Finite Group Databasebeta

← #34003432

Subgroups — #34003432

Order 18688.

698 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625657C212yesmaximal
2C2×#17989432C212yesmaximal
3#22624769C212yesmaximal
4C2×C4⋊C4×Dic73C211yesmaximal, characteristic
5C2×C22⋊#10596833C2214yes—
6C2×#14556646—22no—
7#17989453—22no—
8#17989432C418yescomplemented
9C22×C4⋊Dic73C2211yescharacteristic
10C22×C4×Dic73C411yescharacteristic
11C22×C4×Dic73C411yescharacteristic
12C22×C4×Dic73C2211yescharacteristic
13C4⋊C4×Dic73—22no—
14C4⋊C4×Dic73—22no—
15C2×#14556498C411yescharacteristic
16C2×#14556498C2211yescharacteristic
17C22×C4⋊C4×C73C411yescharacteristic
18C22×#10596833—24no—
19C22⋊#10596833—24no—
20C22⋊#10596833C2×C418yes—
21#14556646—44no—
22C23×Dic73C2×C411yescharacteristic
23C23×Dic73C2×C411yescharacteristic
24C23×Dic73C2×C411yescharacteristic
25C23×Dic73C2311yescharacteristic
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73C2×C414yes—
29C2×C4×Dic73—21no—
30C2×C4×Dic73—21no—
31C2×C4×Dic73—22no—
32C2×C4×Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—22no—
35C2×C4×Dic73—42no—
36C2×C4×Dic73—42no—
37C2×C4×Dic73D412yes—
38C2×C4×Dic73D412yes—
39#14556498—22no—
40#14556498—22no—
41C23×C4×C73C2×C411yescharacteristic
42C23×C4×C73C2×C411yescharacteristic
43C23×C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73—22no—
46C2×C4⋊C4×C73—22no—
47C2×#10596833—28no—
48C2×#10596833—44no—
49C2×#10596833—44no—
50C22×Dic73—21no—
51C22×Dic73—21no—
52C22×Dic73—22no—
53C22×Dic73—22no—
54C22×Dic73—21no—
55C22×Dic73—21no—
56C22×Dic73—21no—
57C22×Dic73—21no—
58C22×Dic73—41no—
59C22×Dic73—41no—
60C22×Dic73—41no—
61C22×Dic73—41no—
62C22×Dic73—41no—
63C22×Dic73—41no—
64C22×Dic73C4212yes—
65C22×Dic73C22⋊C411yescharacteristic
66C22×Dic73C22⋊C411yescharacteristic
67C22×Dic73M1612yes—
68C22×Dic73C22×C412yes—
69C22×Dic73C2×D412yes—
70C22×Dic73#4112yes—
71C4⋊Dic73—28no—
72C4×Dic73—24no—
73C4×Dic73—24no—
74C4×Dic73—24no—
75C4×Dic73—42no—
76C4×Dic73—42no—
77C4×Dic73—42no—
78C24×C73C22×C411yescharacteristic
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—41no—
88C22×C4×C73—41no—
89C22×C4×C73—41no—
90C22×C4×C73—41no—
91C22×C4×C73C4212yes—
92C22×C4×C73C22⋊C412yes—
93C22×C4×C73M1612yes—
94C22×C4×C73M1612yes—
95C22×C4×C73M1612yes—
96C4⋊C4×C73—28no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73C2×C4211yescharacteristic
102C23×C73C2×C22⋊C411yescharacteristic
103C23×C73#11711yescharacteristic
104C23×C73C2×M1611yescharacteristic
105C23×C73C2×M1611yescharacteristic
106C23×C73C2×M1611yescharacteristic
107C23×C73C2×M1611yescharacteristic
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—22no—
117C2×C4×C73—22no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—42no—
124C2×C4×C73#9714yescomplemented
125C2×Dic73—24no—
126C2×Dic73—24no—
127C2×Dic73—21no—
128C2×Dic73—21no—
129C2×Dic73—22no—
130C2×Dic73—21no—
131C2×Dic73—22no—
132C2×Dic73—21no—
133C2×Dic73—22no—
134C2×Dic73—22no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—42no—
138C2×Dic73—42no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73—42no—
144C2×Dic73—42no—
145C2×Dic73#9912yes—
146C2×Dic73#10112yes—
147C2×Dic73C4×D414yes—
148#10596833—48no—
149C22×C73—21no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73C4×C22⋊C411yescharacteristic
162C22×C73C2×#9711yescharacteristic
163C22×C73C4×M1611yescharacteristic
164C22×C73#40711yescharacteristic
165C22×C73#40711yescharacteristic
166C22×C73C2×#9911yescharacteristic
167C22×C73#41011yescharacteristic
168C22×C73C2×#10111yescharacteristic
169C22×C73#41311yescharacteristic
170C22×C73#43211yescharacteristic
171C22×C73#43211yescharacteristic
172C4×C73—24no—
173C4×C73—42no—
174C4×C73—42no—
175Dic73—22no—
176Dic73—22no—
177Dic73—24no—
178Dic73—42no—
179Dic73—42no—
180#8153—731nomaximal, Sylow
181C2×C73—21no—
182C2×C73—21no—
183C2×C73—21no—
184C2×C73—21no—
185C2×C73#146311yescharacteristic
186C2×C73#146411yescharacteristic
