Finite Group Databasebeta

← #13254251

Subgroups — #13254251

Order 1792.

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

normal subgroups only

This group has 610 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
1C7×#2502C211yesmaximal, characteristic, complemented
2#11740181C211yesmaximal, characteristic
3C2×#89740C211yesmaximal, characteristic
4#11729444C211yesmaximal, characteristic, complemented
5#11729308C211yesmaximal, characteristic, complemented
6#11727210C211yesmaximal, characteristic
7#11725423C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C2×#6493C2211yescharacteristic
11C7×#398—22no—
12C7×#391—21no—
13C7×#385—22no—
14C7×#384—21no—
15C4×C7×C22⋊C4C2211yescharacteristic
16#89999—21no—
17#89984C2211yescharacteristic
18C2×C22⋊(C7⋊C8)C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89820—22no—
21#89818—21no—
22#89815—21no—
23#89740—21no—
24#89740—22no—
25#89740—22no—
26C2×#6383—21no—
27C2×#6382—21no—
28C2×#6380—22no—
29#89333—22no—
30#89324—22no—
31#9951—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C23×C4×C7—21no—
34C23×C4×C7C2311yescharacteristic
35C22×C22⋊D7—21no—
36C22×D28D411yescharacteristic
37C7×C22⋊Q8—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—24no—
48C2×C7×C4⋊C4D412yes—
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4D412yes—
52C2×C42×C7—21no—
53C2×C22⋊Dic7—21no—
54C2×#1143—22no—
55#6493—21no—
56#6493—21no—
57#6493—21no—
58#6493—21no—
59C2×(C2×D7)⋊C4—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7D411yescharacteristic
62C22×C7⋊C8—21no—
63C22×C7⋊C8D412yes—
64C7×#95—24no—
65C7×#95—21no—
66#6409—22no—
67C7⋊#95—21no—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70C22⋊(C7⋊C8)—21no—
71C22⋊(C7⋊C8)D412yes—
72#6383—24no—
73#6383—22no—
74#6382—24no—
75#6382—22no—
76#6380—44no—
77#3187—71no—
78C2×#481—71no—
79#2502D711yescharacteristic, complemented
80#1964—71no—
81#1824—71no—
82#1520—71no—
83#1282—71no—
84C24×C7C2×D411yescharacteristic
85C23×D7—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C22×C4×C7—22no—
90C22×C4×C7—21no—
91C22×C4×C7—22no—
92C22×C4×C7—21no—
93C22×C4×C7—42no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7—82no—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7#4111yescharacteristic
103C2×C22⋊D7—41no—
104C2×C22⋊D7—41no—
105C2×C22⋊D7—42no—
106C2×C22⋊D7—42no—
107C22×Dic7—21no—
108C22×Dic7—41no—
109C2×D28—21no—
110C2×D28—21no—
111C2×D28—42no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—84no—
121C7×C22⋊C4—22no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4SD1614yescomplemented
125C42×C7—42no—
126C22⋊Dic7—42no—
127#1143—82no—
128(C2×D7)⋊C4—41no—
129(C2×D7)⋊C4—41no—
130C4⋊Dic7—21no—
131C4⋊Dic7—21no—
132C4⋊Dic7—42no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8—22no—
135C2×C7⋊C8—42no—
136C2×C7⋊C8—42no—
137C2×C22⋊Q8C2×D711yescharacteristic
138C2×#121—71no—
139C22×C4⋊C4C2×D711yescharacteristic
140#483—141no—
141#481—142no—
142#481—142no—
143#481—141no—
144#465—142no—
145#450—141no—
146#448—141no—
147C2×#106—141no—
148C2×#102—141no—
149C2×#99—142no—
150C2×C22⋊C8—71no—
151C2×C22⋊C8—71no—
152#398—22no—
153#391—21no—
154#385—22no—
155#384—21no—
156#380—71no—
157C4×C22⋊C4C2×D711yescharacteristic
158#340—142no—
159#328—142no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7#12011yescharacteristic
163C23×C7#12111yescharacteristic
164C23×C7#12711yescharacteristic
165C23×C7C2×SD1612yes—
166C23×C7#13712yes—
167C22×D7—41no—
168C22×D7—42no—
169C22×D7—81no—
170C22×D7—81no—
171C7×Q8—42no—
172C7×Q8—81no—
173C2×C4×C7—22no—
174C2×C4×C7—22no—
175C2×C4×C7—22no—
176C2×C4×C7—24no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—44no—
184C2×C4×C7—42no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—82no—
189C2×C4×C7—82no—
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C22⋊D7—82no—
195C22⋊D7—81no—
196C22⋊D7—81no—
197C2×Dic7—41no—
198C2×Dic7—42no—
199C2×Dic7—81no—
200C2×Dic7—81no—
201C2×Dic7—81no—
202D28—42no—
