Finite Group Databasebeta

← #33012722

Subgroups — #33012722

Order 18688.

1262 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988455C211yesmaximal, characteristic
2C2×#17988453C211yesmaximal, characteristic
3#22608195C212yesmaximal
4#22608194C212yesmaximal
5#22608190C212yesmaximal
6#22608187C212yesmaximal
7#22608164C214yesmaximal
8C2×C4×C4⋊C4×C73C211yesmaximal, characteristic
9C22×C4⋊Dic73C2211yescharacteristic
10C22×C4⋊Dic73C2211yescharacteristic
11C22×#11924007C2214yes—
12#17988455C418yescomplemented
13#17988453—24no—
14C2×#14556498C2212yes—
15C2×#14556498C2214yes—
16C2×#14556498C2212yes—
17C2×#14556498C2214yes—
18C2×#14556498C2214yes—
19C2×#14556498C2212yes—
20C2×#14556498C2212yes—
21C2×#14556498C2212yes—
22C22×C4⋊C4×C73C2211yescharacteristic
23C22×C4⋊C4×C73C2211yescharacteristic
24C22×C42×C73C2212yes—
25C22×C42×C73C2211yescharacteristic
26C4×C4⋊C4×C73—24no—
27C2×#95×C73C2212yes—
28C23×Dic73C2312yes—
29C23×Dic73C2312yes—
30C23×Dic73C2314yes—
31C2×C4⋊Dic73—22no—
32C2×C4⋊Dic73—22no—
33C2×C4⋊Dic73—42no—
34C2×C4⋊Dic73—42no—
35C2×#11924007—28no—
36C2×#11924007—44no—
37C2×#11924007—44no—
38#14556498—24no—
39#14556498—24no—
40#14556498—28no—
41#14556498—44no—
42#14556498—44no—
43#14556498C2×C418yes—
44#14556498C2×C418yes—
45#14556498C2×C418yes—
46C23×C4×C73C2312yes—
47C23×C4×C73C2312yes—
48C23×C4×C73C2311yescharacteristic
49C23×C4×C73C2311yescharacteristic
50C23×C4×C73C2311yescharacteristic
51C2×C4⋊C4×C73—22no—
52C2×C4⋊C4×C73—22no—
53C2×C4⋊C4×C73—24no—
54C2×C4⋊C4×C73—24no—
55C2×C42×C73—24no—
56C2×C42×C73—44no—
57C2×C42×C73C2×C414yes—
58C2×C42×C73D414yes—
59C2×C42×C73Q814yes—
60#95×C73—24no—
61C22×Dic73—24no—
62C22×Dic73—24no—
63C22×Dic73—24no—
64C22×Dic73—24no—
65C22×Dic73—24no—
66C22×Dic73—24no—
67C22×Dic73—22no—
68C22×Dic73—22no—
69C22×Dic73—24no—
70C22×Dic73—24no—
71C22×Dic73—22no—
72C22×Dic73—24no—
73C22×Dic73—22no—
74C22×Dic73—24no—
75C22×Dic73—24no—
76C22×Dic73C22×C418yes—
77C4⋊Dic73—44no—
78C4⋊Dic73—44no—
79#11924007—416no—
80C24×C73C2411yescharacteristic
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—22no—
88C22×C4×C73—22no—
89C22×C4×C73—22no—
90C22×C4×C73—22no—
91C22×C4×C73—24no—
92C22×C4×C73—24no—
93C22×C4×C73—24no—
94C22×C4×C73—24no—
95C22×C4×C73C22×C412yes—
96C22×C4×C73C22×C412yes—
97C22×C4×C73C22×C412yes—
98C22×C4×C73C2×D412yes—
99C22×C4×C73C2×D412yes—
100C22×C4×C73C2×D412yes—
101C22×C4×C73C2×Q812yes—
102C22×C4×C73C2×Q812yes—
103C22×C4×C73C2×Q812yes—
104C22×C4×C73#4114yes—
105C22×C4×C73#4114yes—
106C4⋊C4×C73—28no—
107C4⋊C4×C73—28no—
108C42×C73—48no—
109C42×C73C4⋊C4116yescomplemented
110C23×C73C23×C411yescharacteristic
111C23×C73C22×D411yescharacteristic
112C23×C73C22×Q811yescharacteristic
113C23×C73C2×#4112yes—
114C23×C73C2×#4112yes—
115C23×C73H(5,𝔽2)12yes—
116C23×C73H(5,𝔽2)12yes—
117C23×C732−1+412yes—
118C23×C732−1+412yes—
119C2×C4×C73—24no—
120C2×C4×C73—24no—
