Finite Group Databasebeta

← #13242004

Subgroups — #13242004

Order 1792.

506 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 101 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#11733924C211yesmaximal, characteristic
2#11726418C211yesmaximal, characteristic
3#11726354C211yesmaximal, characteristic
4C7×#1272C211yesmaximal, characteristic, complemented
5#11725448C211yesmaximal, characteristic, complemented
6#11725284C211yesmaximal, characteristic, complemented
7#11725283C211yesmaximal, characteristic, complemented
8C2×#6509C411yescharacteristic
9C2×Q8×Dic7C411yescharacteristic
10C2×C42×D7C2211yescharacteristic
11C7×#391C2211yescharacteristic
12C2×(C2×D7)⋊C8—21no—
13C2×(C2×D7)⋊C8—21no—
14C2×#6389—21no—
15#89759C411yescharacteristic
16C4⋊C4×Dic7C411yescharacteristic
17#89700C2211yescharacteristic
18#89440—21no—
19#89438—21no—
20#89427—21no—
21#89421—21no—
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7623—71nomaximal, Sylow
28C22×C7×Q8C2×C411yescharacteristic
29C22×C4×D7C2×C411yescharacteristic
30C22×C4×D7D411yescharacteristic
31C22×C4×D7D411yescharacteristic
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C42×C7C2311yescharacteristic
36#6509—22no—
37#6509—22no—
38Q8×Dic7—42no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7C2×C411yescharacteristic
43C2×C4⋊Dic7C2×C411yescharacteristic
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×C4×Dic7—21no—
47C2×C4×Dic7—21no—
48C2×C4×Dic7C2×C411yescharacteristic
49C2×C4×Dic7D411yescharacteristic
50C2×C4×Dic7D411yescharacteristic
51C22×C7⋊C8—21no—
52C22×C7⋊C8—21no—
53C42×D7—42no—
54C7×#95—21no—
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—
62C4×C22⋊Q8—71no—
63#1817—71no—
64#1749—71no—
65#1272—71no—
66#1272D711yescharacteristic, complemented
67#1271—71no—
68#1271—71no—
69C23×D7C22⋊C411yescharacteristic
70C2×C7×Q8—21no—
71C2×C7×Q8—41no—
72C2×C7×Q8—41no—
73C2×C7×Q8M1612yes—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×Dic7—21no—
80C22×Dic7—21no—
81C22×Dic7C22⋊C411yescharacteristic
82C22×Dic7C22⋊C411yescharacteristic
83C22×Dic7C22⋊C411yescharacteristic
84C2×C4×D7—22no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7—21no—
91C2×C4×D7—22no—
92C2×C4×D7—41no—
93C2×C4×D7—41no—
94C2×C4×D7—41no—
95C2×C4×D7—41no—
96C2×C7×C8—22no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C7×C4⋊C4—41no—
103C42×C7—41no—
104(C2×D7)⋊C4—24no—
105(C2×D7)⋊C4—42no—
106C4⋊Dic7—21no—
107C4⋊Dic7—42no—
108C4⋊Dic7—41no—
109C4⋊Dic7M1612yes—
110Dic7⋊C4—24no—
111Dic7⋊C4—41no—
112Dic7⋊C4—41no—
113C4×Dic7—22no—
114C4×Dic7—22no—
115C4×Dic7—41no—
116C4×Dic7—42no—
117C4×Dic7—42no—
118C2×C7⋊C8—22no—
119C2×C7⋊C8—41no—
120C2×C7⋊C8—41no—
121C2×C7⋊C8—41no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×C22⋊Q8—71no—
125C2×C4×Q8—71no—
126C22×C42—71no—
127C2×C22⋊C8—141no—
128C2×C22⋊C8—141no—
129C2×#97—141no—
130#391C2×D711yescharacteristic
131#389—71no—
132#387—141no—
133#385—141no—
134#380—71no—
135C4×C4⋊C4—71no—
136C4×C4⋊C4—141no—
137C4×C22⋊C4—141no—
138#336—71no—
139#336—71no—
140#336—141no—
141#336C2×D711yescharacteristic
142#336C2×D711yescharacteristic
143C23×C7C2×C22⋊C411yescharacteristic
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7—21no—
149C22×D7—21no—
150C22×D7#10412yes—
151C7×Q8—42no—
152C7×Q8—42no—
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—22no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7C2×M1611yescharacteristic
166C2×C4×C7C2×M1611yescharacteristic
167C2×C4×C7#13511yescharacteristic
168C2×C4×C7#13511yescharacteristic
169C2×C4×C7#13711yescharacteristic
170C2×C4×C7#13711yescharacteristic
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—22no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7—41no—
181C2×Dic7—41no—
182C2×Dic7—41no—
183C2×Dic7—41no—
184C2×Dic7—41no—
185C2×Dic7#9912yes—
186C4×D7—42no—
187C4×D7—42no—
188C4×D7—42no—
189C4×D7—42no—
190C4×D7—42no—
191C4×D7—42no—
