Finite Group Databasebeta

← #37508299

Subgroups — #37508299

Order 22912.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#26039189C211yesmaximal, characteristic, complemented
2#26038829C211yesmaximal, characteristic
3#26038804C212yesmaximal, complemented
4#26038180C212yesmaximal, complemented
5#329×C179C211yesmaximal, characteristic, complemented
6C2×#15668246—21no—
7C2×#15668244—21no—
8#19153111—21no—
9#19153110C2211yescharacteristic
10#19153106—22no—
11#19153105C2212yes—
12#19153094—21no—
13#19153090—22no—
14#19153035—22no—
15#19153035—22no—
16#19153024C2211yescharacteristic
17#124×C179C2211yescharacteristic
18#97×C179C2212yes—
19C2×Dic358—21no—
20C2×D716—22no—
21C2×D716—21no—
22C22×Dic179—21no—
23#15668246—42no—
24#15668244—42no—
25C2×C179⋊C8—21no—
26C4⋊Dic179D412yes—
27C4⋊Dic179D412yes—
28Dic179⋊C4—22no—
29Dic179⋊C4—22no—
30Dic179⋊C4—22no—
31C22⋊Dic179—22no—
32C22⋊Dic179—21no—
33C4×Dic179—21no—
34C2×Q8×C179D411yescharacteristic
35C2×D4×C179D411yescharacteristic
36C2×C8×C179—22no—
37C22⋊C4×C179—22no—
38C42×C179C2311yescharacteristic
39C22×D179—41no—
40Dic358—81no—
41D716—41no—
42D716—41no—
43D716—44no—
44C2×Dic179—22no—
45C2×Dic179—22no—
46C2×Dic179—81no—
47C179⋊C8—22no—
48C23×C179—21no—
49Q8×C179—41no—
50D4×C179—41no—
51C2×C4×C179—21no—
52C2×C4×C179C2×D412yes—
53C2×C4×C179C2×D411yescharacteristic
54C8×C179—42no—
55C22×C179—41no—
56C22×C179—42no—
57C22×C179#12011yescharacteristic
58C2×D179—81no—
59C2×D179—82no—
60C4×C179—21no—
61C4×C179—22no—
62C4×C179—41no—
63Dic179—42no—
64Dic179—42no—
65C2×C179—81no—
66C2×C179#45311yescharacteristic
67C2×C179#45412yes—
68D179—161no—
69C179#165711yescharacteristic, complemented, Sylow
70#1657—1791nomaximal, Sylow
71#563—1791no—
72#496—1791no—
73#490—1792no—
74#330—1792no—
75#329D17911yescharacteristic, complemented
76C2×SD16—3581no—
77C2×D8—3581no—
78C4⋊Q8—3581no—
79#127—1791no—
80#126—3582no—
81#125—1792no—
82#124C2×D17911yescharacteristic
83#123—3581no—
84C22⋊Q8—3582no—
85#102—3582no—
86#102—3582no—
87#97—1791no—
88#97C2×D17912yes—
89C2×Q8—3581no—
90C2×Q8D71611yescharacteristic
91C2×D4—3582no—
92C2×D4—3581no—
93C2×D4D71611yescharacteristic
94C22×C4—3581no—
95SD16—7162no—
96D8—7162no—
97C2×C8—22no—
98C2×C8—3581no—
99C4⋊C4—3582no—
100C4⋊C4—3582no—
101C4⋊C4—3582no—
102C4⋊C4—1792no—
103C4⋊C4—1792no—
104C22⋊C4—22no—
105C22⋊C4—3582no—
106C22⋊C4—3581no—
107C42—3581no—
108C42C22×D17911yescharacteristic
109C23—21no—
110C23—7161no—
111Q8—41no—
112Q8—14321no—
113D4—41no—
114D4—7161no—
115D4—7161no—
116D4—7164no—
117C2×C4—21no—
118C2×C4—3582no—
119C2×C4—3582no—
120C2×C4—14321no—
121C2×C4D4×D17912yes—
122C2×C4C2×D71611yescharacteristic
123C8—42no—
124C8—3582no—
125C22—41no—
126C22—42no—
127C22—14321no—
128C22—14322no—
129C22#1915308311yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—7162no—
134C4—7162no—
135C2—81no—
136C2—28641no—
137C2#2603860911yescharacteristic
138C2#2603861312yes—
subgroupG/H
centerC22#19153083
commutator, cosocleC42×C179C23
FrattiniC42C22×D179
Fitting#329×C179C2
socleC22×C179#120
prime radical#26038829C2
hypercenter#329D179
hypocenterC179#1657
LCSG ▷ C42×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #26039189C2semidirect
3. #26038804C2semidirect
4. #26038180C2semidirect
5. #329×C179C2semidirect
69. C179#1657semidirect
75. #329D179semidirect

Maximal subgroups

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

subgroupindex
#260391892
#260388292
#260388042
#260381802
#329×C1792
#1657179