Finite Group Databasebeta

← #13358672

Subgroups — #13358672

Order 1792.

938 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
2#11739505C211yesmaximal, characteristic
3C2×#89698C211yesmaximal, characteristic
4#11729310C212yesmaximal
5#11729309C212yesmaximal
6#11727235C214yesmaximal, complemented
7#11727222C214yesmaximal, complemented
8C22×C7×C4⋊C4C2212yes—
9C22×C42×C7—21no—
10C22×C42×C7C2211yescharacteristic
11C4×C7×C4⋊C4—21no—
12C4×C7×C4⋊C4—21no—
13C4×C7×C4⋊C4C2212yes—
14C4×C7×C4⋊C4C2212yes—
15C2×C7×#95—21no—
16#89984C2211yescharacteristic
17#89984C2211yescharacteristic
18#89983—21no—
19C4×C22⋊Dic7—21no—
20C2×C22⋊(C7⋊C8)C2212yes—
21C2×#6382C2214yes—
22C2×#6381C2214yes—
23#89709—21no—
24#89708C2212yes—
25C4×C4⋊Dic7C2212yes—
26#89704—21no—
27#89698C418yescomplemented
28C2×C4⋊(C7⋊C8)C2212yes—
29C2×#6377C2212yes—
30#89324C2214yes—
31#89324C2214yes—
32#21363—71nomaximal, Sylow
33C23×C4×C7—22no—
34C23×C4×C7D411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—28no—
39C2×C7×C4⋊C4C2314yes—
40C2×C42×C7—22no—
41C2×C42×C7—22no—
42C2×C42×C7—44no—
43C2×C42×C7C2×C414yes—
44C2×C42×C7D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C42×C7Q814yes—
47C2×C42×C7C2311yescharacteristic
48C2×C42×C7C2311yescharacteristic
49C2×C22⋊Dic7—21no—
50C2×C22⋊Dic7—21no—
51C2×C4⋊Dic7C2312yes—
52C2×C4⋊Dic7C2312yes—
53C2×C4×Dic7—22no—
54C22×C7⋊C8C2314yes—
55C7×#95—22no—
56C7×#95—22no—
57C7⋊#95—22no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—22no—
61C7⋊#95—22no—
62C22⋊(C7⋊C8)C2×C418yes—
63#6382—28no—
64#6381—28no—
65C4⋊(C7⋊C8)C2×C418yescomplemented
66#6377C2×C418yescomplemented
67C2×#432—71no—
68#2283—71no—
69C2×C4×C4⋊C4D711yescharacteristic
70#1949—74no—
71#1853—72no—
72#1852—72no—
73#1776—74no—
74C24×C7C2×D411yescharacteristic
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—24no—
78C22×C4×C7—28no—
79C22×C4×C7—24no—
80C22×C4×C7—42no—
81C22×C4×C7—44no—
82C22×C4×C7—44no—
83C22×C4×C7C4⋊C414yes—
84C22×C4×C7C4⋊C414yes—
85C22×C4×C7C22×C412yes—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×Q812yes—
92C22×C4×C7C2411yescharacteristic
93C22×Dic7—22no—
94C22×Dic7—22no—
95C7×C4⋊C4—28no—
96C7×C4⋊C4—24no—
97C7×C4⋊C4#4118yes—
98C42×C7—44no—
99C42×C7—44no—
100C42×C7C4⋊C414yes—
101C42×C7C4⋊C414yes—
102C42×C7C22×C412yes—
103C42×C7C22×C412yes—
104C42×C7C2×Q812yes—
105C42×C7C2×Q812yes—
106C22⋊Dic7—42no—
107C22⋊Dic7—42no—
108C4⋊Dic7—24no—
109C4⋊Dic7—24no—
110C4×Dic7—44no—
111C2×C7⋊C8—24no—
112C2×C7⋊C8—24no—
113C2×C7⋊C8C22×C418yes—
114C22×C4⋊C4C2×D712yes—
115C22×C42—21no—
116C22×C42C2×D711yescharacteristic
117#432—78no—
118C2×#107—74no—
119C2×#106—74no—
120C2×C4⋊C8—72no—
121C2×C22⋊C8—72no—
122C2×#97—72no—
123#383—141no—
124#382—72no—
125#381—141no—
126#380—71no—
127#380—71no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—72no—
131C4×C4⋊C4C2×D712yes—
132C4×C4⋊C4C2×D712yes—
133C4×C22⋊C4—141no—
134#376—141no—
135C2×#95—21no—
136#340—74no—
137#340—74no—
138C23×C7—22no—
139C23×C7—22no—
140C23×C7C2×C4⋊C412yes—
141C23×C7AGL(1,ℤ/8)12yes—
142C23×C7#13712yes—
143C23×C7C22×D411yescharacteristic
144C2×C4×C7—22no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—28no—
152C2×C4×C7—24no—
153C2×C4×C7—24no—
154C2×C4×C7—28no—
155C2×C4×C7—44no—
156C2×C4×C7—416no—
157C2×C4×C7—44no—
158C2×C4×C7—44no—
159C2×C4×C7C2×C4⋊C412yes—
160C2×C4×C7C2×C4⋊C412yes—
161C2×C4×C7C2×C4⋊C412yes—
162C2×C4×C7C2×C4⋊C412yes—
163C2×C4×C7C2×C4⋊C412yes—
164C2×C4×C7C23×C411yescharacteristic
165C2×C4×C7C22×Q811yescharacteristic
166C2×C4×C7C2×#4114yes—
167C2×Dic7—42no—
168C2×Dic7—42no—
169C2×Dic7—42no—
170C2×Dic7—42no—
171C2×Dic7—42no—
172C2×Dic7—42no—
173C7⋊C8—28no—
174C23×C4—22no—
175C23×C4C22⋊D711yescharacteristic
