Finite Group Databasebeta

← #33001989

Subgroups — #33001989

Order 18688.

480 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 101 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#22612644C211yesmaximal, characteristic
2C2×#17988384C211yesmaximal, characteristic
3#22609925C212yesmaximal, complemented
4#22608057C212yesmaximal, complemented
5C2×#355×C73C211yesmaximal, characteristic
6C2×Q8×Dic73C2211yescharacteristic
7C2×D4×Dic73C411yescharacteristic
8#17988610—22no—
9C2×#14556510C2212yes—
10#17988494C2211yescharacteristic
11#17988490C411yescharacteristic
12#17988480—22no—
13#17988462—22no—
14#17988457C411yescharacteristic
15#17988433C411yescharacteristic
16#17988384—22no—
17#17988355—22no—
18C2×#124×C73C2211yescharacteristic
19C2×#101×C73C2212yes—
20#355×C73—22no—
21C23×Dic73—21no—
22C2×#11924012—22no—
23Q8×Dic73—42no—
24D4×Dic73—42no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C4⋊Dic73D412yes—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—22no—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D412yes—
34#14556520—44no—
35#14556510—22no—
36#14556510—22no—
37#14556498—21no—
38#14556498—21no—
39#14556498—22no—
40C22×Q8×C73C2311yescharacteristic
41C22×D4×C73C2×C411yescharacteristic
42C2×M16×C73—22no—
43#124×C73—22no—
44#124×C73D412yes—
45#124×C73Q812yes—
46C2×C22⋊C4×C73—22no—
47C2×C42×C73C2×C411yescharacteristic
48#101×C73—22no—
49#101×C73—22no—
50C22×Dic73—21no—
51C22×Dic73—21no—
52C22×Dic73—42no—
53C22×Dic73—82no—
54C22×Dic73C22⋊C411yescharacteristic
55C22×Dic73C22⋊C412yes—
56C22×Dic73C22⋊C411yescharacteristic
57#11924012—42no—
58#11924012—42no—
59C2×#10596831—42no—
60C4⋊Dic73—41no—
61C4⋊Dic73—42no—
62C4⋊Dic73—41no—
63C4⋊Dic73—82no—
64C22⋊Dic73—42no—
65C22⋊Dic73—82no—
66C4×Dic73—24no—
67C4×Dic73—22no—
68C4×Dic73—41no—
69C24×C73—21no—
70C2×Q8×C73—41no—
71C2×Q8×C73—41no—
72C2×Q8×C73C2×D411yescharacteristic
73C2×Q8×C73C2×Q811yescharacteristic
74C2×Q8×C73#4112yes—
75C2×D4×C73—21no—
76C2×D4×C73—41no—
77C2×D4×C73—41no—
78C2×D4×C73C4⋊C412yes—
79C22×C4×C73—21no—
80C22×C4×C73C22×C411yescharacteristic
81C22×C4×C73C2×D412yes—
82M16×C73—42no—
83M16×C73—42no—
84C2×C8×C73—42no—
85C22⋊C4×C73—24no—
86C22⋊C4×C73—24no—
87C42×C73—21no—
88C42×C73C4⋊C412yes—
89C23×C73—22no—
90C23×C73—42no—
91C23×C73—42no—
92C23×C73—41no—
93C23×C73—41no—
94C23×C73C2×C22⋊C411yescharacteristic
95Q8×C73—41no—
96Q8×C73—41no—
97Q8×C73—42no—
98D4×C73—42no—
99D4×C73—42no—
100C2×C4×C73—22no—
101C2×C4×C73—21no—
102C2×C4×C73—22no—
103C2×C4×C73—21no—
104C2×C4×C73—22no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73C2×C4⋊C411yescharacteristic
108C2×C4×C73#11711yescharacteristic
109C2×C4×C73#12112yes—
110C2×C4×C73C22⋊Q812yes—
111C2×Dic73—21no—
112C2×Dic73—22no—
113C2×Dic73—41no—
114C2×Dic73—42no—
115C2×Dic73—41no—
116C2×Dic73—41no—
117C2×Dic73—42no—
118C2×Dic73—41no—
119C2×Dic73—41no—
120C2×Dic73—41no—
121C2×Dic73—84no—
122C2×Dic73#9912yes—
123C8×C73—82no—
124#10596831—82no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—44no—
128C22×C73—42no—
129C22×C73—82no—
130C22×C73—81no—
131C22×C73—81no—
132C22×C73#38011yescharacteristic
133C22×C73C2×#9911yescharacteristic
134C22×C73#41011yescharacteristic
135C4×C73—22no—
136C4×C73—21no—
137C4×C73—21no—
138C4×C73—22no—
139C4×C73—42no—
140Dic73—22no—
141Dic73—42no—
142Dic73—41no—
143Dic73—81no—
