Finite Group Databasebeta

← #13249705

Subgroups — #13249705

Order 1792.

566 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3077C211yesmaximal, characteristic
2#11739778C211yesmaximal, characteristic
3#11729592C211yesmaximal, characteristic, complemented
4#11729589C211yesmaximal, characteristic, complemented
5#11729351C211yesmaximal, characteristic
6#11725521C211yesmaximal, characteristic, complemented
7#11725519C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C2×D4×Dic7—21no—
11C7×#478—22no—
12C7×#476C412yescomplemented
13C7×#474C412yescomplemented
14C7×#428—21no—
15C7×#423C2211yescharacteristic
16C7×#418—21no—
17#89984—21no—
18C2×C7⋊#95—21no—
19#89757—21no—
20#89722—22no—
21#89708C2211yescharacteristic
22C4×C4⋊Dic7C2211yescharacteristic
23#89697C2211yescharacteristic
24#89697C2211yescharacteristic
25#89333—21no—
26#89333—21no—
27#89333—22no—
28D4⋊(C7⋊C8)—22no—
29#9175—71nomaximal, Sylow
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C23×Dic7—22no—
33C2×C7×M16—21no—
34C7×C4⋊Q8C2×C411yescharacteristic, complemented
35C7×C4⋊Q8C2×C411yescharacteristic, complemented
36C7×#125—21no—
37C7×C22⋊Q8—21no—
38C7×C22⋊Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C4×C7×D4C2×C412yes—
44C4×C7×D4D412yes—
45C7×#117—21no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C22⋊C4—21no—
48C2×C42×C7C2311yescharacteristic
49C2×C22⋊Dic7—22no—
50D4×Dic7—42no—
51D4×Dic7—42no—
52C2×C4⋊Dic7—21no—
53C2×C4⋊Dic7—21no—
54C2×C4⋊Dic7—22no—
55C2×C4×Dic7—21no—
56C7×#107—22no—
57C7×#106—22no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8C2×C412yescomplemented
60C7×#102—22no—
61C7×#102—22no—
62C7⋊#95—22no—
63C7⋊#95—21no—
64C7⋊#95D411yescharacteristic
65C7⋊#95D411yescharacteristic
66C7⋊#95D412yes—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)—21no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)—21no—
71C4×C7⋊C8—21no—
72C4×C7⋊C8—21no—
73#3077D711yescharacteristic
74#2349—71no—
75#1518—71no—
76#1503—71no—
77#1478—71no—
78#1303—71no—
79#1286—71no—
80C24×C7—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4C22⋊C412yes—
87C2×C7×D4SD1612yes—
88C2×C7×D4Q1612yes—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—82no—
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×Dic7—21no—
97C22×Dic7—22no—
98C22×Dic7—21no—
99C22×Dic7—44no—
100C22×Dic7—44no—
101C22×Dic7—82no—
102C7×M16—41no—
103C7×M16—41no—
104C2×C7×C8—22no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4C22⋊C412yes—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—41no—
116C42×C7—41no—
117C42×C7C22×C411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7#4111yescharacteristic
120C42×C7#4111yescharacteristic
121C22⋊Dic7—44no—
122C4⋊Dic7—42no—
123C4⋊Dic7—41no—
124C4⋊Dic7—41no—
125C4⋊Dic7—82no—
126C4×Dic7—41no—
127C4×Dic7—41no—
128C2×C7⋊C8—41no—
129C2×C7⋊C8—41no—
130#533C2×D711yescharacteristic
131C2×C4×D4—141no—
132C2×C4×D4C2×D711yescharacteristic
133#478—22no—
134#476Dic712yescomplemented
135#474Dic712yescomplemented
136#431—71no—
137#431—71no—
138#428—21no—
139#423C2×D711yescharacteristic
140#418—21no—
141#389—141no—
142#385—142no—
143#382—71no—
144#380—141no—
145C4×C4⋊C4—71no—
146C2×#95—141no—
147#328—141no—
148#328—141no—
149#328—142no—
150D4⋊C8—142no—
151C23×C7—22no—
152C23×C7—22no—
153C23×C7—22no—
154C23×C7—42no—
155C23×C7—42no—
156C23×C7#12011yescharacteristic
157C7×Q8—81no—
158C7×Q8—81no—
159C7×D4—22no—
160C7×D4—22no—
161C7×D4—22no—
162C7×D4#10314yes—
163C2×C4×C7—22no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—42no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—82no—
170C2×C4×C7—82no—
171C2×C4×C7C2×C22⋊C411yescharacteristic
172C2×C4×C7C4×D411yescharacteristic
173C2×C4×C7C4×D411yescharacteristic
174C2×C4×C7#12111yescharacteristic
175C2×C4×C7#12111yescharacteristic
176C2×C4×C7#12311yescharacteristic
177C2×C4×C7C2×SD1611yescharacteristic
178C2×C4×C7C2×Q1611yescharacteristic
179C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C7#13711yescharacteristic
181C2×Dic7—22no—
182C2×Dic7—41no—
183C2×Dic7—41no—
184C2×Dic7—41no—
185C2×Dic7—41no—
186C2×Dic7—42no—
187C2×Dic7—42no—
188C2×Dic7—41no—
189C2×Dic7—42no—
190C2×Dic7—88no—
191C7×C8—42no—
192C7⋊C8—81no—
193C7⋊C8—81no—
194C22×D4C22⋊D711yescharacteristic
195C23×C4—21no—
196C23×C4—142no—
197C2×M16—21no—
