Finite Group Databasebeta

← #13250537

Subgroups — #13250537

Order 1792.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2331C211yesmaximal, characteristic, complemented
2#11740658C211yesmaximal, characteristic
3#11729528C211yesmaximal, characteristic, complemented
4#11727299C211yesmaximal, characteristic
5#11727194C211yesmaximal, characteristic
6#11725437C211yesmaximal, characteristic, complemented
7#11725420C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9#90168C2211yescharacteristic
10C7×#391—22no—
11C7×#389—21no—
12C7×#387—22no—
13C7×#382C2211yescharacteristic
14C4×C7×C4⋊C4—21no—
15C7×#376C2211yescharacteristic
16#89999—22no—
17#89857C412yescomplemented
18#89855—22no—
19#89853C412yescomplemented
20C2×#6379—21no—
21C2×#6379—21no—
22#89697C2211yescharacteristic
23#89697C2211yescharacteristic
24#89696C2211yescharacteristic
25#89341—22no—
26#89246—22no—
27#9301—71nomaximal, Sylow
28C22×C7×Q8D412yes—
29C2×#1157—21no—
30C7×C4⋊Q8—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8C2×C412yes—
33C7×C4⋊Q8D412yes—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—22no—
38C2×C42×C7—22no—
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C42×C7C2311yescharacteristic
42C2×#1143—24no—
43#6493—21no—
44#6493—21no—
45C2×#1136—21no—
46#6438—21no—
47#6437C2×C411yescharacteristic
48C4×D28—21no—
49C4×D28—21no—
50#6432C2×C411yescharacteristic
51C7×#95—22no—
52C7×#95—22no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95D412yes—
57#6409—24no—
58C22⋊(C7⋊C8)—21no—
59C22⋊(C7⋊C8)—21no—
60#6379—42no—
61#6379—42no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4×C7⋊C8—21no—
67C4×C7⋊C8C2×C412yes—
68#3206—71no—
69#2331D711yescharacteristic, complemented
70#1963—71no—
71#1548—71no—
72#1517—71no—
73#1286—71no—
74#1274—71no—
75C2×C7×Q8—22no—
76C2×C7×Q8—82no—
77C2×C7×Q8C22⋊C414yes—
78C22×C4×C7—21no—
79C22×C4×C7—22no—
80C22×C4×C7—21no—
81C22×C4×C7—22no—
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C2×C22⋊D7—41no—
86#1157—41no—
87#1157—41no—
88#1157—41no—
89#1157—41no—
90C2×D28—21no—
91C2×D28—21no—
92C2×D28—21no—
93C2×C4×D7—41no—
94C2×Dic14—21no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—82no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—82no—
106C42×C7SD1612yescomplemented
107C42×C7C22×C411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7#4111yescharacteristic
110C42×C7#4111yescharacteristic
111#1143—84no—
112(C2×D7)⋊C4—41no—
113(C2×D7)⋊C4—41no—
114C4⋊Dic7—21no—
115C4⋊Dic7—21no—
116#1136—41no—
117#1136—41no—
118C2×C7⋊C8—22no—
119C2×C7⋊C8—41no—
120C2×C7⋊C8—41no—
121#532—71no—
122C2×C4⋊Q8C2×D711yescharacteristic
123#492—72no—
124#487—72no—
125#461—142no—
126#448—142no—
127#431—71no—
128#431—71no—
129#423—71no—
130C2×#104—141no—
131C2×#104—141no—
132#391—22no—
133#389—21no—
134#387—22no—
135#382C2×D711yescharacteristic
136C4×C4⋊C4—21no—
137#376C2×D711yescharacteristic
138#333—142no—
139#327—142no—
140C23×C7#12011yescharacteristic
141C22×D7—41no—
142C7×Q8—44no—
143C7×Q8—82no—
144C2×C4×C7—24no—
145C2×C4×C7—22no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—41no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—42no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7C2×C22⋊C411yescharacteristic
159C2×C4×C7C4×D411yescharacteristic
160C2×C4×C7C4×D411yescharacteristic
161C2×C4×C7#12111yescharacteristic
162C2×C4×C7#12111yescharacteristic
163C2×C4×C7#12311yescharacteristic
164C2×C4×C7C2×SD1611yescharacteristic
165C2×C4×C7#13511yescharacteristic
166C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C7#13711yescharacteristic
168C22⋊D7—81no—
169C22⋊D7—81no—
170C2×Dic7—41no—
171D28—42no—
172D28—41no—
173D28—41no—
174D28—81no—
175C4×D7—81no—
176C4×D7—81no—
177Dic14—41no—
178Dic14—41no—
179C7⋊C8—42no—
180C7⋊C8—81no—
181C7⋊C8—81no—
182C2×#41—141no—
183C22×Q8C22⋊D712yes—
184C2×SD16—144no—
