Finite Group Databasebeta

← #34010099

Subgroups — #34010099

Order 18688.

418 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 76 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#22625182C212yesmaximal
2#22624872C214yesmaximal, complemented
3#22612555C211yesmaximal, characteristic
4#17989607—22no—
5#17989432C2214yes—
6#17989430—22no—
7C2×#14556550C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988483—22no—
10#17988460—22no—
11#17988456C412yes—
12#17988452C2212yes—
13C22×C4⋊C4×C73C411yescharacteristic
14#14556658—42no—
15C22⋊#10596833—24no—
16#14556646—42no—
17#14556550—24no—
18#14556550—42no—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73C2×C411yescharacteristic
21C2×C4⋊Dic73C2311yescharacteristic
22C2×#11924007—24no—
23C2×C22⋊Dic73—21no—
24C2×C22⋊Dic73D412yes—
25C2×C4×Dic73C2×C412yes—
26C2×C4×Dic73D412yes—
27#14556498—24no—
28C23×C4×C73—21no—
29C23×C4×C73C2×C411yescharacteristic
30C2×C4⋊C4×C73—21no—
31C2×C4⋊C4×C73—42no—
32C2×C4⋊C4×C73C2×C412yes—
33C2×#10596833—44no—
34C22×Dic73—22no—
35C22×Dic73C22⋊C412yes—
36C22×Dic73C2×D412yes—
37C4⋊Dic73—24no—
38C4⋊Dic73—41no—
39C4⋊Dic73—41no—
40#11924007—84no—
41C22⋊Dic73—24no—
42C22⋊Dic73—42no—
43C4×Dic73—22no—
44C4×Dic73—22no—
45C4×Dic73—22no—
46C4×Dic73M1614yes—
47C24×C73C22⋊C411yescharacteristic
48C22×C4×C73—22no—
49C22×C4×C73—22no—
50C22×C4×C73—22no—
51C22×C4×C73—22no—
52C22×C4×C73—42no—
53C22×C4×C73—82no—
54C22×C4×C73C22⋊C411yescharacteristic
55C22×C4×C73C22×C411yescharacteristic
56C4⋊C4×C73—42no—
57C4⋊C4×C73—42no—
58C23×C73—21no—
59C23×C73—21no—
60C23×C73—41no—
61C23×C73—41no—
62C23×C73#10112yes—
63C23×C73C2×C22⋊C411yescharacteristic
64C2×C4×C73—28no—
65C2×C4×C73—21no—
66C2×C4×C73—21no—
67C2×C4×C73—21no—
68C2×C4×C73—21no—
69C2×C4×C73—44no—
70C2×C4×C73—42no—
71C2×C4×C73—84no—
72C2×C4×C73#9912yes—
73C2×C4×C73C2×M1612yes—
74C2×Dic73—24no—
75C2×Dic73—22no—
76C2×Dic73—22no—
77C2×Dic73—42no—
78C2×Dic73—42no—
79C2×Dic73—42no—
80C2×Dic73—42no—
81#10596833—84no—
82C22×C73—21no—
83C22×C73—21no—
84C22×C73—22no—
85C22×C73—42no—
86C22×C73—42no—
87C22×C73—41no—
88C22×C73—41no—
89C22×C73#40711yescharacteristic
90C22×C73C2×#9911yescharacteristic
91C22×C73C2×#10111yescharacteristic
92C4×C73—22no—
93C4×C73—22no—
94C4×C73—82no—
95Dic73—44no—
96Dic73—42no—
97Dic73—42no—
98#11515—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—21no—
101C2×C73—42no—
102C2×C73#146311yescharacteristic
103C2×C73#212911yescharacteristic
104C2×C73#213411yescharacteristic
105#2317—731no—
106#1464—732no—
107#1324—734no—
108C73#1151511yescharacteristic, complemented, Sylow
109C2×#117—731no—
110C2×#117—731no—
111C22×C4⋊C4#6440811yescharacteristic
112#432—1462no—
113#389—1462no—
114#385—1462no—
115#384—732no—
116#382—732no—
117#324—734no—
118#323—1462no—
119C23×C4—21no—
120C23×C4C2×#6440811yescharacteristic
121#117—2922no—
122#117—1464no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—731no—
126C2×C4⋊C4—731no—
127C2×C4⋊C4—1464no—
128C2×C4⋊C4—1461no—
129C2×C4⋊C4C2×#6440812yes—
130C2×C22⋊C4—732no—
131C2×C22⋊C4—1461no—
132C2×C42—732no—
133C2×C42—732no—
134C4⋊C8—2922no—
135C22⋊C8—1464no—
136#97—2922no—
137#95—1464no—
138C24C22⋊#6440811yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—82no—
145C22×C4—732no—
146C22×C4—732no—
147C22×C4—1462no—
148C22×C4C22⋊#6440811yescharacteristic
149C22×C4C22×#6440811yescharacteristic
150C2×C8—2924no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—2921no—
154C4⋊C4—2921no—
155C4⋊C4—5844no—
156C4⋊C4—1464no—
157C22⋊C4—2922no—
158C22⋊C4—1464no—
159C42—734no—
160C42—1462no—
161C42—1462no—
162C42—1462no—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#1455665212yes—
168C23C2×C22⋊#6440811yescharacteristic
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—84no—
177C2×C4—2922no—
178C2×C4—2922no—
179C2×C4—2922no—
180C2×C4—2922no—
181C2×C4—1464no—
182C2×C4—1462no—
183C2×C4—1462no—
184C2×C4#1455664912yes—
185C2×C4D73⋊M1612yes—
186C8—5844no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22#1798955911yescharacteristic
195C22C2×#1455664911yescharacteristic
196C22C2×#1455665211yescharacteristic
197C4—22no—
198C4—22no—
199C4—82no—
200C4—2924no—
201C4—2922no—
202C4—2922no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2#2262518111yescharacteristic
207C2#2262621311yescharacteristic
208C2#2262623011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#11515
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624872C2semidirect
108. C73#11515semidirect

Maximal subgroups

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

subgroupindex
#226251822
#226248722
#226125552
#1151573