Finite Group Databasebeta

← #393×Dic73 #33007790

Subgroups — #393×Dic73 #33007790

Order 18688.

2158 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic73C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic73C213yesmaximal
3#22608202C211yesmaximal, characteristic, complemented
4#22608188C213yesmaximal, complemented
5#22608176C213yesmaximal
6#95×Dic73C213yesmaximal, complemented
7C2×#393×C73C211yesmaximal, characteristic
8C24×Dic73C2211yescharacteristic
9C2×Q8×Dic73—212no—
10C22×C4⋊Dic73C2213yes—
11C22×C22⋊Dic73C2213yes—
12C22×C4×Dic73C2213yes—
13C22×C4×Dic73C2213yes—
14C22⋊C4×Dic73—212no—
15C2×#14556498C2216yes—
16C2×#14556498C2213yes—
17C2×#14556498C2213yes—
18C2×#14556498C2213yes—
19C23×Q8×C73C2211yescharacteristic
20C22×C22⋊C4×C73C2213yes—
21#393×C73C418yescomplemented
22C2×#95×C73C2213yes—
23C23×Dic73—26no—
24C23×Dic73—26no—
25C23×Dic73Q814yescomplemented
26C23×Dic73C2313yes—
27C23×Dic73C2311yescharacteristic
28C23×Dic73C2313yes—
29C23×Dic73C2311yescharacteristic
30Q8×Dic73—232no—
31C2×C4⋊Dic73—212no—
32C2×C4⋊Dic73—26no—
33C2×C22⋊Dic73—26no—
34C2×C22⋊Dic73—26no—
35C2×C22⋊Dic73—26no—
36C2×C4×Dic73—212no—
37C2×C4×Dic73—26no—
38C2×C4×Dic73—212no—
39C2×C4×Dic73D4112yes—
40#14556498—26no—
41#14556498—212no—
42#14556498—26no—
43#14556498—26no—
44C25×C73C2311yescharacteristic
45C22×Q8×C73—26no—
46C22×Q8×C73—26no—
47C22×Q8×C73C2×C414yescomplemented
48C23×C4×C73C2313yes—
49C23×C4×C73C2313yes—
50C2×C22⋊C4×C73—26no—
51C2×C22⋊C4×C73—26no—
52C2×C22⋊C4×C73C2×C4112yes—
53#95×C73C2×C4112yescomplemented
54C22×Dic73—23no—
55C22×Dic73—23no—
56C22×Dic73—26no—
57C22×Dic73—26no—
58C22×Dic73—26no—
59C22×Dic73—26no—
60C22×Dic73—23no—
61C22×Dic73—23no—
62C22×Dic73—23no—
63C22×Dic73—23no—
64C22×Dic73—224no—
65C22×Dic73—44no—
66C22×Dic73—412no—
67C22×Dic73C2×D416yes—
68C22×Dic73C2×D416yes—
69C22×Dic73C2×Q812yes—
70C22×Dic73C2×Q812yes—
71C22×Dic73#4116yes—
72C22×Dic73#4116yes—
73C4⋊Dic73—224no—
74C22⋊Dic73—224no—
75C4×Dic73—224no—
76C4×Dic73—412no—
77C24×C73—23no—
78C24×C73—23no—
79C24×C73—23no—
80C24×C73—23no—
81C24×C73C22×C412yes—
82C24×C73C2×Q812yes—
83C24×C73#4112yes—
84C24×C73C2411yescharacteristic
85C2×Q8×C73—224no—
86C2×Q8×C73—28no—
87C2×Q8×C73—224no—
88C22×C4×C73—26no—
89C22×C4×C73—26no—
90C22×C4×C73—23no—
91C22×C4×C73—23no—
92C22×C4×C73—26no—
93C22×C4×C73—26no—
94C22×C4×C73C22×C416yes—
95C22×C4×C73C22×C416yes—
96C22×C4×C73C2×D416yes—
97C22×C4×C73#4116yes—
98C22⋊C4×C73—224no—
99C23×C73—26no—
100C23×C73—26no—
101C23×C73—26no—
102C23×C73—26no—
103C23×C73—26no—
104C23×C73—26no—
105C23×C73—41no—
106C23×C73—43no—
107C23×C73—43no—
108C23×C73—46no—
109C23×C73—43no—
110C23×C73C4×Q814yescomplemented
111C23×C73C23×C411yescharacteristic
112C23×C73C22×D413yes—
113C23×C73C22×Q811yescharacteristic
114C23×C73C2×#4113yes—
115C23×C73C2×#4113yes—
