Finite Group Databasebeta

← #32977720

Subgroups — #32977720

Order 18688.

706 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 131 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×#17989315C211yesmaximal, characteristic
2#22609927C211yesmaximal, characteristic
3#22608602C212yesmaximal
4#22608361C212yesmaximal
5C73×#1461C211yesmaximal, characteristic
6#17989315—24no—
7#17989315—24no—
8C2×#14556591C2211yescharacteristic
9C2×#14556589—21no—
10C2×#14556583C412yes—
11C2×#14556581—22no—
12C2×C4×Dic146C412yes—
13C2×C4×Dic146C2211yescharacteristic
14C2×#14556556—21no—
15#17988611C2212yes—
16C2×#14556510—21no—
17#17988356—21no—
18#17988332—24no—
19#17988319—24no—
20C2×C4×Q8×C73C2211yescharacteristic
21#431×C73C2212yes—
22#324×C73—22no—
23C22×Dic146—21no—
24C22×Dic146C2×C411yescharacteristic
25C2×#11924012—22no—
26#14556591—22no—
27#14556591D414yes—
28#14556589—28no—
29#14556583—24no—
30#14556583—24no—
31#14556581—216no—
32C4×Dic146—24no—
33C4×Dic146—24no—
34C4×Dic146—22no—
35C4×Dic146D414yes—
36#14556556—28no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73C2×C412yes—
39C2×C4⋊Dic73C2×C411yescharacteristic
40C2×#11924007—22no—
41C2×#11924007—22no—
42C2×#11924007—22no—
43C2×#11924007—21no—
44C2×C4×Dic73—22no—
45C2×C4×Dic73—21no—
46C4⋊#10596831—22no—
47C4⋊#10596831—22no—
48#14556510—42no—
49C22×Q8×C73D411yescharacteristic
50C4×Q8×C73—22no—
51C4×Q8×C73—22no—
52C2×C4⋊C4×C73D411yescharacteristic
53C2×C4⋊C4×C73D412yes—
54C2×C42×C73C2×C412yes—
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—22no—
57C22⋊C8×C73—22no—
58C4×C8×C73—22no—
59C2×Dic146—21no—
60C2×Dic146—24no—
61C2×Dic146—24no—
62C2×Dic146—42no—
63C2×Dic146—41no—
64C2×Dic146—41no—
65C2×Dic146C22⋊C412yes—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73—22no—
69#11924012—42no—
70#11924012—42no—
71C2×#10596831—42no—
72C4⋊Dic73—21no—
73C4⋊Dic73—22no—
74C4⋊Dic73—24no—
75C4⋊Dic73C22⋊C414yes—
76C4⋊Dic73C22⋊C412yes—
77#11924007—28no—
78#11924007—28no—
79#11924007—28no—
80#11924007—24no—
81C4×Dic73—28no—
82C4×Dic73—24no—
83C2×Q8×C73—22no—
84C2×Q8×C73—22no—
85C2×Q8×C73—21no—
86C2×Q8×C73—21no—
87C22×C4×C73C22⋊C412yes—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C2×C8×C73—42no—
93C4⋊C4×C73—22no—
94C4⋊C4×C73—24no—
95C42×C73—22no—
96C42×C73—22no—
97C42×C73C2×D412yes—
98C42×C73C2×D412yes—
99C23×C73C2×C22⋊C411yescharacteristic
100Q8×C73—42no—
101Q8×C73—42no—
102C2×C4×C73—22no—
103C2×C4×C73—22no—
104C2×C4×C73—22no—
105C2×C4×C73—22no—
106C2×C4×C73—22no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73C2×C22⋊C411yescharacteristic
110C2×C4×C73C2×C22⋊C411yescharacteristic
111C2×C4×C73#12012yes—
112C2×C4×C73#12012yes—
113C2×C4×C73#13512yes—
114C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C73#13711yescharacteristic
116C2×Dic73—24no—
117C2×Dic73—22no—
118C2×Dic73—22no—
119C2×Dic73—24no—
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123Dic146—24no—
124Dic146—22no—
125Dic146—44no—
126Dic146—44no—
127Dic146#10114yescomplemented
128C8×C73—82no—
129#10596831—82no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73#37911yescharacteristic
133C22×C73#41611yescharacteristic
134C22×C73#41711yescharacteristic
135C4×C73—22no—
136C4×C73—21no—
137C4×C73—42no—
138C4×C73—42no—
139C4×C73C2×#10112yes—
140C4×C73#44912yes—
141C4×C73#45212yes—
142Dic73—24no—
143Dic73—42no—
144Dic73—44no—
145#8390—731nomaximal, Sylow
146C2×C73—21no—
147C2×C73—21no—
148C2×C73#178411yescharacteristic