198C4⋊Q8C2×Dic711yescharacteristic, complemented
199C4⋊Q8C2×Dic711yescharacteristic, complemented
200#125—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—282no—
208C4×D4—282no—
209C4×D4C2×Dic712yes—
210C4×D4C22⋊D712yes—
211#117—21no—
212C2×C4⋊C4—141no—
213C2×C4⋊C4—141no—
214C2×C4⋊C4—142no—
215C2×C4⋊C4C22⋊D711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—142no—
218C2×C42—141no—
219C2×C42C22×D711yescharacteristic
220#107—22no—
221#106—22no—
222C4⋊C8—21no—
223C4⋊C8—141no—
224C4⋊C8—141no—
225C4⋊C8C2×Dic712yescomplemented
226#102—22no—
227#102—22no—
228C22⋊C8—141no—
229C22⋊C8—141no—
230C4×C8—141no—
231C4×C8—141no—
232#95—71no—
233#95—71no—
234#95—72no—
235#95—142no—
236#95—141no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—24no—
240C22×C7—44no—
241C22×C7—44no—
242C22×C7—42no—
243C22×C7—42no—
244C22×C7#38611yescharacteristic
245C22×C7#45011yescharacteristic
246C22×C7#45111yescharacteristic
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—42no—
252C4×C7—81no—
253C4×C7—81no—
254C4×C7C2×#10311yescharacteristic
255C4×C7#41711yescharacteristic
256C4×C7#48311yescharacteristic
257C4×C7#48411yescharacteristic
258Dic7—42no—
259Dic7—81no—
260Dic7—82no—
261C24—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4#114312yes—
268C2×D4#114412yes—
269C2×D4C22⋊Dic712yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—82no—
274C22×C4—141no—
275C22×C4—142no—
276C22×C4—141no—
277C22×C4—284no—
278C22×C4—284no—
279C22×C4—562no—
280C22×C4D4×D711yescharacteristic
281C22×C4D4×D711yescharacteristic
282C22×C4C2×C22⋊D711yescharacteristic
283M16—41no—
284M16—41no—
285C2×C8—22no—
286C2×C8—281no—
287C2×C8—281no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—282no—
296C4⋊C4—281no—
297C4⋊C4—281no—
298C4⋊C4—562no—
299C4⋊C4C22⋊Dic712yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—284no—
304C42—41no—
305C42—281no—
306C42—281no—
307C42#115911yescharacteristic
308C42#115911yescharacteristic
309C42C22×Dic711yescharacteristic
310C42C2×C22⋊D711yescharacteristic
311C2×C7—21no—
312C2×C7—21no—
313C2×C7—44no—
314C2×C7#187211yescharacteristic
315C2×C7#190411yescharacteristic
316C2×C7#190611yescharacteristic
317C23—22no—
318C23—22no—
319C23—22no—
320C23—42no—
321C23—42no—
322C23#650011yescharacteristic
323Q8—81no—
324Q8—81no—
325D4—22no—
326D4—22no—
327D4—22no—
328D4#640914yes—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—142no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4—281no—
342C2×C4—282no—
343C2×C4—282no—
344C2×C4—281no—
345C2×C4—282no—
346C2×C4—568no—
347C2×C4D4×Dic711yescharacteristic
348C2×C4D4×Dic711yescharacteristic
349C2×C4#649811yescharacteristic
350C2×C4#650211yescharacteristic
351C2×C4#650211yescharacteristic
352C2×C4C2×#114311yescharacteristic
353C2×C4C2×#114411yescharacteristic
354C2×C4#651211yescharacteristic
355C2×C4#651411yescharacteristic
356C2×C4C2×C22⋊Dic711yescharacteristic
357C8—42no—
358C8—561no—
359C8—561no—
360C7#917511yescharacteristic, complemented, Sylow
361C22—21no—
362C22—21no—
363C22—24no—
364C22—44no—
365C22—44no—
366C22—42no—
367C22—42no—
368C22#8981811yescharacteristic
369C22#8982211yescharacteristic
370C22#8999211yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—282no—
379C4—562no—
380C4—561no—
381C4#8981511yescharacteristic
382C4#8981611yescharacteristic
383C4C2×#640911yescharacteristic
384C4#9000411yescharacteristic
385C2—21no—
386C2—21no—
387C2—44no—
388C2#1172944611yescharacteristic
389C2#1172984311yescharacteristic
390C2#1172986811yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#3077C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC2×C42×C7C23
hypercenter#3077D7
hypocenterC7#9175
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
3. #11729592C2semidirect
4. #11729589C2semidirect
6. #11725521C2semidirect
7. #11725519C2semidirect
12. C7×#476C4semidirect
13. C7×#474C4semidirect
34. C7×C4⋊Q8C2×C4semidirect
35. C7×C4⋊Q8C2×C4semidirect
59. C7×C4⋊C8C2×C4semidirect
134. #476Dic7semidirect
135. #474Dic7semidirect
198. C4⋊Q8C2×Dic7semidirect
199. C4⋊Q8C2×Dic7semidirect
225. C4⋊C8C2×Dic7semidirect
360. C7#9175semidirect

Maximal subgroups

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

subgroupindex
C7×#30772
#117397782
#117295922
#117295892
#117293512
#117255212
#117255192
#91757