Finite Group Databasebeta

← #13271236

Subgroups — #13271236

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×#89914C211yesmaximal, characteristic
2C2×#89912C211yesmaximal, characteristic, complemented
3C7×#1816C211yesmaximal, characteristic, complemented
4C2×#89290C211yesmaximal, characteristic, complemented
5C2×#89286C211yesmaximal, characteristic, complemented
6#11725534C211yesmaximal, characteristic, complemented
7#11725529C211yesmaximal, characteristic, complemented
8C2×C7×#131C2211yescharacteristic
9C2×#6512—21no—
10C22×D56C2211yescharacteristic
11C2×C7×#108C2211yescharacteristic
12C2×C7×#107C2211yescharacteristic
13C7×#417—21no—
14#89914—22no—
15#89914C414yescomplemented
16#89912—22no—
17#89912—22no—
18C2×#6397—21no—
19C2×#6393C2211yescharacteristic
20C2×#6386—21no—
21C2×#6386C2211yescharacteristic
22C22×C7⋊C16—21no—
23C22×C7⋊C16C2211yescharacteristic
24C7×#343—21no—
25#89290—22no—
26#89286—21no—
27#89286—21no—
28#13073—71nomaximal, Sylow
29C2×C7×#41D411yescharacteristic
30C22×D28—21no—
31C7×#131—21no—
32C7×#131—21no—
33C7×#131C2×C412yes—
34C7×#131D412yes—
35C2×C7×M16—21no—
36C2×C7×M16—21no—
37C2×C7×M16D411yescharacteristic
38C22×C7×C8—21no—
39C22×C7×C8C2311yescharacteristic
40C2×C7×C4⋊C4—21no—
41#6512—24no—
42#6512—42no—
43C2×#1143—41no—
44C2×#1141—41no—
45C2×#1136—21no—
46C2×D56—21no—
47C2×D56—21no—
48C2×D56—21no—
49C2×D56—21no—
50C2×D56—21no—
51C2×D56C2×C412yes—
52C7×#108—21no—
53C7×#108—21no—
54C7×#107—21no—
55C7×#107—21no—
56C7×#103—41no—
57C7×#102—41no—
58#6397—24no—
59#6393—21no—
60#6393C2×C412yes—
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×#472—71no—
72C2×#361—71no—
73C2×#357—71no—
74#1816D711yescharacteristic, complemented
75#1387—71no—
76#1381—71no—
77C23×D7—41no—
78C7×#41—21no—
79C7×#41—21no—
80C7×#41C22⋊C412yes—
81C7×#41SD1612yes—
82C2×C7×Q8D811yescharacteristic
83C2×C7×D4—21no—
84C2×C7×D4D811yescharacteristic
85C22×C4×C7—21no—
86C22×C4×C7—41no—
87C22×C4×C7C2×D411yescharacteristic
88C2×D28—22no—
89C2×D28—41no—
90C2×D28—41no—
91C2×D28—41no—
92C2×D28—41no—
93C2×D28—41no—
94C7×M16—21no—
95C7×M16—22no—
96C7×M16—22no—
97C7×M16—41no—
98C7×M16—41no—
99C7×M16C22⋊C412yes—
100C2×C7×C8—21no—
101C2×C7×C8—22no—
102C2×C7×C8—22no—
103C2×C7×C8—41no—
104C2×C7×C8C22×C411yescharacteristic
105C2×C7×C8C2×D411yescharacteristic
106C2×C7×C8C2×D411yescharacteristic
107C2×C7×C8C2×D411yescharacteristic
108C2×C7×C8#4111yescharacteristic
109C2×C7×C8#4111yescharacteristic
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112#1143—44no—
113#1141—44no—
114#1136—22no—
115#1136—41no—
116C2×C7⋊C8—41no—
117D56—22no—
118D56—22no—
119D56—41no—
120D56—41no—
121C7⋊C16—21no—
122C7⋊C16—22no—
123C7⋊C16—21no—
124C2×AGL(1,ℤ/8)—141no—
125C22×D8—71no—
126C2×#131C2×D711yescharacteristic
127#504—142no—
128#504—142no—
129C2×M32—71no—
130C2×M32—141no—
131C22×C16—71no—
132C22×C16—141no—
133#472—74no—
134#472—142no—
135C2×#108—71no—
136C2×#108C2×D711yescharacteristic
137C2×#107C2×D711yescharacteristic
138#417—21no—
139C2×#100—141no—
140#361—142no—
141#357—141no—
142#357—141no—
143#343—21no—
144C23×C7—21no—
145C23×C7#13511yescharacteristic
146C22×D7—42no—
147C22×D7—81no—
148C22×D7—81no—
149C22×D7—161no—
150C22×D7—161no—
151C7×Q8—21no—
152C7×Q8#10212yes—
153C7×D4—22no—
154C7×D4—22no—
155C7×D4—21no—
156C7×D4#10212yes—
157C2×C4×C7—21no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—81no—
161C2×C4×C7—161no—
162C2×C4×C7C2×C22⋊C411yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7C2×D811yescharacteristic
166C2×C4×C7C2×SD1611yescharacteristic
167C2×C4×C7#13511yescharacteristic
168D28—42no—
169D28—42no—
170D28—81no—
171D28—81no—
172C7×C8—22no—
173C7×C8—81no—
174C7×C8C4×D411yescharacteristic
175C7×C8C4×D411yescharacteristic
176C7×C8#12011yescharacteristic
177C7×C8#12311yescharacteristic
178C7⋊C8—42no—
179C2×#41C22⋊D711yescharacteristic
180C22×D4—141no—
181AGL(1,ℤ/8)—144no—
182AGL(1,ℤ/8)—282no—
183C2×SD16—281no—
184C2×D8—72no—
