Finite Group Databasebeta

← #32978898

Subgroups — #32978898

Order 18688.

460 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#17988655C211yesmaximal, characteristic
2#22609967C212yesmaximal, complemented
3#22608267C212yesmaximal
4#22607808C211yesmaximal, characteristic, complemented
5C73×#1293C211yesmaximal, characteristic, complemented
6C22×#11924011C411yescharacteristic
7C23×#10596831C2211yescharacteristic
8C2×C4×C22⋊D73C2211yescharacteristic
9#17988655—22no—
10#17988655—22no—
11C2×#14556526—22no—
12C2×#14556520C411yescharacteristic
13#17988551C2212yes—
14C2×#14556503C411yescharacteristic
15C2×#14556501C411yescharacteristic
16#17988351—22no—
17#17988312—22no—
18#17988310—22no—
19#407×C73C2212yes—
20C2×C22⋊C8×C73C2211yescharacteristic
21C22×C22⋊D73Q811yescharacteristic
22C22×C4×D73C2×C411yescharacteristic
23C2×#11924012—22no—
24C2×#11924011—21no—
25C2×#11924011—21no—
26C2×#11924011—42no—
27C22×#10596831—42no—
28C22×#10596831C2×C411yescharacteristic
29C22×#10596831C2×C411yescharacteristic
30C22×#10596831D412yes—
31C22×#10596831D412yes—
32C4×C22⋊D73—22no—
33C4×C22⋊D73—22no—
34C2×#11924007D411yescharacteristic
35C2×C22⋊Dic73Q811yescharacteristic
36C2×#11924005D411yescharacteristic
37C2×C4×Dic73C2×C411yescharacteristic
38#14556526—44no—
39#14556520—22no—
40C22⋊#10596831—22no—
41C22⋊#10596831—22no—
42#14556503—22no—
43#14556501—21no—
44#14556501—21no—
45C23×C4×C73C2311yescharacteristic
46C2×M16×C73—22no—
47C22×C8×C73C2×C411yescharacteristic
48C22×C8×C73C2×C411yescharacteristic
49C22⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51C22⋊C8×C73—22no—
52C23×D73C4⋊C411yescharacteristic
53C2×C22⋊D73—22no—
54C2×C22⋊D73—42no—
55C22×Dic73C4⋊C411yescharacteristic
56C22×Dic73C4⋊C411yescharacteristic
57C22×Dic73C4⋊C411yescharacteristic
58C2×C4×D73—21no—
59C2×C4×D73—21no—
60C2×C4×D73—41no—
61C2×C4×D73—41no—
62#11924012—42no—
63#11924012—42no—
64#11924011—42no—
65#11924011—42no—
66C2×#10596831—24no—
67C2×#10596831—22no—
68C2×#10596831—41no—
69C2×#10596831—42no—
70C2×#10596831—42no—
71C2×#10596831C22⋊C411yescharacteristic
72C2×#10596831C22⋊C412yes—
73C2×#10596831C22⋊C411yescharacteristic
74#11924007—22no—
75C22⋊Dic73—22no—
76#11924005—22no—
77C4×Dic73—21no—
78C4×Dic73—21no—
79C24×C73C2×Q811yescharacteristic
80C22×C4×C73—41no—
81C22×C4×C73—41no—
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D412yes—
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73#4112yes—
86M16×C73—42no—
87M16×C73—42no—
88C2×C8×C73—21no—
89C2×C8×C73—21no—
90C2×C8×C73—41no—
91C2×C8×C73—41no—
92C2×C8×C73—42no—
93C23×C73—22no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73C2×C4⋊C411yescharacteristic
97C23×C73C22⋊Q812yes—
98C22×D73—21no—
99C22×D73—22no—
100C22×D73—41no—
101C22×D73—41no—
102C2×C4×C73—21no—
103C2×C4×C73—22no—
104C2×C4×C73—21no—
105C2×C4×C73—22no—
106C2×C4×C73—41no—
107C2×C4×C73—42no—
108C2×C4×C73C2×C22⋊C411yescharacteristic
109C2×C4×C73#11711yescharacteristic
110C2×C4×C73#12112yes—
111C22⋊D73—44no—
112C2×Dic73—21no—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—22no—
116C2×Dic73—41no—
117C2×Dic73—41no—
118C4×D73—42no—
119C4×D73—41no—
120C4×D73—41no—
121C8×C73—41no—
122C8×C73—41no—
123C8×C73—82no—
124#10596831—22no—
125#10596831—21no—
126#10596831—82no—
127#10596831#9912yes—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—22no—
131C22×C73—22no—
132C22×C73—22no—
133C22×C73—44no—
134C22×C73#38011yescharacteristic
135C22×C73#43012yes—
136C2×D73—41no—
137C2×D73—41no—
138C2×D73—41no—
139C2×D73—82no—
140C4×C73—22no—
141C4×C73—21no—
