Finite Group Databasebeta

← #36352123

Subgroups — #36352123

Order 22144.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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#26011544C211yesmaximal, characteristic, complemented
2#26011357C211yesmaximal, characteristic, complemented
3#26011225C214yesmaximal, complemented
4#532×C173C211yesmaximal, characteristic
5C2×C4⋊Dic173—21no—
6C4⋊#13063436—21no—
7C22⋊#13063436C2212yes—
8C4×#13063436—21no—
9#19149586—21no—
10#19149586C2212yes—
11C2×#41×C173—21no—
12C4⋊Q8×C173C411yescharacteristic, complemented
13#127×C173C411yescharacteristic, complemented
14#124×C173—21no—
15#121×C173—21no—
16#121×C173C2212yes—
17C4×D4×C173—21no—
18C4×D4×C173C412yescomplemented
19C2×C42×C173C2211yescharacteristic
20C22×Dic173—22no—
21C2×#13063436—22no—
22C4⋊Dic173—42no—
23#41×C173—42no—
24#41×C173—42no—
25C2×Q8×C173—21no—
26C2×D4×C173—22no—
27C2×D4×C173—21no—
28C2×D4×C173—21no—
29C2×D4×C173C2×C412yes—
30C22×C4×C173—22no—
31C22×C4×C173D411yescharacteristic
32C22×C4×C173D411yescharacteristic
33C22×C4×C173C2311yescharacteristic
34C4⋊C4×C173—21no—
35C4⋊C4×C173C2×C412yes—
36C22⋊C4×C173—22no—
37C22⋊C4×C173—22no—
38C42×C173C2×C411yescharacteristic
39C42×C173C2×C411yescharacteristic
40C42×C173D411yescharacteristic
41C42×C173D411yescharacteristic
42C2×Dic173—42no—
43C2×Dic173—42no—
44C2×Dic173—42no—
45#13063436—42no—
46C23×C173—22no—
47C23×C173C2×D411yescharacteristic
48Q8×C173—42no—
49D4×C173—42no—
50D4×C173—42no—
51D4×C173—42no—
52D4×C173—42no—
53D4×C173—42no—
54C2×C4×C173—22no—
55C2×C4×C173—41no—
56C2×C4×C173—41no—
57C2×C4×C173—44no—
58C2×C4×C173C22⋊C411yescharacteristic
59C2×C4×C173C22⋊C411yescharacteristic
60C2×C4×C173C22⋊C411yescharacteristic
61C2×C4×C173C22⋊C411yescharacteristic
62C2×C4×C173D812yes—
63C2×C4×C173SD1612yes—
64C2×C4×C173C22×C411yescharacteristic
65C2×C4×C173C2×D411yescharacteristic
66C22×C173—21no—
67C22×C173—21no—
68C22×C173—42no—
69C22×C173—42no—
70C22×C173—42no—
71C22×C173C2×C22⋊C411yescharacteristic
72C22×C173C2×D811yescharacteristic
73C22×C173C2×SD1611yescharacteristic
74C4×C173—21no—
75C4×C173—21no—
76C4×C173—21no—
77C4×C173—21no—
78C4×C173—42no—
79C4×C173#10212yes—
80C4×C173#10212yes—
81Dic173—82no—
82C2×C173—21no—
83C2×C173—21no—
84C2×C173—82no—
85C2×C173#41011yescharacteristic
86C2×C173C2×#10211yescharacteristic
87C2×C173#42111yescharacteristic
88C173#151711yescharacteristic, complemented, Sylow
89#1517—1731nomaximal, Sylow
90#532D17311yescharacteristic
91#431—1731no—
92#382—1731no—
93#327—1734no—
94C2×#41—21no—
95C4⋊Q8Dic17311yescharacteristic, complemented
96#127Dic17311yescharacteristic, complemented
97#124—21no—
98#121—21no—
99#121C2×D17312yes—
100C4×D4—21no—
101C4×D4Dic17312yescomplemented
102C2×C4⋊C4—3461no—
103C2×C42C2×D17311yescharacteristic
104C4⋊C8—3461no—
105C22⋊C8—1732no—
106C4×C8—3461no—
107#95—3461no—
108#95—1732no—
109#41—42no—
110#41—42no—
111C2×Q8—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C2×Dic17312yes—
116C22×C4—22no—
117C22×C4—3462no—
118C22×C4C22⋊D17311yescharacteristic
119C22×C4C22⋊D17311yescharacteristic
120C22×C4C22×D17311yescharacteristic
121C2×C8—3462no—
122C4⋊C4—21no—
123C4⋊C4—6922no—
124C4⋊C4C2×Dic17312yes—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C42C2×Dic17311yescharacteristic
128C42C2×Dic17311yescharacteristic
129C42C22⋊D17311yescharacteristic
130C42C22⋊D17311yescharacteristic
131C23—22no—
132C23C2×C22⋊D17311yescharacteristic
133Q8—42no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—42no—
138D4—42no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4—6922no—
144C2×C4—6922no—
145C2×C4—6922no—
146C2×C4C22⋊Dic17311yescharacteristic
147C2×C4C22⋊Dic17311yescharacteristic
148C2×C4C22⋊Dic17311yescharacteristic
149C2×C4C22⋊Dic17311yescharacteristic
150C2×C4#1458381112yes—
151C2×C4#1458381412yes—
152C2×C4C22×Dic17311yescharacteristic
153C2×C4C2×C22⋊D17311yescharacteristic
154C8—6922no—
155C22—21no—
156C22—21no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22C2×C22⋊Dic17311yescharacteristic
161C22C2×#1458381111yescharacteristic
162C22C2×#1458381411yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—42no—
168C4—13842no—
169C4#1914960012yes—
170C4#1914960012yes—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2601146511yescharacteristic
175C2C2×#1914960011yescharacteristic
176C2#2601150811yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radical#532×C173C2
socleC22×C173C2×C22⋊C4
cosocleC22×C4×C173C23
hypercenter#532D173
hypocenterC173#1517
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #26011544C2semidirect
2. #26011357C2semidirect
3. #26011225C2semidirect
12. C4⋊Q8×C173C4semidirect
13. #127×C173C4semidirect
18. C4×D4×C173C4semidirect
88. C173#1517semidirect
95. C4⋊Q8Dic173semidirect
96. #127Dic173semidirect
101. C4×D4Dic173semidirect

Maximal subgroups

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

subgroupindex
#260115442
#260113572
#260112252
#532×C1732
#1517173