Finite Group Databasebeta

← #34001172

Subgroups — #34001172

Order 18688.

294 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 46 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#22626204C212yesmaximal
2#22610758C211yesmaximal, characteristic
3#17989566—22no—
4#17989494C414yescomplemented
5#17989492—22no—
6#17989137C2211yescharacteristic
7C2×#14556517—22no—
8C42×Dic73C411yescharacteristic, complemented
9#423×C73C411yescharacteristic, complemented
10#14556670—42no—
11#14556650—24no—
12#14556649—42no—
13C2×#11924028D412yes—
14#14556591Q811yescharacteristic
15#14556588D411yescharacteristic
16#14556579C2×C412yes—
17#14556565—41no—
18C4×D292—41no—
19C2×C4×Dic73—21no—
20C2×C4×Dic73—41no—
21#14556517—44no—
22C2×M16×C73—21no—
23C2×C42×C73C2×C411yescharacteristic
24C4⋊C8×C73—41no—
25C4⋊#64408—82no—
26C22⋊#64408—44no—
27C4×#64408—82no—
28#11924028—42no—
29#11924028—42no—
30C2×Dic146C4⋊C412yes—
31C2×D292—41no—
32C2×D292C22⋊C412yes—
33C2×C22⋊D73—22no—
34C22×Dic73—41no—
35C2×C4×D73—22no—
36C4⋊Dic73—41no—
37#11924005—42no—
38C4×Dic73—22no—
39C4×Dic73—41no—
40C4×Dic73—44no—
41C22×C4×C73—21no—
42C22×C4×C73C22⋊C411yescharacteristic
43M16×C73—42no—
44C2×C8×C73—41no—
45C42×C73—21no—
46C42×C73C4211yescharacteristic
47C42×C73C4⋊C411yescharacteristic
48C23×C73#9911yescharacteristic
49C22×D73—22no—
50C2×#64408—84no—
51C2×C4×C73—21no—
52C2×C4×C73—21no—
53C2×C4×C73—21no—
54C2×C4×C73—21no—
55C2×C4×C73—42no—
56C2×C4×C73#9511yescharacteristic
57C2×C4×C73#9911yescharacteristic
58C22⋊D73—82no—
59C2×Dic73—22no—
60C2×Dic73—42no—
61C2×Dic73—44no—
62D292—42no—
63D292—82no—
64C4×D73—42no—
65C4×D73—42no—
66Dic146—42no—
67C8×C73—81no—
68C22×C73—21no—
69C22×C73—41no—
70C22×C73#34211yescharacteristic
71C2×D73—42no—
72C2×D73—82no—
73#64408—164no—
74C4×C73—22no—
75C4×C73—21no—
76C4×C73—41no—
77C4×C73#35311yescharacteristic
78C4×C73#35411yescharacteristic
79Dic73—44no—
80Dic73—42no—
81#7500—731nomaximal, Sylow
82C2×C73—21no—
83C2×C73—41no—
84C2×C73#139311yescharacteristic
85D73—82no—
86#2126—732no—
87#1962—731no—
88C73#750011yescharacteristic, complemented, Sylow
89#532—731no—
90#423#6440811yescharacteristic, complemented
91C2×#104—1462no—
92#410—1462no—
93C43—731no—
94#354—734no—
95#353—1462no—
96C2×#41—732no—
97C2×M16—21no—
98C4⋊Q8—731no—
99#127—731no—
100#124—732no—
101#121—2921no—
102C4×D4—2921no—
103#117—2922no—
104C2×C42—2921no—
105C2×C42—1461no—
106C2×C42C2×#6440811yescharacteristic
107C4⋊C8—41no—
108#104—2924no—
109#99—2922no—
110#99—1464no—
111#41—2922no—
112#41—2922no—
113C2×Q8—732no—
114C2×D4—2921no—
115C2×D4—732no—
116C2×D4—1462no—
117C22×C4—21no—
118C22×C4—2921no—
119C22×C4—1462no—
120C22×C4C22⋊#6440811yescharacteristic
121M16—42no—
122C2×C8—41no—
123C4⋊C4—2921no—
124C4⋊C4—5842no—
125C22⋊C4—2922no—
126C22⋊C4—2924no—
127C42—21no—
128C42—2921no—
129C42—2924no—
130C42—5842no—
131C42—1462no—
132C42C4×#6440811yescharacteristic
133C42C4⋊#6440811yescharacteristic
134C23—1462no—
135C23#1455665011yescharacteristic
136Q8—2922no—
137D4—2922no—
138D4—5842no—
139D4—5842no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—42no—
145C2×C4—2922no—
146C2×C4—2924no—
147C2×C4—2922no—
148C2×C4—2922no—
149C2×C4—5844no—
150C2×C4—1462no—
151C2×C4(C2×#64408)⋊C411yescharacteristic
152C2×C4#1455664911yescharacteristic
153C8—81no—
154C22—21no—
155C22—41no—
156C22—2922no—
157C22—5842no—
158C22#1798946511yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—2924no—
163C4—2922no—
164C4—11684no—
165C4#1798949111yescharacteristic
166C4#1798949311yescharacteristic
167C2—21no—
168C2—41no—
169C2—5842no—
170C2#2262502611yescharacteristic
subgroupG/H
centerC4#17989493
commutatorC42×C73C42
FrattiniC2×C42C2×#64408
Fitting#423×C73C4
socleC2×C73#1393
prime radical, cosocle#17989137C22
hypercenter#423#64408
hypocenterC73#7500
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
4. #17989494C4semidirect
8. C42×Dic73C4semidirect
9. #423×C73C4semidirect
88. C73#7500semidirect
90. #423#64408semidirect

Maximal subgroups

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

subgroupindex
#226262042
#226107582
#750073