Finite Group Databasebeta

← #34009867

Subgroups — #34009867

Order 18688.

434 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 104 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#22625162C212yesmaximal
2#22624872C214yesmaximal, complemented
3#22612555C211yesmaximal, characteristic
4#17989603—22no—
5#17989432C2214yes—
6#17989430—22no—
7C2×#14556550C411yescharacteristic
8C2×#14556550C2211yescharacteristic
9#17988483—22no—
10#17988460—22no—
11#17988456C412yes—
12#17988452C2212yes—
13C22×C4⋊C4×C73C411yescharacteristic
14C4⋊#10596833—42no—
15C22⋊#10596833—24no—
16C4×#10596833—42no—
17#14556550—22no—
18#14556550—42no—
19#14556550C814yescomplemented
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73C2×C411yescharacteristic
22C2×C4⋊Dic73C2311yescharacteristic
23C2×#11924007—24no—
24C2×C22⋊Dic73—21no—
25C2×C22⋊Dic73D412yes—
26C2×C4×Dic73C2×C412yes—
27C2×C4×Dic73D412yes—
28#14556498—24no—
29C23×C4×C73—21no—
30C23×C4×C73C2×C411yescharacteristic
31C2×C4⋊C4×C73—21no—
32C2×C4⋊C4×C73—42no—
33C2×C4⋊C4×C73C2×C412yes—
34C2×#10596833—44no—
35C22×Dic73—22no—
36C22×Dic73C22⋊C412yes—
37C22×Dic73C2×D412yes—
38C4⋊Dic73—24no—
39C4⋊Dic73—41no—
40C4⋊Dic73—41no—
41#11924007—84no—
42C22⋊Dic73—24no—
43C22⋊Dic73—42no—
44C4×Dic73—22no—
45C4×Dic73—22no—
46C4×Dic73—22no—
47C4×Dic73C2×C814yes—
48C24×C73C22⋊C411yescharacteristic
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×C73C2×C812yes—
56C22×C4×C73M1612yes—
57C22×C4×C73C22×C411yescharacteristic
58C4⋊C4×C73—42no—
59C4⋊C4×C73—42no—
60C23×C73—21no—
61C23×C73—21no—
62C23×C73—41no—
63C23×C73—41no—
64C23×C73C22⋊C812yes—
65C23×C73C2×C22⋊C411yescharacteristic
66C2×C4×C73—21no—
67C2×C4×C73—21no—
68C2×C4×C73—24no—
69C2×C4×C73—24no—
70C2×C4×C73—44no—
71C2×C4×C73—42no—
72C2×C4×C73—84no—
73C2×C4×C73C22⋊C812yes—
74C2×C4×C73#9912yes—
75C2×C4×C73#10112yes—
76C2×C4×C73C22×C811yescharacteristic
77C2×C4×C73C2×M1611yescharacteristic
78C2×Dic73—24no—
79C2×Dic73—22no—
80C2×Dic73—22no—
81C2×Dic73—42no—
82C2×Dic73—42no—
83C2×Dic73—42no—
84C2×Dic73—42no—
85#10596833—84no—
86C22×C73—21no—
87C22×C73—21no—
88C22×C73—22no—
89C22×C73—42no—
90C22×C73—42no—
91C22×C73—41no—
92C22×C73—41no—
93C22×C73C2×C22⋊C811yescharacteristic
94C22×C73C2×#9911yescharacteristic
95C22×C73C2×#10111yescharacteristic
96C4×C73—21no—
97C4×C73—21no—
98C4×C73—82no—
99C4×C73#32414yes—
100Dic73—44no—
101Dic73—42no—
102Dic73—42no—
103#11449—731nomaximal, Sylow
104C2×C73—21no—
105C2×C73—21no—
106C2×C73—42no—
107C2×C73C2×#32411yescharacteristic
108C2×C73#212911yescharacteristic
109C2×C73#213411yescharacteristic
110#2317—731no—
111#1458—732no—
112#1324—734no—
113C73#1144911yescharacteristic, complemented, Sylow
114C2×#117—731no—
115C2×#117—731no—
116C22×C4⋊C4#6440811yescharacteristic
117#431—1462no—
118#389—1462no—
119#385—1462no—
120#384—732no—
121#382—732no—
122#324—734no—
123#323—1462no—
124C23×C4—21no—
125C23×C4C2×#6440811yescharacteristic
126#117—2922no—
127#117—734no—
128#117—1462no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—731no—
132C2×C4⋊C4—731no—
133C2×C4⋊C4—1464no—
134C2×C4⋊C4—1461no—
135C2×C4⋊C4C2×#6440812yes—
136C2×C22⋊C4—732no—
137C2×C22⋊C4—1461no—
138C2×C42—732no—
139C2×C42—732no—
140C4⋊C8—2922no—
141C22⋊C8—1464no—
142C4×C8—2922no—
143#95—1464no—
144C24C22⋊#6440811yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—82no—
150C22×C4—732no—
151C22×C4—732no—
152C22×C4—1462no—
153C22×C4C22⋊#6440811yescharacteristic
154C22×C4#1192403612yes—
155C22×C4#1192403812yes—
156C22×C4C22×#6440811yescharacteristic
157C2×C8—2924no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—2921no—
161C4⋊C4—2921no—
162C4⋊C4—5844no—
163C4⋊C4—1464no—
164C22⋊C4—2922no—
165C22⋊C4—1464no—
166C42—734no—
167C42—1462no—
168C42—1462no—
169C42—1462no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23#1455664812yes—
175C23C2×C22⋊#6440811yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4—2922no—
184C2×C4—2922no—
185C2×C4—2922no—
186C2×C4—2922no—
187C2×C4—1464no—
188C2×C4—1462no—
189C2×C4—1462no—
190C2×C4#1455664812yes—
191C2×C4#1455664912yes—
192C2×C4#1455665212yes—
193C2×C4C2×#1192403611yescharacteristic
194C2×C4C2×#1192403811yescharacteristic
195C8—5844no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×#1455664811yescharacteristic
204C22C2×#1455664911yescharacteristic
205C22C2×#1455665211yescharacteristic
206C4—21no—
207C4—21no—
208C4—82no—
209C4—2924no—
210C4—2922no—
211C4—2922no—
212C4#1798943314yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#1798943311yescharacteristic
217C2#2262621311yescharacteristic
218C2#2262623011yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#11449
LCSG ▷ C2×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
19. #14556550C8semidirect
113. C73#11449semidirect

Maximal subgroups

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

subgroupindex
#226251622
#226248722
#226125552
#1144973