Finite Group Databasebeta

← #34010106

Subgroups — #34010106

Order 18688.

446 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 80 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#22624875C214yesmaximal
3#22612415C211yesmaximal, characteristic
4#17989607—22no—
5#17989432—22no—
6#17989430C2214yes—
7C2×#14556550C2211yescharacteristic
8C22×C4×Dic73C411yescharacteristic
9#17988486—22no—
10#17988462C412yes—
11#17988462C2212yes—
12C4×C4⋊Dic73—22no—
13C2×#117×C73C411yescharacteristic
14#14556658—42no—
15C22⋊#10596833—24no—
16#14556646—42no—
17C23×Dic73C2×C412yes—
18#14556550—42no—
19C2×C4⋊Dic73—22no—
20C2×C4⋊Dic73—22no—
21C2×C4⋊Dic73D412yes—
22C2×C22⋊Dic73C2×C411yescharacteristic
23C2×C22⋊Dic73C2311yescharacteristic
24C2×C4×Dic73—21no—
25C2×C4×Dic73—21no—
26C2×C4×Dic73—42no—
27C2×C4×Dic73D412yes—
28#14556498—22no—
29#14556498—22no—
30C23×C4×C73C2×C411yescharacteristic
31#117×C73—24no—
32C2×C4⋊C4×C73—21no—
33C2×C22⋊C4×C73C2×C412yes—
34C2×C42×C73—21no—
35C2×#10596833—44no—
36C22×Dic73—22no—
37C22×Dic73—44no—
38C22×Dic73—42no—
39C22×Dic73C22⋊C412yes—
40C22×Dic73C2×D412yes—
41C4⋊Dic73—42no—
42C4⋊Dic73—44no—
43C4⋊Dic73—44no—
44C22⋊Dic73—41no—
45C22⋊Dic73—41no—
46C4×Dic73—22no—
47C4×Dic73—22no—
48C4×Dic73—42no—
49C4×Dic73—42no—
50C24×C73C22×C411yescharacteristic
51C22×C4×C73—22no—
52C22×C4×C73—22no—
53C22×C4×C73—22no—
54C22×C4×C73—22no—
55C22×C4×C73C22⋊C411yescharacteristic
56C22×C4×C73C22⋊C411yescharacteristic
57C4⋊C4×C73—24no—
58C22⋊C4×C73—22no—
59C22⋊C4×C73M1614yes—
60C42×C73—24no—
61C23×C73—21no—
62C23×C73—21no—
63C23×C73—42no—
64C23×C73C2×C22⋊C411yescharacteristic
65C23×C73C2×M1612yes—
66C2×C4×C73—28no—
67C2×C4×C73—21no—
68C2×C4×C73—21no—
69C2×C4×C73—21no—
70C2×C4×C73—21no—
71C2×C4×C73—24no—
72C2×C4×C73—42no—
73C2×C4×C73—42no—
74C2×C4×C73#9912yes—
75C2×C4×C73#10112yes—
76C2×Dic73—24no—
77C2×Dic73—22no—
78C2×Dic73—22no—
79C2×Dic73—42no—
80C2×Dic73—44no—
81C2×Dic73—42no—
82C2×Dic73—42no—
83C2×Dic73—42no—
84C2×Dic73—44no—
85#10596833—84no—
86C22×C73—22no—
87C22×C73—22no—
88C22×C73—42no—
89C22×C73—42no—
90C22×C73—42no—
91C22×C73#40711yescharacteristic
92C22×C73C2×#9911yescharacteristic
93C22×C73C2×#10111yescharacteristic
94C4×C73—22no—
95C4×C73—22no—
96C4×C73—44no—
97Dic73—42no—
98Dic73—42no—
99Dic73—44no—
100#11518—731nomaximal, Sylow
101C2×C73—21no—
102C2×C73—21no—
103C2×C73—42no—
104C2×C73#146311yescharacteristic
105C2×C73#212012yes—
106#2296—731no—
107#1464—732no—
108#1326—734no—
109C73#1151811yescharacteristic, complemented, Sylow
110C2×#117—731no—
111C2×#117#6440811yescharacteristic
112C22×C42—731no—
113#432—1462no—
114#389—1462no—
115#385—732no—
116#385—732no—
117C4×C4⋊C4—1462no—
118#324—1462no—
119#323—734no—
120C23×C4—732no—
121C23×C4C2×#6440811yescharacteristic
122#117—24no—
123#117—2922no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—732no—
126C2×C4⋊C4—1462no—
127C2×C4⋊C4—1462no—
128C2×C22⋊C4—731no—
129C2×C22⋊C4—731no—
130C2×C22⋊C4C2×#6440812yes—
131C2×C42—21no—
132C2×C42—2922no—
133C2×C42—732no—
134C2×C42—1461no—
135C2×C42—1461no—
136C4⋊C8—2922no—
137C22⋊C8—1464no—
138#97—2922no—
139#95—1462no—
140#95—1462no—
141C24C22×#6440811yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—2922no—
147C22×C4—2924no—
148C22×C4—732no—
149C22×C4—732no—
150C22×C4—1462no—
151C22×C4C22⋊#6440811yescharacteristic
152C22×C4C22⋊#6440811yescharacteristic
153C2×C8—2924no—
154C4⋊C4—24no—
155C4⋊C4—2922no—
156C4⋊C4—2924no—
157C4⋊C4—2924no—
158C22⋊C4—22no—
159C22⋊C4—2921no—
160C22⋊C4—2921no—
161C22⋊C4#1192403714yes—
162C42—24no—
163C42—2922no—
164C42—2922no—
165C42—1462no—
166C42—1462no—
167C23—21no—
168C23—21no—
169C23—42no—
170C23C2×C22⋊#6440811yescharacteristic
171C23C2×#1192403712yes—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—2922no—
181C2×C4—2924no—
182C2×C4—2922no—
183C2×C4—2922no—
184C2×C4—2922no—
185C2×C4—2924no—
186C2×C4—1464no—
187C2×C4—1462no—
188C2×C4—1462no—
189C2×C4#1455665012yes—
190C2×C4#1455665212yes—
191C8—5844no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22#1798955711yescharacteristic
198C22C2×#1455665011yescharacteristic
199C22C2×#1455665211yescharacteristic
200C4—22no—
201C4—22no—
202C4—44no—
203C4—2922no—
204C4—2924no—
205C4—2922no—
206C2—21no—
207C2—21no—
208C2—42no—
209C2#2262517911yescharacteristic
210C2#2262618412yes—
subgroupG/H
centerC22C2×#14556650
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
cosocleC2×C22⋊Dic73C23
hypercenterC2×#117#64408
hypocenterC73#11518
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
109. C73#11518semidirect

Maximal subgroups

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

subgroupindex
#226251822
#226248752
#226124152
#1151873