Finite Group Databasebeta

← #13259250

Subgroups — #13259250

Order 1792.

1162 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90503C211yesmaximal, characteristic
2C22×C7×#101C211yesmaximal, characteristic
3C22×#6410C211yesmaximal, characteristic
4C2×#89354C212yesmaximal
5C2×#89351C212yesmaximal
6C23×C7×Q8C2211yescharacteristic
7C22×C7×M16C2212yes—
8#90503—24no—
9#90503—24no—
10C2×Q8×Dic7—22no—
11C2×Q8×Dic7—22no—
12C2×#6489C414yes—
13C2×#6489C2211yescharacteristic
14C2×#6489C2211yescharacteristic
15C22×#1136C2212yes—
16C2×C7×#101—22no—
17C2×C7×#101—22no—
18C2×C7×#101C414yescomplemented
19C2×#6410—22no—
20C2×#6410—22no—
21C2×#6410C414yescomplemented
22#89354—24no—
23#89351—24no—
24#10770—71nomaximal, Sylow
25C22×C7×Q8—22no—
26C22×C7×Q8—22no—
27C22×C7×Q8—22no—
28C22×C7×Q8—22no—
29C22×C7×Q8C2×C412yes—
30C22×C7×Q8C2×C412yes—
31C22×C7×Q8D412yes—
32C22×C7×Q8D412yes—
33C22×C7×Q8D412yes—
34C22×C7×Q8Q812yes—
35C23×C4×C7C2×C412yes—
36C23×C4×C7C2311yescharacteristic
37C2×C7×M16—22no—
38C2×C7×M16—22no—
39C2×C7×M16—22no—
40C2×C7×M16—22no—
41C2×C7×M16—22no—
42C2×C7×M16C2×C414yes—
43C22×C7×C8—22no—
44C2×C22⋊Dic7C2×C412yes—
45C2×C22⋊Dic7C2×C412yes—
46Q8×Dic7—216no—
47Q8×Dic7—216no—
48#6489—22no—
49#6489—22no—
50#6489—28no—
51#6489—28no—
52#6489D414yes—
53#6489D414yes—
54C2×C4⋊Dic7—24no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4×Dic7—24no—
58C2×C4×Dic7—21no—
59C2×C4×Dic7—21no—
60C2×#1136—22no—
61C2×#1136—22no—
62C2×#1136—22no—
63C2×#1136—22no—
64C2×#1136—22no—
65C2×#1136C2×C414yes—
66C22×C7⋊C8—22no—
67C7×#101—28no—
68#6410—28no—
69C2×#519—71no—
70C22×#101—71no—
71C22×#101D711yescharacteristic
72C2×#344—72no—
73C2×#344—72no—
74C24×C7C22×C411yescharacteristic
75C2×C7×Q8—24no—
76C2×C7×Q8—24no—
77C2×C7×Q8—24no—
78C2×C7×Q8—24no—
79C2×C7×Q8—24no—
80C2×C7×Q8—24no—
81C2×C7×Q8—24no—
82C2×C7×Q8—24no—
83C2×C7×Q8—24no—
84C2×C7×Q8—24no—
85C2×C7×Q8—24no—
86C2×C7×Q8—24no—
87C2×C7×Q8C22⋊C414yes—
88C2×C7×Q8C22⋊C414yes—
89C2×C7×Q8C4⋊C414yes—
90C2×C7×Q8C4⋊C414yes—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7C4214yes—
96C22×C4×C7C22⋊C414yes—
97C22×C4×C7C4⋊C414yes—
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D412yes—
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2×Q811yescharacteristic
106C22×C4×C7#4112yes—
107C22×C4×C7#4112yes—
108C22×Dic7—22no—
109C22×Dic7—22no—
110C7×M16—24no—
111C7×M16—24no—
112C7×M16—42no—
113C7×M16—42no—
114C2×C7×C8—24no—
115C2×C7×C8—42no—
116C2×C7×C8—42no—
117C22⋊Dic7—22no—
118C22⋊Dic7—22no—
119C22⋊Dic7C22⋊C414yes—
120C22⋊Dic7C22⋊C414yes—
121C4⋊Dic7—216no—
122C4⋊Dic7—24no—
123C4⋊Dic7—24no—
124C4×Dic7—216no—
125C4×Dic7—24no—
126C4×Dic7—24no—
127#1136—24no—
128#1136—24no—
129#1136—42no—
130#1136—42no—
131C2×C7⋊C8—24no—
132C2×C7⋊C8—42no—
133C2×C7⋊C8—42no—
134C23×Q8C2×D711yescharacteristic
