Finite Group Databasebeta

← #32974242

Subgroups — #32974242

Order 18688.

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

normal subgroups only

This group has 520 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#22612478C211yesmaximal, characteristic
2#22609783C212yesmaximal, complemented
3#22607756C212yesmaximal, complemented
4#22607752C211yesmaximal, characteristic
5C73×#1270C211yesmaximal, characteristic
6C2×#14556578C411yescharacteristic
7C2×#14556556C411yescharacteristic
8C2×D4×Dic73C411yescharacteristic
9C2×#14556520—22no—
10C2×#14556501—22no—
11#17988493C2211yescharacteristic
12#17988475—22no—
13#17988452C411yescharacteristic
14C22⋊C4×Dic73—22no—
15C42×Dic73C2211yescharacteristic
16#17988342C2212yes—
17#390×C73C2211yescharacteristic
18#336×C73C2212yes—
19C22×Dic146C2×C411yescharacteristic
20C23×Dic73—21no—
21C22×#10596831—22no—
22#14556578—22no—
23#14556578—22no—
24#14556556—42no—
25D4×Dic73—42no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—21no—
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#14556501—44no—
37#14556498—22no—
38#14556498—22no—
39C22×D4×C73C2×C411yescharacteristic
40C22×C8×C73—22no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C22⋊C4×C73—22no—
43C2×C42×C73C2311yescharacteristic
44C2×Dic146—21no—
45C2×Dic146—41no—
46C2×Dic146—41no—
47C2×Dic146M1612yes—
48C22×Dic73—21no—
49C22×Dic73—21no—
50C22×Dic73—42no—
51C22×Dic73—44no—
52C22×Dic73C22⋊C412yes—
53C22×Dic73C22⋊C411yescharacteristic
54C22×Dic73C22⋊C411yescharacteristic
55C2×#10596831—24no—
56C2×#10596831—42no—
57C2×#10596831—42no—
58C4⋊Dic73—41no—
59#11924007—42no—
60C22⋊Dic73—28no—
61C22⋊Dic73—42no—
62C4×Dic73—21no—
63C4×Dic73—21no—
64C4×Dic73—22no—
65C4×Dic73—22no—
66C4×Dic73—22no—
67C4×Dic73—24no—
68C4×Dic73—42no—
69C24×C73—21no—
70C2×D4×C73—21no—
71C2×D4×C73—41no—
72C2×D4×C73—41no—
73C2×D4×C73M1612yes—
74C22×C4×C73—21no—
75C22×C4×C73C22×C411yescharacteristic
76C22×C4×C73C2×D412yes—
77C2×C8×C73—24no—
78C2×C8×C73—42no—
79C2×C8×C73—42no—
80C4⋊C4×C73—41no—
81C22⋊C4×C73—42no—
82C22⋊C4×C73—42no—
83C42×C73—41no—
84C23×C73—22no—
85C23×C73—41no—
86C23×C73—41no—
87C23×C73—42no—
88C23×C73—42no—
89C23×C73C2×C22⋊C411yescharacteristic
90D4×C73—42no—
91D4×C73—42no—
92C2×C4×C73—22no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—22no—
96C2×C4×C73—41no—
97C2×C4×C73—42no—
98C2×C4×C73C2×M1611yescharacteristic
99C2×C4×C73C2×M1611yescharacteristic
100C2×C4×C73AGL(1,ℤ/8)12yes—
101C2×C4×C73#13712yes—
102C2×Dic73—22no—
103C2×Dic73—22no—
104C2×Dic73—21no—
105C2×Dic73—22no—
106C2×Dic73—22no—
107C2×Dic73—22no—
108C2×Dic73—41no—
109C2×Dic73—41no—
110C2×Dic73—41no—
111C2×Dic73—42no—
112C2×Dic73—84no—
113C2×Dic73#10012yes—
114C2×Dic73#10414yes—
115Dic146—42no—
116Dic146—42no—
117C8×C73—44no—
118#10596831—44no—
119C22×C73—42no—
120C22×C73—44no—
121C22×C73—81no—
122C22×C73—82no—
123C22×C73—81no—
124C22×C73#40711yescharacteristic
125C22×C73#41011yescharacteristic
126C22×C73C2×#10011yescharacteristic
127C22×C73#41712yes—
128C22×C73C2×#10412yes—
129C4×C73—21no—
130C4×C73—21no—
131C4×C73—22no—
132C4×C73—22no—
133C4×C73—81no—
134Dic73—41no—
135Dic73—42no—
136Dic73—42no—
137Dic73—81no—
138Dic73#32914yes—
139#7635—731nomaximal, Sylow
140C2×C73—82no—
141C2×C73#146611yescharacteristic
