Finite Group Databasebeta

← #13337658

Subgroups — #13337658

Order 1792.

628 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3113C211yesmaximal, characteristic, complemented
2#11740788C211yesmaximal, characteristic
3#11739889C211yesmaximal, characteristic
4C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
5#11729585C211yesmaximal, characteristic, complemented
6#11728873C212yesmaximal, complemented
7#11728870C212yesmaximal, complemented
8#11728809C212yesmaximal, complemented
9#11728806C212yesmaximal, complemented
10#11728570C211yesmaximal, characteristic
11#11728548C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14#90168C2211yescharacteristic
15C7×#483—21no—
16C7×#475C2212yes—
17C7×#462C2212yes—
18C7×#451—21no—
19C7×#441—22no—
20C7×#432C2211yescharacteristic
21#89999—21no—
22#89860C2211yescharacteristic
23#89857C2211yescharacteristic
24#89853C2211yescharacteristic
25#89851C2211yescharacteristic
26#89820—21no—
27#89820—21no—
28#89799C2212yes—
29#89798—22no—
30C4×#1143—21no—
31C4×#1143—21no—
32Q8×C7⋊C8—21no—
33Q8×C7⋊C8—21no—
34Q8×C7⋊C8C2212yes—
35Q8×C7⋊C8C2212yes—
36#89793C2212yes—
37#89739—21no—
38#89698C2211yescharacteristic
39C2×C4⋊(C7⋊C8)—21no—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×C4×C7⋊C8—21no—
42C2×C4×C7⋊C8C2211yescharacteristic
43#89342C2212yes—
44#89342C2212yes—
45#89280C2212yes—
46#89276C2212yes—
47#89275C2212yes—
48#89274C2212yes—
49#19263—71nomaximal, Sylow
50C22×C7×Q8D411yescharacteristic
51C2×#1157—21no—
52C2×C7×Q16—22no—
53C7×C4⋊Q8C2311yescharacteristic
54C7×C4⋊Q8C2311yescharacteristic
55C7×#125—21no—
56C7×C22⋊Q8—21no—
57C7×C22⋊Q8—21no—
58C4×C7×Q8—21no—
59C4×C7×Q8—21no—
60C4×C7×Q8—21no—
61C4×C7×Q8—21no—
62C4×C7×Q8D412yes—
63C4×C7×Q8C2312yes—
64C7×#117—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4D411yescharacteristic
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×#1143—22no—
70C2×#1143—21no—
71C2×#1143—21no—
72#6493—21no—
73#6493—21no—
74C22×C7⋊C8—21no—
75C22×C7⋊C8D411yescharacteristic
76C22×C7⋊C8D411yescharacteristic
77#6438—21no—
78#6437C2311yescharacteristic
79C4×D28—21no—
80C4×D28—21no—
81#6432C2311yescharacteristic
82C7×#106—22no—
83C7×C4⋊C8C2312yes—
84C7×#103—22no—
85C7×#103—22no—
86C7×C22⋊C8—21no—
87C7×#97—21no—
88#6409—22no—
89#6409—21no—
90#6409—21no—
91C22⋊(C7⋊C8)—21no—
92#6383—22no—
93#6383—21no—
94#6383—21no—
95#6382—21no—
96#6382—21no—
97#6381—22no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)—22no—
100C4⋊(C7⋊C8)—22no—
101C4⋊(C7⋊C8)C2312yes—
102C4⋊(C7⋊C8)C2312yes—
103#6377—21no—
104C4×C7⋊C8—22no—
105C4×C7⋊C8—22no—
106C4×C7⋊C8D411yescharacteristic
107C4×C7⋊C8D411yescharacteristic
108C4×C7⋊C8C2311yescharacteristic
109C4×C7⋊C8C2311yescharacteristic
110#3167—71no—
111#3113D711yescharacteristic, complemented
112#3112—71no—
113C2×C8×Q8—71no—
114#1684—72no—
115#1682—72no—
116#1672—72no—
117#1666—72no—
118#1568—71no—
119#1552—71no—
120#1551—71no—
121C2×C7×Q8—24no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8C2×D412yes—
126C22×C4×C7—21no—
127C22×C4×C7—21no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7#4111yescharacteristic
131C2×C22⋊D7—41no—
132#1157—82no—
133C2×D28—21no—
