Finite Group Databasebeta

← #37510161

Subgroups — #37510161

Order 22912.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#26039070C211yesmaximal, characteristic
2#26038448C212yesmaximal
3#26038208C211yesmaximal, characteristic
4#26038199C212yesmaximal
5#391×C179C211yesmaximal, characteristic
6C2×#15668243C2212yes—
7#19153112—22no—
8#19153104—22no—
9#19153092—22no—
10#19153072C2212yes—
11#19153037—22no—
12#19153037—22no—
13C22×Q8×C179C2211yescharacteristic
14C2×C4⋊C4×C179C2211yescharacteristic
15C2×C42×C179C2211yescharacteristic
16#95×C179—22no—
17C2×Dic358—22no—
18#15668243—22no—
19#15668243—22no—
20C2×C179⋊C8—22no—
21C4⋊Dic179—22no—
22C4⋊Dic179D412yes—
23C4⋊Dic179D412yes—
24Dic179⋊C4—24no—
25C22⋊Dic179—22no—
26C4×Dic179—22no—
27C2×Q8×C179—21no—
28C2×Q8×C179—22no—
29C2×Q8×C179—22no—
30C2×Q8×C179Q812yescomplemented
31C22×C4×C179—21no—
32C22×C4×C179—21no—
33C22×C4×C179D412yes—
34C22×C4×C179C2311yescharacteristic
35C4⋊C4×C179—22no—
36C42×C179—22no—
37Dic358—82no—
38C2×Dic179—22no—
39C2×Dic179—22no—
40C179⋊C8—42no—
41C23×C179C2×D411yescharacteristic
42Q8×C179—24no—
43Q8×C179—42no—
44C2×C4×C179—22no—
45C2×C4×C179—22no—
46C2×C4×C179—22no—
47C2×C4×C179—41no—
48C2×C4×C179—41no—
49C2×C4×C179—41no—
50C2×C4×C179—41no—
51C2×C4×C179—42no—
52C2×C4×C179C2×D412yes—
53C2×C4×C179C2×Q811yescharacteristic
54C2×C4×C179#4112yes—
55C2×C4×C179#4111yescharacteristic
56C22×C179—21no—
57C22×C179—21no—
58C22×C179#12011yescharacteristic
59C22×C179C22⋊Q811yescharacteristic
60C22×C179#12411yescharacteristic
61C4×C179—42no—
62C4×C179—42no—
63C4×C179—42no—
64C4×C179C22⋊Q812yes—
65C4×C179#12412yes—
66Dic179—42no—
67Dic179—42no—
68C2×C179—21no—
69C2×C179—21no—
70C2×C179#39311yescharacteristic
71C2×C179#45511yescharacteristic
72C2×C179#45611yescharacteristic
73C179#202811yescharacteristic, complemented, Sylow
74#2028—1791nomaximal, Sylow
75#544—1791no—
76#419—1792no—
77#391D17911yescharacteristic
78#343—1791no—
79#339—1792no—
80C22×Q8C2×D17911yescharacteristic
81C2×M16—1792no—
82C4⋊Q8—3582no—
83#125—3582no—
84C22⋊Q8—3582no—
85#117—1792no—
86C2×C4⋊C4C2×D17911yescharacteristic
87C2×C42C2×D17911yescharacteristic
88#103—3582no—
89#103—3582no—
90#95—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—3582no—
95C2×Q8Dic35812yescomplemented
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D17912yes—
99C22×C4C22×D17911yescharacteristic
100M16—3582no—
101M16—3582no—
102C2×C8—3582no—
103C4⋊C4—22no—
104C4⋊C4—3584no—
105C4⋊C4—3582no—
106C4⋊C4—1792no—
107C4⋊C4—1792no—
108C22⋊C4—3582no—
109C42—22no—
110C42—3582no—
111C23C2×C22⋊D17911yescharacteristic
112Q8—24no—
113Q8—42no—
114Q8—14322no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—3582no—
124C2×C4—3582no—
125C2×C4D4×D17912yes—
126C2×C4C2×Dic35811yescharacteristic
127C2×C4#1566825812yes—
128C2×C4#1566825911yescharacteristic
129C8—7162no—
130C22—21no—
131C22—21no—
132C22#1915308111yescharacteristic
133C22#1915309211yescharacteristic
134C22#1915310011yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—7162no—
139C4—7162no—
140C4#1915309012yes—
141C4#1915309912yes—
142C2—21no—
143C2—21no—
144C2#2603835111yescharacteristic
145C2#2603861611yescharacteristic
146C2#2603862111yescharacteristic
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
Fitting#391×C179C2
socleC22×C179#120
prime radicalC23×C179C2×D4
hypercenter#391D179
hypocenterC179#2028
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C179Q8semidirect
73. C179#2028semidirect
95. C2×Q8Dic358semidirect

Maximal subgroups

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

subgroupindex
#260390702
#260384482
#260382082
#260381992
#391×C1792
#2028179