Finite Group Databasebeta

← #34005265

Subgroups — #34005265

Order 18688.

514 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 108 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#22625788C212yesmaximal
2#22624809C214yesmaximal, complemented
3#22612337C211yesmaximal, characteristic
4C2×#14556658C2214yes—
5C2×#14556648—22no—
6#17989453—22no—
7C2×#14556549C412yes—
8C2×C42×D73C2211yescharacteristic
9#17988456C2212yes—
10#17988453C411yescharacteristic
11#17988446—22no—
12#17988438—22no—
13#382×C73C411yescharacteristic
14C22×#10596833—24no—
15#14556658—28no—
16#14556648—44no—
17C22×C4×D73C2×C411yescharacteristic
18C22×C4×D73D412yes—
19#14556549—44no—
20C2×C4⋊Dic73C2×C412yes—
21C2×#11924007—22no—
22C2×#11924007—22no—
23C2×#11924005—22no—
24C2×C4×Dic73—22no—
25C2×C4×Dic73D412yes—
26C2×C4×Dic73C2311yescharacteristic
27C42×D73—24no—
28#14556498—22no—
29C2×C4⋊C4×C73C2×C412yes—
30C2×C42×C73C2×C411yescharacteristic
31#95×C73—22no—
32C2×#10596833—44no—
33C2×#10596833—44no—
34C2×#10596833—44no—
35C23×D73C22⋊C411yescharacteristic
36C22×Dic73—22no—
37C22×Dic73C22⋊C411yescharacteristic
38C22×Dic73C2×D412yes—
39C2×C4×D73—22no—
40C2×C4×D73—22no—
41C2×C4×D73—22no—
42C2×C4×D73—24no—
43C2×C4×D73—24no—
44C2×C4×D73—22no—
45C2×C4×D73—22no—
46C4⋊Dic73—42no—
47#11924007—44no—
48#11924007—42no—
49#11924007—42no—
50#11924005—44no—
51C4×Dic73—24no—
52C4×Dic73—44no—
53C4×Dic73#4114yes—
54C22×C4×C73—22no—
55C22×C4×C73C22⋊C412yes—
56C22×C4×C73C22×C411yescharacteristic
57C4⋊C4×C73—42no—
58C42×C73—22no—
59C23×C73C2×C22⋊C411yescharacteristic
60C22×D73—21no—
61C22×D73—21no—
62C22×D73—22no—
63C22×D73—22no—
64C22×D73#10012yes—
65C2×C4×C73—21no—
66C2×C4×C73—21no—
67C2×C4×C73—22no—
68C2×C4×C73—22no—
69C2×C4×C73—22no—
70C2×C4×C73—42no—
71C2×C4×C73—42no—
72C2×C4×C73—42no—
73C2×C4×C73#11712yes—
74C2×Dic73—21no—
75C2×Dic73—21no—
76C2×Dic73—42no—
77C2×Dic73—42no—
78C2×Dic73—42no—
79C2×Dic73#10112yes—
80C2×Dic73#12314yes—
81C2×Dic73#13514yes—
82C2×Dic73AGL(1,ℤ/8)12yes—
83C2×Dic73#13712yes—
84C4×D73—28no—
85C4×D73—44no—
86C4×D73—44no—
87#10596833—84no—
88C22×C73#38111yescharacteristic
89C22×C73C2×#10011yescharacteristic
90C22×C73C2×#10111yescharacteristic
91C22×C73#41612yes—
92C22×C73#41712yes—
93C2×D73—24no—
94C2×D73—22no—
95C2×D73—41no—
96C2×D73—41no—
97C2×D73—42no—
98C4×C73—22no—
99C4×C73—42no—
100C4×C73—82no—
101Dic73—22no—
102Dic73—82no—
103Dic73#48214yes—
104Dic73#48514yes—
105#8729—731nomaximal, Sylow
106C2×C73#154411yescharacteristic
107C2×C73#155112yes—
108C2×C73#156111yescharacteristic
109C2×C73#182111yescharacteristic
110C2×C73#182512yes—
111D73—42no—
112#2285—731no—
113#1797—732no—
114#1296—734no—
115C73#872911yescharacteristic, complemented, Sylow
116C2×#117—732no—
117C22×C42—731no—
118C2×C4⋊C8—734no—
119C2×C22⋊C8—1462no—
120#384—732no—
121#382—731no—
122#382#6440811yescharacteristic
123#380—1462no—
124C4×C4⋊C4—1462no—
125#336—1462no—
126C23×C4—732no—
127C23×C4—731no—
128C22×C8—1464no—
129#117—2924no—
130C2×C4⋊C4—732no—
131C2×C4⋊C4—1462no—
132C2×C4⋊C4—1462no—
133C2×C4⋊C4C2×#6440812yes—
134C2×C22⋊C4—1462no—
135C2×C42—731no—
136C2×C42—732no—
137C2×C42—1464no—
138C2×C42—1462no—
139C2×C42C2×#6440811yescharacteristic
140C4⋊C8—1468no—
141C22⋊C8—2924no—
142#95—22no—
143#95—1462no—
144C24—731no—
145C22×C4—22no—
146C22×C4—732no—
147C22×C4—731no—
148C22×C4—1462no—
149C22×C4—1464no—
150C22×C4—1464no—
151C22×C4—1462no—
152C22×C4—1462no—
153C22×C4—1462no—
154C22×C4—1462no—
155C22×C4—1462no—
156C22×C4C22⋊#6440812yes—
157C22×C4C22×#6440811yescharacteristic
158C2×C8—2924no—
159C2×C8—2924no—
160C2×C8—2924no—
161C4⋊C4—42no—
162C4⋊C4—2922no—
163C4⋊C4—2924no—
164C4⋊C4—2922no—
165C4⋊C4—2922no—
166C22⋊C4—2924no—
167C42—22no—
168C42—2924no—
169C42—734no—
170C42—1464no—
171C23—732no—
172C23—1461no—
173C23—1461no—
174C23—1462no—
175C23—1462no—
176C23C2×C22⋊#6440811yescharacteristic
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—2924no—
186C2×C4—2924no—
187C2×C4—2922no—
188C2×C4—2922no—
189C2×C4—2922no—
190C2×C4—734no—
191C2×C4—734no—
192C2×C4—732no—
193C2×C4—732no—
194C2×C4—732no—
195C2×C4—1468no—
196C2×C4—1461no—
197C2×C4—1461no—
198C2×C4#1455667012yes—
199C8—5844no—
200C22—2922no—
201C22—2921no—
202C22—2921no—
203C22—1462no—
204C22—1464no—
205C22#1798953011yescharacteristic
206C22D73⋊#10011yescharacteristic
207C22D73⋊#10111yescharacteristic
208C22#1798957712yes—
209C22#1798957812yes—
210C4—22no—
211C4—42no—
212C4—82no—
213C4—5842no—
214C4—734no—
215C4—734no—
216C4—1462no—
217C2—2922no—
218C2#2262543711yescharacteristic
219C2#2262545112yes—
220C2#2262547111yescharacteristic
221C2#2262583111yescharacteristic
222C2#2262583612yes—
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#382×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#382#64408
hypocenterC73#8729
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

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

Maximal subgroups

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

subgroupindex
#226257882
#226248092
#226123372
#872973