Finite Group Databasebeta

← #13354337

Subgroups — #13354337

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2283C211yesmaximal, characteristic
2#11739505C211yesmaximal, characteristic
3C22×C4×C7⋊C8C211yesmaximal, characteristic
4#11727303C212yesmaximal, complemented
5#11727301C212yesmaximal, complemented
6#11727266C212yesmaximal
7#11727265C212yesmaximal
8C4×#6381C212yesmaximal, complemented
9C4×#6382C212yesmaximal, complemented
10C22×C42×C7C2211yescharacteristic
11C23×C7⋊C8C2211yescharacteristic
12C23×C7⋊C8C2211yescharacteristic
13C7×#383—21no—
14C7×#382C2212yes—
15C7×#381—21no—
16C7×#380C2212yes—
17C4×C7×C4⋊C4C2212yes—
18C4×C7×C22⋊C4—21no—
19C7×#376—21no—
20#89984C2212yes—
21#89983—21no—
22C4×C22⋊Dic7—21no—
23C2×#6382C2214yes—
24C2×#6381C2214yes—
25#89709—21no—
26#89708C2212yes—
27C4×C4⋊Dic7C2212yes—
28#89704—21no—
29C2×C4×C7⋊C8C418yescomplemented
30C2×C4×C7⋊C8C2211yescharacteristic
31C2×C4×C7⋊C8C2211yescharacteristic
32C2×C4×C7⋊C8C2211yescharacteristic
33C2×C4×C7⋊C8C2211yescharacteristic
34#89324C2214yes—
35#89324C2214yes—
36#20818—71nomaximal, Sylow
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C2×C7×C4⋊C4C2314yes—
41C2×C7×C22⋊C4—22no—
42C2×C42×C7—22no—
43C2×C42×C7C2×C414yes—
44C2×C42×C7D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C42×C7Q814yes—
47C2×C42×C7C2311yescharacteristic
48C2×C42×C7C2311yescharacteristic
49C2×C22⋊Dic7—22no—
50C2×C4⋊Dic7C2314yes—
51C2×C4×Dic7—22no—
52C22×C7⋊C8—24no—
53C22×C7⋊C8—24no—
54C22×C7⋊C8C2×C414yes—
55C22×C7⋊C8C2×C414yes—
56C22×C7⋊C8C2311yescharacteristic
57C22×C7⋊C8C2311yescharacteristic
58C22×C7⋊C8C2311yescharacteristic
59C22×C7⋊C8C2311yescharacteristic
60C7×#95—22no—
61C7×#95—22no—
62C7×#95—22no—
63C7×#95—22no—
64C7⋊#95—22no—
65C7⋊#95—22no—
66C7⋊#95—22no—
67C7⋊#95—22no—
68#6382—28no—
69#6381—28no—
70C4×C7⋊C8C2×C414yes—
71C4×C7⋊C8C2×C414yes—
72C4×C7⋊C8C2×C414yes—
73C4×C7⋊C8C2×C414yes—
74C22×C4×C8—71no—
75#2283—71no—
76#2283D711yescharacteristic
77#1846—72no—
78#1835—72no—
79#1808—72no—
80#1807—72no—
81C4×#107—72no—
82C4×#106—72no—
83C24×C7C2×D411yescharacteristic
84C22×C4×C7—24no—
85C22×C4×C7—24no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—24no—
89C22×C4×C7C4⋊C414yes—
90C22×C4×C7C4⋊C414yes—
91C22×C4×C7C22×C412yes—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×Q812yes—
98C22×C4×C7C2411yescharacteristic
99C22×Dic7—24no—
100C7×C4⋊C4—24no—
101C7×C4⋊C4—24no—
102C7×C22⋊C4—44no—
103C42×C7—44no—
104C42×C7C4⋊C414yes—
105C42×C7C4⋊C414yes—
106C42×C7C22×C412yes—
107C42×C7C22×C412yes—
108C42×C7C2×Q812yes—
109C42×C7C2×Q812yes—
110C22⋊Dic7—44no—
111C4⋊Dic7—24no—
112C4⋊Dic7—24no—
113C4×Dic7—44no—
114C2×C7⋊C8—22no—
115C2×C7⋊C8—22no—
116C2×C7⋊C8—28no—
117C2×C7⋊C8—28no—
118C2×C7⋊C8C22×C412yes—
119C2×C7⋊C8C22×C412yes—
120C2×C7⋊C8C22×C412yes—
121C2×C7⋊C8C22×C412yes—
122C2×C7⋊C8#4114yes—
123C2×C7⋊C8#4114yes—
124C23×C8—71no—
125C23×C8—71no—
126C22×C42C2×D711yescharacteristic
127C2×#107—74no—
128C2×#106—74no—
129C2×C4×C8—78no—
130C2×C4×C8—71no—
131C2×C4×C8—71no—
132C2×C4×C8—71no—
133C2×C4×C8—71no—
134#383—21no—
135#383—141no—
136#382—72no—
137#382C2×D712yes—
138#381—21no—
139#381—141no—
140#380—72no—
141#380C2×D712yes—
142C4×C4⋊C4—72no—
143C4×C4⋊C4C2×D712yes—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—141no—
146#376—21no—
147#376—141no—
148#340—74no—
149#340—74no—
150C23×C7—22no—
151C23×C7—22no—
152C23×C7C2×C4⋊C412yes—
153C23×C7#13514yes—
154C23×C7C22×D411yescharacteristic
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—28no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—24no—
163C2×C4×C7—28no—
164C2×C4×C7—44no—
165C2×C4×C7—44no—
166C2×C4×C7—44no—
167C2×C4×C7C2×C4⋊C412yes—
168C2×C4×C7C2×C4⋊C412yes—
169C2×C4×C7C2×C4⋊C412yes—
170C2×C4×C7C2×C4⋊C412yes—
171C2×C4×C7C2×C4⋊C412yes—
172C2×C4×C7#13514yes—
173C2×C4×C7C23×C411yescharacteristic
174C2×C4×C7C22×Q811yescharacteristic
175C2×C4×C7C2×#4112yes—
176C2×C4×C7C2×#4112yes—
