Finite Group Databasebeta

← #13242229

Subgroups — #13242229

Order 1792.

458 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 97 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#11734051C211yesmaximal, characteristic
2#11726356C211yesmaximal, characteristic, complemented
3#11726351C211yesmaximal, characteristic, complemented
4C7×#1272C211yesmaximal, characteristic, complemented
5#11725448C211yesmaximal, characteristic, complemented
6#11725285C211yesmaximal, characteristic
7#11725285C211yesmaximal, characteristic
8C2×#6510C411yescharacteristic
9C2×#6453C411yescharacteristic
10C7×#391C2211yescharacteristic
11#90002C411yescharacteristic
12C2×#6389—21no—
13C2×Dic7⋊C8—21no—
14C2×C8×Dic7—21no—
15#89760C411yescharacteristic
16#89711C2211yescharacteristic
17C42×Dic7C2211yescharacteristic
18#89442—21no—
19#89442—21no—
20#89441—21no—
21#89441—21no—
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7676—71nomaximal, Sylow
28C22×C7×Q8C2×C411yescharacteristic
29C22×D28C2×C411yescharacteristic
30C22×C7×C8—21no—
31C22×C7×C8—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C42×C7C2311yescharacteristic
34#6510—42no—
35C2×(C2×D7)⋊C4—21no—
36C2×(C2×D7)⋊C4—21no—
37C2×(C2×D7)⋊C4—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7C2×C411yescharacteristic
43C2×Dic7⋊C4—21no—
44C2×C4×Dic7C2×C411yescharacteristic
45C2×C4×Dic7C2×C411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47C2×C4×Dic7D411yescharacteristic
48C2×C4×Dic7D411yescharacteristic
49C2×C4×Dic7D411yescharacteristic
50C22×C7⋊C8—21no—
51C22×C7⋊C8—21no—
52#6453—42no—
53C7×#95—21no—
54C7×#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57#6389—24no—
58Dic7⋊C8—24no—
59C8×Dic7—42no—
60#2861—71no—
61C4×C4⋊C8—71no—
62#1745—71no—
63#1272—71no—
64#1272D711yescharacteristic, complemented
65#1270—71no—
66#1270—71no—
67C23×D7—21no—
68C2×C7×Q8—21no—
69C2×C7×Q8—21no—
70C2×C7×Q8—41no—
71C2×C7×Q8—41no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—21no—
78C22×Dic7C22⋊C411yescharacteristic
79C22×Dic7C22⋊C411yescharacteristic
80C22×Dic7C22⋊C411yescharacteristic
81C22×Dic7C22⋊C411yescharacteristic
82C2×D28—21no—
83C2×D28—21no—
84C2×D28—41no—
85C2×D28—41no—
86C2×C7×C8—22no—
87C2×C7×C8—22no—
88C2×C7×C8—41no—
89C2×C7×C8—41no—
90C2×C7×C8—41no—
91C2×C7×C8—41no—
92C7×C4⋊C4—41no—
93C42×C7—41no—
94(C2×D7)⋊C4—42no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4—41no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4—42no—
100(C2×D7)⋊C4—42no—
101(C2×D7)⋊C4—42no—
102C4⋊Dic7—42no—
103C4⋊Dic7—41no—
104Dic7⋊C4—42no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109C4×Dic7—21no—
110C4×Dic7—21no—
111C4×Dic7—22no—
112C4×Dic7—22no—
113C4×Dic7—22no—
114C2×C7⋊C8—22no—
115C2×C7⋊C8—22no—
116C2×C7⋊C8—41no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119C2×C7⋊C8—41no—
120C2×#125—71no—
121C2×#124—71no—
122C2×C4⋊C8—141no—
123C2×#97—141no—
124C2×C4×C8—141no—
125#391—71no—
126#391C2×D711yescharacteristic
127#390—71no—
128#390—71no—
129#388—141no—
130#388—141no—
131#388—141no—
132#388—141no—
133C43—71no—
134#336—71no—
135#336—71no—
136#336—141no—
137#336C2×D711yescharacteristic
138#336C2×D711yescharacteristic
139C23×C7C2×C22⋊C411yescharacteristic
140C22×D7—41no—
141C22×D7—41no—
142C22×D7—41no—
143C22×D7—41no—
144C22×D7—41no—
145C22×D7—41no—
146C22×D7—41no—
147C7×Q8—81no—
148C7×Q8—81no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7#13111yescharacteristic
162C2×C4×C7#13111yescharacteristic
163C2×C4×C7#13511yescharacteristic
164C2×C4×C7#13511yescharacteristic
165C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C7#13711yescharacteristic
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7#10012yes—
180C2×Dic7#10412yes—
181C2×Dic7#10412yes—
182D28—81no—
183D28—81no—
184C7×C8—42no—
185C7×C8—42no—
186C7⋊C8—42no—
187C7⋊C8—42no—
188C22×Q8C4×D711yescharacteristic
189C22×D4—71no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—141no—
