Finite Group Databasebeta

← #13509363

Subgroups — #13509363

Order 1792.

554 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
1C7×#3169C211yesmaximal, characteristic, complemented
2#11735914C211yesmaximal, characteristic, complemented
3#11729140C211yesmaximal, characteristic, complemented
4#11729137C211yesmaximal, characteristic, complemented
5#11729046C211yesmaximal, characteristic, complemented
6#11729043C211yesmaximal, characteristic, complemented
7#11728828C212yesmaximal, complemented
8#11728827C212yesmaximal, complemented
9C4×#6401C212yesmaximal, complemented
10C4×#6400C212yesmaximal, complemented
11C4×#6386C211yesmaximal, characteristic, complemented
12C7×#532—21no—
13#90170—21no—
14C7×#500C2211yescharacteristic
15C7×#493C2211yescharacteristic
16C7×#486—21no—
17C7×#466—21no—
18C7×#466C2212yes—
19C4×C7×D8—21no—
20C4×C7×D8C2212yes—
21C2×C4×C7×C8C2211yescharacteristic
22C2×#6401C2212yes—
23C2×#6400C2212yes—
24#89888C2211yescharacteristic
25#89887C2211yescharacteristic
26#89886—21no—
27C2×#6386C2212yes—
28C4×Dic28C2211yescharacteristic
29#89469—21no—
30#89468C2211yescharacteristic
31C4×D56C2211yescharacteristic
32C4×#1132—21no—
33#89455C2211yescharacteristic
34#89357C2212yes—
35#89357C2212yes—
36#89287C2212yes—
37#89286C2212yes—
38#89285C2212yes—
39#89284C2212yes—
40#89256C2211yescharacteristic
41#89256C2211yescharacteristic
42C4×C7⋊C16C2212yes—
43#33737—71nomaximal, Sylow
44C2×C7×#41—41no—
45C2×C7×D8—21no—
46C2×C7×D8C2312yes—
47C22×C7×C8D411yescharacteristic
48C22×C7×C8C2311yescharacteristic
49C7×C4⋊Q8—21no—
50C7×#127—21no—
51C7×#124—41no—
52C7×#121—22no—
53C7×#121—41no—
54C4×C7×D4—22no—
55C4×C7×D4—41no—
56C2×C42×C7D411yescharacteristic
57#6493—21no—
58#6492—41no—
59C4×C22⋊D7—41no—
60#6487—21no—
61C2×Dic28C2311yescharacteristic
62C2×D56C2311yescharacteristic
63C2×#1132—21no—
64#6439—41no—
65#6438—21no—
66C4×D28—21no—
67#6435—41no—
68#6433—21no—
69C4×Dic14—21no—
70C7×#107—21no—
71C7×#107C2312yes—
72C7×#102—22no—
73C7×#102—22no—
74C4×C7×C8D411yescharacteristic
75C4×C7×C8D411yescharacteristic
76C4×C7×C8C2311yescharacteristic
77C4×C7×C8C2311yescharacteristic
78#6401—24no—
79#6400—24no—
80#6395—21no—
81#6395—21no—
82#6392C2311yescharacteristic
83#6392C2311yescharacteristic
84#6391—21no—
85#6390—21no—
86#6390—21no—
87#6386—24no—
88C2×C7⋊C16C2312yes—
89C2×C7⋊C16C2312yes—
90#3169D711yescharacteristic, complemented
91#3168—71no—
92#2255—71no—
93#2250—71no—
94#2244—71no—
95#2243—71no—
96#2220—72no—
97#2218—72no—
98C4×SD32—72no—
99C4×D16—72no—
100C4×M32—71no—
101C7×#41—84no—
102C2×C7×Q8—41no—
103C2×C7×D4—22no—
104C2×C7×D4—42no—
105C2×C7×D4—41no—
106C2×C7×D4—41no—
107C22×C4×C7—42no—
108C22×C4×C7D811yescharacteristic
109C22×C4×C7D811yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C2×C22⋊D7—41no—
112C2×D28—21no—
113C2×C4×D7—41no—
114C2×Dic14—21no—
115C7×D8—22no—
116C7×D8—22no—
117C7×D8—42no—
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8C2×D411yescharacteristic
121C2×C7×C8C2×D411yescharacteristic
122C2×C7×C8C2×D411yescharacteristic