121C2×C4×C73—24no—
122C2×C4×C73—22no—
123C2×C4×C73—22no—
124C2×C4×C73—24no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—22no—
128C2×C4×C73—22no—
129C2×C4×C73—24no—
130C2×C4×C73—24no—
131C2×C4×C73—24no—
132C2×C4×C73—24no—
133C2×C4×C73—28no—
134C2×C4×C73—28no—
135C2×C4×C73—42no—
136C2×C4×C73—42no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73—44no—
140C2×C4×C73—44no—
141C2×C4×C73C2×C4⋊C414yes—
142C2×C4×C73C2×C4⋊C414yes—
143C2×C4×C73C2×C4⋊C414yes—
144C2×Dic73—28no—
145C2×Dic73—28no—
146C2×Dic73—28no—
147C2×Dic73—44no—
148C2×Dic73—44no—
149C2×Dic73—44no—
150C2×Dic73—44no—
151C2×Dic73—42no—
152C2×Dic73—42no—
153C2×Dic73—42no—
154C2×Dic73—42no—
155C2×Dic73—42no—
156C2×Dic73—42no—
157C2×Dic73—44no—
158C2×Dic73—44no—
159C2×Dic73—48no—
160C22×C73C22×C4⋊C411yescharacteristic
161C22×C73C4×#4112yes—
162C22×C73#52012yes—
163C22×C73#52012yes—
164C22×C73#52511yescharacteristic
165C22×C73#52511yescharacteristic
166C22×C73#53312yes—
167C22×C73#53511yescharacteristic
168C22×C73#53511yescharacteristic
169C22×C73#53611yescharacteristic
170C22×C73#53611yescharacteristic
171C22×C73#53814yes—
172C22×C73#53914yes—
173C22×C73#54114yes—
174C22×C73#54214yes—
175C22×C73#54412yes—
176C22×C73#54412yes—
177C4×C73—24no—
178C4×C73—24no—
179C4×C73—24no—
180C4×C73—44no—
181C4×C73—44no—
182Dic73—48no—
183Dic73—82no—
184Dic73—82no—
185#14298—731nomaximal, Sylow
186C2×C73#229611yescharacteristic
187C2×C73#231711yescharacteristic
188C2×C73#231711yescharacteristic
189C2×C73#233414yes—
190C2×C73#263111yescharacteristic
191C2×C73#263211yescharacteristic
192C2×C73#263311yescharacteristic
193C2×C73#263411yescharacteristic
194C2×C73#266614yes—
195C2×#383—731no—
196C2×#382—731no—
197C2×C4×C4⋊C4D7311yescharacteristic
198#1444—732no—
199#1444—732no—
200#1442—732no—
201#1442—732no—
202#1436—734no—
203C73#1429811yescharacteristic, complemented, Sylow
204C22×C4⋊C4—734no—
205C22×C4⋊C4—731no—
206C22×C4⋊C4—731no—
207C22×C4⋊C4C2×D7311yescharacteristic
208C22×C4⋊C4C2×D7311yescharacteristic
209C22×C42C2×D7312yes—
210C22×C42C2×D7311yescharacteristic
211#383—738no—
212#382—1464no—
213C4×C4⋊C4—24no—
214C2×#95—734no—
215C2×#95—734no—
216C2×#95—732no—
217C2×#95—732no—
218C2×#95—732no—
219C2×#95—732no—
220C2×#95—734no—
221C2×#95—732no—
222C2×#95C2×D7312yes—
223C23×C4—734no—
224C23×C4—732no—
225C23×C4—732no—
226C23×C4C22×D7312yes—
227C23×C4C22×D7312yes—
228C23×C4C22×D7311yescharacteristic
229C23×C4C22×D7311yescharacteristic
230C23×C4C22×D7311yescharacteristic
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—2924no—
236C2×C4⋊C4—2924no—
237C2×C4⋊C4—2922no—
238C2×C4⋊C4—2922no—
239C2×C4⋊C4—1462no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1468no—
242C2×C42—24no—
243C2×C42—44no—
244C2×C42Dic14614yes—
245C2×C42C4×D7314yes—
246C2×C42C22⋊D7314yes—
247#95—24no—
248#95—2924no—
249#95—2924no—
250#95—738no—
251#95—738no—
252#95—738no—
