Finite Group Databasebeta

← #13319734

Subgroups — #13319734

Order 1792.

1174 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90237C211yesmaximal, characteristic
2#11736709C212yesmaximal, complemented
3C7×#1460C211yesmaximal, characteristic, complemented
4#11729355C211yesmaximal, characteristic, complemented
5#11728536C212yesmaximal, complemented
6#11725857C214yesmaximal, complemented
7#11725856C214yesmaximal, complemented
8C22×#1159C411yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C2×D4×Dic7C411yescharacteristic
11C2×C4×C22⋊D7C412yes—
12C22×Dic7⋊C4C412yes—
13#90237—24no—
14#90237—24no—
15C2×D7×C4⋊C4C411yescharacteristic
16C2×#6450C411yescharacteristic
17C2×(C22⋊D7)⋊C4C2212yes—
18C2×#6440C2212yes—
19C2×#6435C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C7×#431C2212yes—
22#89697C2212yes—
23#89603—24no—
24#89491C2218yes—
25#89451—24no—
26C7×#323C2214yes—
27#89320C2214yes—
28#89264C2214yes—
29#89263C2214yes—
30#17372—71nomaximal, Sylow
31C22×C7×D4C2×C411yescharacteristic
32C23×C4×C7C2×C412yes—
33C22×C22⋊D7C2×C412yes—
34C23×Dic7C2×C412yes—
35C23×Dic7C2×C412yes—
36C2×#1159—24no—
37C2×#1159—24no—
38C2×#1159—24no—
39C22×C4×D7C2×C411yescharacteristic
40C22×C4×D7C2×C412yes—
41C22×Dic14C2×C411yescharacteristic
42C4×C7×D4—22no—
43C4×C7×D4—22no—
44C2×C7×C4⋊C4C2×C411yescharacteristic
45C2×C7×C22⋊C4C2312yes—
46C2×C42×C7C2311yescharacteristic
47C2×C22⋊Dic7C2×C412yes—
48D4×Dic7—24no—
49C4×C22⋊D7—24no—
50C4×C22⋊D7—24no—
51C2×(C2×D7)⋊C4C2×C411yescharacteristic
52C2×(C2×D7)⋊C4C2311yescharacteristic
53C2×C4⋊Dic7C2×C411yescharacteristic
54C2×Dic7⋊C4—24no—
55C2×Dic7⋊C4—24no—
56C2×Dic7⋊C4C2×C411yescharacteristic
57C2×Dic7⋊C4C2×C412yes—
58C2×Dic7⋊C4C2×C412yes—
59C2×Dic7⋊C4C2×C412yes—
60C2×Dic7⋊C4C2311yescharacteristic
61C2×C4×Dic7C2×C411yescharacteristic
62C2×C4×Dic7C2×C412yes—
63C2×C4×Dic7C2312yes—
64D7×C4⋊C4—24no—
65#6450—24no—
66(C22⋊D7)⋊C4—28no—
67#6440—28no—
68#6435—22no—
69#6435—22no—
70C4×Dic14D414yes—
71C4×Dic14D414yes—
72C7×C4⋊C8—22no—
73C7×C22⋊C8C2314yes—
74C4×C7×C8—22no—
75C22⋊(C7⋊C8)C2314yes—
76#6389—28no—
77Dic7⋊C8—28no—
78C4⋊(C7⋊C8)—22no—
79C4×C7⋊C8—22no—
80C2×#520—71no—
81#2983—72no—
82#1466—74no—
83#1464—74no—
84#1460—71no—
85#1460D711yescharacteristic, complemented
86#1458—72no—
87C24×C7C22×C412yes—
88C23×D7C22×C411yescharacteristic
89C2×C7×D4—22no—
90C2×C7×D4—22no—
91C2×C7×D4M1614yes—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—82no—
95C22×C4×C7C22×C412yes—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C22×C411yescharacteristic
98C22×C4×C7C2411yescharacteristic
99C2×C22⋊D7—24no—
100C2×C22⋊D7—24no—
101C2×C22⋊D7—24no—
102C22×Dic7—24no—
103C22×Dic7—24no—
104C22×Dic7—24no—
105C22×Dic7—24no—
106C22×Dic7—82no—
107C22×Dic7C22×C412yes—
108C22×Dic7C22×C411yescharacteristic
109C22×Dic7C22×C411yescharacteristic
110C22×Dic7C22×C411yescharacteristic
111C22×Dic7C22×C412yes—
112#1159—232no—
113C2×C4×D7—22no—
114C2×C4×D7—22no—
115C2×C4×D7—24no—
116C2×C4×D7—82no—
117C2×C4×D7M1614yes—
118C2×Dic14—22no—
119C2×Dic14—22no—
120C2×Dic14C22⋊C414yes—
121C2×C7×C8—24no—
122C7×C4⋊C4—22no—
