Finite Group Databasebeta

← #13509301

Subgroups — #13509301

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739384C211yesmaximal, characteristic
2#11739363C211yesmaximal, characteristic
3C7×#2255C211yesmaximal, characteristic, complemented
4#11729138C211yesmaximal, characteristic, complemented
5#11729108C211yesmaximal, characteristic, complemented
6#11727100C211yesmaximal, characteristic
7#11727099C211yesmaximal, characteristic
8#11727097C211yesmaximal, characteristic, complemented
9#11727095C211yesmaximal, characteristic, complemented
10#11727080C211yesmaximal, characteristic
11D7×#366C211yesmaximal, characteristic
12#11727076C211yesmaximal, characteristic, complemented
13#11727074C211yesmaximal, characteristic, complemented
14#11726826C211yesmaximal, characteristic
15#11726823C211yesmaximal, characteristic, complemented
16#90420—21no—
17#90380—21no—
18C7×#500C2211yescharacteristic
19C7×#499C2211yescharacteristic
20#89852—21no—
21#89837—21no—
22C2×#6372C2211yescharacteristic
23C2×D7×C16C2211yescharacteristic
24#89653—21no—
25#89652C2211yescharacteristic
26#89652C2211yescharacteristic
27D7×#107C2211yescharacteristic
28D7×#107C2211yescharacteristic
29#89650C2211yescharacteristic
30#89650C2211yescharacteristic
31#89648—21no—
32#89646C2211yescharacteristic
33#89646C2211yescharacteristic
34#89633—21no—
35#89628—21no—
36#89461C2211yescharacteristic
37#89455C2211yescharacteristic
38#89454C2211yescharacteristic
39C7×#367C2211yescharacteristic
40C7×#367C2211yescharacteristic
41C7×#366C2211yescharacteristic
42C7×#366C2211yescharacteristic
43C7×#346C2211yescharacteristic
44#89300C2211yescharacteristic
45#89300C2211yescharacteristic
46#89299C2211yescharacteristic
47#89299C2211yescharacteristic
48#89297C2211yescharacteristic
49C16×Dic7C2211yescharacteristic
50#89285C2211yescharacteristic
51#89285C2211yescharacteristic
52#89284C2211yescharacteristic
53#89284C2211yescharacteristic
54#89284C2211yescharacteristic
55#89284C2211yescharacteristic
56#89284C2211yescharacteristic
57#89284C2211yescharacteristic
58C4×C7⋊C16C2211yescharacteristic
59#33729—71nomaximal, Sylow
60C7×C4⋊Q8—21no—
61C7×#125—21no—
62#6509—41no—
63#6507—41no—
64C2×#1131D411yescharacteristic
65C2×C8×D7C2311yescharacteristic
66#6460—41no—
67#6459—41no—
68#6459—41no—
69#6455—21no—
70#6455—21no—
71D7×C4⋊C4—21no—
72D7×C4⋊C4—21no—
73#6453—21no—
74#6453—21no—
75#6452—41no—
76#6452—41no—
77#6451—21no—
78#6451—21no—
79#6451—41no—
80#6435D411yescharacteristic
81#6433—21no—
82#6432—21no—
83C2×C7×C16C2311yescharacteristic
84C2×C7×C16C2311yescharacteristic
85C7×#107—21no—
86C7×#107C2311yescharacteristic
87C7×#107C2311yescharacteristic
88C7×#107C2311yescharacteristic
89C7×#107C2311yescharacteristic
90C7×#106—21no—
91C4×C7×C8C2311yescharacteristic
92#6392—21no—
93#6392C2311yescharacteristic
94#6392C2311yescharacteristic
95#6392C2311yescharacteristic
96#6392C2311yescharacteristic
97#6391—21no—
98#6389D411yescharacteristic
99C8×Dic7C2311yescharacteristic
100C2×C7⋊C16C2311yescharacteristic
101C2×C7⋊C16C2311yescharacteristic
102#6382—21no—
103#6382—21no—
104#6382—21no—
105#6382—21no—
106#6381—21no—
107#6381—21no—
108#6381—21no—
109#6381—21no—
110#6377D411yescharacteristic
111#6372Q814yescomplemented
112D7×C16Q814yescomplemented
113#3164—71no—
114#3164—71no—
115#2255—71no—
116#2255—71no—
