Finite Group Databasebeta

← #13508467

Subgroups — #13508467

Order 1792.

574 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11736610C211yesmaximal, characteristic
2C2×#89646C211yesmaximal, characteristic, complemented
3C7×#2159C211yesmaximal, characteristic, complemented
4#11729193C211yesmaximal, characteristic
5#11729174C211yesmaximal, characteristic, complemented
6#11728403C211yesmaximal, characteristic, complemented
7#11728352C211yesmaximal, characteristic
8#11727109C212yesmaximal, complemented
9#11727107C212yesmaximal, complemented
10#11727089C212yesmaximal, complemented
11#11727087C212yesmaximal, complemented
12C2×#6467C2211yescharacteristic
13#90213—21no—
14C2×#6451—21no—
15C2×C7×M32C2211yescharacteristic
16C7×#427C2211yescharacteristic
17C2×C7×#107C2211yescharacteristic
18C2×#6392C2211yescharacteristic
19C2×C8×Dic7C2211yescharacteristic
20C2×#6386C2211yescharacteristic
21#89774—21no—
22C2×#6381—21no—
23#89646—21no—
24#89646—21no—
25#89646C2212yes—
26#89646C2212yes—
27#89645C2211yescharacteristic
28#89645C2211yescharacteristic
29#89644C2211yescharacteristic
30#89644C2211yescharacteristic
31#89624—21no—
32#89624—21no—
33C7×#367C2212yes—
34C7×#366C2212yes—
35#89304C2212yes—
36#89298C2212yes—
37Dic7⋊C16C2212yes—
38Dic7⋊C16C2212yes—
39#89287C2212yes—
40#89286C2212yes—
41#89285C2212yes—
42#89284C2212yes—
43#33629—71nomaximal, Sylow
44C2×#1157—21no—
45C22×Dic14—41no—
46C22×C7×C8C2311yescharacteristic
47C7×#117—21no—
48C2×C7×C4⋊C4—21no—
49C2×C4⋊Dic7—21no—
50C2×Dic7⋊C4—41no—
51C2×C4×Dic7D411yescharacteristic
52C22×C7⋊C8D411yescharacteristic
53C2×Dic28C2311yescharacteristic
54#6467—22no—
55#6467Q814yescomplemented
56C2×D56C2311yescharacteristic
57C2×#1132—21no—
58#6456—21no—
59#6456—21no—
60#6451—22no—
61#6451—22no—
62#6451—41no—
63#6451—41no—
64#6450—21no—
65#6450—21no—
66(C22⋊D7)⋊C4—41no—
67(C22⋊D7)⋊C4—41no—
68#6435—41no—
69C42×D7—41no—
70C7×M32—22no—
71C2×C7×C16C2312yes—
72C7×#107—21no—
73C7×#107C2311yescharacteristic
74C7×#107C2311yescharacteristic
75C7×#107C2312yes—
76C7×#106—21no—
77#6392—21no—
78#6392C2312yes—
79C8×Dic7D411yescharacteristic
80C8×Dic7D411yescharacteristic
81C8×Dic7C2311yescharacteristic
82C8×Dic7C2311yescharacteristic
83#6386—22no—
84C2×C7⋊C16C2312yes—
85#6384—21no—
86#6384—21no—
87#6383—21no—
88#6383—21no—
89#6381—22no—
90#6381—22no—
91C2×#500—71no—
92C4×#135—71no—
93#2227—72no—
94#2226—72no—
95#2225—72no—
96#2224—72no—
97#2159—71no—
98#2159D711yescharacteristic, complemented
99#2150—71no—
100#2139—71no—
101#2139—71no—
102C22×C4×C7—41no—
103C22×C4×C7C2×D411yescharacteristic
104C2×C22⋊D7—41no—
105C22×Dic7—41no—
106C22×Dic7D811yescharacteristic
107C22×Dic7D811yescharacteristic
108#1157—24no—
109#1157—42no—
110C2×D28—21no—
111C2×C4×D7—41no—
112C2×C4×D7—41no—
113C2×C4×D7—41no—
114C2×Dic14—21no—
115C2×Dic14—42no—
116C2×Dic14—42no—
117C2×Dic14—84no—
118C2×C7×C8C2×D411yescharacteristic
119C2×C7×C8C2×Q812yes—
120C2×C7×C8#4112yes—
121C2×C7×C8C2411yescharacteristic