123C7×C22⋊C4—24no—
124C42×C7C2×D412yes—
125C42×C7C2×D412yes—
126C22⋊Dic7—24no—
127(C2×D7)⋊C4—21no—
128(C2×D7)⋊C4—21no—
129(C2×D7)⋊C4—22no—
130C4⋊Dic7C22⋊C414yes—
131Dic7⋊C4—21no—
132Dic7⋊C4—21no—
133Dic7⋊C4—22no—
134Dic7⋊C4—24no—
135Dic7⋊C4—24no—
136Dic7⋊C4C22⋊C414yes—
137Dic7⋊C4C22⋊C414yes—
138C4×Dic7—22no—
139C4×Dic7—24no—
140C4×Dic7C2×D418yes—
141C2×C7⋊C8—24no—
142C22×#41—71no—
143#520—144no—
144#520—144no—
145C2×C4×Q8—71no—
146C2×C4×Q8—71no—
147C2×C4×D4—71no—
148C2×C4×D4—72no—
149C2×C4×D4—72no—
150C2×C4×D4C2×D711yescharacteristic
151C2×#117—72no—
152C2×#117—71no—
153C22×C4⋊C4—72no—
154C22×C4⋊C4—71no—
155#446—144no—
156#446—144no—
157#432—78no—
158#431—72no—
159#431C2×D712yes—
160#324—74no—
161#323—74no—
162#323—74no—
163#323C2×D714yes—
164C23×C7—22no—
165C23×C7—22no—
166C23×C7—42no—
167C23×C7—42no—
168C23×C7C2×M1614yes—
169C23×C7C23×C411yescharacteristic
170C22×D7—21no—
171C22×D7—21no—
172C22×D7—41no—
173C22×D7—41no—
174C22×D7C2×M1612yes—
175C7×D4—28no—
176C2×C4×C7—22no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—42no—
180C2×C4×C7—82no—
181C2×C4×C7—82no—
182C2×C4×C7C2×C22⋊C411yescharacteristic
183C2×C4×C7C2×C22⋊C411yescharacteristic
184C2×C4×C7C2×C22⋊C412yes—
185C2×C4×C7C2×M1612yes—
186C2×C4×C7C2×M1612yes—
187C2×C4×C7C22×D412yes—
188C22⋊D7—216no—
189C2×Dic7—216no—
190C2×Dic7—21no—
191C2×Dic7—22no—
192C2×Dic7—22no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—21no—
196C2×Dic7—42no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7—84no—
200C2×Dic7C2×C22⋊C414yes—
201C2×Dic7C2×C22⋊C414yes—
202C2×Dic7C2×M1612yes—
203C4×D7—28no—
204C4×D7—84no—
205Dic14—28no—
206C7×C8—44no—
207C7⋊C8—44no—
208C2×#41—144no—
209C2×#41—144no—
210C2×#41—144no—
211C22×Q8—71no—
212C22×D4—72no—
213C22×D4C4×D711yescharacteristic
214C23×C4—71no—
215C23×C4—72no—
216C23×C4—72no—
217C23×C4—72no—
218C23×C4C4×D712yes—
219C4×Q8—74no—
220C4×Q8—74no—
221C4×Q8—144no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—148no—
225C4×D4—144no—
226C4×D4—144no—
227C4×D4—144no—
228#117—148no—
229#117—142no—
230#117—142no—
231C2×C4⋊C4—72no—
232C2×C4⋊C4—71no—
233C2×C4⋊C4—72no—
234C2×C4⋊C4—71no—
235C2×C4⋊C4—72no—
236C2×C4⋊C4—71no—
237C2×C4⋊C4—144no—
238C2×C4⋊C4—144no—
239C2×C4⋊C4—144no—
240C2×C4⋊C4C4×D711yescharacteristic
241C2×C22⋊C4—72no—
242C2×C22⋊C4—71no—
243C2×C22⋊C4—71no—
244C2×C22⋊C4C22×D712yes—
245C2×C42—72no—
246C2×C42—72no—
247C2×C42—71no—
248C2×C42C22×D711yescharacteristic
249C4⋊C8—22no—
250C4⋊C8—148no—
251C4⋊C8—142no—
252C22⋊C8—74no—
253C22⋊C8C22×D714yes—
254#97—148no—
255C4×C8—22no—
256C4×C8—142no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—24no—
260C22×C7—44no—
261C22×C7—44no—
262C22×C7—42no—
263C22×C7—42no—
264C22×C7C22×C22⋊C411yescharacteristic
265C22×C7C22×M1611yescharacteristic
266C22×C7C22×M1611yescharacteristic
267C2×D7—22no—
268C2×D7—42no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—42no—
272C4×C7—22no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—42no—
276C4×C7#40714yes—
277Dic7—24no—
278Dic7—22no—
279Dic7—24no—
280Dic7—41no—
281Dic7—41no—
282C24—71no—
283C24C2×C4×D712yes—