134C2×D28—21no—
135C2×D28—21no—
136C2×C4×D7—41no—
137C2×Dic14—21no—
138C7×Q16—44no—
139C2×C7×C8—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—21no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—41no—
148C7×C4⋊C4—41no—
149C7×C4⋊C4C2×D412yes—
150C7×C22⋊C4—41no—
151C42×C7—22no—
152C42×C7—22no—
153C42×C7—41no—
154C42×C7C2×D411yescharacteristic
155C42×C7C2×D411yescharacteristic
156C42×C7#4111yescharacteristic
157C42×C7C2411yescharacteristic
158#1143—42no—
159#1143—44no—
160#1143—42no—
161(C2×D7)⋊C4—41no—
162(C2×D7)⋊C4—41no—
163C4⋊Dic7—21no—
164C4⋊Dic7—21no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—41no—
169C2×C7⋊C8—42no—
170C2×C7⋊C8C2×D411yescharacteristic
171C2×C7⋊C8C2×D411yescharacteristic
172C2×C7⋊C8C2×D411yescharacteristic
173C2×C7⋊C8C2×D411yescharacteristic
174#533C2×D711yescharacteristic
175#532—71no—
176C2×C4×Q8C2×D711yescharacteristic
177#498—71no—
178#492—71no—
179#487—71no—
180#486—71no—
181#483—21no—
182#480—141no—
183#477—72no—
184#475C2×D712yes—
185#465—141no—
186#465—141no—
187#462C2×D712yes—
188#460—72no—
189#451—21no—
190#448—141no—
191C8×Q8—72no—
192C8×Q8—72no—
193C8×Q8—141no—
194C8×Q8—141no—
195#441—22no—
196#440—142no—
197C4×SD16—141no—
198C4×SD16—141no—
199#432—71no—
200#432C2×D711yescharacteristic
201C2×C4⋊C8—71no—
202C2×C4⋊C8—141no—
203C2×C4×C8—71no—
204C2×C4×C8—141no—
205#335—72no—
206#334—72no—
207#333—72no—
208#332—72no—
209#332—72no—
210#331—72no—
211C23×C7#12111yescharacteristic
212C22×D7—41no—
213C7×Q8—24no—
214C7×Q8—22no—
215C7×Q8—42no—
216C7×Q8—41no—
217C7×Q8—41no—
218C7×Q8#13514yes—
219C2×C4×C7—24no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—22no—
226C2×C4×C7—42no—
227C2×C4×C7—41no—
228C2×C4×C7—41no—
229C2×C4×C7#12111yescharacteristic
230C2×C4×C7#12111yescharacteristic
231C2×C4×C7#12111yescharacteristic
232C2×C4×C7C22×D411yescharacteristic
233C2×C4×C7C22×D411yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C22⋊D7—82no—
236C2×Dic7—41no—
237D28—42no—
238D28—41no—
239D28—41no—
240D28—81no—
241C4×D7—82no—
242Dic14—81no—
243C7×C8—42no—
244C7⋊C8—21no—
245C7⋊C8—21no—
246C7⋊C8—21no—
247C7⋊C8—42no—
248C7⋊C8—42no—
249C7⋊C8#13712yes—
250C2×#41—141no—
251C22×Q8C22⋊D711yescharacteristic
252C2×Q16—22no—
253C2×SD16—141no—
254C2×SD16—141no—
255C2×SD16—142no—
256C22×C8—71no—
257C22×C8—71no—
258C22×C8—141no—
259C4⋊Q8—71no—
260C4⋊Q8C22×D711yescharacteristic
261C4⋊Q8C22×D711yescharacteristic
262#127—71no—
263#125—21no—
264#124—141no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267#121—141no—
268#121—141no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8C22⋊D712yes—
274C4×Q8C22×D712yes—
275C4×D4—141no—
276C4×D4—141no—
277#117—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C22⋊D711yescharacteristic
280C2×C42—21no—
281C2×C42C22×D711yescharacteristic
282#107—142no—
283#106—22no—
284#106—141no—
285#106—141no—
286C4⋊C8—72no—
287C4⋊C8—72no—
288C4⋊C8—142no—
289C4⋊C8—142no—
290C4⋊C8—141no—
291C4⋊C8C22×D712yes—
292#103—22no—
293#103—22no—
294#103—141no—
295#103—141no—
296#103—142no—
297#102—141no—
298#102—141no—
299#102—142no—
300C22⋊C8—21no—
301C22⋊C8—141no—
302#97—21no—
303#97—141no—
304C4×C8—71no—