203D28—42no—
204C7⋊C8—42no—
205C7⋊C8—42no—
206C22×Q8C22⋊D711yescharacteristic
207C22×D4—71no—
208C22×D4—141no—
209C23×C4—21no—
210C23×C4C22×D711yescharacteristic
211C2×SD16—142no—
212C22×C8—72no—
213C22×C8—141no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218#121—141no—
219#121—141no—
220#121—141no—
221#121—141no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—71no—
230C2×C4⋊C4—141no—
231C2×C4⋊C4C22⋊D712yes—
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—141no—
235C2×C22⋊C4—141no—
236C2×C22⋊C4C22⋊D712yes—
237C2×C42—21no—
238#106—144no—
239#106—142no—
240#103—142no—
241#102—144no—
242#102—142no—
243#99—284no—
244C22⋊C8—72no—
245C22⋊C8—141no—
246C22⋊C8—141no—
247C22⋊C8—141no—
248#95—24no—
249#95—21no—
250#95—141no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—42no—
258C22×C7—42no—
259C22×C7#39211yescharacteristic
260C22×C7#44812yes—
261C22×C7#46112yes—
262C22×C7#49211yescharacteristic
263C22×C7#49711yescharacteristic
264C2×D7—81no—
265C2×D7—81no—
266C2×D7—81no—
267C2×D7—162no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—44no—
272C4×C7—81no—
273C4×C7—82no—
274C4×C7—81no—
275C4×C7#45812yes—
276Dic7—81no—
277Dic7—161no—
278C24—141no—
279C24C2×C22⋊D711yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×D4—141no—
284C2×D4—141no—
285C2×D4—282no—
286C2×D4—281no—
287C2×D4—282no—
288C2×D4—282no—
289C2×D4—281no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—82no—
298C22×C4—141no—
299C22×C4—281no—
300C22×C4D4×D711yescharacteristic
301C22×C4D4×D711yescharacteristic
302C22×C4D4×D711yescharacteristic
303C22×C4#115911yescharacteristic
304C22×C4C2×C22⋊D711yescharacteristic
305C22×C4C2×C22⋊D711yescharacteristic
306SD16—562no—
307C2×C8—142no—
308C2×C8—142no—
309C2×C8—282no—
310C2×C8—282no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—84no—
320C4⋊C4—141no—
321C4⋊C4—141no—
322C4⋊C4—282no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—281no—
327C22⋊C4—281no—
328C22⋊C4—282no—
329C22⋊C4#114314yescomplemented
330C42—42no—
331C2×C7—21no—
332C2×C7—22no—
333C2×C7—22no—
334C2×C7—21no—
335C2×C7#200811yescharacteristic
336C2×C7#201311yescharacteristic
337C2×C7#202211yescharacteristic
338D7—161no—
339C23—41no—
340C23—41no—
341C23—281no—
342C23—282no—
343C23—561no—
344C23—561no—
345C23#650111yescharacteristic
346C23#650311yescharacteristic
347C23C2×#114312yes—
348C23#650512yes—
349C23#651611yescharacteristic
350Q8—42no—
351Q8—81no—
352D4—282no—
353D4—282no—
354D4—561no—
355D4—561no—
356D4—562no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—281no—
375C2×C4—282no—
376C2×C4—561no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4#650011yescharacteristic
380C2×C4#650011yescharacteristic
381C2×C4#650211yescharacteristic
382C2×C4#650211yescharacteristic
383C8—282no—
384C8—282no—
385C7#995111yescharacteristic, complemented, Sylow
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—561no—
395C22—561no—
396C22—561no—
397C22—1122no—
398C22#8985512yes—
399C22#8985711yescharacteristic
400C22#8985811yescharacteristic
401C22#8999511yescharacteristic
402C22#8999912yes—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—44no—
407C4—81no—
408C4—82no—
409C4—81no—
410C4—561no—
411C4—1121no—
412C4#8982512yes—
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—1121no—
418C2#1172946011yescharacteristic
419C2#1172954511yescharacteristic
420C2#1172961211yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC7×#2502C2
socleC22×C7#392
prime radicalC2×#6493C22
hypercenter#2502D7
hypocenterC7#9951
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
1. C7×#2502C2semidirect
4. #11729444C2semidirect
5. #11729308C2semidirect
7. #11725423C2semidirect
79. #2502D7semidirect
124. C7×C22⋊C4SD16semidirect
329. C22⋊C4#1143semidirect
385. C7#9951semidirect

Maximal subgroups

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

subgroupindex
C7×#25022
#117401812
C2×#897402
#117294442
#117293082
#117272102
#117254232
#99517