185C2×D8—141no—
186C2×D8—141no—
187C2×D8—141no—
188C2×D8—141no—
189C2×D8—141no—
190C2×D8—281no—
191#131—21no—
192#131—21no—
193#131C4×D712yes—
194#131D2812yes—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—141no—
198C2×M16C22⋊D711yescharacteristic
199C22×C8—21no—
200C22×C8C22×D711yescharacteristic
201C2×C4⋊C4—21no—
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—21no—
213#108—72no—
214#108—141no—
215#107—21no—
216#107—21no—
217#103—41no—
218#102—41no—
219#100—144no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—22no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7#41611yescharacteristic
226C22×C7#46511yescharacteristic
227C22×C7#46611yescharacteristic
228C2×D7—82no—
229C2×D7—162no—
230C2×D7—161no—
231C2×D7—161no—
232C2×D7—162no—
233C4×C7—22no—
234C4×C7—161no—
235C4×C7#38611yescharacteristic
236C4×C7C2×#10211yescharacteristic
237C4×C7#46511yescharacteristic
238C4×C7#46611yescharacteristic
239C24—281no—
240#41—21no—
241#41—21no—
242#41(C2×D7)⋊C412yes—
243#41#114312yes—
244C2×Q8#114111yescharacteristic
245C2×D4—21no—
246C2×D4—142no—
247C2×D4—281no—
248C2×D4—281no—
249C2×D4—281no—
250C2×D4—281no—
251C2×D4—281no—
252C2×D4#114111yescharacteristic
253C22×C4—21no—
254C22×C4—41no—
255C22×C4C2×C22⋊D711yescharacteristic
256SD16—284no—
257D8—142no—
258D8—142no—
259D8—284no—
260D8—281no—
261D8—281no—
262M16—21no—
263M16—22no—
264M16—22no—
265M16—41no—
266M16—41no—
267M16—142no—
268M16—281no—
269M16(C2×D7)⋊C412yes—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—281no—
275C2×C8C2×C4×D711yescharacteristic
276C2×C8C2×D2811yescharacteristic
277C2×C8D4×D711yescharacteristic
278C2×C8D4×D711yescharacteristic
279C2×C8#115911yescharacteristic
280C2×C8#115911yescharacteristic
281C4⋊C4—41no—
282C4⋊C4—41no—
283C16—141no—
284C16—141no—
285C16—142no—
286C2×C7—21no—
287C2×C7—21no—
288C2×C7—42no—
289C2×C7#189311yescharacteristic
290C2×C7#222111yescharacteristic
291C2×C7#222311yescharacteristic
292D7—162no—
293C23—21no—
294C23—282no—
295C23—561no—
296C23—561no—
297C23—1121no—
298C23—1121no—
299C23#651311yescharacteristic
300Q8—21no—
301Q8#638312yes—
302D4—22no—
303D4—22no—
304D4—21no—
305D4—282no—
306D4—282no—
307D4—561no—
308D4—561no—
309D4#638312yes—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—81no—
314C2×C4—161no—
315C2×C4C2×(C2×D7)⋊C411yescharacteristic
316C2×C4C2×#114111yescharacteristic
317C2×C4#650211yescharacteristic
318C2×C4#650211yescharacteristic
319C2×C4C2×#114311yescharacteristic
320C2×C4#651311yescharacteristic
321C8—22no—
322C8—81no—
323C8—282no—
324C8(C22⋊D7)⋊C411yescharacteristic
325C8(C22⋊D7)⋊C411yescharacteristic
326C8#644511yescharacteristic
327C8#644911yescharacteristic
328C7#1307311yescharacteristic, complemented, Sylow
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—562no—
335C22—1122no—
336C22—1122no—
337C22—1121no—
338C22—1121no—
339C22#8977411yescharacteristic
340C22#8981011yescharacteristic
341C22#8982011yescharacteristic
342C4—22no—
343C4—161no—
344C4#8973011yescharacteristic
345C4C2×#638311yescharacteristic
346C4#8981011yescharacteristic
347C4#8982011yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—1122no—
352C2#1172726911yescharacteristic
353C2#1172884611yescharacteristic
354C2#1172885111yescharacteristic
subgroupG/H
centerC22#89810
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#1816C2
socleC22×C7#466
prime radicalC2×#89914C2
cosocleC22×C7×C8C23
hypercenter#1816D7
hypocenterC7#13073
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #1816

Complements

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

NH ≅ G/Ntype
2. C2×#89912C2semidirect
3. C7×#1816C2semidirect
4. C2×#89290C2semidirect
5. C2×#89286C2semidirect
6. #11725534C2semidirect
7. #11725529C2semidirect
15. #89914C4semidirect
74. #1816D7semidirect
328. C7#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#899142
C2×#899122
C7×#18162
C2×#892902
C2×#892862
#117255342
#117255292
#130737