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—41no—
127C42×C7—41no—
128(C2×D7)⋊C4—41no—
129(C2×D7)⋊C4—41no—
130C4⋊Dic7—22no—
131C4⋊Dic7—41no—
132Dic7⋊C4—41no—
133Dic7⋊C4—41no—
134Dic7⋊C4—42no—
135Dic7⋊C4—42no—
136C4×Dic7—41no—
137C4×Dic7—41no—
138C4×Dic7D811yescharacteristic
139C4×Dic7D811yescharacteristic
140C4×Dic7C2×D411yescharacteristic
141C4×Dic7C2×D411yescharacteristic
142C2×C7⋊C8—22no—
143C2×C7⋊C8C2×D411yescharacteristic
144C2×C7⋊C8C2×D411yescharacteristic
145Dic28—21no—
146Dic28C2×Q812yescomplemented
147D56—21no—
148D56C2×Q812yescomplemented
149#1132—24no—
150C7×C16—22no—
151C7⋊C16—22no—
152C2×#135—71no—
153C2×C4⋊Q8—141no—
154C4×#41—141no—
155C2×M32—71no—
156C2×M32C2×D711yescharacteristic
157#500—72no—
158#500—72no—
159#500—141no—
160#500—141no—
161C4×Q16—71no—
162C4×Q16—71no—
163C4×SD16—141no—
164C4×SD16—141no—
165C4×D8—71no—
166C4×D8—71no—
167#427C2×D711yescharacteristic
168C2×#107—71no—
169C2×#107—141no—
170C2×#107C2×D711yescharacteristic
171#416—141no—
172C2×C4×C8—71no—
173#367—72no—
174#367C2×D712yes—
175#366—72no—
176#366C2×D712yes—
177C4⋊C16—72no—
178C4⋊C16—72no—
179#358—72no—
180#358—72no—
181#357—72no—
182#357—72no—
183C23×C7C2×D811yescharacteristic
184C22×D7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—82no—
189C2×C4×C7C22⋊Q812yes—
190C2×C4×C7C2×D811yescharacteristic
191C2×C4×C7#13712yes—
192C2×C4×C7C22×D411yescharacteristic
193C22⋊D7—44no—
194C2×Dic7—22no—
195C2×Dic7—22no—
196C2×Dic7—41no—
197C2×Dic7—42no—
198C2×Dic7—82no—
199C2×Dic7C2×D811yescharacteristic
200C2×Dic7C2×D811yescharacteristic
201C2×Dic7C2×D811yescharacteristic
202C2×Dic7C2×D811yescharacteristic
203D28—22no—
204D28—41no—
205C4×D7—44no—
206C4×D7—82no—
207C4×D7—82no—
208Dic14—22no—
209Dic14—41no—
210Dic14—84no—
211Dic14—84no—
212C7×C8C22⋊Q812yes—
213C7×C8C22×Q811yescharacteristic
214C7×C8C2×#4111yescharacteristic
215C7⋊C8—21no—
216C7⋊C8—21no—
217C2×#41—141no—
218C22×Q8—281no—
219#135—74no—
220#135—142no—
221C2×Q16—71no—
222C2×SD16—141no—
223C2×D8—71no—
224C22×C8—71no—
225C22×C8C22×D711yescharacteristic
226C4⋊Q8—142no—
227C4⋊Q8—142no—
228C4⋊Q8—281no—
229C4⋊Q8—281no—
230C4×Q8—141no—
231C4×Q8—141no—
232C4×D4—141no—
233C4×D4—141no—
234C4×D4—281no—
235C4×D4—281no—
236#117—21no—
237#117—281no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—141no—
240C2×C4⋊C4—281no—
241C2×C42—71no—
242C2×C42—281no—
243M32—22no—
244M32—142no—
245C2×C16—72no—
246C2×C16C22×D712yes—
247#107—21no—
248#107—72no—
249#107—142no—
250#107—142no—
251#107—141no—
252#107C22×D712yes—
253#107C22×D711yescharacteristic
254#107C22×D711yescharacteristic
255#106—21no—
256#103—141no—
257#103—141no—
258#102—141no—
259#102—141no—
260C4×C8—71no—
261C4×C8—71no—
262C4×C8—71no—
263C4×C8—71no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7#47412yes—
267C22×C7C22×D811yescharacteristic
268C2×D7—42no—
269C2×D7—81no—
270C2×D7—81no—
271C4×C7—82no—
272C4×C7—81no—
273C4×C7—81no—
274C4×C7#47412yes—
