Finite Group Databasebeta

← #13250864

Subgroups — #13250864

Order 1792.

470 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739662C211yesmaximal, characteristic
2#11739655C211yesmaximal, characteristic, complemented
3C7×#1825C211yesmaximal, characteristic, complemented
4#11727675C211yesmaximal, characteristic
5#11727263C211yesmaximal, characteristic
6#11725486C211yesmaximal, characteristic, complemented
7#11725432C211yesmaximal, characteristic, complemented
8C2×C7×#117C2211yescharacteristic
9C2×#6493C2211yescharacteristic
10C2×C7×C22⋊C8C2211yescharacteristic
11C7×#380C2211yescharacteristic
12#89979C2211yescharacteristic
13#89979C2211yescharacteristic
14C2×C22⋊(C7⋊C8)C2211yescharacteristic
15#89775—21no—
16#89775—21no—
17#89773C411yescharacteristic, complemented
18#89773C411yescharacteristic, complemented
19#89772—22no—
20#89771—21no—
21#89769—21no—
22#89767C411yescharacteristic, complemented
23#89767C411yescharacteristic, complemented
24#89740—21no—
25#89739—21no—
26#89698—21no—
27#89697—21no—
28C7×#340—21no—
29C7×#340—21no—
30#89263—22no—
31#89246—21no—
32#89246—21no—
33#9348—71nomaximal, Sylow
34C23×C4×C7C2311yescharacteristic
35C22×C22⋊D7—21no—
36C22×D28C2×C411yescharacteristic
37C22×C7×C8—21no—
38C7×#117—21no—
39C7×#117—21no—
40C7×#117C2×C411yescharacteristic
41C7×#117C2×C411yescharacteristic
42C7×#117D411yescharacteristic
43C7×#117D411yescharacteristic
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C42×C7—21no—
50#6493—21no—
51#6493—21no—
52#6493—21no—
53#6493—21no—
54C2×(C2×D7)⋊C4—21no—
55C2×C4⋊Dic7C2×C411yescharacteristic
56C2×#1136—21no—
57C2×#1136C2×C411yescharacteristic
58C2×#1136C2×C411yescharacteristic
59C22×C7⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×C22⋊C8—21no—
62C7×#95—21no—
63C22⋊(C7⋊C8)—21no—
64C22⋊(C7⋊C8)—21no—
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)D411yescharacteristic
68C22⋊(C7⋊C8)D411yescharacteristic
69C22⋊(C7⋊C8)D412yes—
70#6383—21no—
71#6383—21no—
72#6383—21no—
73#6383—21no—
74#6382—21no—
75#6382—21no—
76#6381—21no—
77#6381—21no—
78C4⋊(C7⋊C8)—22no—
79C4⋊(C7⋊C8)—22no—
80#6377—22no—
81C4×C7⋊C8—22no—
82#3094—71no—
83#2924—71no—
84#1825D711yescharacteristic, complemented
85#1519—71no—
86#1471—71no—
87#1305—71no—
88#1298—71no—
89C24×C7#4111yescharacteristic
90C23×D7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C2×C22⋊D7—41no—
103C2×C22⋊D7—41no—
104C2×C22⋊D7—81no—
105C2×C22⋊D7—81no—
106C22×Dic7—21no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—41no—
110C2×D28—41no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8—41no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4C22⋊C411yescharacteristic
119C7×C4⋊C4C22⋊C411yescharacteristic
120C7×C4⋊C4C22⋊C411yescharacteristic
121C7×C4⋊C4C22⋊C411yescharacteristic
122C7×C22⋊C4—21no—
123C7×C22⋊C4—21no—
124C7×C22⋊C4—21no—
125C7×C22⋊C4—21no—
126C7×C22⋊C4—81no—
127C42×C7—21no—
128C42×C7—21no—
129C42×C7—21no—
130C42×C7—21no—
131(C2×D7)⋊C4—41no—
132(C2×D7)⋊C4—41no—
133C4⋊Dic7—21no—
134C4⋊Dic7—21no—
135#1136—41no—
136#1136—41no—
137#1136—41no—
138#1136—41no—
139C2×C7⋊C8—21no—
140C2×C7⋊C8—21no—
141C2×C7⋊C8—21no—
142C2×C7⋊C8—21no—
143C2×C7⋊C8—22no—
144C2×C7⋊C8—41no—
145C2×C7⋊C8—41no—
146C2×#121—71no—
147C2×#117C2×D711yescharacteristic
148#482—141no—
149#482—141no—
150#481—141no—
151#480—141no—
152#433—142no—
153#432—141no—
154#431—141no—
155#428—71no—
156#428—71no—
157#424—141no—
158#418—71no—
159#418—71no—
160#407—71no—
161#407—71no—
162C2×C22⋊C8—71no—
163C2×C22⋊C8C2×D711yescharacteristic
164#405—141no—
165#380C2×D711yescharacteristic
166#340—21no—
167#340—21no—
168#327—141no—
169#327—141no—
170#323—142no—
171C23×C7—21no—
172C23×C7—21no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7C4×D411yescharacteristic
176C23×C7#12111yescharacteristic
177C23×C7#12311yescharacteristic
178C22×D7—41no—
179C22×D7—42no—
180C22×D7—81no—
181C22×D7—81no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—81no—
198C2×C4×C7—81no—
199C2×C4×C7—81no—
200C2×C4×C7—81no—
201C2×C4×C7C2×C22⋊C411yescharacteristic
202C2×C4×C7C4×D411yescharacteristic
203C2×C4×C7#12011yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C22⋊D7—82no—
206C22⋊D7—81no—
207C22⋊D7—81no—
208C2×Dic7—41no—
209C2×Dic7—81no—
210D28—41no—
211D28—41no—
212D28—81no—
213C7×C8—81no—
214C7⋊C8—41no—
215C7⋊C8—41no—
