Finite Group Databasebeta

← #13251582

Subgroups — #13251582

Order 1792.

556 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 109 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#11739642C211yesmaximal, characteristic
2#11736047C211yesmaximal, characteristic
3C7×#1824C211yesmaximal, characteristic, complemented
4#11729308C211yesmaximal, characteristic, complemented
5#11727284C211yesmaximal, characteristic
6#11725482C211yesmaximal, characteristic, complemented
7#11725431C211yesmaximal, characteristic, complemented
8C22×C7×C4⋊C4C2211yescharacteristic
9C2×C4×C22⋊D7C2211yescharacteristic
10C2×C7×C22⋊C8C2211yescharacteristic
11C7×#380C2211yescharacteristic
12#89985—21no—
13#89984C2211yescharacteristic
14#89984C2211yescharacteristic
15#89982—21no—
16C2×C22⋊(C7⋊C8)C2211yescharacteristic
17#89883—22no—
18#89764C412yescomplemented
19#89759—21no—
20#89754C412yescomplemented
21#89745—21no—
22C2×#6380—21no—
23#89502—22no—
24#89498—21no—
25#89491—21no—
26C7×#340—22no—
27#89324—22no—
28#89322—21no—
29#9473—71nomaximal, Sylow
30C23×C4×C7—21no—
31C23×C4×C7C2311yescharacteristic
32C22×C22⋊D7D411yescharacteristic
33C22×C4×D7D411yescharacteristic
34C22×C7×C8D412yes—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4C2×C412yes—
42C2×C7×C22⋊C4—21no—
43C2×C22⋊Dic7—21no—
44C2×C22⋊Dic7—21no—
45C2×C22⋊Dic7D411yescharacteristic
46C4×C22⋊D7—21no—
47C4×C22⋊D7—21no—
48C4×C22⋊D7—21no—
49C4×C22⋊D7—21no—
50C2×(C2×D7)⋊C4—22no—
51C2×(C2×D7)⋊C4C2×C411yescharacteristic
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—21no—
54C2×Dic7⋊C4—22no—
55C2×Dic7⋊C4C2×C411yescharacteristic
56C2×C4×Dic7D411yescharacteristic
57C22×C7⋊C8D412yes—
58C2×#1131—22no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#95—21no—
62C7⋊#95—22no—
63C7⋊#95—22no—
64C7⋊#95—21no—
65C7⋊#95C2×C412yes—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68(C2×D7)⋊C8—22no—
69#6389—22no—
70Dic7⋊C8—22no—
71#6380—42no—
72#2973—71no—
73#2487—71no—
74#1824—71no—
75#1824D711yescharacteristic, complemented
76#1799—71no—
77#1289—71no—
78#1289—71no—
79C24×C7C2×D411yescharacteristic
80C23×D7C22⋊C411yescharacteristic
81C22×C4×C7—22no—
82C22×C4×C7—42no—
83C22×C4×C7—41no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7—82no—
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7#4111yescharacteristic
92C22×C4×C7#4111yescharacteristic
93C2×C22⋊D7—21no—
94C2×C22⋊D7—21no—
95C2×C22⋊D7—21no—
96C2×C22⋊D7—21no—
97C2×C22⋊D7—21no—
98C2×C22⋊D7—21no—
99C22×Dic7—22no—
100C22×Dic7—41no—
101C22×Dic7C22⋊C411yescharacteristic
102C22×Dic7C22⋊C411yescharacteristic
103C22×Dic7C22⋊C411yescharacteristic
104C2×C4×D7—21no—
105C2×C4×D7—21no—
106C2×C4×D7—42no—
107C2×C7×C8—22no—
108C2×C7×C8—42no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—42no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C22⋊Dic7—21no—
117C22⋊Dic7—21no—
118C22⋊Dic7—41no—
119C22⋊Dic7—41no—
120C22⋊Dic7—41no—
121C22⋊Dic7—41no—
122(C2×D7)⋊C4—21no—
123(C2×D7)⋊C4—21no—
124(C2×D7)⋊C4—82no—
125C4⋊Dic7—42no—
126C4⋊Dic7—42no—
127Dic7⋊C4—21no—
128Dic7⋊C4—21no—
129Dic7⋊C4—82no—
130C4×Dic7—21no—
131C4×Dic7—21no—
132C2×C7⋊C8—22no—
133C2×C7⋊C8—42no—
134#1131—82no—
135C2×C4×D4—71no—
136C22×C4⋊C4C2×D711yescharacteristic
137#435—142no—
138#435—142no—
139#432—141no—
140C2×#99—141no—
141#407—141no—
142C2×C22⋊C8—71no—
143C2×C22⋊C8C2×D711yescharacteristic
144#400—72no—
145#399—72no—
146#389—141no—
147#386—141no—
148#385—141no—
149#380—71no—
150#380—71no—
151#380—141no—
152#380C2×D711yescharacteristic
153#342—141no—
154#340—22no—
155#340—142no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7C2×C22⋊C411yescharacteristic
159C23×C7#12111yescharacteristic
160C23×C7#12111yescharacteristic
161C23×C7AGL(1,ℤ/8)11yescharacteristic
162C23×C7#13711yescharacteristic
163C23×C7#13711yescharacteristic
164C23×C7#13711yescharacteristic
165C22×D7—21no—
166C22×D7—21no—
167C22×D7—21no—
168C22×D7—41no—
169C22×D7—41no—
170C2×C4×C7—22no—
171C2×C4×C7—22no—
172C2×C4×C7—22no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—82no—
180C2×C4×C7—82no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7C4×D411yescharacteristic
185C2×C4×C7C4×D411yescharacteristic
186C2×C4×C7#12011yescharacteristic
187C2×C4×C7#12311yescharacteristic
188C22⋊D7—41no—
189C22⋊D7—41no—
190C22⋊D7—41no—
191C22⋊D7—41no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—21no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7—42no—
200C2×Dic7—42no—
201C2×Dic7—42no—
202C2×Dic7—81no—
203C2×Dic7—81no—
204C2×Dic7—81no—
205C4×D7—42no—
