Finite Group Databasebeta

← #32983714

Subgroups — #32983714

Order 18688.

528 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 107 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#22612534C211yesmaximal, characteristic
2#22610011C211yesmaximal, characteristic
3#22608464C212yesmaximal
4#22607784C212yesmaximal
5C73×#2502C211yesmaximal, characteristic
6#17988551C2212yes—
7#17988473—24no—
8#17988450C2212yes—
9#17988447—21no—
10C4×C22⋊Dic73—21no—
11#17988356—22no—
12#17988324—24no—
13C2×C22⋊Q8×C73C2211yescharacteristic
14C22×C4⋊C4×C73C2211yescharacteristic
15#398×C73—22no—
16#391×C73C412yescomplemented
17#385×C73—21no—
18#385×C73—21no—
19#384×C73C412yescomplemented
20C4×C22⋊C4×C73C2211yescharacteristic
21C2×#11924012—22no—
22C2×C4⋊Dic73—22no—
23C2×C22⋊Dic73—22no—
24C2×C4×Dic73—22no—
25C22⋊#10596831—22no—
26C22⋊#10596831—22no—
27#14556498—22no—
28#14556498—22no—
29#14556498—22no—
30#14556498D414yes—
31C22×Q8×C73C2×C411yescharacteristic
32C23×C4×C73D411yescharacteristic
33C23×C4×C73D411yescharacteristic
34C23×C4×C73C2311yescharacteristic
35C22⋊Q8×C73—22no—
36C22⋊Q8×C73—22no—
37C2×C4⋊C4×C73—22no—
38C2×C4⋊C4×C73—22no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—22no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C4⋊C4×C73C2×C412yes—
44C2×C4⋊C4×C73D412yes—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C42×C73C2×C412yes—
48#95×C73—22no—
49#95×C73—22no—
50#95×C73—21no—
51C22×Dic73—24no—
52#11924012—44no—
53C2×#10596831—42no—
54C4⋊Dic73—82no—
55C22⋊Dic73—44no—
56C4×Dic73—82no—
57C24×C73C2×D411yescharacteristic
58C2×Q8×C73—22no—
59C2×Q8×C73—42no—
60C22×C4×C73—22no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—42no—
68C22×C4×C73—81no—
69C22×C4×C73—81no—
70C22×C4×C73C22⋊C412yes—
71C22×C4×C73C22⋊C412yes—
72C22×C4×C73C22×C411yescharacteristic
73C22×C4×C73C2×D412yes—
74C22×C4×C73C2×D411yescharacteristic
75C22×C4×C73#4112yes—
76C4⋊C4×C73—24no—
77C4⋊C4×C73—24no—
78C4⋊C4×C73—22no—
79C4⋊C4×C73—42no—
80C4⋊C4×C73—82no—
81C4⋊C4×C73—82no—
82C22⋊C4×C73—24no—
83C22⋊C4×C73—41no—
84C22⋊C4×C73—41no—
85C42×C73—42no—
86C23×C73—22no—
87C23×C73—22no—
88C23×C73—41no—
89C23×C73C2×C22⋊C411yescharacteristic
90C23×C73#12011yescharacteristic
91C23×C73#12311yescharacteristic
92Q8×C73—82no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—24no—
96C2×C4×C73—22no—
97C2×C4×C73—22no—
98C2×C4×C73—42no—
99C2×C4×C73—42no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—42no—
110C2×C4×C73—81no—
111C2×C4×C73—81no—
112C2×C4×C73C4×D412yes—
113C2×C4×C73#12112yes—
114C2×Dic73—44no—
115C2×Dic73—44no—
116C2×Dic73—44no—
117#10596831—82no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—44no—
126C22×C73#38611yescharacteristic
127C22×C73#41012yes—
128C22×C73#45612yes—
129C4×C73—22no—
130C4×C73—22no—
131C4×C73—44no—
132C4×C73—42no—
133C4×C73—42no—
134C4×C73—81no—
135C4×C73—81no—
136Dic73—84no—
137#9461—731nomaximal, Sylow
138C2×C73—22no—
139C2×C73—22no—
140C2×C73—41no—
141C2×C73#188611yescharacteristic
142C2×C73#189911yescharacteristic
143C2×C73#191111yescharacteristic
144#2502D7311yescharacteristic
145#2313—731no—
146#1821—731no—
147#1511—732no—
148#1284—732no—
149C73#946111yescharacteristic, complemented, Sylow
150C2×C22⋊Q8C2×D7311yescharacteristic
151C22×C4⋊C4C2×D7311yescharacteristic
152#407—732no—
153#398—22no—
154#391Dic7312yescomplemented
155#387—1464no—
156#385—21no—
157#385—21no—
158#384Dic7312yescomplemented
159#381—732no—
160#380—1461no—
161C4×C22⋊C4—1461no—
162C4×C22⋊C4C2×D7311yescharacteristic
163#343—1462no—
164#328—1464no—
165C22×Q8C2×Dic7311yescharacteristic
166C23×C4C22⋊D7311yescharacteristic
167C23×C4C22⋊D7311yescharacteristic
168C23×C4C22×D7311yescharacteristic
169C2×M16—1462no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—1462no—
178C2×C4⋊C4C2×Dic7311yescharacteristic
179C2×C4⋊C4C2×Dic7312yes—
180C2×C4⋊C4C22⋊D7312yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—1462no—
184C2×C42—1462no—
185C2×C42C2×Dic7312yes—
186C22⋊C8—1462no—
187C22⋊C8—1462no—
188#95—22no—
189#95—22no—
190#95—21no—
191#95—734no—
192#95—1462no—
193#95—1462no—
194#95—1462no—
195C24C2×C22⋊D7311yescharacteristic
196C2×Q8—22no—
197C2×Q8—42no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—1464no—
209C22×C4C22⋊Dic7312yes—
210C22×C4C22⋊Dic7312yes—
211C22×C4C22×Dic7311yescharacteristic
212C22×C4D4×D7312yes—
213C22×C4C2×C22⋊D7311yescharacteristic
214C22×C4#1192402712yes—
215M16—2924no—
216C2×C8—2922no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—5842no—
224C22⋊C4—24no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—2924no—
228C42—42no—
229C42—5842no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×C22⋊Dic7311yescharacteristic
234C23#1455655911yescharacteristic
235C23#1455657211yescharacteristic
236Q8—82no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—2924no—
257C2×C4—2924no—
258C2×C4—2924no—
259C2×C4D4×Dic7312yes—
260C2×C4#1455656312yes—
261C8—5842no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1798846711yescharacteristic
271C22#1798856112yes—
272C22#1798876912yes—
273C4—22no—
274C4—22no—
275C4—44no—
276C4—42no—
277C4—42no—
278C4—81no—
279C4—81no—
280C4—5844no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2261032511yescharacteristic
285C2#2261039011yescharacteristic
286C2#2261046411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2502C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2502D73
hypocenterC73#9461
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
16. #391×C73C4semidirect
19. #384×C73C4semidirect
149. C73#9461semidirect
154. #391Dic73semidirect
158. #384Dic73semidirect

Maximal subgroups

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

subgroupindex
#226125342
#226100112
#226084642
#226077842
C73×#25022
#946173