117#2255—71no—
118#2255D711yescharacteristic, complemented
119#2253—71no—
120#2253—71no—
121#2252—71no—
122#2252—71no—
123#2159—71no—
124#2159—71no—
125#2158—71no—
126C2×#366—71no—
127#2108—71no—
128C2×C7×Q8—41no—
129C2×C4×D7—41no—
130C2×C4×D7—41no—
131C2×C4×D7C2×D411yescharacteristic
132C2×Dic14—41no—
133C2×Dic14—41no—
134C2×Dic14—41no—
135C2×C7×C8C2×D411yescharacteristic
136C2×C7×C8C2411yescharacteristic
137C7×C4⋊C4—21no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4—21no—
141C7×C4⋊C4—41no—
142C7×C4⋊C4—41no—
143C7×C4⋊C4—41no—
144C42×C7C2×D411yescharacteristic
145(C2×D7)⋊C4—41no—
146(C2×D7)⋊C4—41no—
147(C2×D7)⋊C4D811yescharacteristic
148(C2×D7)⋊C4D811yescharacteristic
149C4⋊Dic7—21no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153C4⋊Dic7—41no—
154C4⋊Dic7—41no—
155C4⋊Dic7—41no—
156Dic7⋊C4—41no—
157Dic7⋊C4—41no—
158Dic7⋊C4—41no—
159Dic7⋊C4—41no—
160Dic7⋊C4—41no—
161Dic7⋊C4—41no—
162Dic7⋊C4—41no—
163Dic7⋊C4—41no—
164Dic7⋊C4D811yescharacteristic
165Dic7⋊C4D811yescharacteristic
166C4×Dic7—41no—
167C4×Dic7—41no—
168C4×Dic7C2×D411yescharacteristic
169C2×C7⋊C8C2×D411yescharacteristic
170C2×C7⋊C8C2×D411yescharacteristic
171#1131Q1614yes—
172C8×D7C2×Q812yes—
173C8×D7C2×Q812yes—
174C7×C16C2×Q812yes—
175C7×C16C2×Q812yes—
176C7⋊C16C2×Q812yes—
177C7⋊C16C2×Q812yes—
178#544—141no—
179#544—141no—
180C2×M32—71no—
181C22×C16—71no—
182#501—141no—
183#501—141no—
184#501—141no—
185#501—141no—
186#500—71no—
187#500—71no—
188#500—71no—
189#500C2×D711yescharacteristic
190#499—71no—
191#499—71no—
192#499—71no—
193#499C2×D711yescharacteristic
194#428—141no—
195#428—141no—
196#427—71no—
197#427—71no—
198C2×#107—71no—
199C2×#107—71no—
200#405—71no—
201#367—71no—
202#367—71no—
203#367—71no—
204#367—71no—
205#367C2×D711yescharacteristic
206#367C2×D711yescharacteristic
207#366—71no—
208#366—71no—
209#366—71no—
210#366—71no—
211#366—71no—
212#366—71no—
213#366—71no—
214#366—71no—
215#366C2×D711yescharacteristic
216#366C2×D711yescharacteristic
217#346—71no—
218#346C2×D711yescharacteristic
219C4×C16—71no—
220C4×C16—71no—
221C22×D7C2×D811yescharacteristic
222C7×Q8—161no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—41no—
226C2×C4×C7—41no—
227C2×C4×C7C2×D811yescharacteristic
228C2×C4×C7C2×D811yescharacteristic
229C2×C4×C7C22×D411yescharacteristic
230C2×Dic7—41no—
231C2×Dic7—41no—
232C2×Dic7—41no—
233C2×Dic7—41no—
234C2×Dic7C2×D811yescharacteristic
235C2×Dic7C2×D811yescharacteristic
236C2×Dic7C2×D811yescharacteristic
237C4×D7—161no—
238C4×D7—161no—
239C4×D7C4⋊Q812yes—
240C4×D7C2×Q1612yes—
241Dic14—161no—
242Dic14—161no—
243Dic14—161no—
244C7×C8C2×Q1612yes—
245C7×C8C22×Q811yescharacteristic
246C7×C8C22×Q811yescharacteristic
247C7⋊C8C4⋊Q812yes—
248C7⋊C8C2×Q1612yes—
249C2×M16—71no—
250C22×C8—71no—
251C4⋊Q8—21no—
252C4⋊Q8—141no—
253C4⋊Q8—141no—
254C4⋊Q8—141no—
255C4⋊Q8—281no—
256C4⋊Q8—281no—
257#125—21no—
258#125—141no—
259#125—141no—
260#125—141no—
261#125—281no—
262#125—281no—
263C22⋊Q8—281no—
264C22⋊Q8—281no—
265C22⋊Q8—281no—
266C22⋊Q8—281no—
267#117—71no—
268#117—141no—
269#117—141no—
270C2×C4⋊C4—141no—
271C2×C4⋊C4—141no—
272M32—74no—
273C2×C16—71no—
274C2×C16—71no—
275C2×C16—74no—
276C2×C16C22×D711yescharacteristic
277C2×C16C22×D711yescharacteristic
278#107—21no—
279#107—71no—
280#107—71no—
281#107—71no—
282#107—71no—
283#107—141no—
