Finite Group Databasebeta

← #13272008

Subgroups — #13272008

Order 1792.

614 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
1C2×#89923C211yesmaximal, characteristic
2C2×C7×#357C211yesmaximal, characteristic, complemented
3C2×#89357C211yesmaximal, characteristic, complemented
4C2×#89304C211yesmaximal, characteristic, complemented
5C2×C16×Dic7C211yesmaximal, characteristic, complemented
6#11728360C211yesmaximal, characteristic, complemented
7C2×#89286C211yesmaximal, characteristic, complemented
8C22×C7×D8C2211yescharacteristic
9C2×#6503—21no—
10C22×#1141—21no—
11C22×D56C2211yescharacteristic
12C22×C7×C16C2211yescharacteristic
13C2×C7×#107C2211yescharacteristic
14#89923—22no—
15#89923C414yescomplemented
16C2×#6392C2211yescharacteristic
17C2×C8×Dic7C2211yescharacteristic
18C22×C7⋊C16C2211yescharacteristic
19#89748—21no—
20C2×#6381—21no—
21C7×#357—22no—
22#89357—22no—
23#89304—22no—
24C16×Dic7—22no—
25#89286—22no—
26#13202—71nomaximal, Sylow
27C22×C7×D4—21no—
28C22×C22⋊D7—42no—
29C22×D28—21no—
30C2×C7×D8—21no—
31C2×C7×D8—21no—
32C2×C7×D8—21no—
33C2×C7×D8—21no—
34C2×C7×D8—21no—
35C2×C7×D8C2×C412yes—
36C22×C7×C8C2311yescharacteristic
37C2×C7×C4⋊C4—21no—
38#6503—24no—
39#6503—42no—
40C2×#1141—24no—
41C2×#1141—42no—
42C2×#1141—42no—
43C2×C4⋊Dic7—21no—
44C2×Dic7⋊C4—42no—
45C2×C4×Dic7D411yescharacteristic
46C22×C7⋊C8D411yescharacteristic
47C2×D56—21no—
48C2×D56—21no—
49C2×D56—21no—
50C2×D56—21no—
51C2×D56—21no—
52C2×D56C2×C412yes—
53C2×C7×C16—21no—
54C2×C7×C16—21no—
55C2×C7×C16D412yescomplemented
56C7×#107—21no—
57C7×#107—21no—
58#6392—21no—
59#6392—21no—
60C8×Dic7C2×C412yes—
61C8×Dic7D412yes—
62C2×C7⋊C16—21no—
63C2×C7⋊C16—21no—
64C2×C7⋊C16D412yescomplemented
65#6381—42no—
66C2×#493—71no—
67C2×#357—71no—
68C2×#357—71no—
69C2×#357—71no—
70C2×#357D711yescharacteristic, complemented
71C2×C4×C16—71no—
72#1845—71no—
73C24×C7—41no—
74C23×D7—41no—
75C2×C7×D4—22no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C22×C4×C7—41no—
82C22×C4×C7C2×D411yescharacteristic
83C2×C22⋊D7—48no—
84C2×C22⋊D7—84no—
85C2×C22⋊D7—84no—
86C22×Dic7—41no—
87C22×Dic7D812yes—
88C2×D28—22no—
89C2×D28—41no—
90C2×D28—41no—
91C2×D28—41no—
92C2×D28—41no—
93C2×D28—41no—
94C7×D8—22no—
95C7×D8—22no—
96C7×D8—41no—
97C7×D8—41no—
98C2×C7×C8C22×C411yescharacteristic
99C2×C7×C8C2×D411yescharacteristic
100C2×C7×C8C2×D411yescharacteristic
101C2×C7×C8C2×D411yescharacteristic
102C2×C7×C8#4111yescharacteristic
103C2×C7×C8#4111yescharacteristic
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106#1141—48no—
107C4⋊Dic7—41no—
108C4⋊Dic7—41no—
109Dic7⋊C4—84no—
110C4×Dic7C22⋊C412yes—
111C4×Dic7SD1612yes—
112C2×C7⋊C8C22⋊C412yes—
113C2×C7⋊C8D812yes—
114C2×C7⋊C8SD1612yes—
115D56—22no—
116D56—22no—
117D56—41no—
118D56—41no—
119C7×C16—22no—
120C7⋊C16—22no—