123C2×C7×C8C2411yescharacteristic
124C7×C4⋊C4—22no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—42no—
127C7×C22⋊C4—42no—
128C42×C7D811yescharacteristic
129C42×C7D811yescharacteristic
130C42×C7C2×D411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C22⋊Dic7—41no—
133(C2×D7)⋊C4—41no—
134(C2×D7)⋊C4—41no—
135(C2×D7)⋊C4—41no—
136(C2×D7)⋊C4—41no—
137C4⋊Dic7—21no—
138C4⋊Dic7—21no—
139Dic7⋊C4—41no—
140Dic7⋊C4—41no—
141Dic7⋊C4—41no—
142Dic7⋊C4—41no—
143C4×Dic7—41no—
144Dic28—22no—
145D56—22no—
146#1132—42no—
147C7⋊C16—22no—
148C7⋊C16#4114yescomplemented
149#532—21no—
150#529—141no—
151C2×SD32—72no—
152C2×D16—72no—
153C2×M32—72no—
154#500C2×D711yescharacteristic
155#499—71no—
156#495—71no—
157#493C2×D711yescharacteristic
158#488—141no—
159#486—21no—
160#467—71no—
161#466—21no—
162#466—71no—
163#466C2×D712yes—
164#465—141no—
165C4×Q16—71no—
166C4×SD16—141no—
167C4×D8—21no—
168C4×D8—71no—
169C4×D8C2×D712yes—
170C2×C4×C8C2×D711yescharacteristic
171#367—72no—
172#366—72no—
173#358—72no—
174#357—72no—
175#357—72no—
176#357—72no—
177#346—71no—
178#346—71no—
179C4×C16—72no—
180C23×C7—42no—
181C23×C7C2×D811yescharacteristic
182C22×D7—41no—
183C7×Q8—82no—
184C7×D4—42no—
185C7×D4—42no—
186C7×D4—82no—
187C7×D4—82no—
188C7×D4—84no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—22no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—84no—
195C2×C4×C7C2×D811yescharacteristic
196C2×C4×C7C2×D811yescharacteristic
197C2×C4×C7C2×D811yescharacteristic
198C2×C4×C7C2×D811yescharacteristic
199C2×C4×C7C2×D811yescharacteristic
200C2×C4×C7C22×D411yescharacteristic
201C22⋊D7—82no—
202C2×Dic7—41no—
203C2×Dic7—41no—
204C2×Dic7—41no—
205D28—41no—
206D28—41no—
207C4×D7—82no—
208Dic14—41no—
209Dic14—41no—
210C7×C8—21no—
211C7×C8—21no—
212C7×C8#13512yes—
213C7×C8C2×#4112yes—
214C2×#41—41no—
215C2×Q16—71no—
216C2×SD16—141no—
217C2×D8—21no—
218C2×D8—71no—
219C2×D8C22×D712yes—
220C22×C8C22⋊D711yescharacteristic
221C22×C8C22×D711yescharacteristic
222C4⋊Q8—21no—
223#127—21no—
224#126—281no—
225#125—141no—
226#124—41no—
227#124—141no—
228#123—281no—
229C22⋊Q8—141no—
230#121—22no—
231#121—41no—
232#121—141no—
233C4×Q8—141no—
234C4×D4—22no—
235C4×D4—41no—
236C4×D4—141no—
237C4×D4—281no—
238#117—281no—
239C2×C42C22⋊D711yescharacteristic
240SD32—144no—
241D16—144no—
242M32—144no—
243C2×C16—72no—
244C2×C16—72no—
245#107—21no—
246#107—71no—
247#107—71no—
248#107C22×D712yes—
249#106—141no—
250#103—141no—
251#103—141no—
252#102—22no—
253#102—22no—
254#102—141no—
255#102—141no—
256C4×C8C22⋊D711yescharacteristic
257C4×C8C22⋊D711yescharacteristic
258C4×C8C22×D711yescharacteristic
259C4×C8C22×D711yescharacteristic
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—82no—
264C22×C7—82no—
265C22×C7—82no—
266C22×C7C22×D811yescharacteristic
267C2×D7—81no—
268C2×D7—81no—
269C2×D7—81no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—82no—
275C4×C7#50912yes—
276C4×C7#53211yescharacteristic
277C4×C7C2×#13511yescharacteristic
278Dic7—81no—
279Dic7—81no—
280Dic7—81no—
