Finite Group Databasebeta

← #13339163

Subgroups — #13339163

Order 1792.

598 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
1C7×#3104C211yesmaximal, characteristic, complemented
2#11740804C211yesmaximal, characteristic
3#11739889C211yesmaximal, characteristic
4#11739880C211yesmaximal, characteristic, complemented
5#11729586C211yesmaximal, characteristic, complemented
6#11728904C212yesmaximal, complemented
7#11728903C212yesmaximal, complemented
8#11728809C212yesmaximal, complemented
9#11728806C212yesmaximal, complemented
10#11728527C211yesmaximal, characteristic
11#11728501C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14#90168C2211yescharacteristic
15C7×#484—21no—
16C7×#477C2212yes—
17C7×#462C2212yes—
18C7×#451—21no—
19C4×C7×Q16—22no—
20C7×#431C2211yescharacteristic
21#89999—21no—
22#89860C2211yescharacteristic
23#89860C2211yescharacteristic
24#89851C2211yescharacteristic
25#89850C2211yescharacteristic
26#89838—22no—
27#89799C2212yes—
28#89798—22no—
29Q8×C7⋊C8—22no—
30Q8×C7⋊C8C2212yes—
31Q8×C7⋊C8C2212yes—
32#89793C2212yes—
33#89775—22no—
34#89739—21no—
35#89698C2211yescharacteristic
36#89697—22no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89342C2212yes—
40#89341C2212yes—
41#89280C2212yes—
42#89276C2212yes—
43#89275C2212yes—
44#89274C2212yes—
45#19406—71nomaximal, Sylow
46C22×C7×Q8D411yescharacteristic
47C2×#1157—21no—
48C2×C7×Q16—22no—
49C7×C4⋊Q8C2311yescharacteristic
50C7×C4⋊Q8C2311yescharacteristic
51C7×#125—21no—
52C7×C22⋊Q8—21no—
53C7×C22⋊Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C4×C7×Q8—21no—
58C4×C7×Q8D412yes—
59C4×C7×Q8C2312yes—
60C7×#117—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65C2×#1143—22no—
66#6493—21no—
67#6493—21no—
68#6438—21no—
69#6437C2311yescharacteristic
70C4×D28—21no—
71C4×D28—21no—
72#6432C2311yescharacteristic
73C7×#107—22no—
74C7×C4⋊C8C2312yes—
75C7×#103—22no—
76C7×#103—22no—
77C7×C22⋊C8—21no—
78C4×C7×C8—21no—
79#6409—22no—
80C22⋊(C7⋊C8)—21no—
81C22⋊(C7⋊C8)—21no—
82C22⋊(C7⋊C8)D412yes—
83#6383—22no—
84#6383—22no—
85#6383—22no—
86#6382—22no—
87#6381—22no—
88#6381—22no—
89C4⋊(C7⋊C8)—22no—
90C4⋊(C7⋊C8)—22no—
91C4⋊(C7⋊C8)D412yes—
92C4⋊(C7⋊C8)C2312yes—
93C4⋊(C7⋊C8)C2312yes—
94#6377—21no—
95C4×C7⋊C8—21no—
96C4×C7⋊C8—22no—
97C4×C7⋊C8—22no—
98C4×C7⋊C8C2311yescharacteristic
99C4×C7⋊C8C2311yescharacteristic
100#3228—71no—
101#3112—71no—
102#3104D711yescharacteristic, complemented
103#2982—71no—
104#1682—72no—
105#1678—72no—
106#1676—72no—
107#1672—72no—
108#1572—71no—
109#1554—71no—
110#1543—71no—
111C2×C7×Q8—24no—
112C2×C7×Q8—21no—
113C2×C7×Q8—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8C2×D412yes—
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7#4111yescharacteristic
121C2×C22⋊D7—41no—
122#1157—82no—
123C2×D28—21no—
124C2×D28—21no—
125C2×D28—21no—
126C2×C4×D7—41no—
127C2×Dic14—21no—
128C7×Q16—44no—
129C2×C7×C8—22no—
130C7×C4⋊C4—22no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—41no—
138C7×C4⋊C4—41no—
139C7×C4⋊C4C2×D412yes—
140C7×C22⋊C4—41no—
141C42×C7—22no—
142C42×C7—22no—
143C42×C7—41no—
144C42×C7C2×D411yescharacteristic
145C42×C7C2×D411yescharacteristic
146C42×C7#4111yescharacteristic
147C42×C7C2411yescharacteristic
148#1143—44no—
149(C2×D7)⋊C4—41no—
150(C2×D7)⋊C4—41no—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153C2×C7⋊C8—22no—
154C2×C7⋊C8—22no—
155C2×C7⋊C8C2×D412yes—
156C2×C7⋊C8C2×D412yes—
157#533C2×D711yescharacteristic
158#532—71no—
159C2×C4×Q8C2×D711yescharacteristic
160#500—71no—
161#498—71no—
162#486—71no—
163#486—71no—
164#484—21no—
165#482—142no—
166#480—141no—
167#478—142no—
168#477—72no—
169#477C2×D712yes—
170#462C2×D712yes—
171#460—72no—
172#451—21no—
173#448—141no—
174C8×Q8—72no—
175C8×Q8—72no—
176C8×Q8—142no—
177#440—142no—
178C4×Q16—22no—
179#432—71no—
180#431—71no—
181#431—71no—
182#431—142no—
183#431C2×D711yescharacteristic
184#335—72no—
185#334—72no—
186#333—72no—
187#333—72no—
188#332—72no—
189#331—72no—
190C23×C7#12111yescharacteristic
191C22×D7—41no—
192C7×Q8—24no—
193C7×Q8—22no—
194C7×Q8—42no—
195C7×Q8—81no—
196C7×Q8#13514yes—
197C2×C4×C7—24no—
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—22no—
201C2×C4×C7—22no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—81no—