275C4×C7C2×C22⋊Q811yescharacteristic
276C4×C7C2×#13711yescharacteristic
277Dic7—21no—
278Dic7—21no—
279Dic7—42no—
280Dic7—82no—
281Dic7#50912yes—
282Dic7#51012yes—
283#41—144no—
284#41—282no—
285C2×Q8—141no—
286C2×Q8—282no—
287C2×Q8—282no—
288C2×Q8—564no—
289C2×D4—141no—
290C2×D4—281no—
291C22×C4—41no—
292C22×C4—71no—
293C22×C4—71no—
294C22×C4—281no—
295C22×C4—281no—
296C22×C4—281no—
297C22×C4—281no—
298C22×C4D4×D711yescharacteristic
299Q16—72no—
300Q16—141no—
301SD16—144no—
302D8—72no—
303D8—141no—
304C2×C8—71no—
305C2×C8—71no—
306C2×C8—142no—
307C2×C8#115712yes—
308C2×C8D4×D711yescharacteristic
309C2×C8Q8×D712yes—
310C2×C8C23×D711yescharacteristic
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—142no—
316C4⋊C4—282no—
317C4⋊C4—282no—
318C4⋊C4—281no—
319C4⋊C4—281no—
320C4⋊C4—281no—
321C22⋊C4—41no—
322C22⋊C4—281no—
323C22⋊C4—281no—
324C42—41no—
325C42—71no—
326C42—71no—
327C42—71no—
328C42—71no—
329C42—281no—
330C42—281no—
331C16—22no—
332C16—142no—
333C2×C7—22no—
334C2×C7C2×#47411yescharacteristic
335C2×C7C2×#50911yescharacteristic
336C2×C7C2×#51011yescharacteristic
337D7—82no—
338C23—281no—
339C23D7×D811yescharacteristic
340Q8—142no—
341Q8—281no—
342Q8—564no—
343Q8—564no—
344D4—142no—
345D4—284no—
346D4—281no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—71no—
351C2×C4—71no—
352C2×C4—71no—
353C2×C4—71no—
354C2×C4—82no—
355C2×C4—142no—
356C2×C4—142no—
357C2×C4—281no—
358C2×C4—284no—
359C2×C4—282no—
360C2×C4—562no—
361C2×C4—562no—
362C2×C4—562no—
363C2×C4#645912yes—
364C2×C4D7×D811yescharacteristic
365C2×C4#648112yes—
366C2×C4C2×D4×D711yescharacteristic
367C8—141no—
368C8—141no—
369C8#645912yes—
370C8C2×#115711yescharacteristic
371C8C2×Q8×D711yescharacteristic
372C7#3362911yescharacteristic, complemented, Sylow
373C22—21no—
374C22—21no—
375C22—282no—
376C22—561no—
377C22—561no—
378C22#8966212yes—
379C22C2×D7×D811yescharacteristic
380C4—72no—
381C4—72no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4—141no—
386C4—141no—
387C4—282no—
388C4—562no—
389C4#8966212yes—
390C4C2×#645911yescharacteristic
391C4C2×#648111yescharacteristic
392C2—22no—
393C2—562no—
394C2C2×#8966211yescharacteristic
395C2D7×#50911yescharacteristic
396C2D7×#51011yescharacteristic
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2159C2
socleC22×C7C22×D8
prime radicalC2×#6467C22
hypercenter#2159D7
hypocenterC7#33629
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
2. C2×#89646C2semidirect
3. C7×#2159C2semidirect
5. #11729174C2semidirect
6. #11728403C2semidirect
8. #11727109C2semidirect
9. #11727107C2semidirect
10. #11727089C2semidirect
11. #11727087C2semidirect
55. #6467Q8semidirect
98. #2159D7semidirect
146. Dic28C2×Q8semidirect
148. D56C2×Q8semidirect
372. C7#33629semidirect

Maximal subgroups

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

subgroupindex
#117366102
C2×#896462
C7×#21592
#117291932
#117291742
#117284032
#117283522
#117271092
#117271072
#117270892
#117270872
#336297