142C4×C73C2×#9911yescharacteristic
143C4×C73#41011yescharacteristic
144Dic73—41no—
145Dic73—41no—
146Dic73—41no—
147#8613—731nomaximal, Sylow
148C2×C73—22no—
149C2×C73—42no—
150C2×C73#179911yescharacteristic
151C2×C73#181111yescharacteristic
152C2×C73#181311yescharacteristic
153D73—81no—
154C2×#435—731no—
155#1810—732no—
156#1465—732no—
157#1293—731no—
158#1293D7311yescharacteristic, complemented
159C73#861311yescharacteristic, complemented, Sylow
160C22×M16—731no—
161C23×C8—731no—
162C2×C4×D4—731no—
163#435—1462no—
164#435—1462no—
165C2×#108—1462no—
166C2×C4⋊C8—731no—
167#407—732no—
168#407C2×D7312yes—
169C2×C22⋊C8—731no—
170C2×C22⋊C8C2×D7311yescharacteristic
171C2×#97—731no—
172#341—1462no—
173#324—1462no—
174#323—1462no—
175C22×D4—731no—
176C23×C4—731no—
177C23×C4C22×D7311yescharacteristic
178C2×M16—22no—
179C2×M16—2922no—
180C2×M16—1461no—
181C2×M16—1461no—
182C2×M16—1462no—
183C22×C8—2922no—
184C22×C8—731no—
185C22×C8—732no—
186C22×C8—731no—
187C22×C8—732no—
188C22×C8C4×D7311yescharacteristic
189C22×C8C4×D7311yescharacteristic
190C4×D4—1462no—
191C4×D4—1462no—
192C2×C4⋊C4—731no—
193C2×C22⋊C4—731no—
194C2×C22⋊C4—731no—
195C2×C42—731no—
196#108—2924no—
197C4⋊C8—1462no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—1462no—
202C22⋊C8—1462no—
203C22⋊C8—1462no—
204#97—1461no—
205#97—1461no—
206C24—731no—
207C24Q8×D7311yescharacteristic
208C2×D4—2922no—
209C2×D4—1462no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—2921no—
213C22×C4—2921no—
214C22×C4—731no—
215C22×C4—731no—
216C22×C4—731no—
217C22×C4—1461no—
218C22×C4—1461no—
219C22×C4C2×C4×D7311yescharacteristic
220C22×C4D4×D7312yes—
221C22×C4D4×D7311yescharacteristic
222C22×C4#1192402812yes—
223M16—42no—
224M16—42no—
225M16—2922no—
226M16—2922no—
227M16—2922no—
228M16—2922no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—2922no—
235C2×C8—2922no—
236C2×C8—2921no—
237C2×C8—731no—
238C2×C8—731no—
239C2×C8—732no—
240C2×C8—1464no—
241C2×C8—1462no—
242C4⋊C4—1462no—
243C22⋊C4—1462no—
244C22⋊C4—1462no—
245C42—1461no—
246C42—1461no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—2921no—
251C23—2921no—
252C23—1461no—
253C23—1462no—
254C23C4⋊C4×D7311yescharacteristic
255C23#1455656912yes—
256D4—2924no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—2922no—
264C2×C4—2921no—
265C2×C4—2921no—
266C2×C4—2921no—
267C2×C4—2921no—
268C2×C4—1462no—
269C2×C4—1461no—
270C2×C4—1461no—
271C2×C4—1461no—
272C2×C4C22⋊C4×D7311yescharacteristic
273C2×C4#1455654811yescharacteristic
274C2×C4#1455656412yes—
275C8—41no—
276C8—41no—
277C8—82no—
278C8—5842no—
279C8—732no—
280C8—1461no—
281C8—1462no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—44no—
288C22—2921no—
289C22—2921no—
290C22—2921no—
291C22—5842no—
292C22#1798844611yescharacteristic
293C22#1798863412yes—
294C4—22no—
295C4—21no—
296C4—2921no—
297C4—2921no—
298C4—2921no—
299C4#99×D7311yescharacteristic
300C4#1798855911yescharacteristic
301C2—22no—
302C2—42no—
303C2—5841no—
304C2#2260991811yescharacteristic
305C2#2260997211yescharacteristic
306C2#2260998011yescharacteristic
subgroupG/H
centerC22#17988446
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1293C2
socleC22×C73#380
prime radicalC22×C22⋊D73Q8
cosocleC23×C4×C73C23
hypercenter#1293D73
hypocenterC73#8613
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #22609967C2semidirect
4. #22607808C2semidirect
5. C73×#1293C2semidirect
158. #1293D73semidirect
159. C73#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#179886552
#226099672
#226082672
#226078082
C73×#12932
#861373