Finite Group Databasebeta

← #13363309

Subgroups — #13363309

Order 1792.

1066 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6453C211yesmaximal, characteristic
2C2×#89491C211yesmaximal, characteristic
3#11731309C211yesmaximal, characteristic
4C7×#1823C211yesmaximal, characteristic
5#11729307C211yesmaximal, characteristic
6#11727679C211yesmaximal, characteristic
7#11727256C211yesmaximal, characteristic
8#11726559C214yesmaximal, complemented
9#11726555C214yesmaximal, complemented
10C22×C7×C4⋊C4C2211yescharacteristic
11C22×C4⋊Dic7—21no—
12C22×C4⋊Dic7C2211yescharacteristic
13C22×Dic7⋊C4—21no—
14C22×C4×Dic7C2211yescharacteristic
15C2×#6453—21no—
16C2×#6453—21no—
17C2×#6453—24no—
18C2×#6453C418yescomplemented
19C2×#6453C2212yes—
20C2×#6453C2212yes—
21C2×#6440—21no—
22C2×C7×C22⋊C8C2211yescharacteristic
23C7×#380C2211yescharacteristic
24#89984C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2211yescharacteristic
26C2×#6389C2212yes—
27C2×Dic7⋊C8C2212yes—
28#89748C2212yes—
29C4⋊C4×Dic7C2212yes—
30#89714—21no—
31#89491—24no—
32#89422—21no—
33#89421—21no—
34C7×#340C2214yes—
35#89346C2214yes—
36#89324C2214yes—
37#89324C2214yes—
38#21838—71nomaximal, Sylow
39C23×C4×C7—21no—
40C23×C4×C7C2311yescharacteristic
41C23×Dic7—21no—
42C23×Dic7D411yescharacteristic
43C23×Dic7D411yescharacteristic
44C22×C7×C8C2312yes—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—22no—
47C2×C7×C4⋊C4C2×C414yes—
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2312yes—
50C2×C7×C22⋊C4—21no—
51C2×C22⋊Dic7—21no—
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—22no—
54C2×C4⋊Dic7—28no—
55C2×C4⋊Dic7—22no—
56C2×C4⋊Dic7—22no—
57C2×C4⋊Dic7C2×C414yes—
58C2×C4⋊Dic7C2312yes—
59C2×C4⋊Dic7C2312yes—
60C2×Dic7⋊C4—28no—
61C2×Dic7⋊C4—22no—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×C4×Dic7—22no—
66C2×C4×Dic7C2×C414yes—
67C2×C4×Dic7D411yescharacteristic
68C2×C4×Dic7D411yescharacteristic
69C2×C4×Dic7D414yes—
70C2×C4×Dic7C2311yescharacteristic
71C2×C4×Dic7C2311yescharacteristic
72C22×C7⋊C8C2312yes—
73#6453—28no—
74#6453—28no—
75#6453—24no—
76#6453—24no—
77#6453C2×C418yes—
78#6453C2×C418yes—
79#6440—44no—
80C7×C22⋊C8—22no—
81C7×#95—21no—
82C7⋊#95—21no—
83C7⋊#95—22no—
84C22⋊(C7⋊C8)—22no—
85#6389—24no—
86Dic7⋊C8—24no—
87C22×#125—71no—
88C2×#432—71no—
89#2285—71no—
90#1992—74no—
91#1989—74no—
92#1824—71no—
93#1824—71no—
94#1823—71no—
95#1823D711yescharacteristic
96C24×C7C2×D411yescharacteristic
97C22×C4×C7—24no—
98C22×C4×C7—24no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7C22×C412yes—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D412yes—
106C22×C4×C7C2411yescharacteristic
107C22×Dic7—22no—
108C22×Dic7—22no—
109C22×Dic7—24no—
110C22×Dic7—22no—
111C22×Dic7—22no—
112C22×Dic7—42no—
113C22×Dic7—44no—
114C22×Dic7C22⋊C414yes—
115C22×Dic7C22⋊C414yes—
116C22×Dic7C2×D411yescharacteristic