142C2×C73#148912yes—
143C2×C73#150912yes—
144C2×C73C2×#32911yescharacteristic
145C2×C73#156011yescharacteristic
146C4×#124—731no—
147#1767—732no—
148#1272—732no—
149#1270—731no—
150#1270D7311yescharacteristic
151C73#763511yescharacteristic, complemented, Sylow
152C2×#124—731no—
153C2×C4×Q8—731no—
154C2×C4×D4—731no—
155C2×C4⋊C8—1462no—
156C2×#97—1462no—
157#391—731no—
158#390C2×D7311yescharacteristic
159#388—1462no—
160#382—731no—
161C4×C22⋊C4—1462no—
162C43—731no—
163#336—732no—
164#336C2×D7312yes—
165C22×Q8—731no—
166C22×D4C4×D7311yescharacteristic
167C23×C4—1461no—
168C22×C8—22no—
169C22×C8—1462no—
170#124—1462no—
171#124—1462no—
172C4×Q8—2922no—
173C4×D4—2922no—
174C2×C4⋊C4—731no—
175C2×C4⋊C4—1461no—
176C2×C4⋊C4C4×D7311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1461no—
179C2×C22⋊C4—1462no—
180C2×C42—731no—
181C2×C42—731no—
182C2×C42—732no—
183C2×C42—732no—
184C2×C42—1461no—
185C2×C42C22×D7311yescharacteristic
186C4⋊C8—1468no—
187#97—2924no—
188#95—1462no—
189#95—1462no—
190C24—21no—
191C2×Q8—2921no—
192C2×Q8—2921no—
193C2×Q8—732no—
194C2×Q8—1461no—
195C2×D4—21no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4#1192401112yes—
199C22×C4—21no—
200C22×C4—2922no—
201C22×C4—2924no—
202C22×C4—731no—
203C22×C4—732no—
204C22×C4—731no—
205C22×C4—1461no—
206C22×C4—1461no—
207C22×C4C2×C4×D7311yescharacteristic
208C22×C4D4×D7312yes—
209C2×C8—24no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—2922no—
213C2×C8—2922no—
214C2×C8—1464no—
215C4⋊C4—41no—
216C4⋊C4—2921no—
217C4⋊C4—2922no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—2922no—
221C22⋊C4—1468no—
222C42—41no—
223C42—2922no—
224C42—1464no—
225C42—1462no—
226C42—1461no—
227C42—1461no—
228C42—1462no—
229C42—1462no—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—42no—
234C23—42no—
235C23C22⋊C4×D7311yescharacteristic
236Q8—2922no—
237Q8—2922no—
238D4—42no—
239D4—42no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—2921no—
247C2×C4—2921no—
248C2×C4—2921no—
249C2×C4—2922no—
250C2×C4—5844no—
251C2×C4—732no—
252C2×C4—734no—
253C2×C4—1462no—
254C2×C4—1462no—
255C2×C4—1461no—
256C2×C4—1462no—
257C2×C4—1462no—
258C2×C4—1462no—
259C2×C4M16×D7311yescharacteristic
260C2×C4C2×#1192401111yescharacteristic
261C2×C4#1455661512yes—
262C2×C4#1455662012yes—
263C8—44no—
264C8—2924no—
265C22—42no—
266C22—44no—
267C22—81no—
268C22—82no—
269C22—81no—
270C22#1798854911yescharacteristic
271C22#1798855911yescharacteristic
272C22#100×D7311yescharacteristic
273C22#1798858512yes—
274C22#104×D7312yes—
275C4—22no—
276C4—22no—
277C4—21no—
278C4—21no—
279C4—81no—
280C4—2921no—
281C4—2922no—
282C4—2922no—
283C4—5841no—
284C4—734no—
285C2—82no—
286C2#2260827011yescharacteristic
287C2#2260836712yes—
288C2#2260845312yes—
289C2#329×D7311yescharacteristic
290C2#2260866311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1270C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×C73C2×C4
cosocleC2×C42×C73C23
hypercenter#1270D73
hypocenterC73#7635
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #22609783C2semidirect
3. #22607756C2semidirect
151. C73#7635semidirect

Maximal subgroups

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

subgroupindex
#226124782
#226097832
#226077562
#226077522
C73×#12702
#763573