177C2×Dic7—44no—
178C2×Dic7—44no—
179C2×Dic7—44no—
180C7⋊C8—22no—
181C7⋊C8—22no—
182C7⋊C8#11714yes—
183C7⋊C8#11714yes—
184C23×C4C22⋊D711yescharacteristic
185C23×C4C22⋊D711yescharacteristic
186C23×C4C22×D711yescharacteristic
187C22×C8—71no—
188C22×C8—71no—
189C22×C8—74no—
190C22×C8—74no—
191C22×C8—71no—
192C22×C8—71no—
193C22×C8—144no—
194C22×C8—144no—
195C2×C4⋊C4—74no—
196C2×C4⋊C4C22×D714yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—142no—
199C2×C42—22no—
200C2×C42—142no—
201C2×C42Dic1414yes—
202C2×C42C4×D714yes—
203C2×C42C22⋊D711yescharacteristic
204C2×C42C22⋊D711yescharacteristic
205C2×C42C22×D711yescharacteristic
206C2×C42C22×D711yescharacteristic
207#107—148no—
208#106—148no—
209C4×C8—74no—
210C4×C8—74no—
211C4×C8—74no—
212C4×C8—74no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—142no—
218#95—142no—
219#95—142no—
220#95—142no—
221C22×C7—22no—
222C22×C7—28no—
223C22×C7—22no—
224C22×C7#42714yes—
225C22×C7C22×C4⋊C411yescharacteristic
226C22×C7#53212yes—
227C22×C7C2×#13512yes—
228C22×C7C2×#13512yes—
229C4×C7—22no—
230C4×C7—22no—
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—84no—
234C4×C7#42714yes—
235C4×C7C2×#11711yescharacteristic
236C4×C7C2×#11711yescharacteristic
237C4×C7#53312yes—
238Dic7—84no—
239C24C2×C22⋊D711yescharacteristic
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—144no—
246C22×C4Dic7⋊C414yes—
247C22×C4Dic7⋊C414yes—
248C22×C4C2×Dic1412yes—
249C22×C4C2×C4×D712yes—
250C22×C4C2×C22⋊D711yescharacteristic
251C22×C4C2×C22⋊D711yescharacteristic
252C22×C4C2×C22⋊D711yescharacteristic
253C22×C4C2×C22⋊D711yescharacteristic
254C22×C4C2×C22⋊D711yescharacteristic
255C22×C4C23×D711yescharacteristic
256C2×C8—72no—
257C2×C8—72no—
258C2×C8—74no—
259C2×C8—74no—
260C2×C8—72no—
261C2×C8—72no—
262C2×C8—148no—
263C2×C8—148no—
264C2×C8—142no—
265C2×C8—142no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—144no—
269C4⋊C4—144no—
270C22⋊C4—44no—
271C22⋊C4—284no—
272C42—44no—
273C42—284no—
274C42Dic7⋊C414yes—
275C42Dic7⋊C414yes—
276C42C2×Dic1412yes—
277C42C2×Dic1412yes—
278C42C2×C4×D712yes—
279C42C2×C4×D712yes—
280C2×C7—24no—
281C2×C7#228511yescharacteristic
282C2×C7C2×#42711yescharacteristic
283C2×C7C2×#42711yescharacteristic
284C2×C7#316814yes—
285C23—22no—
286C23—22no—
287C23C2×Dic7⋊C412yes—
288C23#651314yes—
289C23C22×C22⋊D711yescharacteristic
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—28no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—284no—
303C2×C4—284no—
304C2×C4—284no—
305C2×C4C2×Dic7⋊C412yes—
306C2×C4C2×Dic7⋊C412yes—
307C2×C4C2×Dic7⋊C412yes—
308C2×C4C2×Dic7⋊C412yes—
309C2×C4C2×Dic7⋊C412yes—
310C2×C4#651314yes—
311C2×C4C22×Dic1411yescharacteristic
312C2×C4C22×C4×D711yescharacteristic
313C2×C4D7×#4112yes—
314C2×C4D7×#4112yes—
315C8—74no—
316C8—74no—
317C8—142no—
318C8—142no—
319C7#2081811yescharacteristic, complemented, Sylow
320C22—22no—
321C22—22no—
322C22—28no—
323C22#8977014yes—
324C22C22×Dic7⋊C411yescharacteristic
325C22C2×#651312yes—
326C22C2×#651312yes—
327C22#9052312yes—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—84no—
333C4—564no—
334C4#8977014yes—
335C4#9020812yes—
336C4D7×#11711yescharacteristic
337C4D7×#11711yescharacteristic
338C2—24no—
339C2C2×#8977011yescharacteristic
340C2C2×#8977011yescharacteristic
341C2#1173966911yescharacteristic
342C2#1174057914yes—
subgroupG/H
centerC22×C4C2×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2283C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#2283D7
hypocenterC7#20818
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
4. #11727303C2semidirect
5. #11727301C2semidirect
8. C4×#6381C2semidirect
9. C4×#6382C2semidirect
29. C2×C4×C7⋊C8C4semidirect
319. C7#20818semidirect

Maximal subgroups

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

subgroupindex
C7×#22832
#117395052
C22×C4×C7⋊C82
#117273032
#117273012
#117272662
#117272652
C4×#63812
C4×#63822
#208187