149C2×C73#178811yescharacteristic
150C2×C73#179611yescharacteristic
151C2×#557—731no—
152#1801—731no—
153#1546—732no—
154#1487—732no—
155#1461D7311yescharacteristic
156C73#839011yescharacteristic, complemented, Sylow
157#557—1464no—
158#557—1464no—
159C2×C4⋊Q8—731no—
160C2×C4⋊Q8—1461no—
161C2×#125—732no—
162C2×#125—1462no—
163C2×C4×Q8—732no—
164C2×C4×Q8—731no—
165C2×C4×Q8—1461no—
166C2×C4×Q8C2×D7311yescharacteristic
167#431C2×D7312yes—
168#423—732no—
169C2×#101—1461no—
170#343—1461no—
171#332—1464no—
172Q8⋊C8—1464no—
173#324—22no—
174C22×Q8—731no—
175C22×Q8—1461no—
176C22×Q8D29211yescharacteristic
177C2×M16—1462no—
178C4⋊Q8—734no—
179C4⋊Q8—1468no—
180C4⋊Q8—1462no—
181#125—1464no—
182#125—1464no—
183#125—14616no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—734no—
187C4×Q8—1464no—
188C4×Q8—1464no—
189C4×Q8—1468no—
190C4×Q8—1462no—
191C2×C4⋊C4—731no—
192C2×C4⋊C4—732no—
193C2×C4⋊C4—1462no—
194C2×C4⋊C4—1462no—
195C2×C4⋊C4—1462no—
196C2×C4⋊C4—1461no—
197C2×C4⋊C4—1461no—
198C2×C4⋊C4D29211yescharacteristic
199C2×C4⋊C4C22⋊D7312yes—
200C2×C42—1462no—
201C2×C42—1461no—
202C2×C42C4×D7312yes—
203C2×C42C22×D7311yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—1462no—
206C4⋊C8—1462no—
207#101—2922no—
208C22⋊C8—22no—
209C4×C8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—2922no—
215C2×Q8—2921no—
216C2×Q8—2921no—
217C2×Q8—732no—
218C2×Q8—1464no—
219C2×Q8—1461no—
220C2×Q8—1464no—
221C22×C4—1461no—
222C22×C4—1462no—
223C22×C4—1461no—
224C22×C4#1192400512yes—
225C22×C4#1192400512yes—
226C22×C4C2×C4×D7311yescharacteristic
227C22×C4C2×C22⋊D7311yescharacteristic
228C22×C4C2×D29211yescharacteristic
229M16—2922no—
230M16—2922no—
231C2×C8—42no—
232C2×C8—2922no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—732no—
236C4⋊C4—734no—
237C4⋊C4—1468no—
238C4⋊C4—1468no—
239C4⋊C4—1468no—
240C4⋊C4—1461no—
241C4⋊C4—1462no—
242C4⋊C4—1464no—
243C4⋊C4—1464no—
244C42—22no—
245C42—22no—
246C42—1468no—
247C42—1464no—
248C42D4×D7312yes—
249C42D4×D7312yes—
250C23C2×#1192400511yescharacteristic
251Q8—42no—
252Q8—42no—
253Q8—2924no—
254Q8—2924no—
255Q8—734no—
256Q8—1464no—
257Q8—1462no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—2922no—
266C2×C4—2922no—
267C2×C4—2922no—
268C2×C4—1462no—
269C2×C4—1464no—
270C2×C4—1464no—
271C2×C4—1462no—
272C2×C4C22⋊C4×D7311yescharacteristic
273C2×C4C22⋊C4×D7311yescharacteristic
274C2×C4#1455655912yes—
275C2×C4#1455656112yes—
276C2×C4#1455661312yes—
277C2×C4#1455661911yescharacteristic
278C2×C4#1455662411yescharacteristic
279C8—82no—
280C8—5842no—
281C22—21no—
282C22—21no—
283C22#1798844211yescharacteristic
284C22#1798858411yescharacteristic
285C22#1798858911yescharacteristic
286C4—21no—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—2924no—
291C4—2922no—
292C4—1464no—
293C4#101×D7312yes—
294C4#1798873412yes—
295C4#1798875112yes—
296C2—21no—
297C2—21no—
298C2#2260985311yescharacteristic
299C2#2260987211yescharacteristic
300C2#2260990411yescharacteristic
subgroupG/H
centerC22#17988442
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1461C2
socleC22×C73#379
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1461D73
hypocenterC73#8390
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
127. Dic146#101semidirect
156. C73#8390semidirect

Maximal subgroups

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

subgroupindex
C2×#179893152
#226099272
#226086022
#226083612
C73×#14612
#839073