135C22×M16—72no—
136C22×M16C2×D712yes—
137#519—144no—
138#519—144no—
139C2×C4×Q8—142no—
140C2×C4×Q8—142no—
141C2×#117—71no—
142C2×#117—71no—
143C2×#117—74no—
144C2×#101—22no—
145C2×#101—22no—
146C2×#101—74no—
147C2×#101—142no—
148C2×#101—142no—
149C2×#101Dic714yescomplemented
150#344—144no—
151#344—144no—
152C23×C7—42no—
153C23×C7C2×C4211yescharacteristic
154C23×C7C2×C22⋊C411yescharacteristic
155C23×C7C2×C22⋊C411yescharacteristic
156C23×C7C2×C22⋊C411yescharacteristic
157C23×C7C2×C4⋊C411yescharacteristic
158C23×C7#11711yescharacteristic
159C23×C7#11711yescharacteristic
160C7×Q8—28no—
161C7×Q8—28no—
162C7×Q8—28no—
163C7×Q8—28no—
164C2×C4×C7—24no—
165C2×C4×C7—24no—
166C2×C4×C7—24no—
167C2×C4×C7—24no—
168C2×C4×C7—24no—
169C2×C4×C7—24no—
170C2×C4×C7#9518yes—
171C2×C4×C7C2×C22⋊C411yescharacteristic
172C2×C4×C7C2×C22⋊C411yescharacteristic
173C2×C4×C7C2×C4⋊C411yescharacteristic
174C2×C4×C7C2×C4⋊C411yescharacteristic
175C2×C4×C7C4×D412yes—
176C2×C4×C7C4×D412yes—
177C2×C4×C7C4×D412yes—
178C2×C4×C7C4×D412yes—
179C2×C4×C7#12012yes—
180C2×C4×C7#12012yes—
181C2×C4×C7#12112yes—
182C2×C4×C7#12112yes—
183C2×C4×C7C22⋊Q812yes—
184C2×C4×C7C22⋊Q812yes—
185C2×C4×C7#12312yes—
186C2×C4×C7#12312yes—
187C2×Dic7—24no—
188C2×Dic7—24no—
189C2×Dic7—44no—
190C2×Dic7—44no—
191C7×C8—44no—
192C7⋊C8—44no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8Dic1412yes—
198C22×Q8C4×D712yes—
199C22×Q8D2812yes—
200C22×Q8C2×Dic712yes—
201C22×Q8C22⋊D712yes—
202C22×Q8C22⋊D712yes—
203C23×C4C4×D712yes—
204C23×C4C22×D711yescharacteristic
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—74no—
211C2×M16—142no—
212C2×M16—142no—
213C2×M16—142no—
214C2×M16—142no—
215C2×M16—142no—
216C2×M16C2×Dic714yes—
217C22×C8—22no—
218C22×C8—142no—
219C4×Q8—1416no—
220C4×Q8—1416no—
221#117—74no—
222#117—74no—
223#117—148no—
224#117—148no—
225#117—142no—
226#117—142no—
227C2×C4⋊C4—144no—
228C2×C4⋊C4—141no—
229C2×C4⋊C4—141no—
230C2×C22⋊C4—72no—
231C2×C22⋊C4—72no—
232C2×C42—144no—
233C2×C42—141no—
234C2×C42—141no—
235#101—28no—
236#101—148no—
237C22×C7—22no—
238C22×C7—22no—
239C22×C7—22no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7C2×#9511yescharacteristic
245C22×C7C4×C22⋊C411yescharacteristic
246C22×C7C4×C22⋊C411yescharacteristic
247C22×C7#37911yescharacteristic
248C22×C7#38011yescharacteristic
249C22×C7#38011yescharacteristic
250C22×C7#38111yescharacteristic
251C4×C7—28no—
252C4×C7#38512yes—
253C4×C7#38512yes—
254C4×C7#38612yes—
255C4×C7#38612yes—
256Dic7—44no—
257Dic7—44no—
258C24C2×C4×D711yescharacteristic
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8Dic7⋊C414yes—
272C2×Q8C4⋊Dic714yes—
273C2×Q8(C2×D7)⋊C414yes—
274C2×Q8C22⋊Dic714yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—142no—
280C22×C4—142no—
281C22×C4C4×Dic714yes—
282C22×C4Dic7⋊C414yes—
