic
2C2×(C2×#10621924)⋊C4C211yesmaximal, characteristic
3#26012344C214yesmaximal, direct product factor
4C2×#19149659C211yesmaximal, characteristic
5C23×#10621924—21no—
6C22×#13063443—21no—
7#19149736—21no—
8#19149736C2212yes—
9(C2×#10621924)⋊C4—21no—
10(C2×#10621924)⋊C4C2212yes—
11C22×Dic346C411yescharacteristic, complemented
12C22×C4×D173C2211yescharacteristic
13#19149659—21no—
14#19149659—21no—
15#19149659C412yescomplemented
16#19149659C2212yes—
17C2×C4⋊Dic173—21no—
18C2×#14583803—21no—
19C2×C4⋊C4×C173C411yescharacteristic, complemented
20C22×#10621924—24no—
21C22×#10621924—22no—
22C22×#10621924—42no—
23C22×#10621924—42no—
24C2×#13063443—22no—
25C2×#13063443—42no—
26C23×D173D411yescharacteristic
27C2×Dic346—21no—
28C2×Dic346—42no—
29C2×Dic346C2×C412yescomplemented
30C22×Dic173—21no—
31C22×Dic173D411yescharacteristic
32C2×C4×D173—42no—
33C2×C4×D173C2×C411yescharacteristic
34C2×C4×D173D411yescharacteristic
35C2×C4×D173D411yescharacteristic
36C2×C4×D173C2311yescharacteristic
37C4⋊Dic173—22no—
38C4⋊Dic173—22no—
39#14583803—22no—
40#14583803—22no—
41C22×C4×C173—21no—
42C22×C4×C173C2×C411yescharacteristic
43C4⋊C4×C173—21no—
44C4⋊C4×C173C2×C412yescomplemented
45C2×#10621924—24no—
46C2×#10621924—48no—
47C2×#10621924—42no—
48C2×#10621924—42no—
49#13063443—42no—
50C22×D173—22no—
51C22×D173—22no—
52C22×D173C22⋊C411yescharacteristic
53C22×D173SD1612yes—
54C22×D173Q1612yes—
55C22×D173C2×D411yescharacteristic
56Dic346—42no—
57Dic346—42no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—42no—
61C2×Dic173C22⋊C411yescharacteristic
62C2×Dic173C2×D411yescharacteristic
63C4×D173—41no—
64C4×D173—41no—
65C4×D173—41no—
66C4×D173—41no—
67C23×C173C22⋊C411yescharacteristic
68C2×C4×C173—22no—
69C2×C4×C173—22no—
70C2×C4×C173—42no—
71C2×C4×C173C22⋊C411yescharacteristic
72C2×C4×C173C22×C411yescharacteristic
73C22×C173#9912yes—
74C22×C173#10312yes—
75C22×C173#10412yes—
76C22×C173C2×C22⋊C411yescharacteristic
77C2×D173—24no—
78C2×D173—24no—
79C2×D173—21no—
80C2×D173—21no—
81C2×D173—21no—
82C2×D173—21no—
83C2×D173#10312yes—
84C2×D173C2×SD1611yescharacteristic
85C2×D173C2×Q1611yescharacteristic
86#10621924—44no—
87C4×C173—21no—
88C4×C173—21no—
89C4×C173—42no—
90Dic173—21no—
91Dic173—21no—
92Dic173—42no—
93C2×C173#32814yescomplemented
94C2×C173C2×#9911yescharacteristic
95C2×C173C2×#10311yescharacteristic
96C2×C173C2×#10411yescharacteristic
97D173—21no—
98D173—22no—
99D173—21no—
100C173C2×#32811yescharacteristic, complemented, Sylow
101C2×#328—1731nomaximal, Sylow
102C2×C22⋊Q8—1731no—
103C2×C22⋊C8—1731no—
104C2×#95—1731no—
105#328—1734no—
106C22×Q8—1731no—
107C23×C4—3461no—
108C23×C4—1731no—
109C22×C8—3461no—
110C22⋊Q8—3461no—
111C22⋊Q8—3461no—
112C22⋊Q8—1732no—
113C22⋊Q8—1732no—
114C2×C4⋊C4—3461no—
115C2×C4⋊C4#1062192411yescharacteristic, complemented
116C2×C22⋊C4—3461no—
117C22⋊C8—3461no—
118C22⋊C8—1732no—
119#95—3461no—
120#95—1732no—
121C24—1731no—
122C2×Q8—3461no—
123C2×Q8—1732no—
124C2×Q8—6922no—
125C22×C4—21no—
126C22×C4—3464no—
127C22×C4—3462no—
128C22×C4—3461no—
129C22×C4—1731no—
130C22×C4—1731no—
131C22×C4—1731no—
132C22×C4—1731no—
133C22×C4—1731no—
134C22×C4—6922no—
135C22×C4—6922no—
136C22×C4—6922no—
137C22×C4C2×#1062192411yescharacteristic
138C2×C8—3462no—
139C2×C8—6922no—
140C4⋊C4—21no—
141C4⋊C4—3462no—
142C4⋊C4—3462no—
143C4⋊C4C2×#1062192412yescomplemented
144C22⋊C4—3462no—
145C22⋊C4—3462no—
146C23—3462no—
147C23—3462no—
148C23—1732no—
149C23—1732no—
150C23—1731no—
151C23—1731no—
152C23C22⋊#1062192411yescharacteristic
153Q8—6922no—
154Q8—6922no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—3462no—
159C2×C4—3464no—
160C2×C4—3462no—
161C2×C4—1731no—
162C2×C4—1731no—
163C2×C4—6921no—
164C2×C4—6921no—
165C2×C4—6921no—
166C2×C4—6921no—
167C2×C4—6922no—
168C2×C4—6928no—
169C2×C4—6922no—
170C2×C4—6922no—
171C2×C4C22⋊#1062192411yescharacteristic
172C2×C4C22×#1062192411yescharacteristic
173C8—6922no—
174C22—3461no—
175C22—3461no—
176C22—3461no—
177C22—3461no—
178C22—3464no—
179C22—3464no—
180C22—1731no—
181C22—1732no—
182C22—1731no—
183C22#1914973812yes—
184C22#1914974212yes—
185C22#1914974312yes—
186C22C2×C22⋊#1062192411yescharacteristic
187C4—21no—
188C4—21no—
189C4—42no—
190C4—3461no—
191C4—3461no—
192C4—6924no—
193C4—6922no—
194C2—3461no—
195C2—3462no—
196C2—3461no—
197C2#2601234414yesdirect product factor
198C2C2×#1914973811yescharacteristic
199C2C2×#1914974211yescharacteristic
200C2C2×#1914974311yescharacteristic
subgroupG/H
centerC23C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC23×C173C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173C2×#328
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012344C2direct (N × H)
11. C22×Dic346C4semidirect
15. #19149659C4semidirect
19. C2×C4⋊C4×C173C4semidirect
29. C2×Dic346C2×C4semidirect
44. C4⋊C4×C173C2×C4semidirect
93. C2×C173#328semidirect
100. C173C2×#328semidirect
115. C2×C4⋊C4#10621924semidirect
143. C4⋊C4C2×#10621924semidirect
197. C2#26012344direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497362
C2×(C2×#10621924)⋊C42
#260123442
C2×#191496592
C2×#328173