Finite Group Databasebeta

← #13242372

Subgroups — #13242372

Order 1792.

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

normal subgroups only

This group has 500 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#11733733C211yesmaximal, characteristic
2#11726418C211yesmaximal, characteristic
3#11726354C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic, complemented
5#11725444C211yesmaximal, characteristic, complemented
6#11725284C211yesmaximal, characteristic, complemented
7#11725283C211yesmaximal, characteristic, complemented
8C2×#6459—21no—
9C2×#6451C411yescharacteristic
10C2×C42×D7C2211yescharacteristic
11C7×#389C2211yescharacteristic
12C2×(C2×D7)⋊C8—21no—
13C2×(C2×D7)⋊C8—21no—
14C2×#6389—21no—
15#89759C411yescharacteristic
16#89759C411yescharacteristic
17#89748C411yescharacteristic
18#89744—21no—
19#89700C2211yescharacteristic
20#89439—21no—
21#89418—21no—
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7708—71nomaximal, Sylow
28C22×C4×D7C2×C411yescharacteristic
29C22×C4×D7D411yescharacteristic
30C22×C4×D7D411yescharacteristic
31C22×Dic14—21no—
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⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×C4×Dic7C2×C411yescharacteristic
47C2×C4×Dic7D411yescharacteristic
48C2×C4×Dic7D411yescharacteristic
49C22×C7⋊C8—21no—
50C22×C7⋊C8—21no—
51#6459—42no—
52#6459—42no—
53#6451—42no—
54C42×D7—42no—
55C7×#95—21no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59(C2×D7)⋊C8—24no—
60(C2×D7)⋊C8—42no—
61#6389—42no—
62#2857—71no—
63#1817—71no—
64#1749—71no—
65#1271—71no—
66#1271—71no—
67#1271—71no—
68#1271D711yescharacteristic, complemented
69C23×D7C22⋊C411yescharacteristic
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×Dic7—21no—
76C22×Dic7—21no—
77C22×Dic7C22⋊C411yescharacteristic
78C22×Dic7C22⋊C411yescharacteristic
79C22×Dic7C22⋊C411yescharacteristic
80C2×C4×D7—22no—
81C2×C4×D7—21no—
82C2×C4×D7—21no—
83C2×C4×D7—21no—
84C2×C4×D7—21no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7—22no—
88C2×C4×D7—22no—
89C2×C4×D7—41no—
90C2×C4×D7—41no—
91C2×C4×D7—41no—
92C2×Dic14—42no—
93C2×Dic14—42no—
94C2×Dic14—42no—
95C2×C7×C8—22no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C42×C7—41no—
107(C2×D7)⋊C4—42no—
108(C2×D7)⋊C4—42no—
109C4⋊Dic7—21no—
110C4⋊Dic7—21no—
111C4⋊Dic7—41no—
112Dic7⋊C4—42no—
113Dic7⋊C4—42no—
114Dic7⋊C4—41no—
115Dic7⋊C4—41no—
116Dic7⋊C4—42no—
117C4×Dic7—22no—
118C4×Dic7—22no—
119C4×Dic7—41no—
120C2×C7⋊C8—22no—
121C2×C7⋊C8—41no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8—41no—
126C2×C4⋊Q8—71no—
127C2×C22⋊Q8—141no—
128C22×C42—71no—
129C2×C22⋊C8—141no—
130C2×C22⋊C8—141no—
131C2×#97—141no—
132#391—141no—
133#389—71no—
134#389—141no—
135#389C2×D711yescharacteristic
136#384—71no—
137#380—71no—
138#380—71no—
139#380—141no—
140#336—71no—
141#336—71no—
142#336—141no—
143#336C2×D711yescharacteristic
144#336C2×D711yescharacteristic
145C23×C7C2×C22⋊C411yescharacteristic
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7#10112yes—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7#13111yescharacteristic
166C2×C4×C7#13111yescharacteristic
167C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C7#13711yescharacteristic
170C2×C4×C7#13711yescharacteristic
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7—41no—
183C2×Dic7—41no—
184C2×Dic7#9912yes—
185C2×Dic7#10412yes—
186C4×D7—42no—
187C4×D7—42no—
188C4×D7—42no—
189C4×D7—42no—
190C4×D7—42no—
191C4×D7—42no—
192Dic14—84no—
193C7×C8—42no—
194C7×C8—81no—
195C7⋊C8—42no—
196C7⋊C8—81no—
197C22×Q8—141no—
198C23×C4—71no—
199C23×C4—71no—
200C23×C4—71no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—141no—
204C22×C8—141no—
205C4⋊Q8—282no—
206C22⋊Q8—282no—
207C22⋊Q8—282no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—71no—