284#41—1432no—
285C2×Q8—74no—
286C2×Q8—142no—
287C2×Q8—142no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—144no—
291C2×D4—144no—
292C2×D4—144no—
293C2×D4#113114yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—72no—
297C22×C4—71no—
298C22×C4—72no—
299C22×C4—74no—
300C22×C4—71no—
301C22×C4—71no—
302C22×C4—82no—
303C22×C4—144no—
304C22×C4—144no—
305C22×C4—144no—
306C22×C4—144no—
307C22×C4—144no—
308C22×C4—142no—
309C22×C4—142no—
310C22×C4—562no—
311C22×C4—562no—
312C22×C4C2×C4×D712yes—
313C22×C4C2×C4×D711yescharacteristic
314C22×C4C2×C4×D711yescharacteristic
315C22×C4C23×D711yescharacteristic
316C2×C8—24no—
317C2×C8—144no—
318C4⋊C4—22no—
319C4⋊C4—74no—
320C4⋊C4—74no—
321C4⋊C4—74no—
322C4⋊C4—144no—
323C4⋊C4—144no—
324C4⋊C4—141no—
325C4⋊C4—141no—
326C4⋊C4—142no—
327C22⋊C4—24no—
328C22⋊C4—144no—
329C22⋊C4—141no—
330C22⋊C4—141no—
331C22⋊C4—142no—
332C42—78no—
333C42—144no—
334C42—142no—
335C42D4×D712yes—
336C42D4×D712yes—
337C2×C7—21no—
338C2×C7—21no—
339C2×C7—44no—
340C2×C7C2×#40711yescharacteristic
341C2×C7#288411yescharacteristic
342C2×C7#288811yescharacteristic
343D7—42no—
344C23—22no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—72no—
349C23—141no—
350C23—141no—
351C23—281no—
352C23—281no—
353C23C2×#113114yes—
354C23C22×C4×D711yescharacteristic
355Q8—148no—
356D4—28no—
357D4—1416no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—74no—
363C2×C4—74no—
364C2×C4—72no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—1416no—
368C2×C4—141no—
369C2×C4—142no—
370C2×C4—141no—
371C2×C4—142no—
372C2×C4—148no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—282no—
376C2×C4—281no—
377C2×C4—281no—
378C2×C4—564no—
379C2×C4—564no—
380C2×C4D7×C22⋊C411yescharacteristic
381C2×C4D7×C22⋊C411yescharacteristic
382C2×C4D7×C22⋊C412yes—
383C2×C4C2×#113112yes—
384C2×C4D7×M1612yes—
385C2×C4C2×D4×D712yes—
386C8—44no—
387C8—284no—
388C7#1737211yescharacteristic, complemented, Sylow
389C22—21no—
390C22—21no—
391C22—24no—
392C22—44no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—142no—
397C22—282no—
398C22—282no—
399C22—281no—
400C22—281no—
401C22C2×D7×C22⋊C411yescharacteristic
402C22C22×#113111yescharacteristic
403C22C2×D7×M1611yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—42no—
408C4—144no—
409C4—142no—
410C4—144no—
411C4—281no—
412C4—281no—
413C4#8949814yes—
414C2—21no—
415C2—21no—
416C2—44no—
417C2—282no—
418C2C2×#8949811yescharacteristic
419C2#1173613411yescharacteristic
420C2#1173614211yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1460C2
socleC22×C7C22×C22⋊C4
prime radicalC22×#1159C4
cosocleC22×C4×C7C24
hypercenter#1460D7
hypocenterC7#17372
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #11736709C2semidirect
3. C7×#1460C2semidirect
4. #11729355C2semidirect
5. #11728536C2semidirect
6. #11725857C2semidirect
7. #11725856C2semidirect
85. #1460D7semidirect
388. C7#17372semidirect

Maximal subgroups

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

subgroupindex
C2×#902372
#117367092
C7×#14602
#117293552
#117285362
#117258572
#117258562
#173727