Finite Group Databasebeta

← #37507590

Subgroups — #37507590

Order 22912.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#26038988C211yesmaximal, characteristic
2#26038391C212yesmaximal, complemented
3#26038179C211yesmaximal, characteristic, complemented
4#26038178C212yesmaximal, complemented
5#329×C179C211yesmaximal, characteristic, complemented
6C2×#15668260C411yescharacteristic
7C2×#15668258C411yescharacteristic
8C2×#15668242—22no—
9#19153111C411yescharacteristic
10#19153109C411yescharacteristic
11#19153101C2211yescharacteristic
12#19153086—22no—
13(C22⋊D179)⋊C4—22no—
14C42×D179C2211yescharacteristic
15#19153024C2212yes—
16C8×Dic179—22no—
17#124×C179C2211yescharacteristic
18#97×C179C2212yes—
19#15668260—42no—
20#15668258—42no—
21C2×Dic358C2×C411yescharacteristic
22C2×D716—21no—
23C2×D716C2×C411yescharacteristic
24C2×C22⋊D179—21no—
25C2×C22⋊D179—22no—
26C22×Dic179—21no—
27C2×C4×D179—21no—
28C2×C4×D179C2×C411yescharacteristic
29C2×C4×D179D412yes—
30#15668242—28no—
31C2×C179⋊C8—22no—
32Dic179⋊C4—22no—
33(C2×D179)⋊C4—22no—
34C4×Dic179C2×C411yescharacteristic
35C4×Dic179D412yes—
36C2×Q8×C179C2×C411yescharacteristic
37C2×D4×C179C2×C411yescharacteristic
38C2×C8×C179—22no—
39C22⋊C4×C179—22no—
40C42×C179C2311yescharacteristic
41C22×D179—21no—
42C22×D179C22⋊C411yescharacteristic
43Dic358—41no—
44D716—41no—
45D716—42no—
46C22⋊D179—42no—
47C22⋊D179—44no—
48C2×Dic179—21no—
49C2×Dic179—42no—
50C2×Dic179C22⋊C411yescharacteristic
51C2×Dic179C22⋊C412yes—
52C4×D179—22no—
53C4×D179—24no—
54C4×D179—42no—
55C179⋊C8—24no—
56C23×C179—21no—
57Q8×C179—41no—
58D4×C179—41no—
59C2×C4×C179—21no—
60C2×C4×C179C22×C411yescharacteristic
61C2×C4×C179C2×D412yes—
62C8×C179—24no—
63C22×C179—41no—
64C22×C179—42no—
65C22×C179C2×C22⋊C411yescharacteristic
66C2×D179—21no—
67C2×D179—22no—
68C2×D179—41no—
69C2×D179—42no—
70C4×C179—21no—
71C4×C179—22no—
72C4×C179—41no—
73Dic179—41no—
74Dic179#10012yes—
75Dic179#10414yes—
76C2×C179—81no—
77C2×C179C2×#10011yescharacteristic
78C2×C179C2×#10412yes—
79D179—41no—
80D179—81no—
81C179#152511yescharacteristic, complemented, Sylow
82#1525—1791nomaximal, Sylow
83#532—1791no—
84C4×M16—1792no—
85#329—1791no—
86#329—1792no—
87#329D17911yescharacteristic, complemented
88C2×#41—1791no—
89C2×#41—1791no—
90C2×M16—3582no—
91C4⋊Q8—1791no—
92#127—1791no—
93#124—1791no—
94#124C2×D17911yescharacteristic
95#121—3582no—
96C4×D4—3582no—
97C2×C42—1791no—
98#97—1792no—
99#97C2×D17912yes—
100C4×C8—3582no—
101#41—7162no—
102#41—7162no—
103C2×Q8—1791no—
104C2×Q8C4×D17911yescharacteristic
105C2×D4—3581no—
106C2×D4—3581no—
107C2×D4—3582no—
108C2×D4—1791no—
109C2×D4C4×D17911yescharacteristic
110C22×C4—3581no—
111C22×C4—3581no—
112C22×C4—1791no—
113C22×C4—1792no—
114M16—3588no—
115C2×C8—22no—
116C2×C8—3582no—
117C4⋊C4—3582no—
118C22⋊C4—22no—
119C22⋊C4—3582no—
120C42—1791no—
121C42—1792no—
122C42C22×D17911yescharacteristic
123C23—21no—
124C23—3581no—
125C23—1791no—
126Q8—41no—
127Q8—7161no—
128D4—41no—
129D4—7161no—
130D4—7162no—
131D4—7164no—
132D4—7162no—
133C2×C4—21no—
134C2×C4—3584no—
135C2×C4—3582no—
136C2×C4—3581no—
137C2×C4—1792no—
138C2×C4—1791no—
139C2×C4—7162no—
140C2×C4—7162no—
141C2×C4C2×C4×D17911yescharacteristic
142C2×C4D4×D17912yes—
143C8—24no—
144C8—3584no—
145C22—41no—
146C22—42no—
147C22—3581no—
148C22—3582no—
149C22—7161no—
150C22—7162no—
151C22C22⋊C4×D17911yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—1792no—
156C4—1794no—
157C4—7161no—
158C2—81no—
159C2—14321no—
160C2—7161no—
161C2#100×D17911yescharacteristic
162C2#104×D17912yes—
subgroupG/H
centerC22C22⋊C4×D179
commutatorC2×C4×C179C22×C4
FrattiniC42C22×D179
Fitting#329×C179C2
socleC22×C179C2×C22⋊C4
prime radical#26038988C2
cosocleC42×C179C23
hypercenter#329D179
hypocenterC179#1525
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26038391C2semidirect
3. #26038179C2semidirect
4. #26038178C2semidirect
5. #329×C179C2semidirect
81. C179#1525semidirect
87. #329D179semidirect

Maximal subgroups

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

subgroupindex
#260389882
#260383912
#260381792
#260381782
#329×C1792
#1525179