Finite Group Databasebeta

← #13264825

Subgroups — #13264825

Order 1792.

522 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2543C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic, complemented
3#11729355C211yesmaximal, characteristic, complemented
4#11725455C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C2×C7×#121—21no—
7C2×C7×#121C411yescharacteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C7×#398C411yescharacteristic, complemented
10C7×#397C2211yescharacteristic
11C7×#394C411yescharacteristic, complemented
12C7×#392C2211yescharacteristic
13C7×#388—21no—
14C7×#386—21no—
15C7×#386—21no—
16C2×C7×#95C411yescharacteristic, complemented
17C2×#6408—21no—
18#89984—21no—
19#89697—21no—
20#89321C2212yes—
21#89320—21no—
22#89320C2212yes—
23#11876—71nomaximal, Sylow
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C7×#121—42no—
32C7×#121—42no—
33C7×#121—41no—
34C7×#121—41no—
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
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×C7D411yescharacteristic
48C2×C22⋊Dic7—22no—
49C2×C4⋊Dic7—41no—
50C7×#95—21no—
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—
59C4×C7⋊C8—41no—
60#2543D711yescharacteristic
61#1799—71no—
62#1460—71no—
63#1344—72no—
64#1343—72no—
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4—42no—
71C2×C7×D4—41no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—42no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—81no—
78C2×C7×D4—81no—
79C2×C7×D4D812yes—
80C2×C7×D4SD1612yes—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—42no—
89C22×C4×C7—42no—
90C22×C4×C7—81no—
91C22×C4×C7—81no—
92C22×C4×C7—81no—
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×Dic7—42no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—42no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—81no—
111C42×C7—21no—
112C42×C7—21no—
113C22⋊Dic7—42no—
114C22⋊Dic7—42no—
115C4⋊Dic7—82no—
116C2×C7⋊C8—42no—
117C2×#121—21no—
118C2×#121Dic711yescharacteristic, complemented
119C2×C4×D4C2×D711yescharacteristic
120#431—141no—
121C2×#99—141no—
122#398Dic711yescharacteristic, complemented
123#397C2×D711yescharacteristic
124#394Dic711yescharacteristic, complemented
125#392C2×D711yescharacteristic
126#388—21no—
127#386—21no—
128#386—21no—
129#380—141no—
130C2×#95Dic711yescharacteristic, complemented
131#342—72no—
132#323—72no—
133#323—141no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7#10212yes—
150C23×C7C2×C22⋊C411yescharacteristic
151C23×C7C2×D811yescharacteristic
152C23×C7C2×SD1611yescharacteristic
153C7×D4—42no—
154C7×D4—42no—
155C7×D4—82no—
156C7×D4—82no—
157C7×D4—82no—
158C7×D4—84no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—22no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—41no—
166C2×C4×C7—42no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7—84no—
182C2×C4×C7#9911yescharacteristic
183C2×C4×C7#9911yescharacteristic
184C2×C4×C7#10212yes—
185C2×Dic7—82no—
186C2×Dic7—82no—
187C2×Dic7—82no—
188C7⋊C8—82no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C2×Dic711yescharacteristic
192C22×D4C22⋊D711yescharacteristic
193C23×C4—21no—
194C23×C4C2×Dic711yescharacteristic
195C23×C4C22⋊D711yescharacteristic
196#121—42no—
197#121—42no—
198#121—41no—
199#121—41no—
200C4×D4—41no—
201C4×D4—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—281no—
204C2×C4⋊C4C2×Dic711yescharacteristic
205C2×C4⋊C4C2×Dic711yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—142no—
212C2×C22⋊C4C22⋊D711yescharacteristic
213C2×C22⋊C4C22×D711yescharacteristic
214C2×C42C22⋊D711yescharacteristic
215C4⋊C8—281no—
216#99—282no—
217C22⋊C8—142no—
218C4×C8—281no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—281no—
223#95C2×Dic711yescharacteristic
224#95C2×Dic711yescharacteristic
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—22no—
229C22×C7—21no—
230C22×C7—42no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—42no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7C2×#9911yescharacteristic
247C22×C7C2×#10211yescharacteristic
248C22×C7#42111yescharacteristic
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—22no—
252C4×C7—42no—
253C4×C7—41no—
254C4×C7—41no—
255C4×C7—41no—
256C4×C7—81no—
257C4×C7—81no—
258Dic7—162no—
259C24—21no—
260C24C22⋊Dic711yescharacteristic
261C24C2×C22⋊D711yescharacteristic
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4#114112yes—
274C2×D4#114212yes—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—282no—
288C22×C4C22⋊Dic711yescharacteristic
289C22×C4C22⋊Dic711yescharacteristic
290C22×C4C22⋊Dic711yescharacteristic
291C22×C4C22×Dic711yescharacteristic
292C22×C4C2×C22⋊D711yescharacteristic
293C2×C8—282no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—562no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—282no—
308C22⋊C4—282no—
309C42—21no—
310C42—21no—
311C2×C7—21no—
312C2×C7—21no—
313C2×C7—41no—
314C2×C7—42no—
315C2×C7—41no—
316C2×C7—161no—
317C2×C7#151911yescharacteristic
318C2×C7#212111yescharacteristic
319C2×C7#212611yescharacteristic
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23#640612yes—
336C23C2×#114111yescharacteristic
337C23C2×#114211yescharacteristic
338C23C2×C22⋊Dic711yescharacteristic
339D4—42no—
340D4—42no—
341D4—82no—
342D4—82no—
343D4—82no—
344D4—84no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—84no—
368C2×C4—562no—
369C2×C4—562no—
370C2×C4—562no—
371C2×C4#640612yes—
372C2×C4#640811yescharacteristic
373C2×C4#640811yescharacteristic
374C8—562no—
375C7#1187611yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×#640611yescharacteristic
398C22C2×#640811yescharacteristic
399C22#9000811yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4—81no—
409C4—1122no—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—161no—
416C2#1172942911yescharacteristic
417C2#1172957311yescharacteristic
418C2#1172960111yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2543C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2543D7
hypocenterC7#11876
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

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

NH ≅ G/Ntype
2. #11729373C2semidirect
3. #11729355C2semidirect
4. #11725455C2semidirect
5. #11725451C2semidirect
7. C2×C7×#121C4semidirect
9. C7×#398C4semidirect
11. C7×#394C4semidirect
16. C2×C7×#95C4semidirect
118. C2×#121Dic7semidirect
122. #398Dic7semidirect
124. #394Dic7semidirect
130. C2×#95Dic7semidirect
375. C7#11876semidirect

Maximal subgroups

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

subgroupindex
C7×#25432
#117293732
#117293552
#117254552
#117254512
#118767