117C22×Dic7C2×D411yescharacteristic
118C22×Dic7C2×D411yescharacteristic
119C22×Dic7C2×D411yescharacteristic
120C2×C7×C8—22no—
121C2×C7×C8—24no—
122C7×C4⋊C4—24no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—24no—
125C7×C4⋊C4C22×C414yes—
126C7×C22⋊C4—42no—
127C22⋊Dic7—42no—
128C4⋊Dic7—24no—
129C4⋊Dic7—22no—
130C4⋊Dic7—28no—
131C4⋊Dic7—28no—
132C4⋊Dic7—24no—
133C4⋊Dic7C22×C414yes—
134Dic7⋊C4—28no—
135Dic7⋊C4—28no—
136Dic7⋊C4—44no—
137Dic7⋊C4—44no—
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—
147C2×#125—78no—
148C2×#125—72no—
149C2×#125—72no—
150C2×#125—141no—
151C2×#125—141no—
152C2×#125—144no—
153C2×#117—141no—
154C22×C4⋊C4—71no—
155C22×C4⋊C4—141no—
156C22×C4⋊C4—141no—
157C22×C4⋊C4C2×D711yescharacteristic
158C22×C42—71no—
159#432—144no—
160C2×C4⋊C8—72no—
161C2×C22⋊C8—71no—
162C2×C22⋊C8C2×D711yescharacteristic
163C2×#97—72no—
164#384—72no—
165#382—141no—
166#380—71no—
167#380—141no—
168#380C2×D711yescharacteristic
169C4×C4⋊C4—72no—
170C4×C4⋊C4—141no—
171#340—74no—
172#340—74no—
173#340—74no—
174#340C2×D714yes—
175C23×C7—22no—
176C23×C7—22no—
177C23×C7C2×C22⋊C412yes—
178C23×C7AGL(1,ℤ/8)12yes—
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—42no—
186C2×C4×C7—42no—
187C2×C4×C7—48no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7C2×C22⋊C412yes—
192C2×C4×C7C23×C411yescharacteristic
193C2×C4×C7C22×D411yescharacteristic
194C2×C4×C72−1+414yes—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—22no—
200C2×Dic7—22no—
201C2×Dic7—28no—
202C2×Dic7—24no—
203C2×Dic7—44no—
204C2×Dic7—48no—
205C2×Dic7—44no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C2×Dic7C2×C22⋊C412yes—
210C2×Dic7C2×C22⋊C412yes—
211C2×Dic7C2×C22⋊C412yes—
212C2×Dic7C2×C22⋊C412yes—
213C7×C8—42no—
214C7⋊C8—42no—
215C23×C4—21no—
216C23×C4—71no—
217C23×C4—71no—
218C23×C4—141no—
219C23×C4C22×D711yescharacteristic
220C22×C8—72no—
221C22×C8C22×D712yes—
222#125—78no—
223#125—78no—
224#125—148no—
225#125—148no—
226#125—144no—
227#125—144no—
228#117—284no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—74no—
232C2×C4⋊C4—72no—
233C2×C4⋊C4—72no—
234C2×C4⋊C4—148no—
235C2×C4⋊C4—142no—
236C2×C4⋊C4—142no—
237C2×C4⋊C4—148no—
238C2×C4⋊C4—142no—
239C2×C4⋊C4—142no—
240C2×C4⋊C4—141no—
241C2×C4⋊C4—142no—
242C2×C4⋊C4—142no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4C4×D714yes—
245C2×C4⋊C4C22×D712yes—
246C2×C4⋊C4C22×D712yes—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—141no—
249C2×C42—74no—
250C2×C42—71no—
251C2×C42—71no—
252C2×C42—74no—
253C2×C42—71no—
254C2×C42—71no—
255C2×C42—142no—
256C4⋊C8—144no—
257C22⋊C8—22no—
258C22⋊C8—142no—
259#97—144no—
260#95—21no—
261#95—142no—
262#95—141no—
263C22×C7—22no—
264C22×C7—22no—
265C22×C7—24no—
266C22×C7—24no—
267C22×C7#41714yes—
268C22×C7C22×C22⋊C411yescharacteristic
269C22×C7#53612yes—
270C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C7C2×#13711yescharacteristic