116C23×C73C2×#4113yes—
117C23×C73C2×#4111yescharacteristic
118Q8×C73—232no—
119C2×C4×C73—212no—
120C2×C4×C73—26no—
121C2×C4×C73—26no—
122C2×C4×C73—212no—
123C2×C4×C73—26no—
124C2×C4×C73—212no—
125C2×C4×C73—43no—
126C2×C4×C73—46no—
127C2×C4×C73—43no—
128C2×C4×C73C4×D4112yes—
129C2×Dic73—26no—
130C2×Dic73—26no—
131C2×Dic73—26no—
132C2×Dic73—26no—
133C2×Dic73—212no—
134C2×Dic73—26no—
135C2×Dic73—41no—
136C2×Dic73—43no—
137C2×Dic73—46no—
138C2×Dic73—43no—
139C2×Dic73—43no—
140C2×Dic73—46no—
141C2×Dic73—46no—
142C2×Dic73—416no—
143C2×Dic73#12014yes—
144C2×Dic73C22⋊Q8112yes—
145C2×Dic73#124112yes—
146C22×C73—212no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—46no—
150C22×C73—46no—
151C22×C73—42no—
152C22×C73—46no—
153C22×C73C2×C4×D413yes—
154C22×C73C2×C4×Q811yescharacteristic
155C22×C73C4×#4113yes—
156C22×C73C2×#12011yescharacteristic
157C22×C73C2×C22⋊Q813yes—
158C22×C73#52513yes—
159C22×C73#52513yes—
160C22×C73C2×#12413yes—
161C22×C73#52913yes—
162C22×C73#52916yes—
163C22×C73#53213yes—
164C22×C73#53313yes—
165C4×C73—212no—
166C4×C73—46no—
167Dic73—42no—
168Dic73—46no—
169Dic73—46no—
170Dic73#39318yesdirect product factor
171C4×#393—731nomaximal, Sylow
172C2×C73—44no—
173C2×C73C4×#12011yescharacteristic
174C2×C73C4×C22⋊Q813yes—
175C2×C73C4×#12413yes—
176C2×C73C2×#39311yescharacteristic
177C2×C73#238811yescharacteristic
178C2×C73#239613yes—
179C2×C73#239813yes—
180C22×C4×Q8—731no—
181C2×#393D7311yescharacteristic
182C2×C4×C22⋊C4—733no—
183#1446—731no—
184#1442—733no—
185#1439—733no—
186C4×#95—733no—
187C73C4×#39311yescharacteristic, complemented, Sylow
188C23×Q8C2×D7311yescharacteristic
189C24×C4—731no—
190C2×C4×Q8—14612no—
191C22×C4⋊C4—733no—
192C22×C22⋊C4—733no—
193C22×C22⋊C4C2×D7313yes—
194C22×C42—733no—
195C22×C42—733no—
196#393Dic7318yesdirect product factor
197C4×C22⋊C4—14612no—
198C2×#95—736no—
199C2×#95—733no—
200C2×#95—733no—
201C2×#95—733no—
202C2×#95C2×D7313yes—
203C25C22×D7311yescharacteristic
204C22×Q8—26no—
205C22×Q8—26no—
206C22×Q8C2×Dic7314yescomplemented
207C23×C4—731no—
208C23×C4—734no—
209C23×C4—733no—
210C23×C4—733no—
211C23×C4—731no—
212C23×C4—1466no—
213C23×C4—1466no—
214C23×C4C22×D7313yes—
215C23×C4C22×D7313yes—
216C4×Q8—14632no—
217C2×C4⋊C4—14612no—
218C2×C4⋊C4—1466no—
219C2×C22⋊C4—26no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—1466no—
222C2×C22⋊C4—1466no—
223C2×C22⋊C4—1466no—
224C2×C22⋊C4C2×Dic73112yes—
225C2×C42—7312no—
226C2×C42—1466no—
227C2×C42—14612no—
228C2×C42—14612no—
229#95—14612no—
230#95—1466no—
231#95—1466no—
232#95—1466no—
233#95C2×Dic73112yescomplemented
234C24—23no—
235C24—23no—
236C24—23no—
237C24—23no—
238C24C22×Dic7312yes—
239C24Q8×D7312yes—
240C24#1192402912yes—
241C24C23×D7311yescharacteristic
242C2×Q8—28no—
243C2×Q8—224no—
244C2×Q8—224no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—23no—
