Finite Group Databasebeta

← #13358106

Subgroups — #13358106

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89641C211yesmaximal, characteristic
2#11733406C212yesmaximal, complemented
3C7×#1941C211yesmaximal, characteristic, complemented
4#11728330C211yesmaximal, characteristic
5#11728316C211yesmaximal, characteristic, complemented
6#11728307C211yesmaximal, characteristic, complemented
7#11727530C211yesmaximal, characteristic
8C4×#6389C211yesmaximal, characteristic, complemented
9#11727369C212yesmaximal, complemented
10#11726611C212yesmaximal, complemented
11#11726536C212yesmaximal, complemented
12C22×D56C2211yescharacteristic
13C2×#6456C2212yes—
14C2×#6455—22no—
15C2×C7×#106C2211yescharacteristic
16C2×C7×#106C2211yescharacteristic
17C2×C4×C7×C8C2211yescharacteristic
18C7×#389C2212yes—
19C2×#6395C2212yes—
20C2×#6392C2211yescharacteristic
21C2×#6389C2211yescharacteristic
22C2×#6389C2211yescharacteristic
23C2×#6389C2211yescharacteristic
24C2×#6389C2211yescharacteristic
25#89762—22no—
26#89760C2212yes—
27#89757C2212yes—
28C4⋊C4×Dic7C2212yes—
29C2×#6383C2212yes—
30C2×#6383C2212yes—
31#89711C2212yes—
32C42×Dic7C2211yescharacteristic
33C2×C4×C7⋊C8C2211yescharacteristic
34#89641C418yescomplemented
35#89442—22no—
36#89438—22no—
37C7×#340C2212yes—
38#89324C2212yes—
39#89324C2212yes—
40#21286—71nomaximal, Sylow
41C22×D28C2312yes—
42C22×C4×D7—22no—
43C22×C7×C8C2311yescharacteristic
44C22×C7×C8C2311yescharacteristic
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4C2312yes—
47C2×C7×C4⋊C4C2312yes—
48C2×C42×C7C2311yescharacteristic
49C2×(C2×D7)⋊C4—22no—
50C2×(C2×D7)⋊C4—22no—
51C2×(C2×D7)⋊C4—22no—
52C2×(C2×D7)⋊C4—22no—
53C2×(C2×D7)⋊C4—22no—
54C2×C4⋊Dic7—22no—
55C2×C4⋊Dic7C2312yes—
56C2×C4×Dic7—22no—
57C2×C4×Dic7D412yes—
58C2×C4×Dic7D412yes—
59C2×C4×Dic7C2311yescharacteristic
60C2×C4×Dic7C2311yescharacteristic
61C22×C7⋊C8C2311yescharacteristic
62C22×C7⋊C8C2311yescharacteristic
63C2×D56—22no—
64C2×D56—22no—
65C2×D56C2×C414yes—
66#6456—24no—
67#6456C2×C418yes—
68#6455—48no—
69C7×#106—22no—
70C7×#106C2×C414yes—
71C4×C7×C8—22no—
72C7×#95—22no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75#6395—24no—
76#6392—22no—
77#6389—22no—
78#6389—22no—
79#6389—22no—
80#6389C2×C414yes—
81#6383—24no—
82#6383C2×C418yes—
83C4×C7⋊C8—22no—
84C2×#442—71no—
85#2293—72no—
86#1972—71no—
87#1970—71no—
88#1944—71no—
89#1941D711yescharacteristic, complemented
90#1936—72no—
91#1934—72no—
92#1776—71no—
93#1762—72no—
94C4×#97—71no—
95C23×D7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7C2×D412yes—
99C22×C4×C7C2411yescharacteristic
100C22×Dic7—22no—
101C22×Dic7C2×D411yescharacteristic
102C22×Dic7C2×D411yescharacteristic
103C22×Dic7C2×D412yes—
104C2×D28—22no—
105C2×D28—22no—
106C2×D28—22no—
107C2×D28—22no—
108C2×D28—22no—
109C2×D28C22×C414yes—
110C2×C4×D7—28no—
111C2×C4×D7—44no—
112C2×C4×D7—44no—
113C2×C7×C8—21no—
114C2×C7×C8—21no—
115C2×C7×C8C22×C412yes—
116C2×C7×C8#4112yes—
117C2×C7×C8#4112yes—
118C2×C7×C8#4112yes—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—44no—
123C7×C4⋊C4C22×C414yes—
124C42×C7—21no—
125C42×C7—21no—