144Dic73—81no—
145#12803—731nomaximal, Sylow
146C2×C73—21no—
147C2×C73—21no—
148C2×C73—82no—
149C2×C73#179911yescharacteristic
150C2×C73#213311yescharacteristic
151C2×C73#213411yescharacteristic
152#2326—731no—
153C2×#355—731no—
154C2×#355D7311yescharacteristic
155#1801—732no—
156#1394—732no—
157C73#1280311yescharacteristic, complemented, Sylow
158C2×#124C2×D7311yescharacteristic
159C2×C4×Q8—731no—
160C2×C4×D4—731no—
161#423—1462no—
162C2×#101—732no—
163C2×#101C2×D7312yes—
164#391—731no—
165#390—731no—
166#388—1462no—
167#385—1462no—
168#384—731no—
169#376—731no—
170#355—22no—
171#355—1462no—
172#343—1462no—
173C22×Q8C22×D7311yescharacteristic
174C22×D4C4×D7311yescharacteristic
175C23×C4—1461no—
176C2×M16—22no—
177C2×M16—1462no—
178#124—22no—
179#124Dic14612yes—
180#124C22⋊D7312yes—
181C4×Q8—2922no—
182C4×D4—2922no—
183C2×C4⋊C4—731no—
184C2×C4⋊C4—732no—
185C2×C4⋊C4—731no—
186C2×C4⋊C4—1462no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1461no—
189C2×C22⋊C4—1462no—
190C2×C42—731no—
191C2×C42—731no—
192C2×C42—732no—
193C2×C42C4×D7311yescharacteristic
194C4⋊C8—2924no—
195#101—22no—
196#101—22no—
197#101—1462no—
198#101—1462no—
199#95—1461no—
200#95—1461no—
201#95—1462no—
202C24—21no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×C22⋊D7311yescharacteristic
206C2×Q8C2×Dic14611yescharacteristic
207C2×Q8#1192402712yes—
208C2×D4—21no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4#1192400712yes—
212C22×C4—21no—
213C22×C4—2922no—
214C22×C4—5842no—
215C22×C4—731no—
216C22×C4—731no—
217C22×C4—732no—
218C22×C4—1461no—
219C22×C4—1461no—
220C22×C4C2×C4×D7311yescharacteristic
221C22×C4D4×D7312yes—
222M16—42no—
223M16—42no—
224M16—2922no—
225M16—2922no—
226C2×C8—42no—
227C2×C8—2922no—
228C4⋊C4—2921no—
229C4⋊C4—2921no—
230C4⋊C4—2922no—
231C4⋊C4—5842no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—2922no—
235C22⋊C4—5842no—
236C42—21no—
237C42—2921no—
238C42—1464no—
239C42—1462no—
240C42#1192400712yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C22⋊C4×D7311yescharacteristic
247Q8—41no—
248Q8—41no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—2922no—
260C2×C4—2921no—
261C2×C4—2921no—
262C2×C4—2922no—
263C2×C4—2921no—
264C2×C4—2921no—
265C2×C4—2921no—
266C2×C4—2921no—
267C2×C4—5844no—
268C2×C4—732no—
269C2×C4—1461no—
270C2×C4—1462no—
271C2×C4C2×#1192400711yescharacteristic
272C2×C4#1455654711yescharacteristic
273C2×C4#1455656312yes—
274C2×C4#1455656812yes—
275C8—82no—
276C8—5842no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1798844511yescharacteristic
285C22#99×D7311yescharacteristic
286C22#1798855911yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—2921no—
293C4—2922no—
294C4—5841no—
295C4—5841no—
296C4—1462no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2260991711yescharacteristic
301C2#2261161711yescharacteristic
302C2#2261162111yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC2×#355×C73C2
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC22×Q8×C73C23
hypercenterC2×#355D73
hypocenterC73#12803
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #22609925C2semidirect
4. #22608057C2semidirect
157. C73#12803semidirect

Maximal subgroups

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

subgroupindex
#226126442
C2×#179883842
#226099252
#226080572
C2×#355×C732
#1280373