Finite Group Databasebeta

← #13271240

Subgroups — #13271240

Order 1792.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#472C211yesmaximal, characteristic
2C2×#89912C211yesmaximal, characteristic, complemented
3C2×#89358C211yesmaximal, characteristic, complemented
4C2×#89357C211yesmaximal, characteristic, complemented
5#11729770C211yesmaximal, characteristic, complemented
6#11725538C211yesmaximal, characteristic, complemented
7#11725534C211yesmaximal, characteristic, complemented
8C2×C7×AGL(1,ℤ/8)—21no—
9C22×C7×D8C2211yescharacteristic
10C2×C7×#131C2211yescharacteristic
11C7×#472—22no—
12C7×#472C414yescomplemented
13C2×C7×#108C2211yescharacteristic
14C2×C7×#100—21no—
15#90004—21no—
16#89912—22no—
17#89912—22no—
18C2×#6393C2211yescharacteristic
19C2×#6392C2211yescharacteristic
20C2×#6386—21no—
21C2×#6386C2211yescharacteristic
22C22×C7⋊C16—21no—
23C22×C7⋊C16C2211yescharacteristic
24#89358—22no—
25#89357—21no—
26#89357—21no—
27#89349—21no—
28#13073—71nomaximal, Sylow
29C2×C7×#41D411yescharacteristic
30C22×C7×D4—21no—
31C7×AGL(1,ℤ/8)—24no—
32C7×AGL(1,ℤ/8)—42no—
33C2×C7×SD16—41no—
34C2×C7×D8—21no—
35C2×C7×D8—21no—
36C2×C7×D8—21no—
37C2×C7×D8—21no—
38C2×C7×D8—21no—
39C2×C7×D8—41no—
40C2×C7×D8C2×C412yes—
41C7×#131—21no—
42C7×#131—21no—
43C7×#131C2×C412yes—
44C7×#131D412yes—
45C2×C7×M16—21no—
46C2×C7×M16—21no—
47C2×C7×M16D411yescharacteristic
48C22×C7×C8—21no—
49C22×C7×C8C2311yescharacteristic
50C2×C4⋊Dic7—21no—
51C2×#1136—21no—
52C7×#108—21no—
53C7×#108C2×C412yes—
54C7×#100—24no—
55#6409—41no—
56#6406—41no—
57#6393—21no—
58#6393—21no—
59#6392—21no—
60#6392—21no—
61#6386—24no—
62#6386—21no—
63#6386D412yes—
64C2×C7⋊C16—24no—
65C2×C7⋊C16—21no—
66C2×C7⋊C16—21no—
67C2×C7⋊C16—21no—
68C2×C7⋊C16D411yescharacteristic
69C2×C7⋊C16D411yescharacteristic
70C2×#504—71no—
71C2×#472D711yescharacteristic
72C2×#361—71no—
73C2×#357—71no—
74#1816—71no—
75#1387—71no—
76#1381—71no—
77C24×C7—41no—
78C7×#41—21no—
79C7×#41—21no—
80C7×#41C22⋊C412yes—
81C7×#41SD1612yes—
82C2×C7×Q8D811yescharacteristic
83C2×C7×D4—21no—
84C2×C7×D4—22no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4D811yescharacteristic
91C22×C4×C7—21no—
92C22×C4×C7C2×D411yescharacteristic
93C22×Dic7—41no—
94C7×SD16—44no—
95C7×D8—22no—
96C7×D8—22no—
97C7×D8—44no—
98C7×D8—41no—
99C7×D8—41no—
100C7×M16—21no—
101C7×M16—22no—
102C7×M16—22no—
103C7×M16—22no—
104C7×M16—41no—
105C7×M16C22⋊C412yes—
106C2×C7×C8—21no—
107C2×C7×C8—22no—
108C2×C7×C8—22no—
109C2×C7×C8—41no—
110C2×C7×C8C22×C411yescharacteristic
111C2×C7×C8C2×D411yescharacteristic
112C2×C7×C8C2×D411yescharacteristic
113C2×C7×C8C2×D411yescharacteristic
114C2×C7×C8#4111yescharacteristic
115C2×C7×C8#4111yescharacteristic
116C4⋊Dic7—41no—
117C4⋊Dic7—41no—
118#1136—41no—
119#1136—41no—
120C2×C7⋊C8—41no—
121C7⋊C16—21no—
122C7⋊C16—22no—
123C7⋊C16—21no—
124C2×AGL(1,ℤ/8)—21no—
125C22×D8C2×D711yescharacteristic
126C2×#131C2×D711yescharacteristic
127#504—142no—
128#504—142no—
129C2×M32—71no—
130C2×M32—141no—
131C22×C16—71no—
132C22×C16—141no—
133#472—22no—
134#472Dic714yescomplemented
135C2×#108—71no—
136C2×#108C2×D711yescharacteristic
137C2×#107—71no—
138#417—141no—
139C2×#100—21no—
140#361—142no—
141#357—141no—
142#357—141no—
143#343—141no—
144C23×C7—21no—
145C23×C7—42no—
146C23×C7—81no—
147C23×C7—81no—
148C23×C7—161no—
149C23×C7—161no—
150C23×C7#13511yescharacteristic
151C7×Q8—21no—
152C7×Q8#10212yes—
153C7×D4—22no—
154C7×D4—22no—
155C7×D4—21no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—81no—
159C7×D4—81no—
160C7×D4#10212yes—
161C2×C4×C7—21no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7C2×C22⋊C411yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7C2×D811yescharacteristic
168C2×C4×C7C2×SD1611yescharacteristic
169C2×C4×C7#13511yescharacteristic
170C2×Dic7—81no—
171C2×Dic7—161no—
172C7×C8—22no—
173C7×C8—42no—
174C7×C8C4×D411yescharacteristic
175C7×C8C4×D411yescharacteristic
176C7×C8#12011yescharacteristic
177C7×C8#12311yescharacteristic