192C7×C8—42no—
193C7×C8—81no—
194C7⋊C8—42no—
195C7⋊C8—81no—
196C22×Q8C4×D711yescharacteristic
197C23×C4—71no—
198C23×C4—71no—
199C23×C4—71no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—141no—
203C22×C8—141no—
204C22⋊Q8—142no—
205C22⋊Q8—142no—
206C4×Q8—282no—
207C2×C4⋊C4—71no—
208C2×C4⋊C4—71no—
209C2×C4⋊C4—141no—
210C2×C4⋊C4—141no—
211C2×C4⋊C4—141no—
212C2×C4⋊C4C4×D711yescharacteristic
213C2×C22⋊C4—141no—
214C2×C22⋊C4—141no—
215C2×C42—71no—
216C2×C42—71no—
217C2×C42—71no—
218C2×C42—141no—
219C2×C42—141no—
220C2×C42—282no—
221C2×C42C22×D711yescharacteristic
222C22⋊C8—144no—
223C22⋊C8—282no—
224#97—282no—
225#95—21no—
226#95—21no—
227#95—141no—
228#95—141no—
229#95—141no—
230C22×C7#40711yescharacteristic
231C22×C7C2×#9911yescharacteristic
232C22×C7#41311yescharacteristic
233C22×C7#41611yescharacteristic
234C22×C7#41911yescharacteristic
235C22×C7C2×#10411yescharacteristic
236C22×C7#42111yescharacteristic
237C2×D7—22no—
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—42no—
251C4×C7—81no—
252Dic7—22no—
253Dic7—41no—
254Dic7—42no—
255Dic7—41no—
256Dic7—81no—
257C24—71no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8#113112yes—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—141no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4—141no—
273C22×C4—141no—
274C22×C4—141no—
275C22×C4—141no—
276C22×C4—141no—
277C22×C4—281no—
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—41no—
297C4⋊C4—72no—
298C4⋊C4—144no—
299C4⋊C4—141no—
300C4⋊C4—281no—
301C4⋊C4—282no—
302C4⋊C4—281no—
303C4⋊C4—281no—
304C22⋊C4—144no—
305C22⋊C4—282no—
306C42—41no—
307C42—142no—
308C42—142no—
309C42—281no—
310C42—282no—
311C42—282no—
312C2×C7#146511yescharacteristic
313C2×C7#148811yescharacteristic
314C2×C7#149111yescharacteristic
315C2×C7#150111yescharacteristic
316C2×C7#151111yescharacteristic
317C2×C7#153211yescharacteristic
318C2×C7#155411yescharacteristic
319D7—42no—
320C23—72no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23—141no—
325C23—141no—
326C23—141no—
327C23D7×C22⋊C411yescharacteristic
328Q8—42no—
329Q8—42no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—72no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—142no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—282no—
353C2×C4—282no—
354C2×C4—282no—
355C2×C4—282no—
356C2×C4—282no—
357C2×C4—282no—
358C2×C4—281no—
359C2×C4—281no—
360C2×C4—281no—
361C2×C4—281no—
362C2×C4—281no—
363C2×C4C2×#113111yescharacteristic
364C2×C4D7×M1611yescharacteristic
365C2×C4#647811yescharacteristic
366C2×C4#647911yescharacteristic
367C2×C4#648111yescharacteristic
368C2×C4#648211yescharacteristic
369C8—42no—
370C8—81no—
371C8—282no—
372C8—561no—
373C7#762311yescharacteristic, complemented, Sylow
374C22—142no—
375C22—142no—
376C22—142no—
377C22—281no—
378C22—281no—
379C22—281no—
380C22—281no—
381C22#8949811yescharacteristic
382C22D7×#9911yescharacteristic
383C22#8951711yescharacteristic
384C22#8957411yescharacteristic
385C22#8957611yescharacteristic
386C22D7×#10411yescharacteristic
387C22#8959311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—142no—
397C4—281no—
398C4—281no—
399C4—282no—
400C4—561no—
401C2—282no—
402C2#1172587511yescharacteristic
403C2#1172599111yescharacteristic
404C2#1172599311yescharacteristic
405C2#1172608711yescharacteristic
406C2#1172609011yescharacteristic
407C2#1172610011yescharacteristic
408C2#1172624011yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1272C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1272D7
hypocenterC7#7623
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
4. C7×#1272C2semidirect
5. #11725448C2semidirect
6. #11725284C2semidirect
7. #11725283C2semidirect
66. #1272D7semidirect
373. C7#7623semidirect

Maximal subgroups

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

subgroupindex
#117339242
#117264182
#117263542
C7×#12722
#117254482
#117252842
#117252832
#76237