Finite Group Databasebeta

← #13323650

Subgroups — #13323650

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11736737C212yesmaximal, complemented
2C4×#6467C211yesmaximal, characteristic
3C7×#1450C211yesmaximal, characteristic, complemented
4#11728509C212yesmaximal, complemented
5C2×#89253C211yesmaximal, characteristic, complemented
6#11725836C214yesmaximal, complemented
7#11725810C212yesmaximal, complemented
8#11725806C212yesmaximal, complemented
9C2×#6467C411yescharacteristic
10C4×#1157C2212yes—
11C7×#431C2211yescharacteristic
12C7×#431C2211yescharacteristic
13C2×C4×C7×C8C2211yescharacteristic
14#89885C412yes—
15#89878C411yescharacteristic
16C2×C4⋊(C7⋊C8)C2212yes—
17#89609—22no—
18#89607—22no—
19#89502—24no—
20C4×Dic28C411yescharacteristic
21C4×Dic28C2211yescharacteristic
22C4×D56C411yescharacteristic
23C4×D56C2211yescharacteristic
24C4×#1132C412yes—
25C4×#1132C2212yes—
26#89460C2214yes—
27#89451C2214yes—
28C7×#322C2212yes—
29C7×C82C2212yes—
30#89254C2214yes—
31#89253C2212yes—
32#89253C2212yes—
33#89251C2214yes—
34#17898—71nomaximal, Sylow
35C2×#1157C2×C412yes—
36C22×C7×C8C2×C411yescharacteristic
37C22×C7×C8C2×C411yescharacteristic
38C2×C42×C7C2311yescharacteristic
39C4×C22⋊D7—22no—
40C4×C22⋊D7—22no—
41#6489C2×C412yes—
42C22×C7⋊C8—22no—
43C2×Dic28C2×C411yescharacteristic
44#6467—24no—
45C2×D56C2×C411yescharacteristic
46C2×#1132C2×C412yes—
47C2×#1131—24no—
48C4×D28C2×C412yes—
49C4×D28C2312yes—
50#6435—22no—
51C42×D7—22no—
52C4×Dic14C2×C412yes—
53C4×Dic14C2312yes—
54C7×C4⋊C8—21no—
55C7×C4⋊C8D411yescharacteristic
56C7×C4⋊C8D411yescharacteristic
57C7×C22⋊C8—21no—
58C7×C22⋊C8D412yes—
59C4×C7×C8C2×C411yescharacteristic
60C4×C7×C8C2×C411yescharacteristic
61C4×C7×C8C2312yes—
62C4×C7×C8C2311yescharacteristic
63C4×C7×C8C2311yescharacteristic
64C4×C7×C8C2312yes—
65#6395C2×C412yes—
66#6395C2×C412yes—
67(C2×D7)⋊C8—24no—
68#6392C2×C411yescharacteristic
69#6392C2×C411yescharacteristic
70#6391C2×C412yes—
71#6390C2×C412yes—
72#6390C2×C412yes—
73#6389—24no—
74Dic7⋊C8—24no—
75C4⋊(C7⋊C8)—22no—
76C4⋊(C7⋊C8)C2314yes—
77#2989—72no—
78C4×#135—71no—
79#1592—74no—
80#1591—72no—
81#1589—72no—
82C2×#335—71no—
83#1477—72no—
84#1450D711yescharacteristic, complemented
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7#4111yescharacteristic
88C2×C22⋊D7—22no—
89#1157—24no—
90#1157—24no—
91C2×D28C22×C412yes—
92C2×C4×D7—22no—
93C2×C4×D7—22no—
94C2×C4×D7—22no—
95C2×Dic14C22×C412yes—
96C2×C7×C8—22no—
97C2×C7×C8—22no—
98C2×C7×C8M1614yescomplemented
99C2×C7×C8C22×C411yescharacteristic
100C2×C7×C8C22×C411yescharacteristic
101C2×C7×C8C22×C411yescharacteristic
102C2×C7×C8C22×C411yescharacteristic
103C2×C7×C8C2×D412yes—
104C2×C7×C8C2×D412yes—
105C42×C7C22×C411yescharacteristic
106C42×C7C2×D411yescharacteristic
