Finite Group Databasebeta

← #13264557

Subgroups — #13264557

Order 1792.

626 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 91 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×#2388C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic, complemented
3#11729357C211yesmaximal, characteristic, complemented
4#11725452C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C2×C4×C7×D4—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#395C411yescharacteristic, complemented
10C7×#394—21no—
11C7×#394C411yescharacteristic, complemented
12C7×#393C2211yescharacteristic
13C7×#392—21no—
14C7×#392C2211yescharacteristic
15C4×C7×C22⋊C4—21no—
16C4×C7×C22⋊C4C411yescharacteristic, complemented
17C2×#6408—21no—
18#89984—21no—
19#89698—21no—
20#89321C2212yes—
21#89320—21no—
22#89320C2212yes—
23#11820—71nomaximal, Sylow
24C2×C7×#41—44no—
25C2×C7×#41—41no—
26C2×C7×#41—41no—
27C22×C7×Q8C2×C411yescharacteristic
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4C2×C411yescharacteristic
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7D411yescharacteristic
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C4×C7×D4—44no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4D411yescharacteristic
46C2×C7×C22⋊C4C2311yescharacteristic
47C2×C42×C7—21no—
48C2×C42×C7D411yescharacteristic
49C2×C22⋊Dic7—22no—
50C2×C4⋊Dic7—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95C2×C411yescharacteristic
54C7×#95C2×C411yescharacteristic
55#6408—42no—
56C7⋊#95—41no—
57C22⋊(C7⋊C8)—22no—
58C4⋊(C7⋊C8)—41no—
59#6377—41no—
60#2388D711yescharacteristic
61#1799—71no—
62#1466—71no—
63#1351—72no—
64#1343—72no—
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C7×#41—44no—
69C7×#41—44no—
70C7×#41—44no—
71C7×#41—44no—
72C2×C7×Q8—42no—
73C2×C7×Q8—41no—
74C2×C7×D4—21no—
75C2×C7×D4—21no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—42no—
87C22×C4×C7—24no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—42no—
96C22×C4×C7—42no—
97C22×C4×C7—42no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C22×C411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×Dic7—42no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—42no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C42×C7—21no—
116C42×C7—21no—
117C42×C7—42no—
118C22⋊Dic7—42no—
119C22⋊Dic7—42no—
120C4⋊Dic7—82no—
121C2×C7⋊C8—42no—
122C22×#41Dic711yescharacteristic, complemented
123C2×C4×D4—21no—
124C2×C4×D4C2×D711yescharacteristic
125#432—141no—
126C2×#99—141no—
127#395Dic711yescharacteristic, complemented
128#394—21no—
129#394Dic711yescharacteristic, complemented
130#393C2×D711yescharacteristic
131#392—21no—
132#392C2×D711yescharacteristic
133#380—141no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4Dic711yescharacteristic, complemented
136#342—72no—
137#323—72no—
138#323—141no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—22no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7C2×C22⋊C411yescharacteristic
156C23×C7AGL(1,ℤ/8)11yescharacteristic
157C23×C7#13711yescharacteristic
158C7×Q8—42no—
159C7×Q8—42no—
160C7×D4—42no—
161C7×D4—42no—
162C7×D4—42no—
163C7×D4—44no—
164C7×D4—44no—
165C7×D4—48no—
166C7×D4—42no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—44no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—44no—
184C2×C4×C7—42no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—81no—
192C2×C4×C7#9911yescharacteristic
193C2×C4×C7#9911yescharacteristic
194C2×Dic7—82no—
195C2×Dic7—82no—
196C2×Dic7—82no—
197C7⋊C8—82no—
198C2×#41—44no—
199C2×#41—41no—
200C2×#41—41no—
201C22×Q8C2×Dic711yescharacteristic
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C2×Dic711yescharacteristic
205C22×D4C22⋊D711yescharacteristic
206C23×C4—21no—
207C23×C4C2×Dic711yescharacteristic
208C23×C4C22⋊D711yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—44no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—281no—
214C2×C4⋊C4C2×Dic711yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—142no—
221C2×C22⋊C4C22⋊D711yescharacteristic
222C2×C22⋊C4C22×D711yescharacteristic
223C2×C42—21no—
224C2×C42C22⋊D711yescharacteristic
225C4⋊C8—281no—
226#99—282no—
227C22⋊C8—142no—
228#97—281no—
229#95—21no—
230#95—21no—
231#95—281no—
232#95C2×Dic711yescharacteristic
233#95C2×Dic711yescharacteristic
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—42no—
241C22×C7—42no—
242C22×C7—41no—
243C22×C7—41no—
244C22×C7—41no—
245C22×C7—42no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7C2×#9911yescharacteristic
256C22×C7#41711yescharacteristic
257C22×C7#42111yescharacteristic
258C4×C7—21no—
259C4×C7—42no—
260C4×C7—42no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7—41no—
264C4×C7—41no—
265C4×C7—81no—
266C4×C7AΣL(1,𝔽16)12yes—
267Dic7—162no—
268C24—21no—
269C24C22⋊Dic711yescharacteristic
270C24C2×C22⋊D711yescharacteristic
271#41—44no—
272#41—44no—
273#41—44no—
274#41—44no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—282no—
304C22×C4C22⋊Dic711yescharacteristic
305C22×C4C22⋊Dic711yescharacteristic
306C22×C4C22⋊Dic711yescharacteristic
307C22×C4C22×Dic711yescharacteristic
308C22×C4C2×C22⋊D711yescharacteristic
309C2×C8—282no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—562no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—282no—
321C22⋊C4—282no—
322C42—21no—
323C42—21no—
324C42—42no—
325C2×C7—21no—
326C2×C7—21no—
327C2×C7—41no—
328C2×C7—41no—
329C2×C7—81no—
330C2×C7—82no—
331C2×C7#151311yescharacteristic
332C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
333C2×C7#212611yescharacteristic
334C23—21no—
335C23—21no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23#651211yescharacteristic
351C23#651411yescharacteristic
352C23C2×C22⋊Dic711yescharacteristic
353Q8—42no—
354Q8—42no—
355D4—42no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—44no—
360D4—48no—
361D4—42no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—81no—
387C2×C4—562no—
388C2×C4—562no—
389C2×C4—562no—
390C2×C4#640811yescharacteristic
391C2×C4#640811yescharacteristic
392C8—562no—
393C7#1182011yescharacteristic, complemented, Sylow
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—41no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22C2×#640811yescharacteristic
416C22#9000411yescharacteristic
417C22#9000811yescharacteristic
418C4—21no—
419C4—42no—
420C4—42no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—1122no—
427C4#8933112yes—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—82no—
433C2—81no—
434C2C2×#8933111yescharacteristic
435C2#1172946311yescharacteristic
436C2#1172957311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2388C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2388D7
hypocenterC7#11820
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2388

Complements

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

NH ≅ G/Ntype
2. #11729373C2semidirect
3. #11729357C2semidirect
4. #11725452C2semidirect
5. #11725451C2semidirect
6. C22×C7×#41C4semidirect
9. C7×#395C4semidirect
11. C7×#394C4semidirect
16. C4×C7×C22⋊C4C4semidirect
122. C22×#41Dic7semidirect
127. #395Dic7semidirect
129. #394Dic7semidirect
135. C4×C22⋊C4Dic7semidirect
393. C7#11820semidirect

Maximal subgroups

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

subgroupindex
C7×#23882
#117293732
#117293572
#117254522
#117254512
#118207