253#95—1464no—
254#95—1464no—
255#95—1468no—
256C24C23×D7311yescharacteristic
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—738no—
272C22×C4—1462no—
273C22×C4—1462no—
274C22×C4—1464no—
275C22×C4—1464no—
276C22×C4—1462no—
277C22×C4—1462no—
278C22×C4—1464no—
279C22×C4—1464no—
280C22×C4—1464no—
281C22×C4—1464no—
282C22×C4—1464no—
283C22×C4—1464no—
284C22×C4—1464no—
285C22×C4—1464no—
286C22×C4—1464no—
287C22×C4C2×C4×D7312yes—
288C22×C4C2×C4×D7312yes—
289C22×C4C2×C4×D7312yes—
290C22×C4C2×C22⋊D7312yes—
291C22×C4C2×C22⋊D7312yes—
292C22×C4C2×C22⋊D7312yes—
293C22×C4C2×Dic14612yes—
294C22×C4C2×Dic14612yes—
295C22×C4C2×Dic14612yes—
296C22×C4#1192402814yes—
297C22×C4#1192402814yes—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—29216no—
301C4⋊C4—2924no—
302C4⋊C4—2924no—
303C42—48no—
304C42#11924007116yescomplemented
305C23C22×C4×D7311yescharacteristic
306C23C22×C22⋊D7311yescharacteristic
307C23C22×Dic14611yescharacteristic
308C23C2×#1192402812yes—
309C23C2×#1192402812yes—
310C23#1455663612yes—
311C23#1455663712yes—
312C23#1455663812yes—
313C23#1455663912yes—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—2924no—
337C2×C4—2924no—
338C2×C4—2924no—
339C2×C4—2924no—
340C2×C4—2922no—
341C2×C4—2922no—
342C2×C4—2922no—
343C2×C4—2922no—
344C2×C4—2924no—
345C2×C4—2924no—
346C2×C4—2928no—
347C2×C4—2922no—
348C2×C4—2922no—
349C2×C4—1468no—
350C2×C4—1468no—
351C2×C4—1468no—
352C2×C4C2×#1192400714yes—
353C2×C4C2×#1192400714yes—
354C2×C4C2×#1192400714yes—
355C22C22×#1192400711yescharacteristic
356C22C4×#1192402812yes—
357C22#1798906712yes—
358C22#1798906812yes—
359C22#1798909811yescharacteristic
360C22#1798909811yescharacteristic
361C22#1798914412yes—
362C22#1798915111yescharacteristic
363C22#1798915411yescharacteristic
364C22#1798915811yescharacteristic
365C22#1798916111yescharacteristic
366C22#1798917014yes—
367C22#1798917914yes—
368C22#1798919514yes—
369C22#1798920914yes—
370C22#1798921512yes—
371C22#1798921812yes—
372C4—24no—
373C4—24no—
374C4—24no—
375C4—44no—
376C4—44no—
377C4—2928no—
378C4—5842no—
379C4—5842no—
380C2#2261241411yescharacteristic
381C2#2261255111yescharacteristic
382C2#2261255511yescharacteristic
383C2#2261270614yes—
384C2#2261578311yescharacteristic
385C2#2261579111yescharacteristic
386C2#2261579911yescharacteristic
387C2#2261580711yescharacteristic
388C2#2261611014yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×C4×C4⋊C4×C73C2
socle, prime radical, cosocleC24×C73C24
hypercenterC2×C4×C4⋊C4D73
hypocenterC73#14298
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. #17988455C4semidirect
109. C42×C73C4⋊C4semidirect
203. C73#14298semidirect
304. C42#11924007semidirect

Maximal subgroups

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

subgroupindex
C2×#179884552
C2×#179884532
#226081952
#226081942
#226081902
#226081872
#226081642
C2×C4×C4⋊C4×C732
#1429873