Finite Group Databasebeta

← #13363329

Subgroups — #13363329

Order 1792.

1238 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6451C211yesmaximal, characteristic
2C2×#89491C211yesmaximal, characteristic
3#11731308C211yesmaximal, characteristic
4C7×#1824C211yesmaximal, characteristic
5#11729308C211yesmaximal, characteristic
6#11727679C211yesmaximal, characteristic
7#11727256C211yesmaximal, characteristic
8#11726560C214yesmaximal, complemented
9#11726554C214yesmaximal, complemented
10C23×Dic14—21no—
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C4⋊Dic7C2211yescharacteristic
13C22×Dic7⋊C4—21no—
14C22×C4×Dic7C2211yescharacteristic
15C2×#6451—21no—
16C2×#6451—21no—
17C2×#6451—24no—
18C2×#6451C418yescomplemented
19C2×#6451C2212yes—
20C2×#6451C2212yes—
21C2×C7×C22⋊C8C2211yescharacteristic
22C7×#380C2211yescharacteristic
23#89984C2211yescharacteristic
24C2×C22⋊(C7⋊C8)C2211yescharacteristic
25C2×#6389C2212yes—
26C2×Dic7⋊C8C2212yes—
27#89749C2212yes—
28C4⋊C4×Dic7C2212yes—
29#89715—21no—
30C22⋊C4×Dic7—21no—
31#89491—24no—
32#89423—21no—
33#89420—21no—
34C7×#340C2214yes—
35#89346C2214yes—
36#89324C2214yes—
37#89324C2214yes—
38#21841—71nomaximal, Sylow
39C23×C4×C7—21no—
40C23×C4×C7C2311yescharacteristic
41C23×Dic7—21no—
42C23×Dic7D411yescharacteristic
43C23×Dic7D411yescharacteristic
44C22×Dic14—28no—
45C22×Dic14—22no—
46C22×Dic14—22no—
47C22×Dic14—48no—
48C22×C7×C8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4C2×C414yes—
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C22⋊C4—21no—
55C2×C22⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—22no—
58C2×C4⋊Dic7C2×C414yes—
59C2×C4⋊Dic7C2312yes—
60C2×C4⋊Dic7C2312yes—
61C2×Dic7⋊C4—28no—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×C4×Dic7—22no—
65C2×C4×Dic7C2×C414yes—
66C2×C4×Dic7D411yescharacteristic
67C2×C4×Dic7D411yescharacteristic
68C2×C4×Dic7D414yes—
69C2×C4×Dic7C2311yescharacteristic
70C2×C4×Dic7C2311yescharacteristic
71C22×C7⋊C8C2312yes—
72#6451—28no—
73#6451—28no—
74#6451—24no—
75#6451—24no—
76#6451C2×C418yes—
77#6451C2×C418yes—
78C7×C22⋊C8—22no—
79C7×#95—21no—
80C7⋊#95—21no—
81C7⋊#95—22no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84C22⋊(C7⋊C8)—22no—
85#6389—24no—
86Dic7⋊C8—24no—
87C22×C4⋊Q8—71no—
88C2×#432—71no—
89#2283—71no—
90#1993—74no—
91#1988—74no—
92#1824—71no—
93#1824—71no—
94#1824—71no—
95#1824D711yescharacteristic
96C24×C7C2×D411yescharacteristic
97C22×C4×C7—24no—
98C22×C4×C7—24no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—44no—
102C22×C4×C7C22×C412yes—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2411yescharacteristic
106C22×Dic7—22no—
107C22×Dic7—22no—
108C22×Dic7—24no—
109C22×Dic7—24no—
110C22×Dic7—22no—
111C22×Dic7—22no—
112C22×Dic7—44no—
113C22×Dic7C22⋊C414yes—
114C22×Dic7C22⋊C414yes—
115C22×Dic7C2×D411yescharacteristic
116C22×Dic7C2×D411yescharacteristic
117C22×Dic7C2×D411yescharacteristic
118C22×Dic7C2×D411yescharacteristic