305C4×C8—71no—
306C4×C8—71no—
307C4×C8—71no—
308C4×C8—142no—
309C4×C8—142no—
310C22×C7—21no—
311C22×C7—21no—
312C22×C7#47112yes—
313C22×C7C2×#12111yescharacteristic
314C2×D7—81no—
315C2×D7—81no—
316C2×D7—81no—
317C4×C7—21no—
318C4×C7—24no—
319C4×C7—21no—
320C4×C7—21no—
321C4×C7—21no—
322C4×C7—42no—
323C4×C7—81no—
324C4×C7—81no—
325C4×C7C2×#13512yes—
326C4×C7C2×#13711yescharacteristic
327C4×C7#57511yescharacteristic
328Dic7—81no—
329#41—562no—
330C2×Q8—24no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—141no—
335C2×Q8C2×C22⋊D712yes—
336C2×D4—141no—
337C2×D4—141no—
338C2×D4—141no—
339C2×D4—281no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—281no—
343C22×C4D4×D711yescharacteristic
344C22×C4#115911yescharacteristic
345C22×C4C2×C22⋊D711yescharacteristic
346Q16—44no—
347SD16—282no—
348SD16—282no—
349SD16—284no—
350C2×C8—22no—
351C2×C8—71no—
352C2×C8—71no—
353C2×C8—71no—
354C2×C8—71no—
355C2×C8—142no—
356C2×C8—142no—
357C2×C8—142no—
358C2×C8—281no—
359C2×C8—282no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—141no—
370C4⋊C4—141no—
371C4⋊C4C2×C22⋊D712yes—
372C22⋊C4—41no—
373C22⋊C4—281no—
374C22⋊C4—281no—
375C42—22no—
376C42—22no—
377C42—41no—
378C42D4×D711yescharacteristic
379C42#115911yescharacteristic
380C42C2×C22⋊D711yescharacteristic
381C42C23×D711yescharacteristic
382C2×C7—22no—
383C2×C7#303711yescharacteristic
384C2×C7#305011yescharacteristic
385C2×C7C2×#47111yescharacteristic
386D7—161no—
387C23—281no—
388C23#650211yescharacteristic
389Q8—24no—
390Q8—22no—
391Q8—41no—
392Q8—41no—
393Q8—42no—
394Q8—561no—
395Q8#651314yes—
396D4—281no—
397D4—281no—
398D4—282no—
399D4—561no—
400D4—562no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—281no—
412C2×C4—562no—
413C2×C4#650211yescharacteristic
414C2×C4#650211yescharacteristic
415C2×C4#650211yescharacteristic
416C2×C4C2×D4×D711yescharacteristic
417C2×C4C2×#115911yescharacteristic
418C2×C4C22×C22⋊D711yescharacteristic
419C8—42no—
420C8—72no—
421C8—141no—
422C8—141no—
423C8—141no—
424C8—282no—
425C8—282no—
426C7#1926311yescharacteristic, complemented, Sylow
427C22—21no—
428C22—21no—
429C22—561no—
430C22—561no—
431C22—561no—
432C22#8997512yes—
433C22C2×#650211yescharacteristic
434C4—24no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—561no—
443C4D7×#13711yescharacteristic
444C4#9046811yescharacteristic
445C4C2×#651312yes—
446C2—22no—
447C2—1121no—
448C2#1174036711yescharacteristic
449C2#1174141111yescharacteristic
450C2C2×#8997511yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3113C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenter#3113D7
hypocenterC7#19263
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
1. C7×#3113C2semidirect
4. C2×Q8×C7⋊C8C2semidirect
5. #11729585C2semidirect
6. #11728873C2semidirect
7. #11728870C2semidirect
8. #11728809C2semidirect
9. #11728806C2semidirect
11. #11728548C2semidirect
111. #3113D7semidirect
426. C7#19263semidirect

Maximal subgroups

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

subgroupindex
C7×#31132
#117407882
#117398892
C2×Q8×C7⋊C82
#117295852
#117288732
#117288702
#117288092
#117288062
#117285702
#117285482
#192637