Finite Group Databasebeta

← #13242049

Subgroups — #13242049

Order 1792.

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

normal subgroups only

This group has 556 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
1#11733410C211yesmaximal, characteristic
2#11726414C211yesmaximal, characteristic
3#11726354C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic, complemented
5#11725444C211yesmaximal, characteristic, complemented
6#11725286C211yesmaximal, characteristic, complemented
7#11725286C211yesmaximal, characteristic, complemented
8C22×#1131—21no—
9C2×#6460C411yescharacteristic
10C2×#6450C411yescharacteristic
11C2×C42×D7C2211yescharacteristic
12C7×#389C2211yescharacteristic
13C2×(C2×D7)⋊C8—21no—
14C2×#6389—21no—
15#89761—21no—
16#89759C411yescharacteristic
17C4⋊C4×Dic7—21no—
18C4⋊C4×Dic7C411yescharacteristic
19#89701C2211yescharacteristic
20#89438—21no—
21#89426—21no—
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7634—71nomaximal, Sylow
28C22×C4×D7C2×C411yescharacteristic
29C22×C4×D7D411yescharacteristic
30C22×C4×D7D411yescharacteristic
31C22×Dic14C2×C411yescharacteristic
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C42×C7C2311yescharacteristic
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×C4⋊Dic7—21no—
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×C4×Dic7—21no—
44C2×C4×Dic7—21no—
45C2×C4×Dic7C2×C411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47C2×C4×Dic7D411yescharacteristic
48C22×C7⋊C8—21no—
49C22×C7⋊C8—21no—
50C2×#1131—24no—
51C2×#1131—42no—
52C2×#1131—42no—
53#6460—22no—
54#6460—22no—
55#6450—42no—
56C42×D7—42no—
57C7×#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—21no—
62(C2×D7)⋊C8—24no—
63#6389—42no—
64C4×C22⋊Q8—71no—
65#1749—71no—
66#1728—71no—
67#1272—71no—
68#1272—71no—
69#1271—71no—
70#1271D711yescharacteristic, complemented
71C23×D7C22⋊C411yescharacteristic
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7C22⋊C411yescharacteristic
80C22×Dic7C22⋊C411yescharacteristic
81C22×Dic7C22⋊C411yescharacteristic
82C2×C4×D7—22no—
83C2×C4×D7—21no—
84C2×C4×D7—21no—
85C2×C4×D7—22no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7—22no—
91C2×C4×D7—41no—
92C2×C4×D7—41no—
93C2×C4×D7—41no—
94C2×Dic14—21no—
95C2×Dic14—41no—
96C2×Dic14—41no—
97C2×Dic14M1612yescomplemented
98C2×C7×C8—22no—
99C2×C7×C8—22no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4M1612yescomplemented
109C42×C7—41no—
110(C2×D7)⋊C4—24no—
111(C2×D7)⋊C4—42no—
112C4⋊Dic7—24no—
113C4⋊Dic7—41no—
114Dic7⋊C4—42no—
115C4×Dic7—22no—
116C4×Dic7—22no—
117C4×Dic7—41no—
118C4×Dic7—42no—
119C4×Dic7—42no—
120C2×C7⋊C8—22no—
121C2×C7⋊C8—22no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8—41no—
126#1131—48no—
127C22×M16—141no—
128C2×C22⋊Q8—71no—
129C2×C4×Q8—71no—
130C22×C42—71no—
131C2×C22⋊C8—141no—
132C2×#97—141no—
133#391—71no—
134#389C2×D711yescharacteristic
135#387—141no—
136#385—141no—
137#380—71no—
138C4×C4⋊C4—71no—
139C4×C4⋊C4—141no—
140C4×C22⋊C4—141no—
141#336—71no—
142#336—71no—
143#336—141no—
144#336C2×D711yescharacteristic
145#336C2×D711yescharacteristic
146C23×C7C2×C22⋊C411yescharacteristic
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7#10412yes—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—22no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7C2×M1611yescharacteristic
167C2×C4×C7C2×M1611yescharacteristic
168C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C7#13711yescharacteristic
170C2×C4×C7#13711yescharacteristic
171C2×C4×C7#13711yescharacteristic
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—22no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7—41no—
183C2×Dic7—41no—
184C2×Dic7—41no—
185C2×Dic7#9912yes—
186C2×Dic7#10412yes—
187C4×D7—24no—
188C4×D7—24no—
189C4×D7—42no—
190C4×D7—42no—
191C4×D7—42no—
192C4×D7—42no—
193Dic14—42no—
194Dic14—42no—
195C7×C8—42no—
196C7×C8—42no—
197C7⋊C8—42no—
198C7⋊C8—42no—
199C22×Q8—71no—
200C23×C4—71no—
201C23×C4—71no—
202C23×C4—71no—
203C2×M16—144no—
204C2×M16—282no—
205C2×M16—282no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—141no—
209C22×C8—141no—