272C22×C7#57512yes—
273C4×C7—22no—
274C4×C7—44no—
275C4×C7—82no—
276C4×C7#51912yes—
277C4×C7#53612yes—
278Dic7—22no—
279Dic7—22no—
280Dic7—22no—
281Dic7—22no—
282Dic7—44no—
283Dic7—82no—
284C24D4×D711yescharacteristic
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—74no—
292C22×C4—71no—
293C22×C4—71no—
294C22×C4—71no—
295C22×C4—71no—
296C22×C4—74no—
297C22×C4—142no—
298C22×C4—142no—
299C22×C4—144no—
300C22×C4—142no—
301C22×C4—142no—
302C22×C4—282no—
303C22×C4—284no—
304C22×C4C2×C4×D712yes—
305C22×C4D4×D711yescharacteristic
306C22×C4D4×D712yes—
307C22×C4C23×D711yescharacteristic
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—142no—
311C2×C8—144no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—74no—
316C4⋊C4—148no—
317C4⋊C4—148no—
318C4⋊C4—148no—
319C4⋊C4—148no—
320C4⋊C4—144no—
321C4⋊C4—142no—
322C4⋊C4—144no—
323C4⋊C4—284no—
324C4⋊C4—284no—
325C4⋊C4C2×C4×D714yes—
326C22⋊C4—42no—
327C22⋊C4—282no—
328C42—74no—
329C42—74no—
330C42—72no—
331C42—72no—
332C42—72no—
333C42—72no—
334C42—284no—
335C2×C7—24no—
336C2×C7#231011yescharacteristic
337C2×C7C2×#41711yescharacteristic
338C2×C7#289611yescharacteristic
339C2×C7#323412yes—
340C2×C7#323612yes—
341C23—22no—
342C23—22no—
343C23D7×C22⋊C412yes—
344C23#647412yes—
345C23#647812yes—
346C23C2×D4×D711yescharacteristic
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—72no—
358C2×C4—72no—
359C2×C4—72no—
360C2×C4—72no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—142no—
366C2×C4—142no—
367C2×C4—148no—
368C2×C4—144no—
369C2×C4—284no—
370C2×C4—288no—
371C2×C4—284no—
372C2×C4—282no—
373C2×C4—282no—
374C2×C4—282no—
375C2×C4D7×C22⋊C412yes—
376C2×C4C22×C4×D711yescharacteristic
377C2×C4C2×D4×D711yescharacteristic
378C2×C4#655414yes—
379C8—42no—
380C8—282no—
381C7#2183811yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—24no—
385C22—24no—
386C22#8954314yes—
387C22C2×D7×C22⋊C411yescharacteristic
388C22#9033912yes—
389C22C2×#647411yescharacteristic
390C22C2×#647811yescharacteristic
391C22#9046812yes—
392C4—22no—
393C4—44no—
394C4—82no—
395C4—142no—
396C4—142no—
397C4—142no—
398C4—142no—
399C4—284no—
400C4—562no—
401C4#9020712yes—
402C4#9033912yes—
403C2—24no—
404C2#1173149811yescharacteristic
405C2C2×#8954311yescharacteristic
406C2#1173640111yescharacteristic
407C2#1173872112yes—
408C2#1173886512yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1823C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1823D7
hypocenterC7#21838
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
8. #11726559C2semidirect
9. #11726555C2semidirect
18. C2×#6453C4semidirect
381. C7#21838semidirect

Maximal subgroups

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

subgroupindex
C22×#64532
C2×#894912
#117313092
C7×#18232
#117293072
#117276792
#117272562
#117265592
#117265552
#218387