Finite Group Databasebeta

← #32999251

Subgroups — #32999251

Order 18688.

310 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 67 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#22621276C211yesmaximal, characteristic
2#22612128C212yesmaximal
3#22607942C212yesmaximal
4#22607871C211yesmaximal, characteristic
5C73×#2086C211yesmaximal, characteristic
6#17989144—21no—
7#17988883—22no—
8#17988828—21no—
9#17988828C2212yes—
10#17988798—22no—
11#17988414—22no—
12#17988390—22no—
13C22⋊#11924003C2212yes—
14C2×#107×C73C2211yescharacteristic
15C2×C4⋊C8×C73C2211yescharacteristic
16#389×C73—21no—
17#340×C73—21no—
18#340×C73C2211yescharacteristic
19C2×Dic292—21no—
20C2×Dic292D412yes—
21#14556591—21no—
22#14556591—21no—
23#14556583—41no—
24#14556570—22no—
25#14556570—41no—
26C4×Dic146—22no—
27C4×Dic146—41no—
28#14556550—41no—
29C2×#11924003—22no—
30#14556525—21no—
31#14556525D412yes—
32#14556516—22no—
33#14556516—22no—
34C22×C8×C73D411yescharacteristic
35C22×C8×C73C2311yescharacteristic
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—41no—
39C2×C42×C73D411yescharacteristic
40#107×C73—22no—
41C4⋊C8×C73—22no—
42#95×C73—41no—
43C2×Dic146—22no—
44C2×Dic146—41no—
45Dic292—42no—
46Dic292—81no—
47C4⋊Dic73—22no—
48C4⋊Dic73—41no—
49C4⋊Dic73—41no—
50C4⋊Dic73—41no—
51#11924007—42no—
52#11924007—42no—
53C22⋊Dic73—42no—
54C4×Dic73—42no—
55#11924003—42no—
56C22×C4×C73—41no—
57C22×C4×C73—41no—
58C22×C4×C73D811yescharacteristic
59C22×C4×C73D811yescharacteristic
60C22×C4×C73C2×D411yescharacteristic
61C2×C8×C73—21no—
62C2×C8×C73—41no—
63C2×C8×C73—41no—
64C2×C8×C73C2×D412yes—
65C4⋊C4×C73—22no—
66C4⋊C4×C73—41no—
67C4⋊C4×C73—41no—
68C4⋊C4×C73—41no—
69C4⋊C4×C73—82no—
70C42×C73—21no—
71C42×C73—21no—
72C23×C73C2×D811yescharacteristic
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—41no—
79C2×C4×C73—42no—
80C2×C4×C73—81no—
81C2×C4×C73—81no—
82C2×C4×C73—81no—
83C2×C4×C73—81no—
84C2×C4×C73—81no—
85C2×C4×C73SD3212yes—
86C2×C4×C73#12011yescharacteristic
87C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
88C2×Dic73—42no—
89C2×Dic73—42no—
90Dic146—42no—
91Dic146—42no—
92Dic146—82no—
93C8×C73—22no—
94C8×C73—41no—
95C22×C73—21no—
96C22×C73—21no—
97C22×C73#44711yescharacteristic
98C22×C73C2×SD3211yescharacteristic
99C22×C73#51011yescharacteristic
100C4×C73—21no—
101C4×C73—22no—
102C4×C73—41no—
103C4×C73—82no—
104C4×C73—161no—
105C4×C73#45611yescharacteristic
106C4×C73#45811yescharacteristic
107Dic73—82no—
108Dic73—82no—
109#12289—731nomaximal, Sylow
110C2×C73—21no—
111C2×C73—21no—
112C2×C73#161011yescharacteristic
113C2×C73#218511yescharacteristic
114C2×C73#219111yescharacteristic
115#3249—731no—
116#2235—732no—
117#2086D7311yescharacteristic
118#1348—732no—
119#1322—731no—
120C73#1228911yescharacteristic, complemented, Sylow
121#533—1461no—
122#477—1462no—
123#467—732no—
124#467—1461no—
125#462—1462no—
126C2×#107C2×D7311yescharacteristic
127C2×C4⋊C8C2×D7311yescharacteristic
128#389—21no—
129#367—1462no—
130#358—1462no—
131C22⋊C16—732no—
132#340—21no—
133#340C2×D7311yescharacteristic
134C2×Q16—732no—
135C2×Q16—1461no—
136C22×C8C22⋊D7311yescharacteristic
137C22×C8C22×D7311yescharacteristic
138C4⋊Q8—1461no—
139C4⋊Q8—1461no—
140#125—2921no—
141C22⋊Q8—2921no—
142C22⋊Q8—1462no—
143C4×Q8—2921no—
144C4×Q8—1462no—
145#117—2921no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C42C22⋊D7311yescharacteristic
150C2×C16—1462no—
151#107—22no—
152#107—732no—
153#107—1461no—
154C4⋊C8—22no—
155#103—1462no—
156#103—1462no—
157#95—41no—
158C2×Q8—2921no—
159C2×Q8—1462no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4#1192401311yescharacteristic
163C22×C4#1192401311yescharacteristic
164C22×C4C2×C22⋊D7311yescharacteristic
165Q16—2922no—
166Q16—5841no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8D4×D7312yes—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—82no—
176C4⋊C4—2922no—
177C4⋊C4—2921no—
178C4⋊C4—2921no—
179C4⋊C4—2921no—
180C4⋊C4—2922no—
181C4⋊C4—1462no—
182C22⋊C4—2922no—
183C42—21no—
184C42—21no—
185C42—2922no—
186C16—2922no—
187C23C2×#1192401311yescharacteristic
188Q8—2922no—
189Q8—2922no—
190Q8—5842no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—2922no—
204C2×C4—2922no—
205C2×C4#1455653512yes—
206C2×C4#1455655911yescharacteristic
207C2×C4#1455661611yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1798872011yescharacteristic
213C22C2×#1455653511yescharacteristic
214C22#1798902711yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—82no—
219C4—161no—
220C4—5842no—
221C4—5842no—
222C4#1798876911yescharacteristic
223C4#1798877611yescharacteristic
224C2—21no—
225C2—21no—
226C2#2260893311yescharacteristic
227C2#2261184711yescharacteristic
228C2#2261188111yescharacteristic
subgroupG/H
centerC22#17988720
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC73×#2086C2
socleC22×C73#447
prime radicalC23×C73C2×D8
hypercenter#2086D73
hypocenterC73#12289
LCSG ▷ C22×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
120. C73#12289semidirect

Maximal subgroups

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

subgroupindex
#226212762
#226121282
#226079422
#226078712
C73×#20862
#1228973