281#41—84no—
282C2×Q8—41no—
283C2×Q8—141no—
284C2×D4—22no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—141no—
289C2×D4—281no—
290C22×C4—42no—
291C22×C4—281no—
292C22×C4#114111yescharacteristic
293C22×C4#114111yescharacteristic
294C22×C4C2×C22⋊D711yescharacteristic
295Q16—142no—
296SD16—282no—
297D8—22no—
298D8—22no—
299D8—42no—
300D8—142no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8C2×C22⋊D711yescharacteristic
304C2×C8C2×C22⋊D711yescharacteristic
305C2×C8C2×C22⋊D711yescharacteristic
306C2×C8C23×D711yescharacteristic
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—141no—
310C4⋊C4—141no—
311C4⋊C4—281no—
312C4⋊C4—281no—
313C4⋊C4—281no—
314C4⋊C4—281no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—281no—
318C22⋊C4—281no—
319C22⋊C4—281no—
320C22⋊C4—281no—
321C22⋊C4—281no—
322C42—281no—
323C42#114111yescharacteristic
324C42#114111yescharacteristic
325C42C2×C22⋊D711yescharacteristic
326C42C2×C22⋊D711yescharacteristic
327C16—74no—
328C16—142no—
329C2×C7—41no—
330C2×C7—162no—
331C2×C7#316911yescharacteristic
332C2×C7C2×#50911yescharacteristic
333C2×C7#341411yescharacteristic
334D7—161no—
335C23—42no—
336C23—281no—
337C23C2×#114111yescharacteristic
338Q8—82no—
339Q8—281no—
340Q8—281no—
341D4—42no—
342D4—42no—
343D4—82no—
344D4—82no—
345D4—84no—
346D4—281no—
347D4—281no—
348D4—562no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—84no—
355C2×C4—281no—
356C2×C4—281no—
357C2×C4—281no—
358C2×C4—562no—
359C2×C4C2×#114111yescharacteristic
360C2×C4C2×#114111yescharacteristic
361C2×C4C2×#114111yescharacteristic
362C2×C4C2×#114111yescharacteristic
363C2×C4C2×#114111yescharacteristic
364C2×C4C22×C22⋊D711yescharacteristic
365C8—21no—
366C8—21no—
367C8#651312yes—
368C8D7×#4112yes—
369C7#3373711yescharacteristic, complemented, Sylow
370C22—21no—
371C22—21no—
372C22—21no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—561no—
377C22—561no—
378C22—561no—
379C22C22×#114111yescharacteristic
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—561no—
386C4—561no—
387C4—561no—
388C4#8991912yes—
389C4C2×#651311yescharacteristic
390C4#9052311yescharacteristic
391C2—41no—
392C2—162no—
393C2—1121no—
394C2#1174073611yescharacteristic
395C2C2×#8991911yescharacteristic
396C2#1174133911yescharacteristic
subgroupG/H
centerC2×C4C2×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#3169C2
socleC22×C7C22×D8
hypercenter#3169D7
hypocenterC7#33737
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C7×#3169C2semidirect
2. #11735914C2semidirect
3. #11729140C2semidirect
4. #11729137C2semidirect
5. #11729046C2semidirect
6. #11729043C2semidirect
7. #11728828C2semidirect
8. #11728827C2semidirect
9. C4×#6401C2semidirect
10. C4×#6400C2semidirect
11. C4×#6386C2semidirect
90. #3169D7semidirect
148. C7⋊C16#41semidirect
369. C7#33737semidirect

Maximal subgroups

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

subgroupindex
C7×#31692
#117359142
#117291402
#117291372
#117290462
#117290432
#117288282
#117288272
C4×#64012
C4×#64002
C4×#63862
#337377