Finite Group Databasebeta

← #33006365

Subgroups — #33006365

Order 18688.

322 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 79 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#22611616C211yesmaximal, characteristic
2#22611596C211yesmaximal, characteristic
3#22608106C214yesmaximal, complemented
4C73×#3142C211yesmaximal, characteristic
5#17988571—21no—
6#17988561—21no—
7#17988386—21no—
8#17988384C2212yes—
9#17988382C2212yes—
10#17988380—21no—
11#532×C73C2211yescharacteristic
12C2×#125×C73C411yescharacteristic, complemented
13#491×C73—21no—
14#490×C73—21no—
15#488×C73C412yescomplemented
16#488×C73C2212yes—
17#417×C73—21no—
18#417×C73—21no—
19C4×M16×C73C411yescharacteristic, complemented
20C2×#11924012—41no—
21#14556550—41no—
22#14556510—22no—
23#14556508—41no—
24#14556506—22no—
25#14556506—41no—
26C2×#41×C73D411yescharacteristic
27C2×#41×C73D411yescharacteristic
28C2×M16×C73—21no—
29C4⋊Q8×C73D411yescharacteristic
30#127×C73D411yescharacteristic
31#125×C73—21no—
32#125×C73—41no—
33#125×C73C2×C412yes—
34#124×C73C2×C411yescharacteristic
35#124×C73C2311yescharacteristic
36#121×C73—41no—
37C4×D4×C73—41no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—41no—
40C2×C42×C73C2×C411yescharacteristic
41#103×C73—22no—
42#103×C73—22no—
43#102×C73—22no—
44#102×C73—22no—
45#97×C73—21no—
46C4×C8×C73C2×C412yes—
47#11924012—42no—
48#11924012—81no—
49C2×#10596831—81no—
50C4⋊Dic73—81no—
51C22⋊Dic73—42no—
52C4×Dic73—81no—
53#41×C73—41no—
54#41×C73—41no—
55#41×C73—41no—
56#41×C73—41no—
57C2×Q8×C73C22⋊C411yescharacteristic
58C2×Q8×C73C2×D411yescharacteristic
59C2×D4×C73—21no—
60C2×D4×C73—21no—
61C2×D4×C73—41no—
62C2×D4×C73C22⋊C411yescharacteristic
63C2×D4×C73C2×D411yescharacteristic
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—41no—
68C22×C4×C73C22⋊C411yescharacteristic
69M16×C73—41no—
70M16×C73—41no—
71C2×C8×C73—22no—
72C4⋊C4×C73—22no—
73C4⋊C4×C73—41no—
74C4⋊C4×C73—42no—
75C4⋊C4×C73—42no—
76C4⋊C4×C73—41no—
77C22⋊C4×C73—41no—
78C22⋊C4×C73—41no—
79C42×C73—21no—
80C42×C73C22⋊C411yescharacteristic
81C42×C73C22×C411yescharacteristic
82C23×C73—21no—
83C23×C73—21no—
84C23×C73#9911yescharacteristic
85Q8×C73—41no—
86Q8×C73—41no—
87D4×C73—41no—
88D4×C73—41no—
89D4×C73—81no—
90D4×C73—81no—
91D4×C73—82no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73—42no—
103C2×C4×C73—81no—
104C2×C4×C73—82no—
105C2×C4×C73#9911yescharacteristic
106C2×C4×C73C2×C22⋊C411yescharacteristic
107C2×Dic73—82no—
108C8×C73—42no—
109#10596831—82no—
110C22×C73—21no—
111C22×C73—41no—
112C22×C73—41no—
113C22×C73—41no—
114C22×C73—81no—
115C22×C73—81no—
116C22×C73C2×#9911yescharacteristic
117C4×C73—21no—
118C4×C73—41no—
119C4×C73—41no—
120C4×C73—41no—
121C4×C73—41no—
122C4×C73—82no—
123C4×C73#35411yescharacteristic
124C4×C73#35411yescharacteristic
125Dic73—162no—
126#13604—731nomaximal, Sylow
127C2×C73—21no—
128C2×C73—41no—
129C2×C73—81no—
130C2×C73—81no—
131C2×C73C2×#35411yescharacteristic
132#3142D7311yescharacteristic
133#2132—731no—
134#2126—731no—
135#1414—734no—
136C73#1360411yescharacteristic, complemented, Sylow
137#532C2×D7311yescharacteristic
138C2×#125Dic7311yescharacteristic, complemented
139#491—21no—
140#490—21no—
141#488Dic7312yescomplemented
142#488C2×D7312yes—
143#417—21no—
144#417—21no—
145#413—1461no—
146#410—1461no—
147C4×M16Dic7311yescharacteristic, complemented
148#356—1461no—
149#355—732no—
150#354—732no—
151#353—1461no—
152C2×#41C22⋊D7311yescharacteristic
153C2×#41C22⋊D7311yescharacteristic
154C2×M16—21no—
155C2×M16—2921no—
156C4⋊Q8C22⋊D7311yescharacteristic
157#127C22⋊D7311yescharacteristic
158#125—21no—
159#125—41no—
160#125C2×Dic7312yes—
161#124C2×Dic7311yescharacteristic
162#124C22×D7311yescharacteristic
163#121—41no—
164C4×D4—41no—
165#117—2921no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C42C2×Dic7311yescharacteristic
169#103—22no—
170#103—22no—
171#102—22no—
172#102—22no—
173#101—1462no—
174#100—2921no—
175#99—2921no—
176#99—1462no—
177#97—21no—
178C4×C8C2×Dic7312yes—
179#41—41no—
180#41—41no—
181#41—41no—
182#41—41no—
183C2×Q8C22⋊Dic7311yescharacteristic
184C2×Q8C2×C22⋊D7311yescharacteristic
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4C22⋊Dic7311yescharacteristic
189C2×D4C2×C22⋊D7311yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4C22⋊Dic7311yescharacteristic
195M16—41no—
196M16—41no—
197M16—2922no—
198M16—5841no—
199C2×C8—22no—
200C2×C8—5841no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—5841no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—2922no—
210C42—21no—
211C42—5841no—
212C42C22⋊Dic7311yescharacteristic
213C42C22×Dic7311yescharacteristic
214C23—21no—
215C23—21no—
216C23#1455650611yescharacteristic
217Q8—41no—
218Q8—41no—
219D4—41no—
220D4—41no—
221D4—81no—
222D4—81no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—5842no—
238C2×C4#1455650611yescharacteristic
239C2×C4C2×C22⋊Dic7311yescharacteristic
240C8—42no—
241C8—5842no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22C2×#1455650611yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—11682no—
256C4#1798838211yescharacteristic
257C4#1798838211yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—81no—
262C2C2×#1798838211yescharacteristic
subgroupG/H
centerC2C2×#17988382
commutatorC42×C73C22×C4
Frattini#124C22×D73
FittingC73×#3142C2
socleC2×C73C2×#354
prime radical#532×C73C22
cosocle#124×C73C23
hypercenter#3142D73
hypocenterC73#13604
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3142

Complements

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

NH ≅ G/Ntype
3. #22608106C2semidirect
12. C2×#125×C73C4semidirect
15. #488×C73C4semidirect
19. C4×M16×C73C4semidirect
136. C73#13604semidirect
138. C2×#125Dic73semidirect
141. #488Dic73semidirect
147. C4×M16Dic73semidirect

Maximal subgroups

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

subgroupindex
#226116162
#226115962
#226081062
C73×#31422
#1360473