Finite Group Databasebeta

← #34005264

Subgroups — #34005264

Order 18688.

514 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 112 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#22625787C212yesmaximal
2#22624809C214yesmaximal, complemented
3#22612334C211yesmaximal, characteristic
4C2×#14556658C2214yes—
5C2×C22⋊#10596833—22no—
6#17989453—22no—
7C2×#14556550C411yescharacteristic
8C22×C4×Dic73C2211yescharacteristic
9#17988456C2212yes—
10#17988452C412yes—
11#17988445—22no—
12C4×#11924007—22no—
13C2×#117×C73C411yescharacteristic
14C22×#10596833—24no—
15#14556658—28no—
16C22⋊#10596833—44no—
17C23×Dic73D412yes—
18#14556550—42no—
19C2×C4⋊Dic73C2×C412yes—
20C2×#11924007—22no—
21C2×#11924007—22no—
22C2×C22⋊Dic73—21no—
23C2×C4×Dic73—24no—
24C2×C4×Dic73—21no—
25C2×C4×Dic73C2×C411yescharacteristic
26C2×C4×Dic73D412yes—
27C2×C4×Dic73C2311yescharacteristic
28#14556498—24no—
29C23×C4×C73C2×C411yescharacteristic
30#117×C73—42no—
31C2×C4⋊C4×C73C2×C412yes—
32C2×C22⋊C4×C73—21no—
33C2×C42×C73—21no—
34C2×#10596833—44no—
35C2×#10596833—44no—
36C2×#10596833—44no—
37C22×Dic73—24no—
38C22×Dic73—24no—
39C22×Dic73—22no—
40C22×Dic73—22no—
41C22×Dic73—22no—
42C22×Dic73C22⋊C412yes—
43C22×Dic73C2×D412yes—
44C4⋊Dic73—42no—
45#11924007—44no—
46#11924007—42no—
47#11924007—42no—
48C22⋊Dic73—42no—
49C4×Dic73—22no—
50C4×Dic73—24no—
51C4×Dic73—42no—
52C4×Dic73#4114yes—
53C24×C73C22⋊C411yescharacteristic
54C22×C4×C73—22no—
55C22×C4×C73—22no—
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73C22⋊C411yescharacteristic
59C22×C4×C73C22×C411yescharacteristic
60C4⋊C4×C73—42no—
61C22⋊C4×C73—42no—
62C42×C73—42no—
63C23×C73—21no—
64C23×C73—21no—
65C23×C73—22no—
66C23×C73—22no—
67C23×C73#10012yes—
68C23×C73C2×C22⋊C411yescharacteristic
69C2×C4×C73—28no—
70C2×C4×C73—21no—
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73—42no—
75C2×C4×C73—42no—
76C2×C4×C73—42no—
77C2×C4×C73#10112yes—
78C2×C4×C73#11712yes—
79C2×Dic73—22no—
80C2×Dic73—22no—
81C2×Dic73—22no—
82C2×Dic73—44no—
83C2×Dic73—44no—
84C2×Dic73—42no—
85C2×Dic73—42no—
86C2×Dic73—42no—
87C2×Dic73#12314yes—
88C2×Dic73#13514yes—
89C2×Dic73AGL(1,ℤ/8)12yes—
90C2×Dic73#13712yes—
91#10596833—84no—
92C22×C73—24no—
93C22×C73—22no—
94C22×C73—41no—
95C22×C73—41no—
96C22×C73—42no—
97C22×C73#38111yescharacteristic
98C22×C73C2×#10011yescharacteristic
99C22×C73C2×#10111yescharacteristic
100C22×C73#41612yes—
101C22×C73#41712yes—
102C4×C73—22no—
103C4×C73—22no—
104C4×C73—82no—
105Dic73—42no—
106Dic73—82no—
107Dic73#48214yes—
108Dic73#48514yes—
109#8729—731nomaximal, Sylow
110C2×C73—42no—
111C2×C73#154411yescharacteristic
112C2×C73#155112yes—
113C2×C73#156111yescharacteristic
114C2×C73#182111yescharacteristic
115C2×C73#182512yes—
116#2285—731no—
117#1797—732no—
118#1296—734no—
119C73#872911yescharacteristic, complemented, Sylow
120C2×#117—731no—
121C2×#117#6440811yescharacteristic
122C22×C42—731no—
123C2×C4⋊C8—734no—
124C2×C22⋊C8—1462no—
125#384—732no—
126#382—732no—
127#380—1462no—
128C4×C4⋊C4—1462no—
129#336—1462no—
130C23×C4—732no—
131C23×C4C2×#6440811yescharacteristic
132C22×C8—1464no—
133#117—42no—
134#117—2922no—
135C2×C4⋊C4—732no—
136C2×C4⋊C4—1462no—
137C2×C4⋊C4—1462no—
138C2×C4⋊C4C2×#6440812yes—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—1461no—
141C2×C42—21no—
142C2×C42—731no—
143C2×C42—732no—
144C2×C42—731no—
145C2×C42—1461no—
146C2×C42—1464no—
147C4⋊C8—1468no—
148C22⋊C8—2924no—
149#95—1464no—
150C24C22⋊#6440811yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—732no—
156C22×C4—732no—
157C22×C4—1462no—
158C22×C4—1462no—
159C22×C4—1464no—
160C22×C4—1464no—
161C22×C4—1462no—
162C22×C4C22⋊#6440811yescharacteristic
163C22×C4C22×#6440811yescharacteristic
164C2×C8—2924no—
165C2×C8—2924no—
166C2×C8—2924no—
167C4⋊C4—42no—
168C4⋊C4—2922no—
169C4⋊C4—2924no—
170C4⋊C4—2922no—
171C4⋊C4—2922no—
172C22⋊C4—42no—
173C22⋊C4—2922no—
174C42—42no—
175C42—2922no—
176C42—734no—
177C42—1464no—
178C42—1462no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23#1455665112yes—
184C23C2×C22⋊#6440811yescharacteristic
185C2×C4—28no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—2924no—
194C2×C4—2924no—
195C2×C4—2922no—
196C2×C4—2922no—
197C2×C4—2922no—
198C2×C4—734no—
199C2×C4—734no—
200C2×C4—732no—
201C2×C4—732no—
202C2×C4—1462no—
203C2×C4—1462no—
204C2×C4—1462no—
205C2×C4#1455665212yes—
206C2×C4#1455667012yes—
207C8—5844no—
208C22—22no—
209C22—24no—
210C22—42no—
211C22—41no—
212C22—41no—
213C22#1798953011yescharacteristic
214C22C2×#1455665111yescharacteristic
215C22C2×#1455665211yescharacteristic
216C22#1798957712yes—
217C22#1798957812yes—
218C4—22no—
219C4—22no—
220C4—82no—
221C4—2922no—
222C4—5842no—
223C4—734no—
224C4—734no—
225C2—42no—
226C2#2262543611yescharacteristic
227C2#2262545012yes—
228C2#2262547011yescharacteristic
229C2#2262583011yescharacteristic
230C2#2262583612yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#8729
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22624809C2semidirect
119. C73#8729semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226248092
#226123342
#872973