121C22×D8—71no—
122C22×D8—141no—
123C22×D8C2×D711yescharacteristic
124C2×#127—141no—
125C22×C16—71no—
126C22×C16C2×D711yescharacteristic
127#493—74no—
128#493—142no—
129C2×#107—71no—
130C2×#107—141no—
131C2×#107C2×D711yescharacteristic
132C2×C4×C8—71no—
133#384—141no—
134#357—22no—
135#357—142no—
136#357—142no—
137#357—142no—
138C4×C16—142no—
139C23×C7—42no—
140C23×C7—81no—
141C23×C7—81no—
142C23×C7—82no—
143C23×C7—82no—
144C23×C7C2×D811yescharacteristic
145C22×D7—42no—
146C22×D7—81no—
147C22×D7—81no—
148C22×D7—82no—
149C22×D7—82no—
150C7×D4—42no—
151C7×D4—42no—
152C7×D4—81no—
153C7×D4—81no—
154C2×C4×C7—81no—
155C2×C4×C7—82no—
156C2×C4×C7C2×C22⋊C411yescharacteristic
157C2×C4×C7#12411yescharacteristic
158C2×C4×C7#12711yescharacteristic
159C2×C4×C7C2×D811yescharacteristic
160C2×C4×C7C2×SD1611yescharacteristic
161C2×C4×C7C2×SD1611yescharacteristic
162C22⋊D7—816no—
163C2×Dic7—81no—
164C2×Dic7—82no—
165C2×Dic7#10214yes—
166C2×Dic7D1614yes—
167C2×Dic7SD3214yes—
168D28—42no—
169D28—42no—
170D28—81no—
171D28—81no—
172C7×C8C4×D411yescharacteristic
173C7×C8C4×D411yescharacteristic
174C7×C8#12111yescharacteristic
175C7×C8#12111yescharacteristic
176C7⋊C8#10214yes—
177C22×D4—21no—
178C22×D4—141no—
179C22×D4—282no—
180C2×D8—21no—
181C2×D8—21no—
182C2×D8—21no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8—72no—
186C2×D8—144no—
187C2×D8—141no—
188C2×D8—141no—
189C2×D8—141no—
190C2×D8—141no—
191C2×D8—141no—
192C2×D8—282no—
193C2×D8—282no—
194C2×D8C4×D712yes—
195C22×C8—71no—
196C22×C8C22×D711yescharacteristic
197#127—144no—
198#127—282no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—141no—
201C2×C4⋊C4—282no—
202C2×C42—71no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—72no—
206C2×C16—141no—
207C2×C16—141no—
208C2×C16C22⋊D712yescomplemented
209#107—21no—
210#107—21no—
211#107—141no—
212#107—141no—
213#107—282no—
214C4×C8—72no—
215C4×C8—72no—
216C22×C7—84no—
217C22×C7—82no—
218C22×C7—162no—
219C22×C7—161no—
220C22×C7—161no—
221C22×C7C2×#10211yescharacteristic
222C22×C7#48611yescharacteristic
223C22×C7#49311yescharacteristic
224C22×C7C2×D1612yes—
225C22×C7C2×SD3212yes—
226C2×D7—84no—
227C2×D7—82no—
228C2×D7—162no—
229C2×D7—161no—
230C2×D7—161no—
231C4×C7—161no—
232C4×C7#39011yescharacteristic
233C4×C7C2×#10211yescharacteristic
234C4×C7#48611yescharacteristic
235C4×C7#49211yescharacteristic
236Dic7—161no—
237Dic7#35718yescomplemented
238C24—41no—
239C24—281no—
240C2×D4—22no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—142no—
247C2×D4—288no—
248C2×D4—281no—
249C2×D4—281no—
250C2×D4—281no—
251C2×D4—281no—
252C2×D4—281no—
253C2×D4—564no—
254C2×D4—564no—
255C22×C4—41no—
256C22×C4—72no—
257C22×C4—281no—
258C22×C4D4×D711yescharacteristic
259D8—22no—
260D8—22no—
261D8—41no—
262D8—41no—
263D8—142no—
264D8—142no—