206C4×D7—81no—
207C7×C8—42no—
208C7⋊C8—42no—
209C22×D4—71no—
210C23×C4—21no—
211C23×C4—71no—
212C23×C4C22×D711yescharacteristic
213C2×M16—142no—
214C22×C8—72no—
215C22×C8C22⋊D712yes—
216C4×D4—141no—
217C4×D4—141no—
218C4×D4—141no—
219C4×D4—141no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—71no—
227C2×C4⋊C4—141no—
228C2×C4⋊C4—141no—
229C2×C4⋊C4—142no—
230C2×C4⋊C4C4×D712yes—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—71no—
233C2×C22⋊C4—71no—
234C2×C22⋊C4—141no—
235C2×C22⋊C4—141no—
236C2×C22⋊C4—142no—
237C2×C42—71no—
238C4⋊C8—142no—
239#99—282no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—142no—
243C22⋊C8—141no—
244C22⋊C8—141no—
245#97—142no—
246#95—21no—
247#95—72no—
248#95—142no—
249#95—142no—
250#95—141no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7#38611yescharacteristic
262C22×C7#41711yescharacteristic
263C22×C7#41911yescharacteristic
264C22×C7#46811yescharacteristic
265C22×C7#46811yescharacteristic
266C22×C7#47011yescharacteristic
267C22×C7#47011yescharacteristic
268C2×D7—42no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—81no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—21no—
276C4×C7—82no—
277C4×C7—161no—
278C4×C7#45812yes—
279Dic7—41no—
280Dic7—41no—
281Dic7—41no—
282Dic7—82no—
283Dic7—161no—
284C24—71no—
285C24D4×D711yescharacteristic
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—141no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—141no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—71no—
298C22×C4—71no—
299C22×C4—71no—
300C22×C4—82no—
301C22×C4—141no—
302C22×C4—141no—
303C22×C4—142no—
304C22×C4—282no—
305C22×C4—281no—
306C22×C4C2×C4×D711yescharacteristic
307C22×C4#115711yescharacteristic
308C22×C4D4×D711yescharacteristic
309C22×C4D4×D711yescharacteristic
310C22×C4#115911yescharacteristic
311C22×C4C2×C22⋊D711yescharacteristic
312M16—562no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—142no—
316C2×C8—282no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—141no—
323C4⋊C4—141no—
324C4⋊C4—282no—
325C4⋊C4—282no—
326C4⋊C4—562no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—141no—
330C22⋊C4—141no—
331C22⋊C4—141no—
332C22⋊C4—141no—
333C22⋊C4—281no—
334C22⋊C4—281no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—562no—
338C42—141no—
339C42—141no—
340C2×C7—21no—
341C2×C7—21no—
342C2×C7—21no—
343C2×C7—21no—
344C2×C7—21no—
345C2×C7—21no—
346C2×C7#189011yescharacteristic
347C2×C7#189411yescharacteristic
348C2×C7#191011yescharacteristic
349D7—81no—
350C23—41no—
351C23—41no—
352C23—141no—
353C23—141no—
354C23—141no—
355C23—281no—
356C23—281no—
357C23D7×C22⋊C411yescharacteristic
358C23#644711yescharacteristic
359C23#647711yescharacteristic
360C23#647811yescharacteristic
361C23#648111yescharacteristic
362C23#648111yescharacteristic
363C23#650111yescharacteristic
364D4—281no—
365D4—281no—
366D4—281no—
367D4—281no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—282no—
388C2×C4—281no—
389C2×C4—281no—
390C2×C4—282no—
391C2×C4—282no—
392C2×C4—282no—
393C2×C4—561no—
394C2×C4—561no—
395C2×C4—561no—
396C2×C4—561no—
397C2×C4(C22⋊D7)⋊C411yescharacteristic
398C2×C4#644611yescharacteristic
399C2×C4C4×C22⋊D711yescharacteristic
400C2×C4#650011yescharacteristic
401C8—42no—
402C8—282no—
403C7#947311yescharacteristic, complemented, Sylow
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—282no—
415C22—281no—
416C22—281no—
417C22—281no—
418C22—561no—
419C22#8957411yescharacteristic
420C22#8957511yescharacteristic
421C22#8958411yescharacteristic
422C22#8958811yescharacteristic
423C22#8972811yescharacteristic
424C22#8994611yescharacteristic
425C22#8996111yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—161no—
431C4—281no—
432C4—281no—
433C4—281no—
434C4—562no—
435C4—1121no—
436C4#8982512yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—561no—
444C2#1172729211yescharacteristic
445C2#1172802511yescharacteristic
446C2#1172808411yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1824C2
socleC22×C7#386
prime radicalC22×C22⋊D7D4
cosocleC23×C4×C7C23
hypercenter#1824D7
hypocenterC7#9473
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
3. C7×#1824C2semidirect
4. #11729308C2semidirect
6. #11725482C2semidirect
7. #11725431C2semidirect
18. #89764C4semidirect
20. #89754C4semidirect
75. #1824D7semidirect
403. C7#9473semidirect

Maximal subgroups

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

subgroupindex
#117396422
#117360472
C7×#18242
#117293082
#117272842
#117254822
#117254312
#94737