107C42×C7#4111yescharacteristic
108C42×C7C2411yescharacteristic
109C22⋊Dic7—22no—
110(C2×D7)⋊C4—22no—
111(C2×D7)⋊C4—22no—
112C4⋊Dic7C22×C412yes—
113C4⋊Dic7C22×C412yes—
114Dic7⋊C4—22no—
115Dic7⋊C4—22no—
116C4×Dic7—22no—
117C4×Dic7—22no—
118C4×Dic7—22no—
119C2×C7⋊C8—24no—
120C2×C7⋊C8—82no—
121Dic28—22no—
122D56—22no—
123#1132—24no—
124#1131—84no—
125C2×#135—71no—
126C4×#41—72no—
127#446—74no—
128C4×Q16—71no—
129C4×Q16—71no—
130C4×SD16—72no—
131C4×SD16—72no—
132C4×D8—71no—
133C4×D8—71no—
134#436—74no—
135#435—144no—
136#432—142no—
137#431C2×D711yescharacteristic
138#431C2×D711yescharacteristic
139#427—71no—
140#423—142no—
141C2×C4⋊C8—72no—
142#416—72no—
143C2×C4×C8C2×D711yescharacteristic
144#335—72no—
145#335—72no—
146Q8⋊C8—74no—
147D4⋊C8—74no—
148#322C2×D712yes—
149C82C2×D712yes—
150C23×C7C4×D411yescharacteristic
151C22×D7—22no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—41no—
155C2×C4×C7C4×D411yescharacteristic
156C2×C4×C7C4×D411yescharacteristic
157C2×C4×C7C4×D411yescharacteristic
158C2×C4×C7C2×M1612yes—
159C2×C4×C7#13112yes—
160C2×C4×C7C23×C411yescharacteristic
161C2×C4×C7C22×D411yescharacteristic
162C2×C4×C7C2×#4111yescharacteristic
163C22⋊D7—44no—
164C2×Dic7—22no—
165C2×Dic7—22no—
166C2×Dic7—22no—
167D28—22no—
168D28—22no—
169C4×D7—42no—
170C4×D7—42no—
171C4×D7—44no—
172C4×D7—82no—
173Dic14—22no—
174Dic14—22no—
175C7×C8—21no—
176C7×C8—21no—
177C7×C8—22no—
178C7×C8—22no—
179C7×C8—42no—
180C7×C8C2×M1612yes—
181C7×C8C2×M1612yes—
182C7⋊C8—44no—
183C2×#41—72no—
184#135—144no—
185C2×Q16—71no—
186C2×SD16—72no—
187C2×D8—71no—
188C2×M16—144no—
189C22×C8—142no—
190C22×C8C4×D711yescharacteristic
191C22×C8C4×D711yescharacteristic
192C4×Q8—72no—
193C4×Q8—72no—
194C4×D4—72no—
195C4×D4—72no—
196C4×D4—142no—
197C4×D4—142no—
198#117—72no—
199#117—142no—
200C2×C42—142no—
201C2×C42C22×D711yescharacteristic
202#107—71no—
203#107—71no—
204#106—72no—
205C4⋊C8—21no—
206C4⋊C8—74no—
207C4⋊C8—144no—
208C4⋊C8—142no—
209C4⋊C8D2811yescharacteristic
210C4⋊C8D2811yescharacteristic
211#103—72no—
212#103—72no—
213#102—72no—
214#102—72no—
215C22⋊C8—21no—
216C22⋊C8—144no—
217C22⋊C8D2812yes—
218#97—144no—
219C4×C8C4×D711yescharacteristic
220C4×C8C4×D711yescharacteristic
221C4×C8C22×D712yes—
222C4×C8C22×D711yescharacteristic
223C4×C8C22×D711yescharacteristic
224C4×C8C22×D712yes—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7#43612yes—
228C22×C7C2×C4×D411yescharacteristic
229C2×D7—42no—
230C2×D7—42no—
231C2×D7—42no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7#43612yes—
236C4×C7C22×M1611yescharacteristic
237C4×C7C2×#13111yescharacteristic
238C4×C7#57611yescharacteristic
239C4×C7#57811yescharacteristic
240Dic7—42no—
241Dic7—42no—
242Dic7—42no—
243#41—144no—
244#41—144no—
245C2×Q8—72no—
246C2×D4—72no—
