Finite Group Databasebeta

← #32978745

Subgroups — #32978745

Order 18688.

540 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 125 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#124×Dic73C211yesmaximal, characteristic
2#22609785C212yesmaximal, complemented
3C2×#17988327C211yesmaximal, characteristic
4#22607813C212yesmaximal, complemented
5C2×#329×C73C211yesmaximal, characteristic
6C2×Q8×Dic73C411yescharacteristic
7C2×D4×Dic73C411yescharacteristic
8C2×#14556520—22no—
9C2×#14556502C2212yes—
10#17988494C411yescharacteristic
11#17988490C411yescharacteristic
12#17988480—22no—
13#17988453C2211yescharacteristic
14C22⋊C4×Dic73—22no—
15C42×Dic73C2211yescharacteristic
16#17988341—22no—
17#17988327—22no—
18C2×#124×C73C2211yescharacteristic
19C2×#97×C73C2212yes—
20#329×C73—22no—
21C23×Dic73—21no—
22C22×#10596831—22no—
23Q8×Dic73—42no—
24D4×Dic73—42no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C22⋊Dic73—22no—
29C2×C22⋊Dic73—21no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D412yes—
34C2×C4×Dic73D412yes—
35#14556520—28no—
36#14556502—22no—
37#14556502—22no—
38#14556498—22no—
39#14556498—22no—
40C22×Q8×C73C2×C411yescharacteristic
41C22×D4×C73C2×C411yescharacteristic
42C22×C8×C73—22no—
43#124×C73—22no—
44#124×C73D412yes—
45#124×C73Q812yes—
46C2×C22⋊C4×C73—22no—
47C2×C42×C73C2311yescharacteristic
48#97×C73—22no—
49#97×C73—22no—
50C22×Dic73—21no—
51C22×Dic73—21no—
52C22×Dic73—42no—
53C22×Dic73—44no—
54C22×Dic73C22⋊C412yes—
55C22×Dic73C22⋊C411yescharacteristic
56C22×Dic73C22⋊C411yescharacteristic
57C2×#10596831—24no—
58C2×#10596831—42no—
59C2×#10596831—42no—
60C4⋊Dic73—42no—
61C4⋊Dic73—41no—
62C4⋊Dic73—41no—
63C22⋊Dic73—44no—
64C22⋊Dic73—42no—
65C4×Dic73—22no—
66C4×Dic73—24no—
67C4×Dic73—41no—
68C4×Dic73—42no—
69C4×Dic73—42no—
70C24×C73—21no—
71C2×Q8×C73—21no—
72C2×Q8×C73—41no—
73C2×Q8×C73—41no—
74C2×Q8×C73C4⋊C412yes—
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—
82C2×C8×C73—24no—
83C2×C8×C73—42no—
84C2×C8×C73—42no—
85C22⋊C4×C73—24no—
86C22⋊C4×C73—24no—
87C42×C73C2×D411yescharacteristic
88C42×C73C2×Q811yescharacteristic
89C42×C73#4112yes—
90C23×C73—22no—
91C23×C73—41no—
92C23×C73—41no—
93C23×C73—42no—
94C23×C73—42no—
95C23×C73C2×C22⋊C411yescharacteristic
96Q8×C73—42no—
97Q8×C73—42no—
98D4×C73—42no—
99D4×C73—42no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—22no—
103C2×C4×C73—22no—
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—22no—
114C2×Dic73—22no—
115C2×Dic73—22no—
116C2×Dic73—41no—
117C2×Dic73—41no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73—84no—
121C2×Dic73#10012yes—
122C2×Dic73#10414yes—
123C8×C73—44no—
124#10596831—44no—
125C22×C73—42no—
126C22×C73—44no—
127C22×C73—81no—
128C22×C73—81no—
129C22×C73—82no—
130C22×C73#38011yescharacteristic
131C22×C73C2×#10011yescharacteristic
132C22×C73#41311yescharacteristic
133C22×C73C2×#10412yes—
134C22×C73#42112yes—
135C4×C73—21no—
136C4×C73—21no—
137C4×C73—22no—
138C4×C73—22no—
139C4×C73—42no—
140Dic73—41no—
141Dic73—42no—
142Dic73—81no—
143Dic73—81no—
144Dic73#32914yescomplemented
145#8579—731nomaximal, Sylow
146C2×C73—82no—
147C2×C73C2×#32911yescharacteristic
148C2×C73#152912yes—
149C2×C73#153611yescharacteristic
150C2×C73#180011yescharacteristic
151C2×C73#180612yes—