206C2×C4×C7#12111yescharacteristic
207C2×C4×C7#12111yescharacteristic
208C2×C4×C7#12111yescharacteristic
209C2×C4×C7C22×D411yescharacteristic
210C2×C4×C7C22×D411yescharacteristic
211C2×C4×C7C2×#4111yescharacteristic
212C22⋊D7—82no—
213C2×Dic7—41no—
214D28—42no—
215D28—42no—
216D28—81no—
217C4×D7—82no—
218Dic14—81no—
219C7×C8—42no—
220C7⋊C8—22no—
221C7⋊C8—22no—
222C7⋊C8—42no—
223C7⋊C8—42no—
224C2×#41—141no—
225C22×Q8C22⋊D711yescharacteristic
226C2×Q16—22no—
227C2×SD16—142no—
228C4⋊Q8—71no—
229C4⋊Q8C22×D711yescharacteristic
230C4⋊Q8C22×D711yescharacteristic
231#127—71no—
232#125—21no—
233#124—141no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—141no—
237#121—141no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8C22⋊D712yes—
243C4×Q8C22×D712yes—
244C4×D4—141no—
245C4×D4—141no—
246#117—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22⋊D711yescharacteristic
249C2×C42—21no—
250C2×C42C22×D711yescharacteristic
251#107—22no—
252#107—142no—
253#107—142no—
254#106—142no—
255C4⋊C8—72no—
256C4⋊C8—72no—
257C4⋊C8—72no—
258C4⋊C8—142no—
259C4⋊C8—142no—
260C4⋊C8C22×D712yes—
261#103—22no—
262#103—22no—
263#103—142no—
264#102—142no—
265#102—142no—
266#102—142no—
267C22⋊C8—21no—
268C22⋊C8—72no—
269C22⋊C8—141no—
270C22⋊C8—141no—
271#97—141no—
272C4×C8—21no—
273C4×C8—71no—
274C4×C8—71no—
275C4×C8—142no—
276C4×C8—142no—
277C4×C8—141no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7—21no—
281C22×C7C2×#12111yescharacteristic
282C2×D7—81no—
283C2×D7—81no—
284C2×D7—81no—
285C4×C7—21no—
286C4×C7—24no—
287C4×C7—21no—
288C4×C7—21no—
289C4×C7—21no—
290C4×C7—42no—
291C4×C7—81no—
292C4×C7—81no—
293C4×C7C2×#13512yes—
294C4×C7#57812yes—
295Dic7—81no—
296#41—562no—
297C2×Q8—24no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—141no—
302C2×Q8C2×C22⋊D712yes—
303C2×D4—141no—
304C2×D4—141no—
305C2×D4—141no—
306C2×D4—281no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—281no—
310C22×C4D4×D711yescharacteristic
311C22×C4#115911yescharacteristic
312C22×C4C2×C22⋊D711yescharacteristic
313Q16—44no—
314SD16—284no—
315C2×C8—22no—
316C2×C8—72no—
317C2×C8—72no—
318C2×C8—142no—
319C2×C8—142no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—141no—
330C4⋊C4—141no—
331C4⋊C4C2×C22⋊D712yes—
332C22⋊C4—41no—
333C22⋊C4—281no—
334C22⋊C4—281no—
335C42—22no—
336C42—22no—
337C42—41no—
338C42D4×D711yescharacteristic
339C42#115911yescharacteristic
340C42C2×C22⋊D711yescharacteristic
341C42C23×D711yescharacteristic
342C2×C7—41no—
343C2×C7#304411yescharacteristic
344C2×C7#305011yescharacteristic
345C2×C7#306811yescharacteristic
346D7—161no—
347C23—281no—
348C23#650211yescharacteristic
349Q8—24no—
350Q8—22no—
351Q8—42no—
352Q8—81no—
353Q8—561no—
354Q8#651314yes—
355D4—282no—
356D4—282no—
357D4—561no—
358D4—562no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—281no—
369C2×C4—562no—
370C2×C4#650211yescharacteristic
371C2×C4#650211yescharacteristic
372C2×C4#650211yescharacteristic
373C2×C4C2×D4×D711yescharacteristic
374C2×C4C2×#115911yescharacteristic
375C2×C4C22×C22⋊D711yescharacteristic
376C8—42no—
377C8—142no—
378C8—142no—
379C8—282no—
380C8—282no—
381C7#1940611yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—21no—
385C22—561no—
386C22—561no—
387C22—561no—
388C22C2×#650211yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—24no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—561no—
398C4#9045712yes—
399C4C2×#651312yes—
400C2—41no—
401C2—1121no—
402C2#1174036711yescharacteristic
403C2#1174132211yescharacteristic
404C2#1174144711yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3104C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenter#3104D7
hypocenterC7#19406
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
1. C7×#3104C2semidirect
4. #11739880C2semidirect
5. #11729586C2semidirect
6. #11728904C2semidirect
7. #11728903C2semidirect
8. #11728809C2semidirect
9. #11728806C2semidirect
11. #11728501C2semidirect
102. #3104D7semidirect
381. C7#19406semidirect

Maximal subgroups

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

subgroupindex
C7×#31042
#117408042
#117398892
#117398802
#117295862
#117289042
#117289032
#117288092
#117288062
#117285272
#117285012
#194067