Finite Group Databasebeta

← #13339148

Subgroups — #13339148

Order 1792.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2982C211yesmaximal, characteristic, complemented
2C2×#89803C211yesmaximal, characteristic, complemented
3#11739886C211yesmaximal, characteristic
4#11736713C211yesmaximal, characteristic
5#11728698C214yesmaximal, complemented
6#11728695C214yesmaximal, complemented
7#11728532C211yesmaximal, characteristic, complemented
8#11728526C211yesmaximal, characteristic
9#11728504C211yesmaximal, characteristic, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11C22×#1144—21no—
12C2×#6505—21no—
13#90168C2211yescharacteristic
14C2×#6432C2211yescharacteristic
15C7×C8×Q8—22no—
16C7×C8×Q8C2214yes—
17C7×#431—22no—
18C7×#431C2211yescharacteristic
19C7×#431C2211yescharacteristic
20#89803—22no—
21#89803C2214yes—
22#89803C2212yes—
23#89800—22no—
24#89799C2212yes—
25C2×#6384—21no—
26#89738—21no—
27#89696C2211yescharacteristic
28C2×C4⋊(C7⋊C8)C2211yescharacteristic
29#89618—22no—
30#89508—22no—
31#89466C2212yes—
32#89454C2212yes—
33#89281C2214yes—
34#89281C2212yes—
35#89280C2212yes—
36#89253C2214yes—
37#19404—71nomaximal, Sylow
38C22×C7×Q8D411yescharacteristic
39C2×#1157—21no—
40C22×Dic14—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8D412yes—
44C4×C7×Q8C2312yes—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C42×C7—21no—
48C2×C42×C7C2311yescharacteristic
49C2×#1144—22no—
50C2×#1144—24no—
51C2×#1144—44no—
52#6505—44no—
53C2×#1143—22no—
54#6493—21no—
55#6493—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×#1136—21no—
59C22×C7⋊C8—21no—
60#6438—21no—
61#6437C2311yescharacteristic
62C4×D28—21no—
63C4×D28—21no—
64#6432—21no—
65#6432—21no—
66#6432—21no—
67#6432C2312yes—
68#6432C2311yescharacteristic
69C7×C4⋊C8—24no—
70C7×C4⋊C8D412yes—
71C7×C4⋊C8C2312yes—
72C7×C22⋊C8—21no—
73C7×C22⋊C8D412yes—
74C4×C7×C8—24no—
75C4×C7×C8—21no—
76C4×C7×C8C2312yes—
77#6395—24no—
78#6392—24no—
79#6384—22no—
80#6384—24no—
81#6383—22no—
82C4⋊(C7⋊C8)—21no—
83C4⋊(C7⋊C8)—21no—
84C4⋊(C7⋊C8)C2312yes—
85C4⋊(C7⋊C8)C2312yes—
86#3226—71no—
87#3037—71no—
88C2×#462—71no—
89#2982D711yescharacteristic, complemented
90#1682—74no—
91#1674—74no—
92#1571—71no—
93#1546—71no—
94#1540—71no—
95C2×C7×Q8—21no—
96C2×C7×Q8—22no—
97C2×C7×Q8D814yes—
98C2×C7×Q8C2×D412yes—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7#4111yescharacteristic
104C2×C22⋊D7—41no—
105C22×Dic7—41no—
106#1157—42no—
107#1157—42no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28—21no—
111C2×C4×D7—41no—
112C2×Dic14—22no—
113C2×Dic14—21no—
114C2×Dic14—42no—
115C2×Dic14—42no—
116C2×Dic14—41no—
117C2×C7×C8—22no—
118C2×C7×C8C2×D414yes—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4C2×D412yes—
123C42×C7—22no—
124C42×C7—22no—
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128C42×C7C2411yescharacteristic
129#1144—44no—
130#1144—48no—
131#1143—44no—
132(C2×D7)⋊C4—41no—
133(C2×D7)⋊C4—41no—
134C4⋊Dic7—22no—
135C4⋊Dic7—21no—
136C4⋊Dic7—22no—
137C4⋊Dic7—21no—
138C4⋊Dic7—41no—
139C4⋊Dic7—41no—
140#1136—42no—
141C2×C7⋊C8—22no—
142C2×C7⋊C8—22no—
143C2×C7⋊C8—42no—
144C2×#137—141no—
145C22×Q16—141no—
146#532—71no—
147C2×C4⋊Q8—71no—
148C2×C4×Q8C2×D711yescharacteristic
149#500—72no—
150#486—72no—
151#480—142no—
152#474—142no—
153#464—142no—
154#462—72no—
155#462—74no—
156#462—142no—
157#460—72no—
158C8×Q8—22no—
159C8×Q8C2×D714yes—
160#431—22no—
161#431C2×D711yescharacteristic
162#431C2×D711yescharacteristic
163#423—71no—
164C2×C4⋊C8—71no—
165#417—141no—
166C2×#103—141no—
167#335—74no—
168#332—72no—
169#332—74no—
170#331—72no—
171C23×C7#12111yescharacteristic
172C22×D7—41no—
173C7×Q8—22no—
174C7×Q8—24no—
175C7×Q8C2×D814yes—
176C2×C4×C7—22no—
177C2×C4×C7—24no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—41no—
181C2×C4×C7—81no—
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7C2×D812yes—
186C2×C4×C7#13512yes—
187C2×C4×C7C22×D411yescharacteristic
188C2×C4×C7C22×D411yescharacteristic
189C2×C4×C7C2×#4111yescharacteristic
190C22⋊D7—82no—
191C2×Dic7—42no—
192C2×Dic7—41no—
193C2×Dic7—82no—
194D28—41no—
195D28—41no—
196D28—42no—
197D28—81no—
198C4×D7—82no—
199Dic14—42no—