126(C2×D7)⋊C4—28no—
127(C2×D7)⋊C4—44no—
128(C2×D7)⋊C4—44no—
129(C2×D7)⋊C4—44no—
130(C2×D7)⋊C4—44no—
131C4⋊Dic7—22no—
132C4⋊Dic7—22no—
133C4⋊Dic7—44no—
134C4×Dic7—24no—
135C4×Dic7—21no—
136C4×Dic7—21no—
137C4×Dic7—42no—
138C4×Dic7—44no—
139C4×Dic7C22×C412yes—
140C4×Dic7#4112yes—
141C2×C7⋊C8—21no—
142C2×C7⋊C8—21no—
143C2×C7⋊C8—21no—
144C2×C7⋊C8—21no—
145C2×C7⋊C8C22×C412yes—
146C2×C7⋊C8#4112yes—
147D56—28no—
148C22×D8—71no—
149C2×C4×D4—72no—
150C2×#117—142no—
151#442—78no—
152C2×#107—71no—
153C2×#106C2×D711yescharacteristic
154C2×#106C2×D711yescharacteristic
155C2×#102—72no—
156C2×#102—72no—
157C2×#102—72no—
158C2×#97—71no—
159C2×#97—71no—
160C2×#97—71no—
161C2×#97—71no—
162C2×C4×C8—71no—
163C2×C4×C8C2×D711yescharacteristic
164#390—72no—
165#390—72no—
166#389—72no—
167#389C2×D712yes—
168#388—142no—
169#388—142no—
170C4×C4⋊C4—72no—
171C4×C22⋊C4—142no—
172C43—71no—
173#340—72no—
174#340—72no—
175#340C2×D712yes—
176C23×C7C22×D411yescharacteristic
177C22×D7—24no—
178C22×D7—42no—
179C22×D7—42no—
180C22×D7—42no—
181C22×D7—42no—
182C22×D7—42no—
183C22×D7—42no—
184C2×C4×C7—22no—
185C2×C4×C7—22no—
186C2×C4×C7—24no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7AGL(1,ℤ/8)14yes—
193C2×C4×C7C23×C411yescharacteristic
194C2×C4×C7C2×#4111yescharacteristic
195C2×C4×C7C2×#4111yescharacteristic
196C2×C4×C7C2×#4111yescharacteristic
197C2×C4×C7C2×#4111yescharacteristic
198C2×C4×C7C2×#4111yescharacteristic
199C2×Dic7—22no—
200C2×Dic7—22no—
201C2×Dic7—21no—
202C2×Dic7—21no—
203C2×Dic7—22no—
204C2×Dic7—22no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7C4×D414yes—
209D28—24no—
210D28—24no—
211D28—42no—
212D28—42no—
213C4×D7—416no—
214C7×C8—22no—
215C7×C8#11714yescomplemented
216C7⋊C8—22no—
217C7⋊C8—22no—
218C22×D4—72no—
219C23×C4—142no—
220C2×D8—74no—
221C2×D8—142no—
222C2×D8—142no—
223C22×C8—71no—
224C22×C8—71no—
225C22×C8C22×D711yescharacteristic
226C22×C8C22×D711yescharacteristic
227C4×D4—78no—
228C4×D4—144no—
229#117—288no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—72no—
232C2×C4⋊C4—142no—
233C2×C4⋊C4C22×D712yes—
234C2×C4⋊C4C22×D712yes—
235C2×C22⋊C4—142no—
236C2×C22⋊C4—142no—
237C2×C22⋊C4—142no—
238C2×C22⋊C4—142no—
239C2×C22⋊C4—142no—
240C2×C42—71no—
241C2×C42—72no—
242C2×C42—71no—
243C2×C42—72no—
244C2×C42—142no—
245C2×C42C22×D711yescharacteristic
246#107—142no—
247#106—22no—
248#106C4×D714yes—
249#102—78no—
250#102—144no—
251#102—144no—
252#97—74no—
253#97—142no—
254#97—142no—
255#97—142no—
256C4×C8—22no—
257C4×C8—142no—
258#95—22no—
259#95—142no—
260#95—142no—
261C22×C7C2×C4×D411yescharacteristic
262C22×C7#53211yescharacteristic
263C22×C7#53211yescharacteristic
264C22×C7C2×AGL(1,ℤ/8)12yes—
265C22×C7#57512yes—
266C2×D7—44no—
267C2×D7—44no—
268C2×D7—44no—
269C2×D7—82no—
270C2×D7—82no—
271C2×D7—82no—
272C2×D7—82no—
273C4×C7—24no—
274C4×C7—44no—
275C4×C7—82no—
276C4×C7C2×#11711yescharacteristic
277C4×C7C4×#4111yescharacteristic
278C4×C7#52911yescharacteristic
279C4×C7#52911yescharacteristic
280Dic7—24no—
281Dic7—41no—
282Dic7—41no—
283Dic7—82no—
284C24—142no—
285C2×D4—74no—
286C2×D4—142no—
287C2×D4—142no—
288C2×D4—142no—