284#107—141no—
285#107—141no—
286#107—141no—
287#107—141no—
288#107C22×D711yescharacteristic
289#107C22×D711yescharacteristic
290#107C22×D711yescharacteristic
291#107C22×D711yescharacteristic
292#106—21no—
293#106—141no—
294#106—141no—
295#106—141no—
296#106—141no—
297#106—141no—
298#97—71no—
299#97—71no—
300C4×C8—71no—
301C4×C8C22×D711yescharacteristic
302C22×C7C22×D811yescharacteristic
303C2×D7—21no—
304C2×D7—21no—
305C2×D7#50012yes—
306C4×C7—21no—
307C4×C7—21no—
308C4×C7—81no—
309C4×C7—81no—
310C4×C7—81no—
311C4×C7—81no—
312C4×C7C2×C4⋊Q811yescharacteristic
313C4×C7C22×Q1611yescharacteristic
314Dic7—21no—
315Dic7—21no—
316Dic7—81no—
317Dic7—81no—
318Dic7—81no—
319Dic7—81no—
320Dic7#50012yes—
321C2×Q8—41no—
322C2×Q8—281no—
323C2×Q8—281no—
324C2×Q8—281no—
325C22×C4—71no—
326C22×C4—281no—
327C22×C4—281no—
328M16—74no—
329C2×C8—71no—
330C2×C8—71no—
331C2×C8—72no—
332C2×C8—72no—
333C2×C8D4×D711yescharacteristic
334C2×C8C23×D711yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—71no—
343C4⋊C4—71no—
344C4⋊C4—141no—
345C4⋊C4—141no—
346C4⋊C4—141no—
347C4⋊C4—141no—
348C4⋊C4—281no—
349C4⋊C4—281no—
350C4⋊C4—281no—
351C4⋊C4—281no—
352C4⋊C4—281no—
353C4⋊C4—281no—
354C4⋊C4—281no—
355C4⋊C4—281no—
356C4⋊C4—281no—
357C4⋊C4—281no—
358C4⋊C4—281no—
359C22⋊C4—71no—
360C22⋊C4—71no—
361C22⋊C4—281no—
362C22⋊C4—281no—
363C42—71no—
364C42—281no—
365C42—281no—
366C42D4×D711yescharacteristic
367C16—72no—
368C16—72no—
369C16Q8×D712yes—
370C16Q8×D712yes—
371C2×C7C2×#50011yescharacteristic
372C2×C7#341511yescharacteristic
373C2×C7#341711yescharacteristic
374D7—22no—
375C23—71no—
376Q8—161no—
377Q8—1121no—
378Q8—1121no—
379Q8—1121no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—71no—
385C2×C4—72no—
386C2×C4—72no—
387C2×C4—71no—
388C2×C4—71no—
389C2×C4—281no—
390C2×C4—281no—
391C2×C4—281no—
392C2×C4—281no—
393C2×C4—1121no—
394C2×C4—1121no—
395C2×C4D7×D811yescharacteristic
396C2×C4D7×D811yescharacteristic
397C2×C4C2×D4×D711yescharacteristic
398C8—72no—
399C8—72no—
400C8D7×Q1612yes—
401C8C2×Q8×D711yescharacteristic
402C8C2×Q8×D711yescharacteristic
403C7#3372911yescharacteristic, complemented, Sylow
404C22—72no—
405C22—141no—
406C22—141no—
407C22C2×D7×D811yescharacteristic
408C4—21no—
409C4—21no—
410C4—72no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—141no—
416C4—141no—
417C4—561no—
418C4—561no—
419C4—561no—
420C4—561no—
421C4D7×C4⋊Q811yescharacteristic
422C4C2×D7×Q1611yescharacteristic
423C2—142no—
424C2D7×#50011yescharacteristic
425C2#1173948111yescharacteristic
426C2#1173948611yescharacteristic
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2255C2
socleC22×C7C22×D8
prime radicalC22×D7C2×D8
hypercenter#2255D7
hypocenterC7#33729
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. C7×#2255C2semidirect
4. #11729138C2semidirect
5. #11729108C2semidirect
8. #11727097C2semidirect
9. #11727095C2semidirect
12. #11727076C2semidirect
13. #11727074C2semidirect
15. #11726823C2semidirect
111. #6372Q8semidirect
112. D7×C16Q8semidirect
118. #2255D7semidirect
403. C7#33729semidirect

Maximal subgroups

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

subgroupindex
#117393842
#117393632
C7×#22552
#117291382
#117291082
#117271002
#117270992
#117270972
#117270952
#117270802
D7×#3662
#117270762
#117270742
#117268262
#117268232
#337297