Finite Group Databasebeta

← #13250523

Subgroups — #13250523

Order 1792.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2328C211yesmaximal, characteristic, complemented
2#11740611C211yesmaximal, characteristic
3#11729513C211yesmaximal, characteristic, complemented
4#11727299C211yesmaximal, characteristic, complemented
5#11727194C211yesmaximal, characteristic, complemented
6#11725437C211yesmaximal, characteristic, complemented
7#11725420C211yesmaximal, characteristic, complemented
8C2×C7×#127C2211yescharacteristic
9C2×C4×C7×D4—21no—
10#90168C2211yescharacteristic
11C7×#390—21no—
12C7×#386—22no—
13C7×#386—22no—
14C7×#382C2211yescharacteristic
15C7×#376C2211yescharacteristic
16#89989—22no—
17#89845C412yescomplemented
18#89844—22no—
19#89842C412yescomplemented
20C2×#6379—21no—
21C2×#6379—21no—
22#89697C2211yescharacteristic
23#89697C2211yescharacteristic
24#89696C2211yescharacteristic
25#89339—22no—
26#89246—22no—
27#9299—71nomaximal, Sylow
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4D412yes—
31C23×C4×C7—22no—
32C2×#1157—21no—
33C7×#127—21no—
34C7×#127—21no—
35C7×#127C2×C412yes—
36C7×#127D412yes—
37C4×C7×D4—42no—
38C4×C7×D4—42no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C42×C7D411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C2×C42×C7C2311yescharacteristic
46C2×#1141—24no—
47#6493—21no—
48#6493—21no—
49C2×#1136—21no—
50#6438—21no—
51#6437C2×C411yescharacteristic, complemented
52C4×D28—21no—
53C4×D28—21no—
54#6432C2×C411yescharacteristic, complemented
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95D412yes—
59#6406—24no—
60C22⋊(C7⋊C8)—21no—
61C22⋊(C7⋊C8)—21no—
62#6379—42no—
63#6379—42no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68C4×C7⋊C8—21no—
69C4×C7⋊C8C2×C412yescomplemented
70#3205—71no—
71#2328D711yescharacteristic, complemented
72#1963—71no—
73#1545—71no—
74#1517—71no—
75#1286—71no—
76#1274—71no—
77C24×C7—22no—
78C2×C7×D4—22no—
79C2×C7×D4—22no—
80C2×C7×D4—22no—
81C2×C7×D4—22no—
82C2×C7×D4—22no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—82no—
88C2×C7×D4—82no—
89C2×C7×D4C22⋊C414yes—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—22no—
93C22×C4×C7—44no—
94C22×C4×C7—82no—
95C22×C4×C7—82no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C2×C22⋊D7—41no—
100#1157—41no—
101#1157—41no—
102#1157—41no—
103#1157—41no—
104C2×D28—21no—
105C2×D28—21no—
106C2×D28—21no—
107C2×C4×D7—41no—
108C2×Dic14—21no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C22⋊C4—44no—
112C7×C22⋊C4—82no—
113C7×C22⋊C4—82no—
114C42×C7—21no—
115C42×C7—21no—
116C42×C7—21no—
117C42×C7D812yescomplemented
118C42×C7C22×C411yescharacteristic
119C42×C7C2×D411yescharacteristic
120C42×C7#4111yescharacteristic
121C42×C7#4111yescharacteristic
122#1141—84no—
123(C2×D7)⋊C4—41no—
124(C2×D7)⋊C4—41no—
125C4⋊Dic7—21no—
126C4⋊Dic7—21no—
127#1136—41no—
128#1136—41no—
129C2×C7⋊C8—22no—
130C2×C7⋊C8—41no—
131C2×C7⋊C8—41no—
132#532—71no—
133C2×#127C2×D711yescharacteristic
134C2×C4×D4—21no—
135#493—72no—
136#486—72no—
137#459—142no—
138#447—142no—
139#431—71no—
140#431—71no—
141#423—71no—
142C2×#104—141no—
143C2×#104—141no—
144#390—21no—
145#386—22no—
146#386—22no—
147#382C2×D711yescharacteristic
148#376C2×D711yescharacteristic
149#331—142no—
150#327—142no—
151C23×C7—24no—
152C23×C7—42no—
153C23×C7—42no—
154C23×C7—42no—
155C23×C7—42no—
156C23×C7—42no—
157C23×C7—42no—
158C23×C7#12011yescharacteristic
159C22×D7—41no—
160C7×D4—44no—
161C7×D4—42no—
162C7×D4—42no—
163C7×D4—42no—
164C7×D4—42no—
165C7×D4—82no—
166C7×D4—82no—
167C7×D4—82no—
168C2×C4×C7—22no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—88no—
180C2×C4×C7C2×C22⋊C411yescharacteristic
181C2×C4×C7C4×D411yescharacteristic
182C2×C4×C7C4×D411yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12311yescharacteristic
186C2×C4×C7C2×D811yescharacteristic
187C2×C4×C7#13511yescharacteristic
188C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
190C22⋊D7—81no—
191C22⋊D7—81no—
192C2×Dic7—41no—
193D28—42no—
194D28—41no—
195D28—41no—
196D28—81no—
197C4×D7—81no—
198C4×D7—81no—
199Dic14—41no—
200Dic14—41no—
201C7⋊C8—42no—
202C7⋊C8—81no—
203C7⋊C8—81no—
204C2×#41—141no—
205C22×D4—21no—
206C22×D4—21no—
207C22×D4C22⋊D712yes—
208C23×C4—22no—
209C2×D8—144no—
210C2×M16—141no—
211C4⋊Q8—71no—
212#127—21no—
