Finite Group Databasebeta

← #13364337

Subgroups — #13364337

Order 1792.

850 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89457C211yesmaximal, characteristic
2#11731763C211yesmaximal, characteristic
3#11731763C211yesmaximal, characteristic
4C7×#1834C211yesmaximal, characteristic, complemented
5#11727519C211yesmaximal, characteristic, complemented
6#11727330C211yesmaximal, characteristic, complemented
7#11727328C211yesmaximal, characteristic, complemented
8#11726631C214yesmaximal
9#11726552C214yesmaximal, complemented
10C2×#6455—21no—
11C2×#6455C2212yes—
12C2×C42×D7C2211yescharacteristic
13C2×C4×C7×C8C2211yescharacteristic
14C7×#384C2211yescharacteristic
15C7×#384C2211yescharacteristic
16C2×(C2×D7)⋊C8C2212yes—
17C2×#6392C2214yes—
18C2×Dic7⋊C8C2212yes—
19#89759—21no—
20#89759—21no—
21#89759—21no—
22#89759C2212yes—
23#89749—21no—
24#89749C2212yes—
25C4⋊C4×Dic7—21no—
26C4⋊C4×Dic7—21no—
27C4⋊C4×Dic7—21no—
28C4⋊C4×Dic7C2212yes—
29C2×#6382C2214yes—
30#89707C2211yescharacteristic
31#89707C2211yescharacteristic
32C2×C4×C7⋊C8C2211yescharacteristic
33#89457C418yescomplemented
34C7×#340C2214yes—
35#89324C2214yes—
36#21984—71nomaximal, Sylow
37C22×C4×D7D411yescharacteristic
38C22×C4×D7D411yescharacteristic
39C22×C4×D7C2311yescharacteristic
40C22×C7×C8C2312yes—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4C2312yes—
46C2×C7×C4⋊C4C2312yes—
47C2×C42×C7C2311yescharacteristic
48C2×(C2×D7)⋊C4—22no—
49C2×(C2×D7)⋊C4—22no—
50C2×C4⋊Dic7—21no—
51C2×C4⋊Dic7—21no—
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—21no—
54C2×C4⋊Dic7C2312yes—
55C2×C4⋊Dic7C2312yes—
56C2×C4×Dic7—22no—
57C2×C4×Dic7—22no—
58C2×C4×Dic7D411yescharacteristic
59C2×C4×Dic7D411yescharacteristic
60C2×C4×Dic7C2311yescharacteristic
61C22×C7⋊C8C2312yes—
62#6455—44no—
63#6455—44no—
64C42×D7C2×C414yes—
65C42×D7Q814yes—
66C4×C7×C8C2×C414yes—
67C7⋊#95—22no—
68C7⋊#95—22no—
69C7⋊#95—22no—
70C7⋊#95—22no—
71(C2×D7)⋊C8C2×C418yes—
72#6392—44no—
73Dic7⋊C8C2×C418yes—
74#6382—44no—
75C4×C7⋊C8C2×C414yes—
76C2×#431—71no—
77#2285—71no—
78#2285—71no—
79#1949—74no—
80#1854—74no—
81#1845—71no—
82#1844—71no—
83#1834—71no—
84#1834D711yescharacteristic, complemented
85C23×D7C2×D411yescharacteristic
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2411yescharacteristic
91C22×Dic7—22no—
92C22×Dic7—22no—
93C22×Dic7C2×D411yescharacteristic
94C22×Dic7C2×D411yescharacteristic
95C22×Dic7C2×D411yescharacteristic
96C2×C4×D7—22no—
97C2×C4×D7—22no—
98C2×C4×D7—22no—
99C2×C4×D7—22no—
100C2×C4×D7—22no—
101C2×C4×D7—22no—
102C2×C4×D7C4⋊C414yes—
103C2×C4×D7C4⋊C414yes—
104C2×C4×D7C22×C412yes—
105C2×C4×D7C2×Q812yes—
106C2×C7×C8—22no—
107C2×C7×C8—22no—
108C2×C7×C8C22×C414yes—
109C7×C4⋊C4—24no—
110C7×C4⋊C4—24no—
111C7×C4⋊C4—42no—
112C7×C4⋊C4—42no—
113C7×C4⋊C4—42no—
114C7×C4⋊C4—42no—
115C42×C7C22×C412yes—
116C42×C7C2×Q812yes—
117(C2×D7)⋊C4—44no—
118(C2×D7)⋊C4—44no—
119C4⋊Dic7—24no—
120C4⋊Dic7—24no—