185C2×M16—141no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—71no—
189C4⋊Q8C4×D712yes—
190C4⋊Q8D2812yes—
191#127—71no—
192#124—141no—
193#121—141no—
194#121—141no—
195C4×D4—141no—
196C4×D4—141no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C42—22no—
202C2×C42C22⋊D711yescharacteristic
203C2×C42C22⋊D711yescharacteristic
204C2×C42C22×D711yescharacteristic
205C4⋊C8—141no—
206C4⋊C8—141no—
207C4⋊C8—141no—
208C4⋊C8—141no—
209#104—282no—
210#104—282no—
211#103—144no—
212C22⋊C8—141no—
213C22⋊C8—141no—
214C4×C8—72no—
215C4×C8—141no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C22⋊D712yes—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7#38611yescharacteristic
225C22×C7#44811yescharacteristic
226C22×C7#44911yescharacteristic
227C2×D7—81no—
228C2×D7—81no—
229C2×D7—81no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—42no—
235C4×C7—44no—
236C4×C7—82no—
237C4×C7—81no—
238C4×C7C4×SD1611yescharacteristic
239C4×C7#44011yescharacteristic
240C4×C7#46011yescharacteristic
241C4×C7#46411yescharacteristic
242Dic7—81no—
243#41—281no—
244#41—281no—
245#41—281no—
246#41—281no—
247C2×Q8—22no—
248C2×Q8—82no—
249C2×Q8—141no—
250C2×Q8(C2×D7)⋊C414yes—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—281no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—281no—
260C22×C4C2×C22⋊D711yescharacteristic
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263SD16—564no—
264M16—281no—
265M16—281no—
266C2×C8—142no—
267C2×C8—281no—
268C2×C8—281no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—82no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C22⋊C4—281no—
279C22⋊C4—281no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—82no—
284C42#114312yescomplemented
285C42C2×C4×D711yescharacteristic
286C42C2×D2811yescharacteristic
287C42#115711yescharacteristic
288C42#115711yescharacteristic
289C2×C7—21no—
290C2×C7—21no—
291C2×C7#188011yescharacteristic
292C2×C7#190511yescharacteristic
293C2×C7#191111yescharacteristic
294D7—161no—
295C23—281no—
296C23#651611yescharacteristic
297Q8—44no—
298Q8—82no—
299Q8—281no—
300Q8—281no—
301D4—282no—
302D4—281no—
303D4—281no—
304D4—561no—
305D4—561no—
306D4—561no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—281no—
322C2×C4—561no—
323C2×C4—561no—
324C2×C4C2×(C2×D7)⋊C411yescharacteristic
325C2×C4C4×C22⋊D711yescharacteristic
326C2×C4C4×C22⋊D711yescharacteristic
327C2×C4#649211yescharacteristic
328C2×C4#649311yescharacteristic
329C2×C4#649311yescharacteristic
330C2×C4C2×#114311yescharacteristic
331C2×C4#650511yescharacteristic
332C2×C4#651211yescharacteristic
333C2×C4#651311yescharacteristic
334C8—282no—
335C8—561no—
336C8—561no—
337C7#930111yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—561no—
341C22—561no—
342C22—561no—
343C22#8998511yescharacteristic
344C22#8999911yescharacteristic
345C22#9001011yescharacteristic
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—42no—
351C4—44no—
352C4—82no—
353C4—81no—
354C4—561no—
355C4C4×#114311yescharacteristic
356C4#8979811yescharacteristic
357C4#8979911yescharacteristic
358C4#8980011yescharacteristic
359C2—21no—
360C2—21no—
361C2—1121no—
362C2#1172938711yescharacteristic
363C2#1172940611yescharacteristic
364C2#1172941611yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2331C2
socleC22×C7#386
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#2331D7
hypocenterC7#9301
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C7×#2331C2semidirect
3. #11729528C2semidirect
6. #11725437C2semidirect
7. #11725420C2semidirect
17. #89857C4semidirect
19. #89853C4semidirect
69. #2331D7semidirect
106. C42×C7SD16semidirect
284. C42#1143semidirect
337. C7#9301semidirect

Maximal subgroups

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

subgroupindex
C7×#23312
#117406582
#117295282
#117272992
#117271942
#117254372
#117254202
#93017