265D8—288no—
266D8—281no—
267D8—281no—
268C2×C8—72no—
269C2×C8—72no—
270C2×C8—72no—
271C2×C8C2×C4×D711yescharacteristic
272C2×C8#115711yescharacteristic
273C2×C8D4×D711yescharacteristic
274C2×C8D4×D711yescharacteristic
275C2×C8#115911yescharacteristic
276C2×C8C2×C22⋊D711yescharacteristic
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—281no—
280C4⋊C4—281no—
281C4⋊C4—564no—
282C42—72no—
283C42—72no—
284C16—22no—
285C16—142no—
286C2×C7—162no—
287C2×C7#195711yescharacteristic
288C2×C7C2×#35712yes—
289C2×C7#224012yes—
290C2×C7#224611yescharacteristic
291C2×C7#224811yescharacteristic
292D7—162no—
293C23—42no—
294C23—81no—
295C23—82no—
296C23—82no—
297C23—81no—
298C23—282no—
299C23—561no—
300C23—562no—
301C23—562no—
302C23—561no—
303C23D7×D811yescharacteristic
304D4—42no—
305D4—42no—
306D4—81no—
307D4—81no—
308D4—282no—
309D4—282no—
310D4—5616no—
311D4—561no—
312D4—561no—
313C2×C4—74no—
314C2×C4—74no—
315C2×C4—74no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—561no—
319C2×C4—562no—
320C2×C4D7×C22⋊C411yescharacteristic
321C2×C4#644811yescharacteristic
322C2×C4D7×D811yescharacteristic
323C2×C4D7×SD1611yescharacteristic
324C2×C4D7×SD1611yescharacteristic
325C2×C4#650311yescharacteristic
326C8—74no—
327C8(C22⋊D7)⋊C411yescharacteristic
328C8#644711yescharacteristic
329C8C4×C22⋊D711yescharacteristic
330C8#650211yescharacteristic
331C7#1320211yescharacteristic, complemented, Sylow
332C22—82no—
333C22—84no—
334C22—161no—
335C22—161no—
336C22—162no—
337C22—562no—
338C22—564no—
339C22—1121no—
340C22—1121no—
341C22—1122no—
342C22#8953211yescharacteristic
343C22D7×#10211yescharacteristic
344C22D7×D1612yes—
345C22D7×SD3212yes—
346C22#8992311yescharacteristic
347C4—78no—
348C4—161no—
349C4—1121no—
350C4#8953211yescharacteristic
351C4D7×#10211yescharacteristic
352C4#8972911yescharacteristic
353C4#8994711yescharacteristic
354C2—162no—
355C2—1122no—
356C2#1172693712yes—
357C2D7×#35712yes—
358C2#1172791111yescharacteristic
359C2#1172921811yescharacteristic
360C2#1172923311yescharacteristic
subgroupG/H
centerC23D7×D8
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC2×C7×#357C2
socleC23×C7C2×D8
prime radicalC2×#89923C2
cosocleC22×C7×C8C23
hypercenterC2×#357D7
hypocenterC7#13202
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×C7×#357C2semidirect
3. C2×#89357C2semidirect
4. C2×#89304C2semidirect
5. C2×C16×Dic7C2semidirect
6. #11728360C2semidirect
7. C2×#89286C2semidirect
15. #89923C4semidirect
55. C2×C7×C16D4semidirect
64. C2×C7⋊C16D4semidirect
70. C2×#357D7semidirect
208. C2×C16C22⋊D7semidirect
237. Dic7#357semidirect
331. C7#13202semidirect

Maximal subgroups

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

subgroupindex
C2×#899232
C2×C7×#3572
C2×#893572
C2×#893042
C2×C16×Dic72
#117283602
C2×#892862
#132027