216C7⋊C8—42no—
217C7⋊C8—41no—
218C7⋊C8—41no—
219C22×D4—71no—
220C22×D4—141no—
221C23×C4C22×D711yescharacteristic
222C2×M16—71no—
223C2×M16—71no—
224C2×M16—141no—
225C22×C8—21no—
226C22×C8—141no—
227#121—141no—
228#121—141no—
229#121—141no—
230#121—141no—
231#117—21no—
232#117—21no—
233#117C4×D711yescharacteristic
234#117C4×D711yescharacteristic
235#117D2811yescharacteristic
236#117D2811yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—71no—
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C4⋊C4C22⋊D711yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—141no—
244C2×C42—21no—
245#107—141no—
246#107—141no—
247#106—141no—
248#106—141no—
249C4⋊C8—142no—
250C4⋊C8—142no—
251#102—141no—
252#102—141no—
253#102—141no—
254#102—141no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—72no—
258C22⋊C8—71no—
259C22⋊C8—71no—
260C22⋊C8—141no—
261C22⋊C8—141no—
262C22⋊C8—141no—
263C22⋊C8—141no—
264#97—142no—
265C4×C8—142no—
266#95—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—81no—
276C22×C7—81no—
277C22×C7#38611yescharacteristic
278C22×C7#47111yescharacteristic
279C22×C7#47211yescharacteristic
280C2×D7—81no—
281C2×D7—81no—
282C2×D7—81no—
283C2×D7—162no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—21no—
287C4×C7—41no—
288C4×C7—41no—
289C4×C7—41no—
290C4×C7—41no—
291C4×C7—161no—
292C4×C7#41011yescharacteristic
293C4×C7#41011yescharacteristic
294Dic7—81no—
295C24—141no—
296C24#115911yescharacteristic
297C2×D4—141no—
298C2×D4—141no—
299C2×D4—281no—
300C2×D4—281no—
301C2×D4—281no—
302C2×D4—281no—
303C2×D4—561no—
304C2×D4—561no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—141no—
311C22×C4C2×C4×D711yescharacteristic
312C22×C4C2×D2811yescharacteristic
313C22×C4D4×D711yescharacteristic
314C22×C4D4×D711yescharacteristic
315C22×C4#115911yescharacteristic
316C22×C4C2×C22⋊D711yescharacteristic
317M16—281no—
318M16—281no—
319M16—281no—
320M16—281no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—141no—
325C2×C8—142no—
326C2×C8—141no—
327C2×C8—141no—
328C2×C8—141no—
329C2×C8—281no—
330C2×C8—281no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—141no—
336C4⋊C4—141no—
337C4⋊C4(C2×D7)⋊C411yescharacteristic
338C4⋊C4(C2×D7)⋊C411yescharacteristic
339C4⋊C4(C2×D7)⋊C411yescharacteristic
340C4⋊C4(C2×D7)⋊C411yescharacteristic
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—81no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C2×C7—21no—
353C2×C7—21no—
354C2×C7—41no—
355C2×C7—41no—
356C2×C7#188111yescharacteristic
357C2×C7#189911yescharacteristic
358C2×C7#190711yescharacteristic
359D7—161no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23—282no—
365C23—281no—
366C23—561no—
367C23—561no—
368C23(C22⋊D7)⋊C411yescharacteristic
369C23#644911yescharacteristic
370C23#650211yescharacteristic
371D4—281no—
372D4—281no—
373D4—561no—
374D4—562no—
375D4—561no—
376D4—561no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—281no—
397C2×C4—561no—
398C2×C4(C22⋊D7)⋊C411yescharacteristic
399C2×C4#644511yescharacteristic
400C2×C4C2×(C2×D7)⋊C411yescharacteristic
401C2×C4#650211yescharacteristic
402C8—81no—
403C8—281no—
404C8—281no—
405C8—282no—
406C8—281no—
407C8—281no—
408C7#934811yescharacteristic, complemented, Sylow
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—561no—
420C22—561no—
421C22—561no—
422C22—1122no—
423C22#8973011yescharacteristic
424C22#8997111yescharacteristic
425C22#8997511yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—161no—
434C4—561no—
435C4#8977811yescharacteristic
436C4#8977811yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—1121no—
442C2#1172728911yescharacteristic
443C2#1172804711yescharacteristic
444C2#1172805711yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1825C2
socleC22×C7#386
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#1825D7
hypocenterC7#9348
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. #11739655C2semidirect
3. C7×#1825C2semidirect
6. #11725486C2semidirect
7. #11725432C2semidirect
17. #89773C4semidirect
18. #89773C4semidirect
22. #89767C4semidirect
23. #89767C4semidirect
84. #1825D7semidirect
408. C7#9348semidirect

Maximal subgroups

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

subgroupindex
#117396622
#117396552
C7×#18252
#117276752
#117272632
#117254862
#117254322
#93487