251C22×C4—2924no—
252C22×C4—29212no—
253C22×C4—736no—
254C22×C4—732no—
255C22×C4—736no—
256C22×C4—736no—
257C22×C4—732no—
258C22×C4—736no—
259C22×C4—14624no—
260C22×C4—1463no—
261C22×C4—1463no—
262C22×C4—1466no—
263C22×C4—1466no—
264C22×C4—1466no—
265C22×C4—1463no—
266C22×C4—1463no—
267C22×C4—1466no—
268C22×C4—1463no—
269C22×C4—1463no—
270C22×C4C22×Dic7316yes—
271C22×C4C22×Dic7316yes—
272C22×C4D4×D7316yes—
273C22×C4#1192402716yes—
274C4⋊C4—14624no—
275C22⋊C4—224no—
276C22⋊C4—14624no—
277C42—29212no—
278C42—14624no—
279C23—26no—
280C23—26no—
281C23—26no—
282C23—26no—
283C23—26no—
284C23—26no—
285C23—43no—
286C23—41no—
287C23—43no—
288C23—43no—
289C23—46no—
290C23Q8×Dic7314yescomplemented
291C23C23×Dic7311yescharacteristic
292C23C2×D4×D7313yes—
293C23C2×Q8×D7311yescharacteristic
294C23#41×D7313yes—
295C23#41×D7313yes—
296C23C2×#1192402713yes—
297C23C2×#1192402911yescharacteristic
298Q8—232no—
299C2×C4—212no—
300C2×C4—26no—
301C2×C4—26no—
302C2×C4—212no—
303C2×C4—26no—
304C2×C4—212no—
305C2×C4—43no—
306C2×C4—43no—
307C2×C4—46no—
308C2×C4—2926no—
309C2×C4—2926no—
310C2×C4—2921no—
311C2×C4—2923no—
312C2×C4—2926no—
313C2×C4—2923no—
314C2×C4—2923no—
315C2×C4—29216no—
316C2×C4—734no—
317C2×C4—7312no—
318C2×C4—7312no—
319C2×C4—1466no—
320C2×C4—1466no—
321C2×C4—1466no—
322C2×C4—1466no—
323C2×C4—1466no—
324C2×C4—14612no—
325C2×C4D4×Dic73112yes—
326C22—212no—
327C22—42no—
328C22—42no—
329C22—46no—
330C22—42no—
331C22—46no—
332C22—46no—
333C22C2×D4×Dic7313yes—
334C22C2×Q8×Dic7311yescharacteristic
335C22#41×Dic7313yes—
336C22#120×D7311yescharacteristic
337C22C22⋊Q8×D7313yes—
338C22#1798909313yes—
339C22#1798910013yes—
340C22#124×D7313yes—
341C22#1798911616yes—
342C22#1798912313yes—
343C22#1798913613yes—
344C22#1798913913yes—
345C4—212no—
346C4—46no—
347C4—2922no—
348C4—2926no—
349C4—2926no—
350C4—738no—
351C2—44no—
352C2#120×Dic7311yescharacteristic
353C2C22⋊Q8×Dic7313yes—
354C2#124×Dic7313yes—
355C2#393×D7311yescharacteristic
356C2#2261314811yescharacteristic
357C2#2261319813yes—
358C2#2261321313yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#393×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#393D73
hypocenterC73C4×#393
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×Dic73C2semidirect
3. #22608202C2semidirect
4. #22608188C2semidirect
6. #95×Dic73C2semidirect
21. #393×C73C4semidirect
25. C23×Dic73Q8semidirect
47. C22×Q8×C73C2×C4semidirect
53. #95×C73C2×C4semidirect
110. C23×C73C4×Q8semidirect
170. Dic73#393direct (N × H)
187. C73C4×#393semidirect
196. #393Dic73direct (N × H)
206. C22×Q8C2×Dic73semidirect
233. #95C2×Dic73semidirect
290. C23Q8×Dic73semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic732
C2×C22⋊C4×Dic732
#226082022
#226081882
#226081762
#95×Dic732
C2×#393×C732
C4×#39373