178C7⋊C8—81no—
179C2×#41C22⋊D711yescharacteristic
180C22×D4—21no—
181AGL(1,ℤ/8)—24no—
182AGL(1,ℤ/8)—42no—
183C2×SD16—41no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—41no—
190C2×D8C2×Dic712yes—
191#131—21no—
192#131—21no—
193#131C2×Dic712yes—
194#131C22⋊D712yes—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—141no—
198C2×M16C22⋊D711yescharacteristic
199C22×C8—21no—
200C22×C8C22×D711yescharacteristic
201C2×C4⋊C4—141no—
202M32—72no—
203M32—144no—
204M32—141no—
205C2×C16—71no—
206C2×C16—71no—
207C2×C16—144no—
208C2×C16—141no—
209C2×C16—141no—
210C2×C16—141no—
211#108—21no—
212#108—141no—
213#108—141no—
214#108C2×Dic712yes—
215#107—141no—
216#107—141no—
217#103—281no—
218#102—281no—
219#100—24no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—22no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—82no—
226C22×C7—162no—
227C22×C7—162no—
228C22×C7—161no—
229C22×C7—161no—
230C22×C7#41611yescharacteristic
231C22×C7#46511yescharacteristic
232C22×C7#46611yescharacteristic
233C4×C7—22no—
234C4×C7#38611yescharacteristic
235C4×C7C2×#10211yescharacteristic
236C4×C7#46511yescharacteristic
237C4×C7#46611yescharacteristic
238Dic7—161no—
239C24—41no—
240#41—21no—
241#41—21no—
242#41#114212yes—
243#41C22⋊Dic712yes—
244C2×Q8#114111yescharacteristic
245C2×D4—21no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4#114111yescharacteristic
253C22×C4—21no—
254C22×C4—281no—
255C22×C4C2×C22⋊D711yescharacteristic
256SD16—44no—
257D8—22no—
258D8—22no—
259D8—44no—
260D8—41no—
261D8—41no—
262M16—21no—
263M16—22no—
264M16—22no—
265M16—22no—
266M16—41no—
267M16—281no—
268M16—281no—
269M16C22⋊Dic712yes—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—281no—
275C2×C8D4×D711yescharacteristic
276C2×C8D4×D711yescharacteristic
277C2×C8#115911yescharacteristic
278C2×C8#115911yescharacteristic
279C2×C8C22×Dic711yescharacteristic
280C2×C8C2×C22⋊D711yescharacteristic
281C4⋊C4—281no—
282C4⋊C4—281no—
283C16—141no—
284C16—141no—
285C16—142no—
286C2×C7—21no—
287C2×C7—21no—
288C2×C7—42no—
289C2×C7—162no—
290C2×C7#189311yescharacteristic
291C2×C7#222111yescharacteristic
292C2×C7#222311yescharacteristic
293C23—21no—
294C23—42no—
295C23—81no—
296C23—81no—
297C23—161no—
298C23—161no—
299C23#651311yescharacteristic
300Q8—21no—
301Q8#640612yes—
302D4—22no—
303D4—22no—
304D4—21no—
305D4—42no—
306D4—42no—
307D4—81no—
308D4—81no—
309D4#640612yes—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—561no—
314C2×C4—1121no—
315C2×C4C2×#114111yescharacteristic
316C2×C4C2×#114211yescharacteristic
317C2×C4#650211yescharacteristic
318C2×C4#650211yescharacteristic
319C2×C4#651311yescharacteristic
320C2×C4C2×C22⋊Dic711yescharacteristic
321C8—22no—
322C8—42no—
323C8—561no—
324C8D4×Dic711yescharacteristic
325C8D4×Dic711yescharacteristic
326C8#649811yescharacteristic
327C8#650011yescharacteristic
328C7#1307311yescharacteristic, complemented, Sylow
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#8981011yescharacteristic
340C22#8981311yescharacteristic
341C22#9000511yescharacteristic
342C4—22no—
343C4—1121no—
344C4#8981011yescharacteristic
345C4#8981311yescharacteristic
346C4C2×#640611yescharacteristic
347C4#8999211yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—162no—
352C2#1172884611yescharacteristic
353C2#1172884711yescharacteristic
354C2#1172943511yescharacteristic
subgroupG/H
centerC22#89810
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
Fitting, prime radicalC2×C7×#472C2
socleC22×C7#466
cosocleC22×C7×C8C23
hypercenterC2×#472D7
hypocenterC7#13073
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
2. C2×#89912C2semidirect
3. C2×#89358C2semidirect
4. C2×#89357C2semidirect
5. #11729770C2semidirect
6. #11725538C2semidirect
7. #11725534C2semidirect
12. C7×#472C4semidirect
134. #472Dic7semidirect
328. C7#13073semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4722
C2×#899122
C2×#893582
C2×#893572
#117297702
#117255382
#117255342
#130737