210C2×C4⋊C4—71no—
211C2×C4⋊C4—141no—
212C2×C4⋊C4—141no—
213C2×C4⋊C4—141no—
214C2×C4⋊C4—141no—
215C2×C4⋊C4C4×D711yescharacteristic
216C2×C4⋊C4C4×D711yescharacteristic
217C2×C22⋊C4—141no—
218C2×C22⋊C4—141no—
219C2×C42—71no—
220C2×C42—71no—
221C2×C42—71no—
222C2×C42—282no—
223C2×C42C22×D711yescharacteristic
224C22⋊C8—144no—
225C22⋊C8—282no—
226#97—282no—
227#95—21no—
228#95—141no—
229#95—141no—
230#95—141no—
231C22×C7#40811yescharacteristic
232C22×C7C2×#9911yescharacteristic
233C22×C7C2×#10111yescharacteristic
234C22×C7#41811yescharacteristic
235C22×C7#41911yescharacteristic
236C22×C7C2×#10411yescharacteristic
237C22×C7#42111yescharacteristic
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—22no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—81no—
252C4×C7—81no—
253Dic7—22no—
254Dic7—22no—
255Dic7—41no—
256Dic7—81no—
257Dic7—81no—
258C24—71no—
259C2×Q8—282no—
260C2×Q8—282no—
261C2×Q8—282no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4—141no—
273C22×C4—141no—
274C22×C4—141no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4—141no—
278C22×C4—281no—
279C22×C4—281no—
280C22×C4—281no—
281C22×C4C2×C4×D711yescharacteristic
282C22×C4D4×D711yescharacteristic
283C22×C4D4×D711yescharacteristic
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—142no—
291C2×C8—281no—
292C2×C8—281no—
293C2×C8—281no—
294C2×C8—281no—
295C2×C8—281no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—141no—
302C4⋊C4—141no—
303C4⋊C4—281no—
304C4⋊C4—282no—
305C4⋊C4—282no—
306C4⋊C4—282no—
307C4⋊C4—281no—
308C4⋊C4—281no—
309C22⋊C4—282no—
310C22⋊C4—282no—
311C42—41no—
312C42—142no—
313C42—142no—
314C42—281no—
315C2×C7#147211yescharacteristic
316C2×C7#148411yescharacteristic
317C2×C7#149711yescharacteristic
318C2×C7#150811yescharacteristic
319C2×C7#151011yescharacteristic
320C2×C7#153111yescharacteristic
321C2×C7#155611yescharacteristic
322D7—42no—
323C23—72no—
324C23—141no—
325C23—141no—
326C23—141no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23D7×C22⋊C411yescharacteristic
331Q8—564no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—72no—
345C2×C4—72no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—282no—
354C2×C4—282no—
355C2×C4—282no—
356C2×C4—282no—
357C2×C4—282no—
358C2×C4—282no—
359C2×C4—281no—
360C2×C4—281no—
361C2×C4—281no—
362C2×C4—281no—
363C2×C4—281no—
364C2×C4—281no—
365C2×C4#646411yescharacteristic
366C2×C4#647011yescharacteristic
367C2×C4#647411yescharacteristic
368C2×C4#647711yescharacteristic
369C2×C4#647811yescharacteristic
370C2×C4#648111yescharacteristic
371C8—42no—
372C8—81no—
373C8—282no—
374C8—561no—
375C7#770811yescharacteristic, complemented, Sylow
376C22—142no—
377C22—142no—
378C22—142no—
379C22—281no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22#8949911yescharacteristic
384C22D7×#9911yescharacteristic
385C22D7×#10111yescharacteristic
386C22#8954211yescharacteristic
387C22#8957411yescharacteristic
388C22D7×#10411yescharacteristic
389C22#8959311yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—142no—
399C4—142no—
400C4—281no—
401C4—561no—
402C4—561no—
403C2—282no—
404C2#1172590111yescharacteristic
405C2#1172592511yescharacteristic
406C2#1172599711yescharacteristic
407C2#1172605811yescharacteristic
408C2#1172608911yescharacteristic
409C2#1172611911yescharacteristic
410C2#1172618211yescharacteristic
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#7708
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. #11725284C2semidirect
7. #11725283C2semidirect
68. #1271D7semidirect
375. C7#7708semidirect

Maximal subgroups

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

subgroupindex
#117337332
#117264182
#117263542
C7×#12712
#117254442
#117252842
#117252832
#77087