213#127—21no—
214#127—71no—
215#127C4×D712yes—
216#127D2812yes—
217#124—141no—
218#121—141no—
219#121—141no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—141no—
223C4×D4—141no—
224C2×C4⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C42C22⋊D711yescharacteristic
229C2×C42C22⋊D711yescharacteristic
230C2×C42C22×D711yescharacteristic
231C4⋊C8—141no—
232C4⋊C8—141no—
233C4⋊C8—141no—
234C4⋊C8—141no—
235#104—282no—
236#104—282no—
237#102—144no—
238C22⋊C8—141no—
239C22⋊C8—141no—
240C4×C8—72no—
241C4×C8—141no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95C22⋊D712yes—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—44no—
249C22×C7—44no—
250C22×C7—44no—
251C22×C7—82no—
252C22×C7—82no—
253C22×C7—82no—
254C22×C7—82no—
255C22×C7#38611yescharacteristic
256C22×C7#44711yescharacteristic
257C22×C7#44911yescharacteristic
258C2×D7—81no—
259C2×D7—81no—
260C2×D7—81no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—42no—
266C4×C7—81no—
267C4×C7—82no—
268C4×C7C4×D811yescharacteristic
269C4×C7#44211yescharacteristic
270C4×C7#45911yescharacteristic
271C4×C7#46311yescharacteristic
272Dic7—81no—
273C24—22no—
274#41—281no—
275#41—281no—
276#41—281no—
277#41—281no—
278C2×Q8—141no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—141no—
291C2×D4—141no—
292C2×D4—141no—
293C2×D4—281no—
294C2×D4(C2×D7)⋊C414yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—44no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—281no—
302C22×C4C2×C22⋊D711yescharacteristic
303C22×C4C2×C22⋊D711yescharacteristic
304C22×C4C2×C22⋊D711yescharacteristic
305D8—564no—
306M16—281no—
307M16—281no—
308C2×C8—142no—
309C2×C8—281no—
310C2×C8—281no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—141no—
314C4⋊C4—141no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C22⋊C4—281no—
319C22⋊C4—281no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42#114112yescomplemented
324C42C2×C4×D711yescharacteristic
325C42C2×D2811yescharacteristic
326C42#115711yescharacteristic
327C42#115711yescharacteristic
328C2×C7—21no—
329C2×C7—21no—
330C2×C7—84no—
331C2×C7#187911yescharacteristic
332C2×C7#190411yescharacteristic
333C2×C7#191111yescharacteristic
334D7—161no—
335C23—24no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—281no—
343C23#651611yescharacteristic
344Q8—281no—
345Q8—281no—
346D4—44no—
347D4—42no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—282no—
355D4—281no—
356D4—281no—
357D4—561no—
358D4—561no—
359D4—561no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—88no—
372C2×C4—281no—
373C2×C4—561no—
374C2×C4—561no—
375C2×C4C2×(C2×D7)⋊C411yescharacteristic
376C2×C4C4×C22⋊D711yescharacteristic
377C2×C4C4×C22⋊D711yescharacteristic
378C2×C4#649211yescharacteristic
379C2×C4#649311yescharacteristic
380C2×C4#649311yescharacteristic
381C2×C4C2×#114111yescharacteristic
382C2×C4#649511yescharacteristic
383C2×C4#651211yescharacteristic
384C2×C4#651311yescharacteristic
385C8—282no—
386C8—561no—
387C8—561no—
388C7#929911yescharacteristic, complemented, Sylow
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—561no—
399C22—561no—
400C22—561no—
401C22#8998511yescharacteristic
402C22#8998911yescharacteristic
403C22#9001011yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—82no—
410C4—81no—
411C4—561no—
412C4C4×#114111yescharacteristic
413C4#8978611yescharacteristic
414C4#8978711yescharacteristic
415C4#8978811yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—1121no—
420C2#1172936711yescharacteristic
421C2#1172940511yescharacteristic
422C2#1172940611yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2328C2
socleC22×C7#386
prime radical#11740611C2
cosocleC2×C42×C7C23
hypercenter#2328D7
hypocenterC7#9299
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. C7×#2328C2semidirect
3. #11729513C2semidirect
4. #11727299C2semidirect
5. #11727194C2semidirect
6. #11725437C2semidirect
7. #11725420C2semidirect
17. #89845C4semidirect
19. #89842C4semidirect
51. #6437C2×C4semidirect
54. #6432C2×C4semidirect
69. C4×C7⋊C8C2×C4semidirect
71. #2328D7semidirect
117. C42×C7D8semidirect
323. C42#1141semidirect
388. C7#9299semidirect

Maximal subgroups

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

subgroupindex
C7×#23282
#117406112
#117295132
#117272992
#117271942
#117254372
#117254202
#92997