Finite Group Databasebeta

← #13250530

Subgroups — #13250530

Order 1792.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2327C211yesmaximal, characteristic, complemented
2#11740660C211yesmaximal, characteristic
3#11729527C211yesmaximal, characteristic, complemented
4#11727299C211yesmaximal, characteristic
5#11727194C211yesmaximal, characteristic
6#11725437C211yesmaximal, characteristic, complemented
7#11725420C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×C7×Q8—21no—
10#90168C2211yescharacteristic
11C7×#391—21no—
12C7×#387—22no—
13C7×#387—22no—
14C7×#382C2211yescharacteristic
15C7×#376C2211yescharacteristic
16#89860C412yescomplemented
17#89859—22no—
18#89850C412yescomplemented
19#89739—22no—
20C2×#6379—21no—
21C2×#6379—21no—
22#89697C2211yescharacteristic
23#89697C2211yescharacteristic
24#89696C2211yescharacteristic
25#89342—22no—
26#89246—22no—
27#9300—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C2×#1157—21no—
30C7×C4⋊Q8—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8C2×C412yes—
33C7×C4⋊Q8D412yes—
34C4×C7×Q8—42no—
35C4×C7×Q8—42no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4D412yes—
40C2×C42×C7—22no—
41C2×C42×C7D411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7C2311yescharacteristic
44#6493—21no—
45#6493—21no—
46C2×#1136—21no—
47#6438—21no—
48#6437C2×C411yescharacteristic
49C4×D28—21no—
50C4×D28—21no—
51#6432C2×C411yescharacteristic
52C7×#95—24no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95D412yes—
57C22⋊(C7⋊C8)—21no—
58C22⋊(C7⋊C8)—21no—
59#6383—24no—
60#6381—24no—
61#6379—42no—
62#6379—42no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)—21no—
67C4×C7⋊C8—21no—
68C4×C7⋊C8C2×C412yes—
69#3226—71no—
70#2327D711yescharacteristic, complemented
71#1963—71no—
72#1546—71no—
73#1517—71no—
74#1286—71no—
75#1274—71no—
76C2×C7×Q8—22no—
77C2×C7×Q8—22no—
78C2×C7×Q8—41no—
79C2×C7×Q8—41no—
80C2×C7×Q8—41no—
81C2×C7×Q8—41no—
82C22×C4×C7—21no—
83C22×C4×C7—22no—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C2×C22⋊D7—41no—
90#1157—41no—
91#1157—41no—
92#1157—41no—
93#1157—41no—
94C2×D28—21no—
95C2×D28—21no—
96C2×D28—21no—
97C2×C4×D7—41no—
98C2×Dic14—21no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—44no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4C22⋊C414yes—
106C42×C7—21no—
107C42×C7—21no—
108C42×C7—21no—
109C42×C7—44no—
110C42×C7D812yes—
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115(C2×D7)⋊C4—41no—
116(C2×D7)⋊C4—41no—
117C4⋊Dic7—21no—
118C4⋊Dic7—21no—
119#1136—41no—
120#1136—41no—
121C2×C7⋊C8—22no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124#532—71no—
125C2×C4⋊Q8C2×D711yescharacteristic
126C2×C4×Q8—21no—
127#500—72no—
128#486—72no—
129#480—142no—
130#474—142no—
131#431—71no—
132#431—71no—
133#423—71no—
134C2×#104—141no—
135C2×#104—141no—
136#391—21no—
137#387—22no—
138#387—22no—
139#382C2×D711yescharacteristic
140#376C2×D711yescharacteristic
141#332—142no—
142#327—142no—
143C23×C7#12011yescharacteristic
144C22×D7—41no—
145C7×Q8—42no—
146C7×Q8—42no—
147C7×Q8—42no—
148C7×Q8—42no—
149C2×C4×C7—24no—
150C2×C4×C7—22no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—42no—
156C2×C4×C7—42no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—41no—
163C2×C4×C7C2×C22⋊C411yescharacteristic
164C2×C4×C7C4×D411yescharacteristic
165C2×C4×C7C4×D411yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7#12111yescharacteristic
168C2×C4×C7#12311yescharacteristic
169C2×C4×C7C2×D811yescharacteristic
170C2×C4×C7#13511yescharacteristic
171C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
173C22⋊D7—81no—
174C22⋊D7—81no—
175C2×Dic7—41no—
176D28—42no—
177D28—41no—
178D28—41no—
179D28—81no—
180C4×D7—81no—
181C4×D7—81no—
182Dic14—41no—
183Dic14—41no—
184C7⋊C8—42no—
185C7⋊C8—81no—
186C7⋊C8—81no—
187C2×#41—141no—
188C22×Q8—21no—
189C2×M16—141no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—71no—