152C4×#124—731no—
153#1767—732no—
154C2×#329—731no—
155C2×#329D7311yescharacteristic
156#1295—732no—
157C73#857911yescharacteristic, complemented, Sylow
158C2×#124C2×D7311yescharacteristic
159C2×C4×Q8—731no—
160C2×C4×D4—731no—
161C2×C4⋊C8—1462no—
162C2×#97—732no—
163C2×#97C2×D7312yes—
164#391—731no—
165#390—731no—
166#388—1462no—
167#382—731no—
168C4×C22⋊C4—1462no—
169C43—731no—
170#336—1462no—
171#329—22no—
172#329—1462no—
173C22×Q8C4×D7311yescharacteristic
174C22×D4C4×D7311yescharacteristic
175C23×C4—1461no—
176C22×C8—22no—
177C22×C8—1462no—
178#124—22no—
179#124Dic14612yes—
180#124C22⋊D7312yes—
181C4×Q8—2922no—
182C4×D4—2922no—
183C2×C4⋊C4—731no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—1461no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—1461no—
188C2×C22⋊C4—1462no—
189C2×C42—731no—
190C2×C42—732no—
191C2×C42—731no—
192C2×C42—732no—
193C2×C42—1461no—
194C2×C42C22×D7311yescharacteristic
195C4⋊C8—1468no—
196#97—22no—
197#97—22no—
198#97—1462no—
199#97—1462no—
200#95—1462no—
201#95—1462no—
202C24—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8#1192400712yes—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4#1192400712yes—
211C22×C4—21no—
212C22×C4—2922no—
213C22×C4—2924no—
214C22×C4—731no—
215C22×C4—732no—
216C22×C4—731no—
217C22×C4—1461no—
218C22×C4—1461no—
219C22×C4C2×C4×D7311yescharacteristic
220C22×C4D4×D7312yes—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—2922no—
225C2×C8—2922no—
226C2×C8—1464no—
227C4⋊C4—2921no—
228C4⋊C4—2921no—
229C4⋊C4—2922no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—2924no—
233C22⋊C4—2922no—
234C42—2921no—
235C42—2922no—
236C42—2922no—
237C42—1464no—
238C42—1462no—
239C42C2×C22⋊D7311yescharacteristic
240C42C2×Dic14611yescharacteristic
241C42#1192402712yes—
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C22⋊C4×D7311yescharacteristic
248Q8—42no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—2921no—
260C2×C4—2921no—
261C2×C4—2922no—
262C2×C4—2922no—
263C2×C4—5844no—
264C2×C4—732no—
265C2×C4—734no—
266C2×C4—1461no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4—1462no—
270C2×C4—1462no—
271C2×C4C2×#1192400711yescharacteristic
272C2×C4#1455654711yescharacteristic
273C2×C4#1455656312yes—
274C2×C4#1455656812yes—
275C8—44no—
276C8—2924no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22#1798844511yescharacteristic
283C22#100×D7311yescharacteristic
284C22#1798856811yescharacteristic
285C22#104×D7312yes—
286C22#1798860212yes—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—2922no—
293C4—2921no—
294C4—5841no—
295C4—5841no—
296C4—734no—
297C2—82no—
298C2#329×D7311yescharacteristic
299C2#2260853512yes—
300C2#2260856411yescharacteristic
301C2#2260992111yescharacteristic
302C2#2260994912yes—
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×#329×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×C73C2×C4
cosocleC2×C42×C73C23
hypercenterC2×#329D73
hypocenterC73#8579
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. #22609785C2semidirect
4. #22607813C2semidirect
144. Dic73#329semidirect
157. C73#8579semidirect

Maximal subgroups

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

subgroupindex
#124×Dic732
#226097852
C2×#179883272
#226078132
C2×#329×C732
#857973