193C22×C8—141no—
194#125—282no—
195#124—282no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—141no—
199C2×C4⋊C4C4×D711yescharacteristic
200C2×C22⋊C4—141no—
201C2×C22⋊C4—141no—
202C2×C22⋊C4—141no—
203C2×C22⋊C4—141no—
204C2×C22⋊C4—141no—
205C2×C22⋊C4—141no—
206C2×C42—71no—
207C2×C42—71no—
208C2×C42—71no—
209C2×C42—71no—
210C2×C42—71no—
211C2×C42—71no—
212C2×C42C22×D711yescharacteristic
213C4⋊C8—144no—
214#97—144no—
215C4×C8—282no—
216#95—21no—
217#95—21no—
218#95—141no—
219#95—141no—
220C22×C7#40811yescharacteristic
221C22×C7#41011yescharacteristic
222C22×C7C2×#10011yescharacteristic
223C22×C7#41611yescharacteristic
224C22×C7#41711yescharacteristic
225C22×C7C2×#10411yescharacteristic
226C22×C7C2×#10411yescharacteristic
227C2×D7—81no—
228C2×D7—81no—
229C2×D7—81no—
230C2×D7—81no—
231C2×D7—81no—
232C2×D7—81no—
233C2×D7—81no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—21no—
240C4×C7—81no—
241C4×C7—81no—
242Dic7—22no—
243Dic7—22no—
244Dic7—22no—
245Dic7—41no—
246Dic7—81no—
247C24—141no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—281no—
255C2×D4—281no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—141no—
263C22×C4C2×C4×D711yescharacteristic
264C22×C4D4×D711yescharacteristic
265C22×C4D4×D711yescharacteristic
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—142no—
273C2×C8—142no—
274C2×C8—281no—
275C2×C8—281no—
276C2×C8—281no—
277C2×C8—281no—
278C4⋊C4—41no—
279C4⋊C4—281no—
280C4⋊C4—282no—
281C4⋊C4—282no—
282C22⋊C4—282no—
283C22⋊C4—282no—
284C22⋊C4—282no—
285C22⋊C4—282no—
286C22⋊C4—281no—
287C22⋊C4—281no—
288C22⋊C4—281no—
289C22⋊C4—281no—
290C42—41no—
291C42—142no—
292C42—142no—
293C42—142no—
294C42—141no—
295C42—141no—
296C42—141no—
297C42—141no—
298C42—141no—
299C42—141no—
300C2×C7#146911yescharacteristic
301C2×C7#148311yescharacteristic
302C2×C7#149411yescharacteristic
303C2×C7#150011yescharacteristic
304C2×C7#150911yescharacteristic
305C2×C7#152511yescharacteristic
306C2×C7#155111yescharacteristic
307D7—161no—
308C23—281no—
309C23—281no—
310C23—281no—
311C23—281no—
312C23—281no—
313C23—281no—
314C23—281no—
315C23D7×C22⋊C411yescharacteristic
316Q8—81no—
317Q8—81no—
318D4—561no—
319D4—561no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—72no—
333C2×C4—72no—
334C2×C4—72no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—281no—
345C2×C4—281no—
346C2×C4—281no—
347C2×C4#646411yescharacteristic
348C2×C4#647011yescharacteristic
349C2×C4#647711yescharacteristic
350C2×C4#647911yescharacteristic
351C2×C4#648111yescharacteristic
352C2×C4#648211yescharacteristic
353C8—42no—
354C8—42no—
355C8—282no—
356C8—282no—
357C7#767611yescharacteristic, complemented, Sylow
358C22—561no—
359C22—561no—
360C22—561no—
361C22—561no—
362C22—561no—
363C22—561no—
364C22—561no—
365C22#8949911yescharacteristic
366C22#8951211yescharacteristic
367C22D7×#10011yescharacteristic
368C22#8957511yescharacteristic
369C22#8957611yescharacteristic
370C22D7×#10411yescharacteristic
371C22D7×#10411yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—81no—
379C4—81no—
380C4—142no—
381C4—142no—
382C4—142no—
383C4—281no—
384C4—561no—
385C2—1121no—
386C2#1172585811yescharacteristic
387C2#1172598811yescharacteristic
388C2#1172599511yescharacteristic
389C2#1172606211yescharacteristic
390C2#1172606511yescharacteristic
391C2#1172609611yescharacteristic
392C2#1172623911yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1272C2
socleC23×C7C2×C22⋊C4
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#1272D7
hypocenterC7#7676
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #11726356C2semidirect
3. #11726351C2semidirect
4. C7×#1272C2semidirect
5. #11725448C2semidirect
64. #1272D7semidirect
357. C7#7676semidirect

Maximal subgroups

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

subgroupindex
#117340512
#117263562
#117263512
C7×#12722
#117254482
#117252852
#76767