119C2×Dic14—28no—
120C2×Dic14—28no—
121C2×Dic14—48no—
122C2×Dic14—432no—
123C2×Dic14—48no—
124C2×C7×C8—22no—
125C2×C7×C8—24no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—24no—
129C7×C4⋊C4C22×C414yes—
130C7×C22⋊C4—42no—
131C22⋊Dic7—42no—
132C4⋊Dic7—24no—
133C4⋊Dic7—22no—
134C4⋊Dic7—24no—
135C4⋊Dic7C22×C414yes—
136Dic7⋊C4—28no—
137Dic7⋊C4—28no—
138C4×Dic7—44no—
139C4×Dic7C22⋊C414yes—
140C4×Dic7C22⋊C414yes—
141C4×Dic7C22×C412yes—
142C4×Dic7C22×C412yes—
143C4×Dic7C2×D412yes—
144C4×Dic7C2×D412yes—
145C2×C7⋊C8—22no—
146C2×C7⋊C8—24no—
147C23×Q8—141no—
148C2×C4⋊Q8—78no—
149C2×C4⋊Q8—72no—
150C2×C4⋊Q8—72no—
151C2×C4⋊Q8—141no—
152C2×C4⋊Q8—141no—
153C2×C4⋊Q8—144no—
154C22×C4⋊C4—71no—
155C22×C4⋊C4—141no—
156C22×C4⋊C4C2×D711yescharacteristic
157C22×C42—71no—
158#432—144no—
159C2×C4⋊C8—72no—
160C2×C22⋊C8—71no—
161C2×C22⋊C8C2×D711yescharacteristic
162C2×#97—72no—
163#383—141no—
164#382—72no—
165#381—141no—
166#380—71no—
167#380C2×D711yescharacteristic
168C4×C4⋊C4—72no—
169C4×C22⋊C4—141no—
170#376—141no—
171#340—74no—
172#340—74no—
173#340—74no—
174#340C2×D714yes—
175C23×C7—22no—
176C23×C7—22no—
177C23×C7C2×C22⋊C412yes—
178C23×C7#13712yes—
179C23×C7#13712yes—
180C23×C7C22×D411yescharacteristic
181C2×C4×C7—22no—
182C2×C4×C7—24no—
183C2×C4×C7—24no—
184C2×C4×C7—24no—
185C2×C4×C7—48no—
186C2×C4×C7—44no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7C2×C22⋊C412yes—
191C2×C4×C7C23×C411yescharacteristic
192C2×C4×C7C22×D411yescharacteristic
193C2×C4×C72−1+414yes—
194C2×Dic7—22no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—28no—
200C2×Dic7—28no—
201C2×Dic7—24no—
202C2×Dic7—48no—
203C2×Dic7—44no—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7C2×C22⋊C412yes—
208C2×Dic7C2×C22⋊C412yes—
209C2×Dic7C2×C22⋊C412yes—
210C2×Dic7C2×C22⋊C412yes—
211C2×Dic7AGL(1,ℤ/8)14yes—
212Dic14—416no—
213Dic14—416no—
214C7×C8—42no—
215C7⋊C8—42no—
216C22×Q8—148no—
217C22×Q8—142no—
218C22×Q8—142no—
219C22×Q8—288no—
220C23×C4—21no—
221C23×C4—71no—
222C23×C4—71no—
223C23×C4—141no—
224C23×C4C22×D711yescharacteristic
225C22×C8—72no—
226C22×C8C22×D712yes—
227C4⋊Q8—78no—
228C4⋊Q8—78no—
229C4⋊Q8—148no—
230C4⋊Q8—148no—
231C4⋊Q8—144no—
232C4⋊Q8—144no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—74no—
236C2×C4⋊C4—72no—
237C2×C4⋊C4—72no—
238C2×C4⋊C4—148no—
239C2×C4⋊C4—142no—
240C2×C4⋊C4—142no—
241C2×C4⋊C4—141no—
242C2×C4⋊C4—142no—
243C2×C4⋊C4C4×D714yes—
244C2×C4⋊C4C22×D712yes—
245C2×C4⋊C4C22×D712yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—141no—
248C2×C42—74no—
249C2×C42—71no—
250C2×C42—71no—
251C2×C42—74no—
252C2×C42—71no—
253C2×C42—71no—
254C2×C42—142no—
255C4⋊C8—144no—
256C22⋊C8—22no—
257C22⋊C8—142no—
258#97—144no—
259#95—21no—
260#95—142no—
261#95—142no—
262#95—142no—
263#95—141no—
264C22×C7—22no—
265C22×C7—22no—
266C22×C7—24no—
267C22×C7—24no—
268C22×C7#41914yes—
269C22×C7C22×C22⋊C411yescharacteristic
270C22×C7#53612yes—
271C22×C7C2×AGL(1,ℤ/8)12yes—
272C22×C7C2×#13711yescharacteristic
