Finite Group Databasebeta

← #32995398

Subgroups — #32995398

Order 18688.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#22608265C212yesmaximal
2#22607877C214yesmaximal, complemented
3C73×#2317C211yesmaximal, characteristic
4#17988644—22no—
5#17988313C2214yes—
6#17988311—22no—
7C2×#117×C73C411yescharacteristic
8C2×#117×C73C2211yescharacteristic
9C22×C4⋊C4×C73C411yescharacteristic
10#389×C73—22no—
11#385×C73—22no—
12#384×C73C2212yes—
13#382×C73C412yes—
14C4⋊#10596831—42no—
15C22⋊#10596831—24no—
16#14556502—42no—
17C23×C4×C73—21no—
18C23×C4×C73C2×C411yescharacteristic
19#117×C73—24no—
20#117×C73—42no—
21C2×C4⋊C4×C73—21no—
22C2×C4⋊C4×C73—21no—
23C2×C4⋊C4×C73—24no—
24C2×C4⋊C4×C73—42no—
25C2×C4⋊C4×C73C2×C412yes—
26C2×C4⋊C4×C73C2×C411yescharacteristic
27C2×C4⋊C4×C73C2311yescharacteristic
28C2×C22⋊C4×C73—21no—
29C2×C22⋊C4×C73D412yes—
30C2×C42×C73C2×C412yes—
31C2×C42×C73D412yes—
32#95×C73—24no—
33C2×#10596831—44no—
34C24×C73C22⋊C411yescharacteristic
35C22×C4×C73—22no—
36C22×C4×C73—22no—
37C22×C4×C73—22no—
38C22×C4×C73—22no—
39C22×C4×C73—22no—
40C22×C4×C73—42no—
41C22×C4×C73—82no—
42C22×C4×C73C22⋊C412yes—
43C22×C4×C73C22⋊C411yescharacteristic
44C22×C4×C73C22×C411yescharacteristic
45C22×C4×C73C2×D412yes—
46C4⋊C4×C73—24no—
47C4⋊C4×C73—41no—
48C4⋊C4×C73—41no—
49C4⋊C4×C73—42no—
50C4⋊C4×C73—42no—
51C4⋊C4×C73—84no—
52C22⋊C4×C73—24no—
53C22⋊C4×C73—42no—
54C42×C73—22no—
55C42×C73—22no—
56C42×C73—22no—
57C42×C73M1614yes—
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—24no—
70C2×C4×C73—22no—
71C2×C4×C73—22no—
72C2×C4×C73—42no—
73C2×C4×C73—42no—
74C2×C4×C73—42no—
75C2×C4×C73—42no—
76C2×C4×C73—44no—
77C2×C4×C73—42no—
78C2×C4×C73—84no—
79C2×C4×C73#9912yes—
80C2×C4×C73C2×M1612yes—
81#10596831—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—44no—
95C4×C73—42no—
96C4×C73—42no—
97C4×C73—82no—
98#11516—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—21no—
101C2×C73—42no—
102C2×C73#146311yescharacteristic
103C2×C73#212811yescharacteristic
104C2×C73#213511yescharacteristic
105#2317D7311yescharacteristic
106#1464—732no—
107#1325—734no—
108C73#1151611yescharacteristic, complemented, Sylow
109C2×#117Dic7311yescharacteristic
110C2×#117C2×D7311yescharacteristic
111C22×C4⋊C4Dic7311yescharacteristic
112#432—1462no—
113#389—22no—
114#385—22no—
115#384C2×D7312yes—
116#382Dic7312yes—
117#324—734no—
118#323—1462no—
119C23×C4—21no—
120C23×C4C2×Dic7311yescharacteristic
121#117—24no—
122#117—42no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×Dic7312yes—
128C2×C4⋊C4C2×Dic7311yescharacteristic
129C2×C4⋊C4C22×D7311yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4C22⋊D7312yes—
132C2×C42C2×Dic7312yes—
133C2×C42C22⋊D7312yes—
134C4⋊C8—2922no—
135C22⋊C8—1464no—
136#97—2922no—
137#95—24no—
138C24C22⋊Dic7311yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—82no—
146C22×C4C22⋊Dic7312yes—
147C22×C4C22⋊Dic7311yescharacteristic
148C22×C4C22×Dic7311yescharacteristic
149C22×C4C2×C22⋊D7312yes—
150C2×C8—2924no—
151C4⋊C4—24no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42#1192401214yes—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#1455651012yes—
168C23C2×C22⋊Dic7311yescharacteristic
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4#1455650612yes—
185C2×C4C2×#1192401212yes—
186C8—5844no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22#1798855111yescharacteristic
195C22C2×#1455650611yescharacteristic
196C22C2×#1455651011yescharacteristic
197C4—22no—
198C4—22no—
199C4—44no—
200C4—42no—
201C4—42no—
202C4—82no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2#2260826111yescharacteristic
207C2#2261160311yescharacteristic
208C2#2261162611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2317C2
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊C4×C73C23
hypercenter#2317D73
hypocenterC73#11516
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #22607877C2semidirect
108. C73#11516semidirect

Maximal subgroups

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

subgroupindex
#226082652
#226078772
C73×#23172
#1151673