Finite Group Databasebeta

← #13271226

Subgroups — #13271226

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3067C211yesmaximal, characteristic
2C2×#89912C211yesmaximal, characteristic, complemented
3C2×#89362C211yesmaximal, characteristic, complemented
4#11729830C211yesmaximal, characteristic, complemented
5#11729767C211yesmaximal, characteristic, complemented
6#11725540C211yesmaximal, characteristic, complemented
7#11725535C211yesmaximal, characteristic, complemented
8C7×#575—21no—
9C2×C7×#135C2211yescharacteristic
10C2×C7×#131C2211yescharacteristic
11C7×#473C412yescomplemented
12C7×#472C412yescomplemented
13C7×#471—22no—
14C2×C7×#108C2211yescharacteristic
15C7×#413—21no—
16#90008—21no—
17#89912—22no—
18#89912—22no—
19C2×#6393C2211yescharacteristic
20#89878C2211yescharacteristic
21C2×#6386—21no—
22C2×#6386C2211yescharacteristic
23C22×C7⋊C16—21no—
24C22×C7⋊C16C2211yescharacteristic
25#89362—21no—
26#89362—21no—
27#89361—22no—
28#89354—21no—
29#13071—71nomaximal, Sylow
30C2×C7×#41—21no—
31C2×C7×#41D411yescharacteristic
32C7×#137—22no—
33C7×#137—41no—
34C7×AGL(1,ℤ/8)—22no—
35C7×AGL(1,ℤ/8)—41no—
36C7×#135—21no—
37C7×#135—21no—
38C7×#135—21no—
39C7×#135—21no—
40C7×#135—42no—
41C2×C7×Q16C2×C411yescharacteristic
42C2×C7×SD16—21no—
43C2×C7×D8C2×C411yescharacteristic
44C7×#131—21no—
45C7×#131—21no—
46C7×#131C2×C412yes—
47C7×#131D412yes—
48C2×C7×M16—21no—
49C2×C7×M16—21no—
50C2×C7×M16D411yescharacteristic
51C22×C7×C8—21no—
52C22×C7×C8C2311yescharacteristic
53#6489—21no—
54C2×#1136—21no—
55C7×#108—21no—
56C7×#108C2×C412yes—
57C7×#101—22no—
58C7×#100—22no—
59#6411—42no—
60#6393—21no—
61#6393—21no—
62#6392—21no—
63#6391—21no—
64#6386—24no—
65#6386—21no—
66#6386D412yes—
67C2×C7⋊C16—24no—
68C2×C7⋊C16—21no—
69C2×C7⋊C16—21no—
70C2×C7⋊C16—21no—
71C2×C7⋊C16D411yescharacteristic
72C2×C7⋊C16D411yescharacteristic
73C2×#504—71no—
74#3067D711yescharacteristic
75#2145—71no—
76C2×#359—71no—
77#1814—71no—
78#1388—71no—
79#1382—71no—
80C7×#41—21no—
81C7×#41—21no—
82C7×#41—41no—
83C7×#41—41no—
84C7×#41—41no—
85C7×#41—41no—
86C7×#41C22⋊C412yes—
87C7×#41D812yes—
88C2×C7×Q8—21no—
89C2×C7×Q8SD1611yescharacteristic
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—41no—
93C2×C7×D4SD1611yescharacteristic
94C22×C4×C7—21no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C7×Q16—21no—
98C7×Q16—21no—
99C7×Q16—42no—
100C7×SD16—42no—
101C7×SD16—42no—
102C7×SD16—41no—
103C7×SD16—41no—
104C7×D8—21no—
105C7×D8—21no—
106C7×D8—42no—
107C7×M16—21no—
108C7×M16—22no—
109C7×M16—22no—
110C7×M16—22no—
111C7×M16—41no—
112C7×M16C22⋊C412yes—
113C2×C7×C8—21no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—41no—
117C2×C7×C8C22×C411yescharacteristic
118C2×C7×C8C2×D411yescharacteristic
119C2×C7×C8C2×D411yescharacteristic
120C2×C7×C8C2×D411yescharacteristic
121C2×C7×C8#4111yescharacteristic
122C2×C7×C8#4111yescharacteristic
123C22⋊Dic7—41no—
124C4⋊Dic7—41no—
125C4×Dic7—41no—
126#1136—41no—
127#1136—41no—
128C2×C7⋊C8—41no—
129C7⋊C16—21no—
130C7⋊C16—22no—
131C7⋊C16—21no—
132#575—21no—
133C2×#135C2×D711yescharacteristic
134C2×#131C2×D711yescharacteristic
135#504—142no—
136#504—142no—
137C2×M32—71no—
138C2×M32—141no—
139C22×C16—71no—
140C22×C16—141no—
141#473Dic712yescomplemented
142#472Dic712yescomplemented
143#471—22no—
144C2×#108—71no—
145C2×#108C2×D711yescharacteristic
146#427—71no—
147#421—141no—
148#413—21no—
149#360—142no—
150#359—141no—
151#359—141no—
152#344—141no—
153C23×C7—21no—
154C23×C7—41no—
155C23×C7#13511yescharacteristic
156C7×Q8—21no—
157C7×Q8—41no—
158C7×Q8—41no—
159C7×Q8#10212yes—
160C7×D4—22no—
161C7×D4—22no—
162C7×D4—21no—
163C7×D4—41no—
164C7×D4—41no—
165C7×D4—81no—
166C7×D4—81no—
167C7×D4#10212yes—
168C2×C4×C7—21no—
169C2×C4×C7—22no—
170C2×C4×C7—22no—
171C2×C4×C7—41no—
172C2×C4×C7—81no—
173C2×C4×C7—81no—
174C2×C4×C7C2×C22⋊C411yescharacteristic
175C2×C4×C7#12111yescharacteristic
176C2×C4×C7#12111yescharacteristic
177C2×C4×C7C2×D811yescharacteristic
178C2×C4×C7C2×SD1611yescharacteristic
179C2×C4×C7#13511yescharacteristic
180C2×Dic7—81no—
181C7×C8—22no—
182C7×C8—42no—
183C7×C8C4×D411yescharacteristic
