Finite Group Databasebeta

← #34014334

Subgroups — #34014334

Order 18688.

406 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 60 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#22625065C214yesmaximal, complemented
3#22620390C211yesmaximal, characteristic
4#17989566—22no—
5#17989494—22no—
6#17989492C2214yes—
7C2×#14556624—22no—
8#17989137C2211yescharacteristic
9C2×#14556591C411yescharacteristic, complemented
10#17988981—22no—
11#17988967C412yescomplemented
12#17988960C2212yes—
13C4×M16×C73C411yescharacteristic, complemented
14#14556670—42no—
15#14556650—24no—
16#14556649—42no—
17C2×#11924028D412yes—
18C22×Dic146—21no—
19#14556624—44no—
20#14556624—44no—
21C2×Dic292—24no—
22C2×#11924017—24no—
23#14556591—21no—
24#14556591—41no—
25#14556591C2×C412yes—
26#14556591C2×C411yescharacteristic
27#14556588C2311yescharacteristic
28#14556579D412yes—
29#14556565—41no—
30C4×D292—41no—
31C2×C4⋊Dic73—41no—
32C2×M16×C73—21no—
33C2×C42×C73C2×C411yescharacteristic
34#97×C73—21no—
35C4×C8×C73C2×C412yes—
36C4⋊#64408—82no—
37C22⋊#64408—44no—
38C4×#64408—82no—
39#11924028—42no—
40#11924028—42no—
41C2×Dic146—22no—
42C2×Dic146—42no—
43C2×Dic146—42no—
44C2×Dic146—41no—
45C2×Dic146C22⋊C412yes—
46C2×D292—41no—
47C2×D292C2×D412yes—
48C2×C22⋊D73—22no—
49C22×Dic73—41no—
50C2×C4×D73—22no—
51Dic292—48no—
52#11924017—48no—
53C4⋊Dic73—41no—
54C4⋊Dic73—42no—
55C4⋊Dic73—42no—
56#11924005—42no—
57C22×C4×C73—21no—
58C22×C4×C73C22⋊C411yescharacteristic
59M16×C73—24no—
60C2×C8×C73—22no—
61C42×C73—21no—
62C42×C73C22⋊C411yescharacteristic
63C42×C73C22×C411yescharacteristic
64C23×C73#9911yescharacteristic
65C22×D73—22no—
66C2×#64408—84no—
67C2×C4×C73—21no—
68C2×C4×C73—22no—
69C2×C4×C73—21no—
70C2×C4×C73—42no—
71C2×C4×C73#9911yescharacteristic
72C2×C4×C73C2×C22⋊C411yescharacteristic
73C22⋊D73—82no—
74C2×Dic73—22no—
75C2×Dic73—42no—
76C2×Dic73—81no—
77C2×Dic73—81no—
78D292—42no—
79D292—82no—
80C4×D73—42no—
81C4×D73—42no—
82Dic146—42no—
83Dic146—44no—
84Dic146—82no—
85C8×C73—24no—
86C22×C73—21no—
87C22×C73—41no—
88C22×C73C2×#9911yescharacteristic
89C2×D73—42no—
90C2×D73—82no—
91#64408—164no—
92C4×C73—22no—
93C4×C73—21no—
94C4×C73—41no—
95C4×C73#35312yes—
96Dic73—42no—
97Dic73—82no—
98#13586—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—41no—
101C2×C73C2×#35311yescharacteristic
102D73—82no—
103#3152—731no—
104#2126—732no—
105#1409—734no—
106C73#1358611yescharacteristic, complemented, Sylow
107C2×#137—1462no—
108#532—731no—
109C2×C4⋊Q8—731no—
110#497—1462no—
111#494—732no—
112#492—732no—
113#410—1462no—
114C4×M16#6440811yescharacteristic, complemented
115#354—1462no—
116#353—734no—
117C2×#41—732no—
118C22×Q8—1461no—
119#137—2924no—
120#137—2924no—
121C2×Q16—1464no—
122C2×SD16—1464no—
123C2×M16—21no—
124C4⋊Q8—2921no—
125C4⋊Q8—731no—
126C4⋊Q8—732no—
127C4⋊Q8—1461no—
128#127—731no—
129#124—732no—
130#121—2921no—
131C4×D4—2921no—
132#117—2922no—
133C2×C4⋊C4—2921no—
134C2×C42C2×#6440811yescharacteristic
135#99—2922no—
136#99—1464no—
137#97—21no—
138C4×C8C2×#6440812yes—
139#41—2922no—
140#41—2922no—
141C2×Q8—2922no—
142C2×Q8—2922no—
143C2×Q8—2921no—
144C2×Q8—732no—
145C2×Q8—1462no—
146C2×D4—2921no—
147C2×D4—732no—
148C2×D4—1462no—
149C22×C4—21no—
150C22×C4—2921no—
151C22×C4—1462no—
152C22×C4C22⋊#6440811yescharacteristic
153Q16—2928no—
154SD16—2928no—
155M16—24no—
156C2×C8—22no—
157C4⋊C4—2922no—
158C4⋊C4—2922no—
159C4⋊C4—2921no—
160C4⋊C4—5842no—
161C22⋊C4—2922no—
162C22⋊C4—2924no—
163C42—21no—
164C42—5842no—
165C42C22⋊#6440811yescharacteristic
166C42C22×#6440811yescharacteristic
167C23—1462no—
168C23#1455665011yescharacteristic
169Q8—2922no—
170Q8—2924no—
171Q8—5842no—
172D4—2922no—
173D4—5842no—
174D4—5842no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—2922no—
180C2×C4—2922no—
181C2×C4—2922no—
182C2×C4—5841no—
183C2×C4—5841no—
184C2×C4—5844no—
185C2×C4—1462no—
186C2×C4#1455665011yescharacteristic
187C2×C4C2×C22⋊#6440811yescharacteristic
188C8—24no—
189C22—21no—
190C22—41no—
191C22—2922no—
192C22—5842no—
193C22C2×#1455665011yescharacteristic
194C4—22no—
195C4—21no—
196C4—41no—
197C4—2922no—
198C4—5842no—
199C4—11684no—
200C4#1798949212yes—
201C2—21no—
202C2—41no—
203C2—5842no—
204C2C2×#1798949211yescharacteristic
subgroupG/H
centerC2C2×#17989492
commutatorC42×C73C22×C4
FrattiniC42C22×#64408
FittingC4×M16×C73C4
socleC2×C73C2×#353
prime radical#17989137C22
cosocle#14556588C23
hypercenterC4×M16#64408
hypocenterC73#13586
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #22625065C2semidirect
9. C2×#14556591C4semidirect
11. #17988967C4semidirect
13. C4×M16×C73C4semidirect
106. C73#13586semidirect
114. C4×M16#64408semidirect

Maximal subgroups

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

subgroupindex
#226262042
#226250652
#226203902
#1358673