210C22⋊Q8—142no—
211C22⋊Q8—142no—
212C4×Q8—282no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—71no—
215C2×C4⋊C4—141no—
216C2×C4⋊C4—141no—
217C2×C4⋊C4C4×D711yescharacteristic
218C2×C4⋊C4C4×D711yescharacteristic
219C2×C22⋊C4—141no—
220C2×C22⋊C4—141no—
221C2×C42—71no—
222C2×C42—71no—
223C2×C42—71no—
224C2×C42—141no—
225C2×C42—141no—
226C2×C42—282no—
227C2×C42C22×D711yescharacteristic
228C22⋊C8—144no—
229#97—282no—
230#95—21no—
231#95—141no—
232#95—141no—
233#95—141no—
234#95—141no—
235C22×C7#40711yescharacteristic
236C22×C7C2×#9911yescharacteristic
237C22×C7#41311yescharacteristic
238C22×C7#41711yescharacteristic
239C22×C7#41911yescharacteristic
240C22×C7C2×#10411yescharacteristic
241C22×C7C2×#10411yescharacteristic
242C2×D7—22no—
243C2×D7—22no—
244C2×D7—22no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—42no—
256C4×C7—81no—
257Dic7—22no—
258Dic7—22no—
259Dic7—42no—
260Dic7—41no—
261Dic7—81no—
262C24—71no—
263C2×Q8—72no—
264C2×Q8—141no—
265C2×Q8—281no—
266C2×Q8—281no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—142no—
273C22×C4—142no—
274C22×C4—142no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4—141no—
278C22×C4—141no—
279C22×C4—141no—
280C22×C4—141no—
281C22×C4—141no—
282C22×C4—141no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4—281no—
286C22×C4C2×C4×D711yescharacteristic
287C22×C4D4×D711yescharacteristic
288C22×C4D4×D711yescharacteristic
289M16—288no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—142no—
297C2×C8—142no—
298C2×C8—281no—
299C2×C8—281no—
300C2×C8—281no—
301C2×C8—281no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—144no—
307C4⋊C4—281no—
308C4⋊C4—282no—
309C4⋊C4#113112yescomplemented
310C22⋊C4—144no—
311C22⋊C4—282no—
312C42—41no—
313C42—142no—
314C42—142no—
315C42—281no—
316C42—282no—
317C42—282no—
318C2×C7#146511yescharacteristic
319C2×C7#148911yescharacteristic
320C2×C7#149111yescharacteristic
321C2×C7#150411yescharacteristic
322C2×C7#150811yescharacteristic
323C2×C7#152411yescharacteristic
324C2×C7#155911yescharacteristic
325D7—42no—
326C23—72no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23—141no—
333C23D7×C22⋊C411yescharacteristic
334Q8—282no—
335Q8—282no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—72no—
349C2×C4—72no—
350C2×C4—144no—
351C2×C4—144no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—142no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—282no—
361C2×C4—282no—
362C2×C4—282no—
363C2×C4—282no—
364C2×C4—281no—
365C2×C4—281no—
366C2×C4—281no—
367C2×C4—281no—
368C2×C4—281no—
369C2×C4C2×#113111yescharacteristic
370C2×C4D7×M1611yescharacteristic
371C2×C4#647411yescharacteristic
372C2×C4#647811yescharacteristic
373C2×C4#647811yescharacteristic
374C2×C4#648111yescharacteristic
375C8—42no—
376C8—42no—
377C8—282no—
378C8—282no—
379C7#763411yescharacteristic, complemented, Sylow
380C22—142no—
381C22—142no—
382C22—142no—
383C22—281no—
384C22—281no—
385C22—281no—
386C22—281no—
387C22#8949811yescharacteristic
388C22D7×#9911yescharacteristic
389C22#8952311yescharacteristic
390C22#8954311yescharacteristic
391C22#8957411yescharacteristic
392C22D7×#10411yescharacteristic
393C22D7×#10411yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—142no—
403C4—142no—
404C4—281no—
405C4—282no—
406C4—561no—
407C2—282no—
408C2#1172589911yescharacteristic
409C2#1172592811yescharacteristic
410C2#1172599311yescharacteristic
411C2#1172605611yescharacteristic
412C2#1172608911yescharacteristic
413C2#1172611511yescharacteristic
414C2#1172618411yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1271C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1271D7
hypocenterC7#7634
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. C7×#1271C2semidirect
5. #11725444C2semidirect
6. #11725286C2semidirect
7. #11725286C2semidirect
70. #1271D7semidirect
97. C2×Dic14M16semidirect
108. C7×C4⋊C4M16semidirect
309. C4⋊C4#1131semidirect
379. C7#7634semidirect

Maximal subgroups

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

subgroupindex
#117334102
#117264142
#117263542
C7×#12712
#117254442
#117252862
#76347