Finite Group Databasebeta

← #36369003

Subgroups — #36369003

Order 22144.

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

normal subgroups only

This group has 288 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
1D173⋊(C22⋊C8)C211yesmaximal, characteristic
2#26012433C211yesmaximal, characteristic
3#26012345C214yesmaximal, complemented
4C2×#19149655C211yesmaximal, characteristic
5C2×D173⋊C8—21no—
6C2×C4⋊#10621924—21no—
7C2×C22⋊#10621924—21no—
8#19149736C2212yes—
9C22⋊#13063443—21no—
10(C2×#10621924)⋊C4C2212yes—
11C22×C4×D173C2211yescharacteristic
12#19149655—21no—
13#19149655—21no—
14#19149655C412yescomplemented
15#19149655C2212yes—
16C2×C4⋊Dic173C411yescharacteristic, complemented
17C2×#14583805—21no—
18C2×#14583803—21no—
19C22×Q8×C173C411yescharacteristic, complemented
20C22×#10621924—22no—
21D173⋊C8—42no—
22C2×#13063443—22no—
23C4⋊#10621924—42no—
24C22⋊#10621924—42no—
25C23×D173D411yescharacteristic
26C22×Dic173—21no—
27C22×Dic173D411yescharacteristic
28C2×C4×D173—42no—
29C2×C4×D173C2×C411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C2×C4×D173D411yescharacteristic
32C2×C4×D173C2311yescharacteristic
33C4⋊Dic173—21no—
34C4⋊Dic173C2×C412yes—
35#14583805—22no—
36#14583805—22no—
37#14583803—22no—
38#14583803—22no—
39C2×Q8×C173—21no—
40C2×Q8×C173—42no—
41C2×Q8×C173C2×C412yes—
42C22×C4×C173—21no—
43C22×C4×C173C2×C411yescharacteristic
44C2×#10621924—42no—
45C2×#10621924—42no—
46C2×#10621924—42no—
47#13063443—42no—
48C22×D173—41no—
49C22×D173—41no—
50C22×D173C22⋊C411yescharacteristic
51C22×D173SD1612yes—
52C22×D173Q1612yes—
53C22×D173C2×D411yescharacteristic
54C2×Dic173—22no—
55C2×Dic173—41no—
56C2×Dic173—42no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C4×D173—22no—
60C4×D173—22no—
61C4×D173—41no—
62C4×D173—41no—
63C23×C173C22⋊C411yescharacteristic
64Q8×C173—42no—
65Q8×C173—42no—
66C2×C4×C173—22no—
67C2×C4×C173—22no—
68C2×C4×C173—42no—
69C2×C4×C173C22⋊C411yescharacteristic
70C2×C4×C173C22×C411yescharacteristic
71C22×C173—21no—
72C22×C173—21no—
73C22×C173#10312yes—
74C22×C173C2×C22⋊C411yescharacteristic
75C2×D173—21no—
76C2×D173—21no—
77C2×D173—21no—
78C2×D173—21no—
79C2×D173—42no—
80C2×D173—42no—
81C2×D173#10312yes—
82C2×D173C2×SD1611yescharacteristic
83C2×D173C2×Q1611yescharacteristic
84#10621924—82no—
85C4×C173—21no—
86C4×C173—42no—
87C4×C173#9912yes—
88Dic173—21no—
89Dic173—21no—
90Dic173—42no—
91C2×C173—22no—
92C2×C173C2×#9911yescharacteristic
93C2×C173C2×#10311yescharacteristic
94C2×C173#42111yescharacteristic
95D173—21no—
96D173—22no—
97D173—21no—
98C173#152011yescharacteristic, complemented, Sylow
99#1520—1731nomaximal, Sylow
100C2×C22⋊Q8—1731no—
101C2×C22⋊C8—1731no—
102#380—1731no—
103#328—1734no—
104C22×Q8#1062192411yescharacteristic, complemented
105C23×C4—1731no—
106C22×C8—3461no—
107C22⋊Q8—3461no—
108C22⋊Q8—3461no—
109C22⋊Q8—1732no—
110C22⋊Q8—1732no—
111C2×C4⋊C4—3461no—
112C2×C4⋊C4—3461no—
113C2×C4⋊C4—1731no—
114C2×C22⋊C4—3461no—
115C2×C22⋊C4—3461no—
116C22⋊C8—3461no—
117C22⋊C8—1732no—
118#95—1732no—
119C24—1731no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8C2×#1062192412yes—
123C22×C4—21no—
124C22×C4—3461no—
125C22×C4—3462no—
126C22×C4—1731no—
127C22×C4—1731no—
128C22×C4—1731no—
129C22×C4—1731no—
130C22×C4—1731no—
131C22×C4—6922no—
132C22×C4C2×#1062192411yescharacteristic
133C2×C8—3462no—
134C2×C8—6922no—
135C4⋊C4—3462no—
136C4⋊C4—3462no—
137C4⋊C4—3461no—
138C4⋊C4—1732no—
139C4⋊C4—6922no—
140C22⋊C4—3462no—
141C22⋊C4—3462no—
142C22⋊C4—6922no—
143C23—1731no—
144C23—1731no—
145C23—1732no—
146C23—1732no—
147C23—6921no—
148C23—6921no—
149C23C22⋊#1062192411yescharacteristic
150Q8—42no—
151Q8—42no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—3462no—
156C2×C4—3462no—
157C2×C4—3462no—
158C2×C4—1731no—
159C2×C4—1731no—
160C2×C4—6921no—
161C2×C4—6921no—
162C2×C4—6921no—
163C2×C4—6922no—
164C2×C4—6922no—
165C2×C4—6922no—
166C2×C4—6922no—
167C2×C4C22⋊#1062192411yescharacteristic
168C2×C4C22×#1062192411yescharacteristic
169C8—6922no—
170C22—21no—
171C22—21no—
172C22—3461no—
173C22—3461no—
174C22—3461no—
175C22—3461no—
176C22—1731no—
177C22—1732no—
178C22—1731no—
179C22—6922no—
180C22—6922no—
181C22#1914974212yes—
182C22C2×C22⋊#1062192411yescharacteristic
183C4—21no—
184C4—42no—
185C4—3461no—
186C4—3461no—
187C4—13842no—
188C4—6922no—
189C4#1914973712yes—
190C2—22no—
191C2—3461no—
192C2—3462no—
193C2—3461no—
194C2C2×#1914973711yescharacteristic
195C2C2×#1914974211yescharacteristic
196C2#2601249011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC22×Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173D4
cosocleC2×C4×D173C23
hypercenterC22×Q8#10621924
hypocenterC173#1520
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #26012345C2semidirect
14. #19149655C4semidirect
16. C2×C4⋊Dic173C4semidirect
19. C22×Q8×C173C4semidirect
98. C173#1520semidirect
104. C22×Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C22⋊C8)2
#260124332
#260123452
C2×#191496552
#1520173