Finite Group Databasebeta

← #32988276

Subgroups — #32988276

Order 18688.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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#22621072C212yesmaximal
2#22610751C211yesmaximal, characteristic
3#22608495C212yesmaximal
4#22607776C211yesmaximal, characteristic
5C73×#3162C211yesmaximal, characteristic
6C2×#14556591C2211yescharacteristic
7#17988983—22no—
8#17988937—22no—
9#17988915—24no—
10#17988872—24no—
11#17988640C2212yes—
12C2×#14556516—22no—
13#17988453C2211yescharacteristic
14#17988324—24no—
15#533×C73C2212yes—
16#501×C73—22no—
17#501×C73—22no—
18#427×C73—22no—
19C2×#97×C73C2211yescharacteristic
20C22×Dic146—21no—
21#14556591—22no—
22#14556591—21no—
23#14556591D412yes—
24C2×C4⋊Dic73—22no—
25C2×C4⋊Dic73—21no—
26#14556522—28no—
27C4⋊#10596831—22no—
28#14556516—44no—
29#14556515—28no—
30C22⋊#10596831D414yes—
31C4×#10596831—22no—
32#14556498—22no—
33C22×C8×C73D412yes—
34C4⋊Q8×C73D412yes—
35C4⋊Q8×C73D412yes—
36#125×C73—22no—
37C22⋊Q8×C73—22no—
38C22⋊Q8×C73—22no—
39C4×Q8×C73—22no—
40C4×Q8×C73—22no—
41#117×C73—22no—
42C2×C42×C73C2311yescharacteristic
43#107×C73—24no—
44#106×C73—22no—
45#106×C73—22no—
46#97×C73—22no—
47C2×Dic146—22no—
48C2×Dic146—42no—
49C2×Dic146—42no—
50C2×Dic146—41no—
51C22×Dic73—41no—
52C22×Dic73—41no—
53C2×#10596831—24no—
54C4⋊Dic73—24no—
55C4⋊Dic73—42no—
56C4⋊Dic73—42no—
57C2×Q8×C73—22no—
58C2×Q8×C73—22no—
59C22×C4×C73C2×D412yes—
60C22×C4×C73C2×D411yescharacteristic
61C2×C8×C73—22no—
62C2×C8×C73—22no—
63C2×C8×C73SD1614yes—
64C4⋊C4×C73—22no—
65C4⋊C4×C73—22no—
66C4⋊C4×C73—22no—
67C4⋊C4×C73—22no—
68C4⋊C4×C73—42no—
69C4⋊C4×C73—42no—
70C22⋊C4×C73—42no—
71C42×C73—42no—
72C42×C73C2×D411yescharacteristic
73C42×C73C2×D412yes—
74C42×C73#4111yescharacteristic
75C23×C73#12711yescharacteristic
76Q8×C73—44no—
77Q8×C73—82no—
78C2×C4×C73—42no—
79C2×C4×C73—42no—
80C2×C4×C73—42no—
81C2×C4×C73#12012yes—
82C2×C4×C73#12011yescharacteristic
83C2×C4×C73#12111yescharacteristic
84C2×C4×C73#12112yes—
85C2×C4×C73C2×SD1612yes—
86C2×C4×C73#13712yes—
87C2×Dic73—42no—
88C2×Dic73—81no—
89C2×Dic73—81no—
90C2×Dic73—82no—
91C2×Dic73—81no—
92Dic146—44no—
93Dic146—82no—
94C8×C73—24no—
95#10596831—44no—
96C22×C73—22no—
97C22×C73#39211yescharacteristic
98C22×C73#49211yescharacteristic
99C22×C73#49711yescharacteristic
100C4×C73—22no—
101C4×C73—22no—
102C4×C73—82no—
103C4×C73—82no—
104C4×C73#44812yes—
105C4×C73#46112yes—
106Dic73—82no—
107Dic73—161no—
108#10227—731nomaximal, Sylow
109C2×C73—21no—
110C2×C73—21no—
111C2×C73#200912yes—
112C2×C73#201311yescharacteristic
113#3229—732no—
114#3162D7311yescharacteristic
115#1960—731no—
116#1518—732no—
117#1281—731no—
118C73#1022711yescharacteristic, complemented, Sylow
119#533C2×D7312yes—
120C2×C4⋊Q8—731no—
121#501—22no—
122#501—22no—
123#498—1462no—
124#487—1462no—
125#483—1464no—
126#475—1464no—
127#431—732no—
128#427—22no—
129C2×#103—1462no—
130C2×#97C2×D7311yescharacteristic
131#382—731no—
132#328—1464no—
133C22×Q8—1461no—
134C22×C8C22⋊D7312yes—
135C4⋊Q8—732no—
136C4⋊Q8—1462no—
137C4⋊Q8—1461no—
138C4⋊Q8C22⋊D7312yes—
139C4⋊Q8C22⋊D7312yes—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—22no—
146C2×C4⋊C4—1461no—
147C2×C4⋊C4—1462no—
148C2×C42C22×D7311yescharacteristic
149#107—24no—
150#106—22no—
151#106—22no—
152#106—1468no—
153C4⋊C8—1462no—
154#103—2924no—
155#103—1468no—
156C22⋊C8—734no—
157#97—22no—
158C4×C8—1462no—
159#95—1462no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—2921no—
163C2×Q8—2922no—
164C2×Q8—2922no—
165C2×Q8—1462no—
166C22×C4—2921no—
167C22×C4—2921no—
168C22×C4D4×D7312yes—
169C22×C4C2×C22⋊D7311yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—1464no—
173C2×C8#1192401514yes—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—2922no—
181C4⋊C4—2922no—
182C4⋊C4—1464no—
183C22⋊C4—42no—
184C42—42no—
185C42D4×D7311yescharacteristic
186C42C2×C22⋊D7312yes—
187C42#1192402711yescharacteristic
188C23#1455658711yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—2924no—
192Q8—5842no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—2922no—
197C2×C4—5841no—
198C2×C4—5841no—
199C2×C4—5842no—
200C2×C4—5841no—
201C2×C4#1455655912yes—
202C2×C4#1455656011yescharacteristic
203C2×C4#1455656211yescharacteristic
204C2×C4#1455656312yes—
205C2×C4C2×#1192401512yes—
206C2×C4#1455662212yes—
207C8—24no—
208C8—2924no—
209C22—22no—
210C22#1798849511yescharacteristic
211C22#1798896111yescharacteristic
212C22#1798897911yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—5842no—
218C4—11681no—
219C4#1798872612yes—
220C4#1798878912yes—
221C2—21no—
222C2—21no—
223C2#2261098812yes—
224C2#2261101411yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#3162C2
socleC22×C73#392
prime radicalC23×C73#127
hypercenter#3162D73
hypocenterC73#10227
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
118. C73#10227semidirect

Maximal subgroups

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

subgroupindex
#226210722
#226107512
#226084952
#226077762
C73×#31622
#1022773