Finite Group Databasebeta

← #34014364

Subgroups — #34014364

Order 18688.

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

normal subgroups only

This group has 388 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
1C2×#17989492C212yesmaximal
2#22625072C214yesmaximal, complemented
3#22620306C211yesmaximal, characteristic
4C2×#14556650—22no—
5#17989492—22no—
6#17989492C2214yes—
7#17989144C411yescharacteristic, complemented
8C2×#14556588C2211yescharacteristic
9#17988952C412yescomplemented
10#17988952C2212yes—
11#17988934—22no—
12#17988593—22no—
13C4×M16×C73C411yescharacteristic, complemented
14C2×C22⋊#64408—42no—
15#14556650—24no—
16#14556650—42no—
17C22×D292—41no—
18C22×D292D412yes—
19#14556591—21no—
20#14556588—41no—
21#14556588C2×C411yescharacteristic
22#14556588D412yes—
23#14556588C2311yescharacteristic
24#14556583C2×C412yescomplemented
25#14556570—41no—
26C4×Dic146—41no—
27#14556550—21no—
28#14556513—24no—
29#14556513—24no—
30C2×M16×C73—21no—
31C2×C42×C73C2×C411yescharacteristic
32#97×C73C2×C412yescomplemented
33C4×C8×C73—21no—
34C22×#64408—82no—
35C22⋊#64408—44no—
36C22⋊#64408—82no—
37C23×D73—22no—
38C2×Dic146—41no—
39C2×D292—22no—
40C2×D292—42no—
41C2×D292—42no—
42C2×D292—41no—
43C2×D292—42no—
44C2×D292—41no—
45C2×D292—84no—
46C2×D292C22⋊C412yes—
47C2×D292C2×D412yes—
48C4⋊Dic73—22no—
49C4⋊Dic73—41no—
50#11924007—42no—
51C22⋊Dic73—41no—
52C4×Dic73—41no—
53C22×C4×C73—21no—
54C22×C4×C73C22⋊C411yescharacteristic
55M16×C73—42no—
56C2×C8×C73—22no—
57C42×C73—21no—
58C42×C73C22⋊C411yescharacteristic
59C42×C73C22×C411yescharacteristic
60C23×C73#9911yescharacteristic
61C22×D73—22no—
62C22×D73—22no—
63C22×D73—42no—
64C22×D73—42no—
65C22×D73—84no—
66C2×#64408—84no—
67C2×#64408—164no—
68C2×C4×C73—22no—
69C2×C4×C73—21no—
70C2×C4×C73—21no—
71C2×C4×C73—42no—
72C2×C4×C73#9911yescharacteristic
73C2×C4×C73C2×C22⋊C411yescharacteristic
74C2×Dic73—42no—
75D292—42no—
76D292—42no—
77D292—82no—
78D292—84no—
79D292—82no—
80D292—82no—
81Dic146—82no—
82C8×C73—42no—
83C22×C73—41no—
84C22×C73#35412yes—
85C22×C73C2×#9911yescharacteristic
86C2×D73—42no—
87C2×D73—42no—
88C2×D73—84no—
89C2×D73—82no—
90C2×D73—82no—
91C2×D73—162no—
92#64408—164no—
93C4×C73—22no—
94C4×C73—21no—
95C4×C73—21no—
96C4×C73—41no—
97Dic73—82no—
98#13595—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—22no—
101C2×C73C2×#35411yescharacteristic
102D73—82no—
103D73—82no—
104#3138—731no—
105C2×#353—732no—
106#1412—734no—
107C73#1359511yescharacteristic, complemented, Sylow
108#533—731no—
109C2×#127—731no—
110#490—732no—
111#490—732no—
112#486—1462no—
113#418—1462no—
114C2×#99—1462no—
115C4×M16#6440811yescharacteristic, complemented
116#353—734no—
117#353—1462no—
118C22×D4—2921no—
119C22×D4—732no—
120C2×M16—21no—
121C4⋊Q8—1461no—
122#127—2921no—
123#127—732no—
124#127—731no—
125#127—731no—
126#125—732no—
127C22⋊Q8—2921no—
128C4×Q8—2921no—
129#117—1461no—
130C2×C22⋊C4—2922no—
131C2×C42C2×#6440811yescharacteristic
132#102—1464no—
133#102—1464no—
134#99—2922no—
135#99—1464no—
136#97C2×#6440812yescomplemented
137C4×C8—21no—
138C24—1462no—
139C2×Q8—2921no—
140C2×D4—2922no—
141C2×D4—2922no—
142C2×D4—2922no—
143C2×D4—2921no—
144C2×D4—2921no—
145C2×D4—5844no—
146C2×D4—732no—
147C2×D4—732no—
148C2×D4—1462no—
149C22×C4—21no—
150C22×C4—5842no—
151C22×C4C22⋊#6440811yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—2922no—
155C4⋊C4—2921no—
156C4⋊C4—1462no—
157C22⋊C4—2921no—
158C22⋊C4—2924no—
159C22⋊C4—5842no—
160C42—21no—
161C42—2921no—
162C42C22⋊#6440811yescharacteristic
163C42C22×#6440811yescharacteristic
164C23—2922no—
165C23—2922no—
166C23—5844no—
167C23—1462no—
168C23—1462no—
169C23#1455665011yescharacteristic
170Q8—5842no—
171D4—2922no—
172D4—2922no—
173D4—5842no—
174D4—5844no—
175D4—5842no—
176D4—5842no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—2922no—
182C2×C4—5844no—
183C2×C4—11684no—
184C2×C4#1455665011yescharacteristic
185C2×C4C2×C22⋊#6440811yescharacteristic
186C8—42no—
187C22—41no—
188C22—2922no—
189C22—2922no—
190C22—5842no—
191C22—5842no—
192C22—5844no—
193C22—11682no—
194C22#1798949412yes—
195C22C2×#1455665011yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—5842no—
201C4—11684no—
202C2—21no—
203C2—22no—
204C2—5842no—
205C2—5842no—
206C2C2×#1798949411yescharacteristic
subgroupG/H
centerC2C2×#17989494
commutatorC42×C73C22×C4
FrattiniC42C22×#64408
FittingC4×M16×C73C4
socleC2×C73C2×#354
prime radicalC2×#14556588C22
cosocle#14556588C23
hypercenterC4×M16#64408
hypocenterC73#13595
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #22625072C2semidirect
7. #17989144C4semidirect
9. #17988952C4semidirect
13. C4×M16×C73C4semidirect
24. #14556583C2×C4semidirect
32. #97×C73C2×C4semidirect
107. C73#13595semidirect
115. C4×M16#64408semidirect
136. #97C2×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179894922
#226250722
#226203062
#1359573