121C4⋊Dic7—42no—
122C4⋊Dic7—42no—
123C4⋊Dic7—42no—
124C4⋊Dic7—42no—
125C4×Dic7—44no—
126C4×Dic7—44no—
127C4×Dic7C4⋊C414yes—
128C4×Dic7C4⋊C414yes—
129C4×Dic7C22×C412yes—
130C4×Dic7C2×Q812yes—
131C2×C7⋊C8—22no—
132C2×C7⋊C8—22no—
133C2×C7⋊C8C22×C414yes—
134C2×#117—72no—
135C2×#117—141no—
136C22×C42—71no—
137#431—78no—
138C2×#107—74no—
139C2×#106—74no—
140C2×C4⋊C8—72no—
141C2×C22⋊C8—72no—
142C2×C4×C8—71no—
143C2×C4×C8C2×D711yescharacteristic
144#384—71no—
145#384—71no—
146#384C2×D711yescharacteristic
147#384C2×D711yescharacteristic
148#382—72no—
149#382—141no—
150#380—72no—
151#380—141no—
152#380—141no—
153#380—141no—
154C4×C4⋊C4—72no—
155C4×C4⋊C4—141no—
156C4×C4⋊C4—141no—
157C4×C4⋊C4—141no—
158#340—74no—
159#340C2×D714yes—
160C23×C7C22×D411yescharacteristic
161C22×D7—21no—
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—21no—
167C22×D7C2×C4⋊C412yes—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7C2×C4⋊C412yes—
175C2×C4×C7C2×C4⋊C412yes—
176C2×C4×C7#13512yes—
177C2×C4×C7#13512yes—
178C2×C4×C7AGL(1,ℤ/8)12yes—
179C2×C4×C7#13712yes—
180C2×C4×C7C23×C411yescharacteristic
181C2×C4×C7C22×Q811yescharacteristic
182C2×C4×C7H(5,𝔽2)12yes—
183C2×C4×C72−1+412yes—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C2×Dic7—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—42no—
191C2×Dic7—42no—
192C2×Dic7—42no—
193C2×Dic7—42no—
194C2×Dic7—42no—
195C2×Dic7—42no—
196C2×Dic7C2×C4⋊C412yes—
197C2×Dic7C2×C4⋊C412yes—
198C2×Dic7C2×C4⋊C412yes—
199C4×D7—24no—
200C4×D7—24no—
201C4×D7—28no—
202C4×D7—28no—
203C7×C8—24no—
204C7⋊C8—24no—
205C23×C4—71no—
206C23×C4—71no—
207C23×C4—71no—
208C22×C8—72no—
209C22×C8C22×D712yes—
210#117—284no—
211#117—284no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—72no—
217C2×C4⋊C4—72no—
218C2×C4⋊C4—141no—
219C2×C4⋊C4—141no—
220C2×C4⋊C4—141no—
221C2×C4⋊C4—141no—
222C2×C4⋊C4C22×D712yes—
223C2×C4⋊C4C22×D712yes—
224C2×C22⋊C4—142no—
225C2×C22⋊C4—142no—
226C2×C42—71no—
227C2×C42—71no—
228C2×C42—71no—
229C2×C42—74no—
230C2×C42—74no—
231C2×C42—142no—
232C2×C42—142no—
233C2×C42C22×D711yescharacteristic
234#107—284no—
235#106—284no—
236C4⋊C8—78no—
237C22⋊C8—78no—
238C4×C8—74no—
239C4×C8C4×D714yes—
240#95—142no—
241#95—142no—
242#95—142no—
243#95—142no—
244C22×C7C22×C4⋊C411yescharacteristic
245C22×C7#53511yescharacteristic
246C22×C7#53611yescharacteristic
247C22×C7C2×#13511yescharacteristic
248C22×C7C2×#13511yescharacteristic
249C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
250C22×C7C2×#13711yescharacteristic
251C2×D7—22no—
252C2×D7—22no—
253C2×D7—22no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C4×C7—82no—
259C4×C7—82no—
260C4×C7#42714yes—
261C4×C7#42814yes—
262C4×C7#52012yes—
263C4×C7#54412yes—
264Dic7—22no—
265Dic7—22no—
266Dic7—22no—
267Dic7—82no—
268Dic7—82no—
269C24—71no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—71no—
275C22×C4—74no—
276C22×C4—74no—
277C22×C4—72no—
278C22×C4—72no—