273C22×C7C2×#13711yescharacteristic
274C4×C7—22no—
275C4×C7—44no—
276C4×C7—82no—
277C4×C7#51912yes—
278C4×C7#53612yes—
279Dic7—22no—
280Dic7—22no—
281Dic7—22no—
282Dic7—44no—
283Dic7—82no—
284Dic7#41814yes—
285C24D4×D711yescharacteristic
286C2×Q8—148no—
287C2×Q8—148no—
288C2×Q8—288no—
289C2×Q8—2832no—
290C2×Q8—288no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—44no—
296C22×C4—71no—
297C22×C4—74no—
298C22×C4—74no—
299C22×C4—71no—
300C22×C4—71no—
301C22×C4—71no—
302C22×C4—144no—
303C22×C4—142no—
304C22×C4—142no—
305C22×C4—144no—
306C22×C4—142no—
307C22×C4—142no—
308C22×C4—284no—
309C22×C4C2×C4×D712yes—
310C22×C4D4×D711yescharacteristic
311C22×C4D4×D712yes—
312C22×C4C23×D711yescharacteristic
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—142no—
316C2×C8—144no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—74no—
321C4⋊C4—148no—
322C4⋊C4—148no—
323C4⋊C4—144no—
324C4⋊C4—142no—
325C4⋊C4—144no—
326C4⋊C4C2×C4×D714yes—
327C22⋊C4—42no—
328C22⋊C4—282no—
329C42—74no—
330C42—74no—
331C42—72no—
332C42—72no—
333C42—72no—
334C42—72no—
335C42—284no—
336C2×C7—24no—
337C2×C7#231011yescharacteristic
338C2×C7C2×#41811yescharacteristic
339C2×C7C2×#41911yescharacteristic
340C2×C7#323912yes—
341C2×C7#323912yes—
342C23—22no—
343C23—22no—
344C23D7×C22⋊C412yes—
345C23#647812yes—
346C23#648112yes—
347C23C2×D4×D711yescharacteristic
348Q8—2816no—
349Q8—2816no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—72no—
360C2×C4—74no—
361C2×C4—72no—
362C2×C4—72no—
363C2×C4—72no—
364C2×C4—148no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—142no—
370C2×C4—148no—
371C2×C4—144no—
372C2×C4—288no—
373C2×C4—284no—
374C2×C4—282no—
375C2×C4—282no—
376C2×C4—282no—
377C2×C4D7×C22⋊C412yes—
378C2×C4C22×C4×D711yescharacteristic
379C2×C4C2×D4×D711yescharacteristic
380C2×C4#655414yes—
381C8—42no—
382C8—282no—
383C7#2184111yescharacteristic, complemented, Sylow
384C22—24no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22#8957414yes—
389C22C2×D7×C22⋊C411yescharacteristic
390C22#9033912yes—
391C22C2×#647811yescharacteristic
392C22C2×#648111yescharacteristic
393C22D7×AGL(1,ℤ/8)12yes—
394C4—22no—
395C4—44no—
396C4—74no—
397C4—82no—
398C4—142no—
399C4—142no—
400C4—142no—
401C4—284no—
402C4—562no—
403C4#9020712yes—
404C4#9033912yes—
405C2—24no—
406C2#1173149811yescharacteristic
407C2C2×#8957411yescharacteristic
408C2D7×#41811yescharacteristic
409C2#1173876012yes—
410C2#1173890412yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1824C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1824D7
hypocenterC7#21841
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
8. #11726560C2semidirect
9. #11726554C2semidirect
18. C2×#6451C4semidirect
383. C7#21841semidirect

Maximal subgroups

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

subgroupindex
C22×#64512
C2×#894912
#117313082
C7×#18242
#117293082
#117276792
#117272562
#117265602
#117265542
#218417