Finite Group Databasebeta

← #13242160

Subgroups — #13242160

Order 1792.

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

normal subgroups only

This group has 482 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#11733732C211yesmaximal, characteristic
2#11726416C211yesmaximal, characteristic
3#11726353C211yesmaximal, characteristic
4C7×#1271C211yesmaximal, characteristic, complemented
5#11725444C211yesmaximal, characteristic, complemented
6#11725286C211yesmaximal, characteristic, complemented
7#11725286C211yesmaximal, characteristic, complemented
8C2×#6460C411yescharacteristic
9C2×C42×D7C2211yescharacteristic
10C7×#389C2211yescharacteristic
11C2×(C2×D7)⋊C8—21no—
12C2×(C2×D7)⋊C8—21no—
13C2×Dic7⋊C8—21no—
14C2×C8×Dic7—21no—
15#89761—21no—
16#89759C411yescharacteristic
17#89753—21no—
18#89746C411yescharacteristic
19#89745C411yescharacteristic
20#89701C2211yescharacteristic
21#89436—21no—
22#89427—21no—
23C7×#336C2211yescharacteristic
24C7×#336C2211yescharacteristic
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7661—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×Dic7C2×C411yescharacteristic
44C2×C4×Dic7D411yescharacteristic
45C2×C4×Dic7D411yescharacteristic
46C22×C7⋊C8—21no—
47C22×C7⋊C8—21no—
48#6460—42no—
49C42×D7—42no—
50C7×#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57(C2×D7)⋊C8—24no—
58(C2×D7)⋊C8—42no—
59Dic7⋊C8—42no—
60C8×Dic7—42no—
61#2855—71no—
62#1797—71no—
63C4×C22⋊C8—71no—
64#1272—71no—
65#1272—71no—
66#1271—71no—
67#1271D711yescharacteristic, complemented
68C23×D7C22⋊C411yescharacteristic
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—21no—
75C22×Dic7—21no—
76C22×Dic7C22⋊C411yescharacteristic
77C22×Dic7C22⋊C411yescharacteristic
78C22×Dic7C22⋊C411yescharacteristic
79C2×C4×D7—21no—
80C2×C4×D7—21no—
81C2×C4×D7—21no—
82C2×C4×D7—21no—
83C2×C4×D7—22no—
84C2×C4×D7—22no—
85C2×C4×D7—41no—
86C2×C4×D7—41no—
87C2×C4×D7—41no—
88C2×C4×D7—41no—
89C2×C4×D7SD1612yes—
90C2×C4×D7Q1612yes—
91C2×Dic14—21no—
92C2×Dic14—21no—
93C2×Dic14—41no—
94C2×Dic14—41no—
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—42no—
110C4⋊Dic7—41no—
111Dic7⋊C4—42no—
112C4×Dic7—22no—
113C4×Dic7—22no—
114C4×Dic7—41no—
115C2×C7⋊C8—22no—
116C2×C7⋊C8—41no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119C2×C7⋊C8—41no—
120C2×C7⋊C8—41no—
121C2×C22⋊Q8—71no—
122C22×C42—71no—
123C2×C4⋊C8—141no—
124C2×C22⋊C8—141no—
125C2×C22⋊C8—141no—
126C2×C4×C8—141no—
127#391—71no—
128#391—71no—
129#389—71no—
130#389C2×D711yescharacteristic
131#387—141no—
132#387—141no—
133#385—141no—
134#381—141no—
135#380—71no—
136#336—71no—
137#336—71no—
138#336C2×D711yescharacteristic
139#336C2×D711yescharacteristic
140C23×C7C2×C22⋊C411yescharacteristic
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7#10312yes—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7#13111yescharacteristic
161C2×C4×C7#13111yescharacteristic
162C2×C4×C7C2×SD1611yescharacteristic
163C2×C4×C7C2×Q1611yescharacteristic
164C2×C4×C7#13511yescharacteristic
165C2×C4×C7#13511yescharacteristic
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7#9912yes—
179C2×Dic7#10312yes—
180C2×Dic7#10412yes—
181C4×D7—42no—
182C4×D7—42no—
183C4×D7—42no—
184C4×D7—42no—
185C4×D7—42no—
186C4×D7—42no—
187Dic14—81no—
188Dic14—81no—
189C7×C8—42no—
190C7×C8—81no—
191C7⋊C8—42no—
192C7⋊C8—81no—
193C22×Q8—71no—
194C23×C4—71no—
195C23×C4—71no—
196C23×C4—71no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—141no—
200C22×C8—141no—
201C22⋊Q8—282no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—141no—
205C2×C4⋊C4—141no—
206C2×C4⋊C4C4×D711yescharacteristic
207C2×C4⋊C4C4×D711yescharacteristic
