Finite Group Databasebeta

← #36369470

Subgroups — #36369470

Order 22144.

306 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#26012462C211yesmaximal, characteristic
2C4×C22⋊#10621924C211yesmaximal, characteristic
3#26012361C214yesmaximal, complemented
4C2×#19149637C211yesmaximal, characteristic
5C2×#14583836C2212yes—
6C2×C22⋊#10621924—21no—
7C2×C4×#10621924C2212yes—
8#19149736—21no—
9C22⋊#13063443—21no—
10(C2×#10621924)⋊C4—21no—
11C22×C4×D173C2211yescharacteristic
12#19149637—21no—
13#19149637—21no—
14#19149637C412yescomplemented
15#19149637C2212yes—
16C2×C4⋊Dic173C411yescharacteristic, complemented
17C2×#14583803—21no—
18C2×C4×Dic173—21no—
19C2×C4⋊C4×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21#14583836—24no—
22C2×#13063443—22no—
23C22⋊#10621924—24no—
24C4×#10621924—24no—
25C23×D173D411yescharacteristic
26C22×Dic173—21no—
27C22×Dic173D411yescharacteristic
28C2×C4×D173—24no—
29C2×C4×D173C2×C411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C2×C4×D173D411yescharacteristic
32C2×C4×D173C2311yescharacteristic
33C4⋊Dic173—21no—
34C4⋊Dic173C2×C412yes—
35#14583803—22no—
36#14583803—22no—
37C4×Dic173—22no—
38C4×Dic173—22no—
39C22×C4×C173—21no—
40C22×C4×C173C2×C411yescharacteristic
41C4⋊C4×C173—21no—
42C4⋊C4×C173C2×C412yes—
43C2×#10621924—24no—
44C2×#10621924—42no—
45C2×#10621924—42no—
46#13063443—42no—
47C22×D173—22no—
48C22×D173—22no—
49C22×D173—41no—
50C22×D173C22⋊C411yescharacteristic
51C22×D173C2×D411yescharacteristic
52C2×Dic173—22no—
53C2×Dic173—22no—
54C2×Dic173—42no—
55C2×Dic173C22⋊C411yescharacteristic
56C2×Dic173C2×D411yescharacteristic
57C4×D173—22no—
58C4×D173—22no—
59C4×D173—22no—
60C4×D173#4114yes—
61C23×C173C22⋊C411yescharacteristic
62C2×C4×C173—22no—
63C2×C4×C173—22no—
64C2×C4×C173—42no—
65C2×C4×C173C22⋊C411yescharacteristic
66C2×C4×C173C22×C411yescharacteristic
67C22×C173—21no—
68C22×C173—21no—
69C22×C173#10412yes—
70C22×C173C2×C22⋊C411yescharacteristic
71C2×D173—21no—
72C2×D173—21no—
73C2×D173—21no—
74C2×D173—21no—
75C2×D173—21no—
76C2×D173—44no—
77C2×D173#12312yes—
78#10621924—44no—
79C4×C173—21no—
80C4×C173—42no—
81C4×C173#11712yes—
82Dic173—21no—
83Dic173—42no—
84Dic173#12312yes—
85C2×C173—22no—
86C2×C173#38111yescharacteristic
87C2×C173C2×#10411yescharacteristic
88C2×C173#42111yescharacteristic
89D173—22no—
90D173—41no—
91C173#182611yescharacteristic, complemented, Sylow
92#1826—1731nomaximal, Sylow
93C2×#117—1731no—
94#407—1731no—
95C4×C22⋊C4—1731no—
96#339—1734no—
97C23×C4—1731no—
98C2×M16—1732no—
99#117—3461no—
100#117—3461no—
101#117—1732no—
102#117—1732no—
103C2×C4⋊C4—1731no—
104C2×C4⋊C4#1062192411yescharacteristic, complemented
105C2×C22⋊C4—3461no—
106C2×C22⋊C4—3461no—
107C2×C42—3461no—
108C2×C42—1732no—
109C22⋊C8—3461no—
110C22⋊C8—3461no—
111#95—3461no—
112C24—1731no—
113C22×C4—21no—
114C22×C4—3464no—
115C22×C4—3462no—
116C22×C4—3461no—
117C22×C4—1731no—
118C22×C4—1731no—
119C22×C4—1731no—
120C22×C4—1731no—
121C22×C4—1731no—
122C22×C4C2×#1062192411yescharacteristic
123M16—3464no—
124C2×C8—3462no—
125C4⋊C4—21no—
126C4⋊C4—3461no—
127C4⋊C4—1732no—
128C4⋊C4C2×#1062192412yes—
129C22⋊C4—3462no—
130C22⋊C4—3462no—
131C22⋊C4—3464no—
132C42—3462no—
133C42—3462no—
134C42—3464no—
135C23—3462no—
136C23—3462no—
137C23—1731no—
138C23—1731no—
139C23—6921no—
140C23C22⋊#1062192411yescharacteristic
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—42no—
144C2×C4—3462no—
145C2×C4—3462no—
146C2×C4—3462no—
147C2×C4—3462no—
148C2×C4—3464no—
149C2×C4—3462no—
150C2×C4—1731no—
151C2×C4—1731no—
152C2×C4—1734no—
153C2×C4—6922no—
154C2×C4—6922no—
155C2×C4—6922no—
156C2×C4C22⋊#1062192411yescharacteristic
157C2×C4C22×#1062192411yescharacteristic
158C8—6922no—
159C22—21no—
160C22—21no—
161C22—3461no—
162C22—3461no—
163C22—3461no—
164C22—3461no—
165C22—3461no—
166C22—1732no—
167C22—6924no—
168C22#1914974412yes—
169C22C2×C22⋊#1062192411yescharacteristic
170C4—21no—
171C4—42no—
172C4—3461no—
173C4—1732no—
174C4—6924no—
175C4—6922no—
176C4#1914975812yes—
177C2—22no—
178C2—3462no—
179C2—6921no—
180C2#2601243411yescharacteristic
181C2C2×#1914974411yescharacteristic
182C2#2601248911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173#1826
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012361C2semidirect
14. #19149637C4semidirect
16. C2×C4⋊Dic173C4semidirect
19. C2×C4⋊C4×C173C4semidirect
91. C173#1826semidirect
104. C2×C4⋊C4#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124622
C4×C22⋊#106219242
#260123612
C2×#191496372
#1826173