Finite Group Databasebeta

← #36354724

Subgroups — #36354724

Order 22144.

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#26012122C211yesmaximal, characteristic
2#26011500C212yesmaximal
3#26011260C211yesmaximal, characteristic
4#26011251C212yesmaximal
5#391×C173C211yesmaximal, characteristic
6C2×#14583810C2212yes—
7#19149678—22no—
8#19149670—22no—
9#19149658—22no—
10#19149638C2212yes—
11#19149603—22no—
12#19149603—22no—
13C22×Q8×C173C2211yescharacteristic
14C2×C4⋊C4×C173C2211yescharacteristic
15C2×C42×C173C2211yescharacteristic
16#95×C173—22no—
17C2×Dic346—22no—
18#14583810—22no—
19#14583810—22no—
20C2×#13063436—22no—
21C4⋊Dic173—22no—
22C4⋊Dic173D412yes—
23C4⋊Dic173D412yes—
24#14583805—24no—
25C22⋊Dic173—22no—
26C4×Dic173—22no—
27C2×Q8×C173—21no—
28C2×Q8×C173—22no—
29C2×Q8×C173—22no—
30C2×Q8×C173Q812yescomplemented
31C22×C4×C173—21no—
32C22×C4×C173—21no—
33C22×C4×C173D412yes—
34C22×C4×C173C2311yescharacteristic
35C4⋊C4×C173—22no—
36C42×C173—22no—
37Dic346—82no—
38C2×Dic173—22no—
39C2×Dic173—22no—
40#13063436—42no—
41C23×C173C2×D411yescharacteristic
42Q8×C173—24no—
43Q8×C173—42no—
44C2×C4×C173—22no—
45C2×C4×C173—22no—
46C2×C4×C173—22no—
47C2×C4×C173—41no—
48C2×C4×C173—41no—
49C2×C4×C173—41no—
50C2×C4×C173—41no—
51C2×C4×C173—42no—
52C2×C4×C173C2×D412yes—
53C2×C4×C173C2×Q811yescharacteristic
54C2×C4×C173#4112yes—
55C2×C4×C173#4111yescharacteristic
56C22×C173—21no—
57C22×C173—21no—
58C22×C173#12011yescharacteristic
59C22×C173C22⋊Q811yescharacteristic
60C22×C173#12411yescharacteristic
61C4×C173—42no—
62C4×C173—42no—
63C4×C173—42no—
64C4×C173C22⋊Q812yes—
65C4×C173#12412yes—
66Dic173—42no—
67Dic173—42no—
68C2×C173—21no—
69C2×C173—21no—
70C2×C173#39311yescharacteristic
71C2×C173#45511yescharacteristic
72C2×C173#45611yescharacteristic
73C173#202811yescharacteristic, complemented, Sylow
74#2028—1731nomaximal, Sylow
75#544—1731no—
76#419—1732no—
77#391D17311yescharacteristic
78#343—1731no—
79#339—1732no—
80C22×Q8C2×D17311yescharacteristic
81C2×M16—1732no—
82C4⋊Q8—3462no—
83#125—3462no—
84C22⋊Q8—3462no—
85#117—1732no—
86C2×C4⋊C4C2×D17311yescharacteristic
87C2×C42C2×D17311yescharacteristic
88#103—3462no—
89#103—3462no—
90#95—22no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—3462no—
95C2×Q8Dic34612yescomplemented
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D17312yes—
99C22×C4C22×D17311yescharacteristic
100M16—3462no—
101M16—3462no—
102C2×C8—3462no—
103C4⋊C4—22no—
104C4⋊C4—3462no—
105C4⋊C4—3464no—
106C4⋊C4—1732no—
107C4⋊C4—1732no—
108C22⋊C4—3462no—
109C42—22no—
110C42—3462no—
111C23C2×C22⋊D17311yescharacteristic
112Q8—24no—
113Q8—42no—
114Q8—13842no—
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—3462no—
124C2×C4—3462no—
125C2×C4D4×D17312yes—
126C2×C4C2×Dic34611yescharacteristic
127C2×C4#1458382512yes—
128C2×C4#1458382611yescharacteristic
129C8—6922no—
130C22—21no—
131C22—21no—
132C22#1914964711yescharacteristic
133C22#1914965811yescharacteristic
134C22#1914966611yescharacteristic
135C4—42no—
136C4—42no—
137C4—42no—
138C4—6922no—
139C4—6922no—
140C4#1914965612yes—
141C4#1914966512yes—
142C2—21no—
143C2—21no—
144C2#2601140311yescharacteristic
145C2#2601166811yescharacteristic
146C2#2601167311yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#391×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#391D173
hypocenterC173#2028
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C173Q8semidirect
73. C173#2028semidirect
95. C2×Q8Dic346semidirect

Maximal subgroups

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

subgroupindex
#260121222
#260115002
#260112602
#260112512
#391×C1732
#2028173