289C2×D4—142no—
290C2×D4—142no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—72no—
294C22×C4—71no—
295C22×C4—71no—
296C22×C4—148no—
297C22×C4—142no—
298C22×C4—284no—
299C22×C4—284no—
300C22×C4D4×D712yes—
301C22×C4C23×D711yescharacteristic
302D8—148no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—72no—
306C2×C8—72no—
307C2×C8—141no—
308C2×C8—141no—
309C2×C8—141no—
310C2×C8—141no—
311C2×C8C2×C4×D712yes—
312C2×C8#115912yes—
313C2×C8#116112yes—
314C2×C8#116112yes—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—44no—
319C4⋊C4—142no—
320C4⋊C4—142no—
321C4⋊C4—284no—
322C4⋊C4C2×C4×D714yes—
323C22⋊C4—148no—
324C22⋊C4—284no—
325C22⋊C4—284no—
326C22⋊C4—284no—
327C22⋊C4—284no—
328C42—21no—
329C42—21no—
330C42—72no—
331C42—72no—
332C42—144no—
333C42—141no—
334C42—141no—
335C42—282no—
336C42—284no—
337C2×C7#229311yescharacteristic
338C2×C7C4×AGL(1,ℤ/8)12yes—
339C2×C7#318012yes—
340C2×C7#318311yescharacteristic
341C2×C7#318411yescharacteristic
342D7—84no—
343C23—144no—
344C23—282no—
345C23—282no—
346C23—282no—
347C23—282no—
348C23—282no—
349C23—282no—
350C23C2×D4×D711yescharacteristic
351D4—144no—
352D4—144no—
353D4—282no—
354D4—282no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—74no—
364C2×C4—142no—
365C2×C4—142no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—142no—
369C2×C4—142no—
370C2×C4—2816no—
371C2×C4—282no—
372C2×C4—282no—
373C2×C4—282no—
374C2×C4#647714yes—
375C2×C4C22×C4×D711yescharacteristic
376C2×C4C2×#115911yescharacteristic
377C2×C4C2×#116111yescharacteristic
378C2×C4C2×#116111yescharacteristic
379C2×C4D7×#4111yescharacteristic
380C2×C4D7×#4111yescharacteristic
381C8—22no—
382C8—142no—
383C8—142no—
384C8#645514yescomplemented
385C7#2128611yescharacteristic, complemented, Sylow
386C22—284no—
387C22—284no—
388C22—284no—
389C22—562no—
390C22—562no—
391C22—562no—
392C22—562no—
393C22C4×D4×D711yescharacteristic
394C22#9028711yescharacteristic
395C22#9041611yescharacteristic
396C22C2×#647712yes—
397C22#9045912yes—
398C4—24no—
399C4—44no—
400C4—82no—
401C4—144no—
402C4—281no—
403C4—281no—
404C4—562no—
405C4C2×#645511yescharacteristic
406C4C4×#116111yescharacteristic
407C4#9031511yescharacteristic
408C4#9038211yescharacteristic
409C2—564no—
410C2#1173340611yescharacteristic
411C2C4×#647712yes—
412C2#1173808312yes—
413C2#1173933311yescharacteristic
414C2#1173935611yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1941C2
socleC23×C7C22×D4
prime radicalC22×D56C22
cosocleC22×C4×C7C24
hypercenter#1941D7
hypocenterC7#21286
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. #11733406C2semidirect
3. C7×#1941C2semidirect
5. #11728316C2semidirect
6. #11728307C2semidirect
8. C4×#6389C2semidirect
9. #11727369C2semidirect
10. #11726611C2semidirect
11. #11726536C2semidirect
34. #89641C4semidirect
89. #1941D7semidirect
215. C7×C8#117semidirect
384. C8#6455semidirect
385. C7#21286semidirect

Maximal subgroups

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

subgroupindex
C2×#896412
#117334062
C7×#19412
#117283302
#117283162
#117283072
#117275302
C4×#63892
#117273692
#117266112
#117265362
#212867