200Dic14—41no—
201Dic14—42no—
202Dic14—41no—
203Dic14—82no—
204C7×C8—24no—
205C7×C8—42no—
206C7⋊C8—42no—
207C7⋊C8—42no—
208C2×#41—141no—
209C22×Q8—141no—
210C22×Q8D2811yescharacteristic
211#137—284no—
212C2×Q16—144no—
213C2×Q16—142no—
214C2×Q16—284no—
215C2×SD16—142no—
216C2×M16—141no—
217C22×C8—141no—
218C4⋊Q8—72no—
219C4⋊Q8—71no—
220C4⋊Q8—141no—
221C4⋊Q8—141no—
222C4⋊Q8—141no—
223#127—71no—
224#124—141no—
225#121—141no—
226#121—141no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8D2812yes—
230C4×Q8C22×D712yes—
231C4×D4—141no—
232C4×D4—141no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—141no—
235C2×C4⋊C4—141no—
236C2×C4⋊C4D2811yescharacteristic
237C2×C42—21no—
238C2×C42C22×D711yescharacteristic
239#107—144no—
240C4⋊C8—24no—
241C4⋊C8—72no—
242C4⋊C8—72no—
243C4⋊C8—141no—
244C4⋊C8—141no—
245C4⋊C8C22⋊D712yes—
246C4⋊C8C22×D712yes—
247#103—144no—
248#103—142no—
249#102—144no—
250#102—142no—
251C22⋊C8—21no—
252C22⋊C8C22⋊D712yes—
253C4×C8—24no—
254C4×C8—21no—
255C4×C8C22×D712yes—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7#46612yes—
259C22×C7C2×#12111yescharacteristic
260C2×D7—81no—
261C2×D7—81no—
262C2×D7—81no—
263C4×C7—21no—
264C4×C7—24no—
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—42no—
268C4×C7#46612yes—
269C4×C7C22×D811yescharacteristic
270C4×C7C2×#13511yescharacteristic
271C4×C7#57812yes—
272Dic7—82no—
273Dic7—81no—
274#41—282no—
275#41—282no—
276C2×Q8—22no—
277C2×Q8—21no—
278C2×Q8—141no—
279C2×Q8—142no—
280C2×Q8—281no—
281C2×Q8—282no—
282C2×Q8—282no—
283C2×Q8D5614yes—
284C2×Q8C2×D2812yes—
285C2×D4—141no—
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—281no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—281no—
292C22×C4—281no—
293C22×C4C2×D2811yescharacteristic
294C22×C4#115711yescharacteristic
295C22×C4C2×C22⋊D711yescharacteristic
296Q16—288no—
297Q16—284no—
298SD16—284no—
299M16—282no—
300C2×C8—22no—
301C2×C8—142no—
302C2×C8—142no—
303C2×C8—282no—
304C2×C8C2×C22⋊D714yes—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—142no—
309C4⋊C4—141no—
310C4⋊C4—142no—
311C4⋊C4—141no—
312C4⋊C4—281no—
313C4⋊C4—281no—
314C4⋊C4C2×D2812yes—
315C22⋊C4—281no—
316C22⋊C4—281no—
317C42—22no—
318C42—22no—
319C42C2×D2811yescharacteristic
320C42#115711yescharacteristic
321C42C2×C22⋊D711yescharacteristic
322C42C23×D711yescharacteristic
323C2×C7—22no—
324C2×C7#304411yescharacteristic
325C2×C7C2×#46611yescharacteristic
326C2×C7#306911yescharacteristic
327D7—161no—
328C23—281no—
329C23#649311yescharacteristic
330Q8—24no—
331Q8—22no—
332Q8—282no—
333Q8—281no—
334Q8—282no—
335Q8—281no—
336Q8—562no—
337Q8C2×D5614yes—
338D4—281no—
339D4—281no—
340D4—282no—
341D4—561no—
342D4—562no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—281no—
350C2×C4—282no—
351C2×C4—562no—
352C2×C4—562no—
353C2×C4C2×D5612yes—
354C2×C4#646712yes—
355C2×C4#649311yescharacteristic
356C2×C4#649311yescharacteristic
357C2×C4#649311yescharacteristic
358C2×C4C22×D2811yescharacteristic
359C2×C4C2×#115711yescharacteristic
360C2×C4C22×C22⋊D711yescharacteristic
361C8—24no—
362C8—42no—
363C8—282no—
364C8—282no—
365C7#1940411yescharacteristic, complemented, Sylow
366C22—21no—
367C22—21no—
368C22—561no—
369C22—561no—
370C22—561no—
371C22#8988712yes—
372C22C2×#649311yescharacteristic
373C4—24no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—561no—
379C4—562no—
380C4#8988712yes—
381C4C22×D5611yescharacteristic
382C4C2×#646711yescharacteristic
383C4#9052912yes—
384C2—22no—
385C2—1121no—
386C2#1173999911yescharacteristic
387C2C2×#8988711yescharacteristic
388C2#1174126111yescharacteristic
subgroupG/H
centerC22C2×#6493
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#2982C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenter#2982D7
hypocenterC7#19404
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C7×#2982C2semidirect
2. C2×#89803C2semidirect
5. #11728698C2semidirect
6. #11728695C2semidirect
7. #11728532C2semidirect
9. #11728504C2semidirect
89. #2982D7semidirect
365. C7#19404semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
C2×#898032
#117398862
#117367132
#117286982
#117286952
#117285322
#117285262
#117285042
#194047