184C7×C8C4×D411yescharacteristic
185C7×C8#12011yescharacteristic
186C7×C8#12311yescharacteristic
187C7⋊C8—81no—
188C2×#41—21no—
189C2×#41C22⋊D711yescharacteristic
190#137—22no—
191#137—41no—
192AGL(1,ℤ/8)—22no—
193AGL(1,ℤ/8)—41no—
194#135—21no—
195#135—21no—
196#135—21no—
197#135—21no—
198#135—42no—
199C2×Q16C2×Dic711yescharacteristic
200C2×SD16—21no—
201C2×D8C2×Dic711yescharacteristic
202#131—21no—
203#131—21no—
204#131C2×Dic712yes—
205#131C22⋊D712yes—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—141no—
209C2×M16C22⋊D711yescharacteristic
210C22×C8—21no—
211C22×C8C22×D711yescharacteristic
212#117—141no—
213M32—72no—
214M32—144no—
215M32—141no—
216C2×C16—71no—
217C2×C16—71no—
218C2×C16—144no—
219C2×C16—141no—
220C2×C16—141no—
221C2×C16—141no—
222#108—21no—
223#108—141no—
224#108—141no—
225#108C2×Dic712yes—
226#107—141no—
227#106—141no—
228#104—282no—
229#101—22no—
230#100—22no—
231C22×C7—22no—
232C22×C7—41no—
233C22×C7—42no—
234C22×C7—81no—
235C22×C7—161no—
236C22×C7#41611yescharacteristic
237C22×C7#46511yescharacteristic
238C22×C7#46611yescharacteristic
239C4×C7—22no—
240C4×C7—81no—
241C4×C7#38611yescharacteristic
242C4×C7C2×#10211yescharacteristic
243C4×C7#46511yescharacteristic
244C4×C7#46611yescharacteristic
245Dic7—161no—
246#41—21no—
247#41—21no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41#114112yes—
253#41C22⋊Dic712yes—
254C2×Q8—21no—
255C2×Q8#114211yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4#114211yescharacteristic
260C22×C4—21no—
261C22×C4—41no—
262C22×C4C2×C22⋊D711yescharacteristic
263Q16—21no—
264Q16—21no—
265Q16—42no—
266SD16—42no—
267SD16—42no—
268SD16—41no—
269SD16—41no—
270D8—21no—
271D8—21no—
272D8—42no—
273M16—21no—
274M16—22no—
275M16—22no—
276M16—22no—
277M16—41no—
278M16—281no—
279M16—281no—
280M16C22⋊Dic712yes—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—281no—
286C2×C8D4×D711yescharacteristic
287C2×C8D4×D711yescharacteristic
288C2×C8#115911yescharacteristic
289C2×C8#115911yescharacteristic
290C2×C8C22×Dic711yescharacteristic
291C2×C8C2×C22⋊D711yescharacteristic
292C4⋊C4—281no—
293C22⋊C4—281no—
294C42—281no—
295C16—141no—
296C16—141no—
297C16—142no—
298C2×C7—21no—
299C2×C7—21no—
300C2×C7—21no—
301C2×C7—42no—
302C2×C7—161no—
303C2×C7#189311yescharacteristic
304C23—21no—
305C23—41no—
306C23#651311yescharacteristic
307Q8—21no—
308Q8—41no—
309Q8—41no—
310Q8#640612yes—
311D4—22no—
312D4—22no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—81no—
317D4—81no—
318D4#640612yes—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—561no—
326C2×C4C2×#114111yescharacteristic
327C2×C4C2×#114211yescharacteristic
328C2×C4#650211yescharacteristic
329C2×C4#650211yescharacteristic
330C2×C4#651311yescharacteristic
331C2×C4C2×C22⋊Dic711yescharacteristic
332C8—22no—
333C8—42no—
334C8—561no—
335C8D4×Dic711yescharacteristic
336C8D4×Dic711yescharacteristic
337C8#649811yescharacteristic
338C8#650011yescharacteristic
339C7#1307111yescharacteristic, complemented, Sylow
340C22—22no—
341C22—41no—
342C22—42no—
343C22—81no—
344C22—161no—
345C22#8981011yescharacteristic
346C22#8981311yescharacteristic
347C22#9000511yescharacteristic
348C4—22no—
349C4—81no—
350C4—1121no—
351C4#8981011yescharacteristic
352C4#8981311yescharacteristic
353C4C2×#640611yescharacteristic
354C4#8999211yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—161no—
360C2#1172943511yescharacteristic
subgroupG/H
centerC4#89810
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
Fitting, prime radicalC7×#3067C2
socleC2×C7#1893
cosocleC22×C7×C8C23
hypercenter#3067D7
hypocenterC7#13071
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
2. C2×#89912C2semidirect
3. C2×#89362C2semidirect
4. #11729830C2semidirect
5. #11729767C2semidirect
6. #11725540C2semidirect
7. #11725535C2semidirect
11. C7×#473C4semidirect
12. C7×#472C4semidirect
141. #473Dic7semidirect
142. #472Dic7semidirect
339. C7#13071semidirect

Maximal subgroups

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

subgroupindex
C7×#30672
C2×#899122
C2×#893622
#117298302
#117297672
#117255402
#117255352
#130717