208C2×C22⋊C4—141no—
209C2×C22⋊C4—141no—
210C2×C42—71no—
211C2×C42—71no—
212C2×C42—71no—
213C2×C42—282no—
214C2×C42C22×D711yescharacteristic
215C4⋊C8—282no—
216C22⋊C8—144no—
217C22⋊C8—282no—
218C4×C8—282no—
219#95—21no—
220#95—141no—
221#95—141no—
222#95—141no—
223#95—141no—
224#95—141no—
225#95—141no—
226C22×C7#40811yescharacteristic
227C22×C7C2×#9911yescharacteristic
228C22×C7#41311yescharacteristic
229C22×C7C2×#10311yescharacteristic
230C22×C7#41611yescharacteristic
231C22×C7C2×#10411yescharacteristic
232C22×C7#42111yescharacteristic
233C2×D7—22no—
234C2×D7—22no—
235C2×D7—22no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C2×D7—41no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—21no—
245C4×C7—21no—
246C4×C7—81no—
247C4×C7—81no—
248Dic7—22no—
249Dic7—22no—
250Dic7—22no—
251Dic7—81no—
252Dic7—81no—
253C24—71no—
254C2×Q8—141no—
255C2×Q8—141no—
256C2×Q8—281no—
257C2×Q8—281no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—72no—
264C22×C4—72no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—141no—
268C22×C4—141no—
269C22×C4—141no—
270C22×C4—141no—
271C22×C4—141no—
272C22×C4—141no—
273C22×C4—281no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4—281no—
277C22×C4C2×C4×D711yescharacteristic
278C22×C4D4×D711yescharacteristic
279C22×C4D4×D711yescharacteristic
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—142no—
287C2×C8—281no—
288C2×C8—281no—
289C2×C8—281no—
290C2×C8—281no—
291C2×C8—281no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—281no—
298C4⋊C4—282no—
299C4⋊C4—282no—
300C22⋊C4—282no—
301C22⋊C4—282no—
302C42—41no—
303C42—142no—
304C42—142no—
305C42—281no—
306C2×C7#146811yescharacteristic
307C2×C7#148211yescharacteristic
308C2×C7#149311yescharacteristic
309C2×C7#150111yescharacteristic
310C2×C7#152011yescharacteristic
311C2×C7#152911yescharacteristic
312C2×C7#154211yescharacteristic
313D7—42no—
314C23—72no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23D7×C22⋊C411yescharacteristic
322Q8—561no—
323Q8—561no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—72no—
337C2×C4—72no—
338C2×C4—72no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—282no—
346C2×C4—282no—
347C2×C4—282no—
348C2×C4—282no—
349C2×C4—282no—
350C2×C4—282no—
351C2×C4—281no—
352C2×C4—281no—
353C2×C4—281no—
354C2×C4—281no—
355C2×C4—281no—
356C2×C4—281no—
357C2×C4#646411yescharacteristic
358C2×C4#647011yescharacteristic
359C2×C4#647511yescharacteristic
360C2×C4D7×SD1611yescharacteristic
361C2×C4#647911yescharacteristic
362C2×C4D7×Q1611yescharacteristic
363C8—42no—
364C8—81no—
365C8—282no—
366C8—561no—
367C7#766111yescharacteristic, complemented, Sylow
368C22—142no—
369C22—142no—
370C22—142no—
371C22—281no—
372C22—281no—
373C22—281no—
374C22—281no—
375C22#8949911yescharacteristic
376C22D7×#9911yescharacteristic
377C22#8952311yescharacteristic
378C22#8954411yescharacteristic
379C22D7×#10311yescharacteristic
380C22D7×#10411yescharacteristic
381C22#8959311yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—142no—
391C4—142no—
392C4—142no—
393C4—561no—
394C4—561no—
395C2—282no—
396C2#1172590011yescharacteristic
397C2#1172592311yescharacteristic
398C2#1172599411yescharacteristic
399C2#1172605511yescharacteristic
400C2#1172609311yescharacteristic
401C2#1172611811yescharacteristic
402C2#1172617211yescharacteristic
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#7661
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
67. #1271D7semidirect
367. C7#7661semidirect

Maximal subgroups

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

subgroupindex
#117337322
#117264162
#117263532
C7×#12712
#117254442
#117252862
#76617