193C4⋊Q8C4×D712yes—
194C4⋊Q8D2812yes—
195#127—71no—
196#124—141no—
197#121—141no—
198#121—141no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×D4—141no—
202C4×D4—141no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22⋊D712yes—
207C2×C42—22no—
208C2×C42C22⋊D711yescharacteristic
209C2×C42C22⋊D711yescharacteristic
210C2×C42C22×D711yescharacteristic
211#107—144no—
212C4⋊C8—141no—
213C4⋊C8—141no—
214C4⋊C8—141no—
215C4⋊C8—141no—
216#104—282no—
217#104—282no—
218#102—144no—
219C22⋊C8—141no—
220C22⋊C8—141no—
221C4×C8—72no—
222C4×C8—141no—
223#95—24no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C22⋊D712yes—
228C22×C7—21no—
229C22×C7—21no—
230C22×C7#38611yescharacteristic
231C22×C7#44711yescharacteristic
232C22×C7#44911yescharacteristic
233C2×D7—81no—
234C2×D7—81no—
235C2×D7—81no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—21no—
240C4×C7—42no—
241C4×C7—44no—
242C4×C7—82no—
243C4×C7—81no—
244C4×C7C4×D811yescharacteristic
245C4×C7#44211yescharacteristic
246C4×C7#45911yescharacteristic
247C4×C7#46311yescharacteristic
248Dic7—81no—
249#41—281no—
250#41—281no—
251#41—281no—
252#41—281no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—141no—
260C2×D4—141no—
261C2×D4—141no—
262C2×D4—141no—
263C2×D4—281no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—281no—
269C22×C4C2×C22⋊D711yescharacteristic
270C22×C4C2×C22⋊D711yescharacteristic
271C22×C4C2×C22⋊D711yescharacteristic
272M16—281no—
273M16—281no—
274C2×C8—142no—
275C2×C8—281no—
276C2×C8—281no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—44no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—141no—
284C4⋊C4—141no—
285C4⋊C4(C2×D7)⋊C414yes—
286C22⋊C4—281no—
287C22⋊C4—281no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—44no—
292C42#114112yes—
293C42C2×C4×D711yescharacteristic
294C42C2×D2811yescharacteristic
295C42#115711yescharacteristic
296C42#115711yescharacteristic
297C2×C7—21no—
298C2×C7—21no—
299C2×C7#187911yescharacteristic
300C2×C7#190511yescharacteristic
301C2×C7#191211yescharacteristic
302D7—161no—
303C23—281no—
304C23#651611yescharacteristic
305Q8—42no—
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—281no—
310Q8—281no—
311D4—282no—
312D4—281no—
313D4—281no—
314D4—561no—
315D4—561no—
316D4—561no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—281no—
332C2×C4—561no—
333C2×C4—561no—
334C2×C4C2×(C2×D7)⋊C411yescharacteristic
335C2×C4C4×C22⋊D711yescharacteristic
336C2×C4C4×C22⋊D711yescharacteristic
337C2×C4#649211yescharacteristic
338C2×C4#649311yescharacteristic
339C2×C4#649311yescharacteristic
340C2×C4C2×#114111yescharacteristic
341C2×C4#649511yescharacteristic
342C2×C4#651211yescharacteristic
343C2×C4#651311yescharacteristic
344C8—282no—
345C8—561no—
346C8—561no—
347C7#930011yescharacteristic, complemented, Sylow
348C22—21no—
349C22—21no—
350C22—561no—
351C22—561no—
352C22—561no—
353C22#8998511yescharacteristic
354C22#8998911yescharacteristic
355C22#9001011yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—44no—
362C4—82no—
363C4—81no—
364C4—561no—
365C4C4×#114111yescharacteristic
366C4#8978611yescharacteristic
367C4#8978711yescharacteristic
368C4#8978811yescharacteristic
369C2—21no—
370C2—21no—
371C2—1121no—
372C2#1172936711yescharacteristic
373C2#1172941611yescharacteristic
374C2#1172941711yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2327C2
socleC22×C7#386
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#2327D7
hypocenterC7#9300
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
1. C7×#2327C2semidirect
3. #11729527C2semidirect
6. #11725437C2semidirect
7. #11725420C2semidirect
16. #89860C4semidirect
18. #89850C4semidirect
70. #2327D7semidirect
347. C7#9300semidirect

Maximal subgroups

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

subgroupindex
C7×#23272
#117406602
#117295272
#117272992
#117271942
#117254372
#117254202
#93007