176C23×C4C22⋊D711yescharacteristic
177C23×C4C22×D711yescharacteristic
178C22×C8—74no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—72no—
182C2×C4⋊C4—72no—
183C2×C4⋊C4C22×D714yes—
184C2×C22⋊C4—141no—
185C2×C22⋊C4—141no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—44no—
189C2×C42—142no—
190C2×C42Dic1414yes—
191C2×C42C4×D714yes—
192C2×C42C22⋊D711yescharacteristic
193C2×C42C22⋊D711yescharacteristic
194C2×C42C22×D711yescharacteristic
195C2×C42C22×D711yescharacteristic
196#107—148no—
197#106—148no—
198C4⋊C8—78no—
199C22⋊C8—78no—
200#97—78no—
201#95—22no—
202#95—22no—
203#95—141no—
204#95—141no—
205#95—142no—
206#95—142no—
207#95—142no—
208C22×C7—24no—
209C22×C7—24no—
210C22×C7—22no—
211C22×C7—22no—
212C22×C7#42814yes—
213C22×C7C22×C4⋊C411yescharacteristic
214C22×C7#52512yes—
215C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
216C22×C7C2×#13711yescharacteristic
217C22×C7#57512yes—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—22no—
221C4×C7—22no—
222C4×C7—22no—
223C4×C7—48no—
224C4×C7C4×#4112yes—
225C4×C7#53312yes—
226Dic7—82no—
227Dic7—82no—
228C24C2×C22⋊D711yescharacteristic
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—142no—
238C22×C4—142no—
239C22×C4Dic7⋊C414yes—
240C22×C4Dic7⋊C414yes—
241C22×C4C2×Dic1412yes—
242C22×C4C2×C4×D712yes—
243C22×C4C2×C22⋊D711yescharacteristic
244C22×C4C2×C22⋊D711yescharacteristic
245C22×C4C2×C22⋊D711yescharacteristic
246C22×C4C2×C22⋊D711yescharacteristic
247C22×C4C2×C22⋊D711yescharacteristic
248C22×C4C23×D711yescharacteristic
249C2×C8—78no—
250C2×C8—144no—
251C2×C8—144no—
252C4⋊C4—28no—
253C4⋊C4—24no—
254C4⋊C4—144no—
255C4⋊C4—144no—
256C4⋊C4#115718yes—
257C22⋊C4—282no—
258C22⋊C4—282no—
259C42—44no—
260C42—44no—
261C42—284no—
262C42Dic7⋊C414yes—
263C42Dic7⋊C414yes—
264C42C2×Dic1412yes—
265C42C2×Dic1412yes—
266C42C2×C4×D712yes—
267C42C2×C4×D712yes—
268C2×C7—24no—
269C2×C7#229611yescharacteristic
270C2×C7C2×#42811yescharacteristic
271C2×C7#291111yescharacteristic
272C2×C7#311412yes—
273C2×C7#311712yes—
274C23—22no—
275C23—22no—
276C23C2×Dic7⋊C412yes—
277C23#649512yes—
278C23#650512yes—
279C23C22×C22⋊D711yescharacteristic
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—28no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—44no—
292C2×C4—416no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4—282no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4C2×Dic7⋊C412yes—
302C2×C4C2×Dic7⋊C412yes—
303C2×C4C2×Dic7⋊C412yes—
304C2×C4C2×Dic7⋊C412yes—
305C2×C4C2×Dic7⋊C412yes—
306C2×C4C22×Dic1411yescharacteristic
307C2×C4C22×C4×D711yescharacteristic
308C2×C4C2×#115714yes—
309C8—148no—
310C7#2136311yescharacteristic, complemented, Sylow
311C22—22no—
312C22—24no—
313C22—22no—
314C22—24no—
315C22#8973614yes—
316C22C22×Dic7⋊C411yescharacteristic
317C22#9048212yes—
318C22C2×#649511yescharacteristic
319C22C2×#650511yescharacteristic
320C22#9051312yes—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—48no—
327C4—562no—
328C4—562no—
329C4#9016112yes—
330C4C4×#115712yes—
331C2—24no—
332C2#1173951411yescharacteristic
333C2C2×#8973611yescharacteristic
334C2#1173965211yescharacteristic
335C2#1173975612yes—
336C2#1173989012yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C4×C7×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC2×C4×C4⋊C4D7
hypocenterC7#21363
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #11727235C2semidirect
7. #11727222C2semidirect
27. #89698C4semidirect
65. C4⋊(C7⋊C8)C2×C4semidirect
66. #6377C2×C4semidirect
310. C7#21363semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×C4⋊C42
#117395052
C2×#896982
#117293102
#117293092
#117272352
#117272222
#213637