187C2×C73#146511yescharacteristic
188C2×C73#146711yescharacteristic
189C2×C73#174911yescharacteristic
190C2×C73C4×#9911yescharacteristic
191C2×C73C4×#10111yescharacteristic
192C2×C4×C4⋊C4—731no—
193#1749—732no—
194C2×#324—732no—
195#1280—732no—
196C73#815311yescharacteristic, complemented, Sylow
197C22×C4⋊C4—731no—
198C22×C4⋊C4#6440811yescharacteristic
199C22×C42—731no—
200C22×C42—731no—
201C22×C42—731no—
202C2×C22⋊C8—734no—
203C2×#97—1462no—
204C4×C4⋊C4—1462no—
205C4×C4⋊C4—1462no—
206C2×#95—731no—
207C2×#95—731no—
208#336—1462no—
209#324—738no—
210C23×C4—731no—
211C23×C4—731no—
212C23×C4—731no—
213C23×C4—731no—
214C23×C4C2×#6440811yescharacteristic
215C23×C4C2×#6440811yescharacteristic
216C23×C4C2×#6440811yescharacteristic
217C22×C8—1464no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—734no—
222C2×C4⋊C4—1462no—
223C2×C4⋊C4—1462no—
224C2×C42—2922no—
225C2×C42—2922no—
226C2×C42—732no—
227C2×C42—732no—
228C2×C42—1461no—
229C2×C42—1461no—
230C2×C42—1461no—
231C2×C42—1461no—
232C2×C42—1462no—
233C2×C42—1462no—
234C22⋊C8—738no—
235C22⋊C8—1464no—
236#97—2924no—
237#95—1462no—
238#95—1462no—
239C24C22×#6440811yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—2921no—
253C22×C4—2921no—
254C22×C4—2921no—
255C22×C4—2921no—
256C22×C4—2921no—
257C22×C4—2921no—
258C22×C4—732no—
259C22×C4—732no—
260C22×C4—732no—
261C22×C4—732no—
262C22×C4—731no—
263C22×C4—731no—
264C22×C4—732no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1461no—
268C22×C4—1461no—
269C22×C4—1462no—
270C22×C4—1462no—
271C22×C4—1461no—
272C22×C4—1461no—
273C22×C4C4×#6440812yes—
274C22×C4C22⋊#6440812yes—
275C22×C4#1192403712yes—
276C22×C4#1192403712yes—
277C22×C4#1192403812yes—
278C2×C8—2924no—
279C2×C8—2924no—
280C2×C8—1468no—
281C4⋊C4—28no—
282C4⋊C4—1468no—
283C42—2922no—
284C42—2922no—
285C42—2922no—
286C42—1464no—
287C42—1464no—
288C42—1464no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×C4×#6440811yescharacteristic
294C23C2×C22⋊#6440811yescharacteristic
295C23#1455667011yescharacteristic
296C23C2×#1192403711yescharacteristic
297C23C2×#1192403711yescharacteristic
298C23D73⋊M1611yescharacteristic
299C23C2×#1192403811yescharacteristic
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—2922no—
317C2×C4—2922no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—2922no—
323C2×C4—2922no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—734no—
327C2×C4—732no—
328C2×C4—732no—
329C2×C4—1462no—
330C2×C4—1462no—
331C2×C4—1464no—
332C2×C4—1464no—
333C2×C4—1461no—
334C2×C4—1461no—
335C2×C4—1461no—
336C2×C4—1462no—
337C2×C4—1462no—
338C2×C4—1461no—
339C2×C4#1455664614yescomplemented
340C8—2928no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22C4×C22⋊#6440811yescharacteristic
354C22C2×#1455664611yescharacteristic
355C22C4×#1192403711yescharacteristic
356C22#1798955711yescharacteristic
357C22#1798955811yescharacteristic
358C22D73⋊#9911yescharacteristic
359C22#1798956811yescharacteristic
360C22D73⋊#10111yescharacteristic
361C22#1798957411yescharacteristic
362C22#1798960711yescharacteristic
363C22#1798960811yescharacteristic
364C4—24no—
365C4—42no—
366C4—42no—
367C4—2922no—
368C4—2922no—
369C4—1464no—
370C4—1462no—
371C4—1462no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2262518011yescharacteristic
377C2#2262518511yescharacteristic
378C2#2262518711yescharacteristic
379C2#2262519411yescharacteristic
380C2#2262565711yescharacteristic
381C2#2262567611yescharacteristic
382C2#2262568411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC23×C73C2×C42
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8153
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #17989432C4semidirect
124. C2×C4×C73#97semidirect
196. C73#8153semidirect
339. C2×C4#14556646semidirect

Maximal subgroups

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

subgroupindex
#226256572
C2×#179894322
#226247692
C2×C4⋊C4×Dic732
#815373