283C22×C4(C2×D7)⋊C414yes—
284C22×C4C2×Dic1411yescharacteristic
285C22×C4C2×C4×D711yescharacteristic
286C22×C4C2×D2811yescharacteristic
287C22×C4D4×D712yes—
288C22×C4D4×D712yes—
289C22×C4#115912yes—
290C22×C4#115912yes—
291C22×C4C22×Dic711yescharacteristic
292C22×C4C2×C22⋊D711yescharacteristic
293C22×C4C2×C22⋊D711yescharacteristic
294M16—24no—
295M16—24no—
296M16—42no—
297M16—42no—
298M16—144no—
299M16—144no—
300M16—282no—
301M16—282no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—144no—
306C2×C8—282no—
307C2×C8—282no—
308C4⋊C4—1416no—
309C4⋊C4—144no—
310C4⋊C4—144no—
311C22⋊C4—74no—
312C22⋊C4—74no—
313C22⋊C4—142no—
314C22⋊C4—142no—
315C42—1416no—
316C42—144no—
317C42—144no—
318C2×C7—21no—
319C2×C7—21no—
320C2×C7—21no—
321C2×C7—21no—
322C2×C7—21no—
323C2×C7—21no—
324C2×C7C22⋊#9511yescharacteristic
325C2×C7#183012yes—
326C23—42no—
327C23#644011yescharacteristic
328C23#644011yescharacteristic
329C23D7×C22⋊C411yescharacteristic
330C23D7×C22⋊C411yescharacteristic
331C23C2×C4×Dic711yescharacteristic
332C23C2×Dic7⋊C411yescharacteristic
333C23C2×(C2×D7)⋊C411yescharacteristic
334Q8—28no—
335Q8—28no—
336Q8—28no—
337Q8—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—144no—
345C2×C4—144no—
346C2×C4—284no—
347C2×C4—284no—
348C2×C4C7⋊#9518yes—
349C2×C4#644112yes—
350C2×C4#644112yes—
351C2×C4(C22⋊D7)⋊C412yes—
352C2×C4(C22⋊D7)⋊C412yes—
353C2×C4#644512yes—
354C2×C4#644912yes—
355C2×C4C2×Dic7⋊C411yescharacteristic
356C2×C4C2×C4⋊Dic711yescharacteristic
357C2×C4C2×(C2×D7)⋊C411yescharacteristic
358C2×C4D4×Dic712yes—
359C2×C4D4×Dic712yes—
360C2×C4#649812yes—
361C2×C4#650012yes—
362C2×C4#650212yes—
363C2×C4#650212yes—
364C2×C4C2×C22⋊Dic711yescharacteristic
365C8—44no—
366C8—284no—
367C7#1077011yescharacteristic, complemented, Sylow
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22C22⋊C4×Dic711yescharacteristic
376C22C22⋊C4×Dic711yescharacteristic
377C22#8971411yescharacteristic
378C22#8971411yescharacteristic
379C22#8971511yescharacteristic
380C22#8972711yescharacteristic
381C22C2×C7⋊#9511yescharacteristic
382C4—28no—
383C4—284no—
384C4—284no—
385C4#8971812yes—
386C4#8972212yes—
387C4#8973012yes—
388C4#8999212yes—
389C2—21no—
390C2—21no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2#1172785512yes—
396C2#1172929811yescharacteristic
subgroupG/H
centerC22#89715
commutatorC23×C7C2×C42
FrattiniC23×C4C22×D7
FittingC22×C7×#101C2
socleC22×C7#381
prime radicalC24×C7C22×C4
cosocleC23×C4×C7C23
hypercenterC22×#101D7
hypocenterC7#10770
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
18. C2×C7×#101C4semidirect
21. C2×#6410C4semidirect
149. C2×#101Dic7semidirect
367. C7#10770semidirect

Maximal subgroups

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

subgroupindex
C2×#905032
C22×C7×#1012
C22×#64102
C2×#893542
C2×#893512
#107707