247C2×D4—142no—
248C22×C4—142no—
249C22×C4—142no—
250C22×C4—142no—
251C22×C4C2×C4×D711yescharacteristic
252C22×C4C2×D2811yescharacteristic
253C22×C4#115711yescharacteristic
254Q16—142no—
255SD16—144no—
256D8—142no—
257M16—564no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—144no—
261C2×C8—562no—
262C2×C8#113114yescomplemented
263C2×C8C2×C4×D711yescharacteristic
264C2×C8C2×C4×D711yescharacteristic
265C2×C8C2×C4×D711yescharacteristic
266C2×C8C2×C4×D711yescharacteristic
267C2×C8C2×D2812yes—
268C2×C8C2×D2812yes—
269C4⋊C4—72no—
270C4⋊C4—72no—
271C4⋊C4—142no—
272C4⋊C4—142no—
273C22⋊C4—142no—
274C22⋊C4—142no—
275C22⋊C4—142no—
276C42—142no—
277C42—142no—
278C42—142no—
279C42C2×C4×D711yescharacteristic
280C42C2×D2811yescharacteristic
281C42#115711yescharacteristic
282C42C23×D711yescharacteristic
283C2×C7—22no—
284C2×C7C2×#43611yescharacteristic
285C2×C7#295111yescharacteristic
286C2×C7#295711yescharacteristic
287D7—82no—
288C23—142no—
289C23C4×D2811yescharacteristic
290Q8—142no—
291Q8—142no—
292D4—142no—
293D4—142no—
294D4—284no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—142no—
299C2×C4—142no—
300C2×C4—142no—
301C2×C4—282no—
302C2×C4—282no—
303C2×C4—284no—
304C2×C4—562no—
305C2×C4C4×D2811yescharacteristic
306C2×C4C4×D2811yescharacteristic
307C2×C4C4×D2811yescharacteristic
308C2×C4C2×#113112yes—
309C2×C4#646412yes—
310C2×C4C22×C4×D711yescharacteristic
311C2×C4C22×D2811yescharacteristic
312C2×C4C2×#115711yescharacteristic
313C8—21no—
314C8—21no—
315C8—22no—
316C8—22no—
317C8—42no—
318C8—284no—
319C8C2×#113112yes—
320C8C2×#113112yes—
321C7#1789811yescharacteristic, complemented, Sylow
322C22—21no—
323C22—21no—
324C22—282no—
325C22—282no—
326C22—282no—
327C22#8946012yes—
328C22C2×C4×D2811yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—282no—
333C4—282no—
334C4—282no—
335C4#8946012yes—
336C4C22×#113111yescharacteristic
337C4C2×#646411yescharacteristic
338C4#9044311yescharacteristic
339C4#9044411yescharacteristic
340C2—22no—
341C2—562no—
342C2C2×#8946011yescharacteristic
343C2#1173600711yescharacteristic
344C2#1173602511yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1450C2
socleC22×C7C2×C4×D4
prime radicalC2×#6467C4
cosocleC42×C7C24
hypercenter#1450D7
hypocenterC7#17898
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
1. #11736737C2semidirect
3. C7×#1450C2semidirect
4. #11728509C2semidirect
5. C2×#89253C2semidirect
6. #11725836C2semidirect
7. #11725810C2semidirect
8. #11725806C2semidirect
84. #1450D7semidirect
98. C2×C7×C8M16semidirect
262. C2×C8#1131semidirect
321. C7#17898semidirect

Maximal subgroups

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

subgroupindex
#117367372
C4×#64672
C7×#14502
#117285092
C2×#892532
#117258362
#117258102
#117258062
#178987