279C22×C4—142no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4—142no—
283C22×C4—142no—
284C22×C4—142no—
285C22×C4—142no—
286C22×C4—142no—
287C22×C4D4×D711yescharacteristic
288C22×C4D4×D711yescharacteristic
289C22×C4C23×D711yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—74no—
293C2×C8—142no—
294C2×C8—142no—
295C2×C8C2×C4×D714yes—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—144no—
303C4⋊C4—144no—
304C4⋊C4—282no—
305C4⋊C4—282no—
306C4⋊C4—282no—
307C4⋊C4—282no—
308C22⋊C4—284no—
309C22⋊C4—284no—
310C42—74no—
311C42—74no—
312C42—72no—
313C42—72no—
314C42—284no—
315C42—284no—
316C42C2×C4×D712yes—
317C42Q8×D712yes—
318C2×C7#231711yescharacteristic
319C2×C7C2×#42711yescharacteristic
320C2×C7C2×#42811yescharacteristic
321C2×C7#320711yescharacteristic
322C2×C7#321011yescharacteristic
323C2×C7#322811yescharacteristic
324C2×C7#323111yescharacteristic
325D7—42no—
326C23—72no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23—141no—
333C23C2×D4×D711yescharacteristic
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—72no—
341C2×C4—72no—
342C2×C4—72no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—144no—
350C2×C4—148no—
351C2×C4—148no—
352C2×C4—144no—
353C2×C4—282no—
354C2×C4—282no—
355C2×C4—282no—
356C2×C4—282no—
357C2×C4—282no—
358C2×C4—282no—
359C2×C4D7×C4⋊C412yes—
360C2×C4D7×C4⋊C412yes—
361C2×C4#647512yes—
362C2×C4#647712yes—
363C2×C4#647812yes—
364C2×C4#648212yes—
365C2×C4C22×C4×D711yescharacteristic
366C2×C4C2×Q8×D711yescharacteristic
367C2×C4#655312yes—
368C2×C4#655412yes—
369C8—24no—
370C8—144no—
371C7#2198411yescharacteristic, complemented, Sylow
372C22—142no—
373C22—142no—
374C22—142no—
375C22—281no—
376C22—281no—
377C22—281no—
378C22—281no—
379C22C2×D7×C4⋊C411yescharacteristic
380C22#9036511yescharacteristic
381C22#9036611yescharacteristic
382C22C2×#647511yescharacteristic
383C22C2×#647711yescharacteristic
384C22C2×#647811yescharacteristic
385C22C2×#648211yescharacteristic
386C4—82no—
387C4—82no—
388C4—142no—
389C4—142no—
390C4—142no—
391C4—562no—
392C4—562no—
393C4#8963314yes—
394C4#8965214yes—
395C4#9021412yes—
396C4#9038012yes—
397C2—282no—
398C2#1173167011yescharacteristic
399C2C2×#8963311yescharacteristic
400C2C2×#8965211yescharacteristic
401C2#1173901411yescharacteristic
402C2#1173901811yescharacteristic
403C2#1173904811yescharacteristic
404C2#1173905511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1834C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1834D7
hypocenterC7#21984
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
4. C7×#1834C2semidirect
5. #11727519C2semidirect
6. #11727330C2semidirect
7. #11727328C2semidirect
9. #11726552C2semidirect
33. #89457C4semidirect
84. #1834D7semidirect
371. C7#21984semidirect

Maximal subgroups

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

subgroupindex
C2×#894572
#117317632
C7×#18342
#117275192
#117273302
#117273282
#117266312
#117265522
#219847