Finite Group Databasebeta

← #13285137

Subgroups — #13285137

Order 1792.

2294 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6438C211yesmaximal, characteristic
2C2×C7×#382C211yesmaximal, characteristic, complemented
3C2×#89746C212yesmaximal, complemented
4C2×#89760C212yesmaximal
5C2×C42×Dic7C211yesmaximal, characteristic, complemented
6#11727162C218yesmaximal, complemented
7C23×D28C2211yescharacteristic
8C23×Dic14C2211yescharacteristic
9C22×C7×C4⋊C4C2212yes—
10C22×C42×C7C2211yescharacteristic
11C22×(C2×D7)⋊C4C2218yes—
12C22×(C2×D7)⋊C4C2214yes—
13C22×C4×Dic7C2214yes—
14C22×C4×Dic7C2212yes—
15C2×#6438—24no—
16C2×#6438—24no—
17C2×#6438C418yescomplemented
18C7×#382—24no—
19C2×C7×#95C2214yes—
20C2×C7⋊#95C2218yes—
21#89760—28no—
22#89746—28no—
23C42×Dic7—24no—
24#14708—71nomaximal, Sylow
25C24×D7C2312yes—
26C23×C4×C7C2314yes—
27C23×C4×C7C2312yes—
28C23×C4×C7C2311yescharacteristic
29C23×Dic7C2312yes—
30C23×Dic7C2314yes—
31C22×D28—22no—
32C22×D28—22no—
33C22×D28—24no—
34C22×D28—24no—
35C22×D28C2×C414yes—
36C22×Dic14—22no—
37C22×Dic14—22no—
38C22×Dic14—24no—
39C22×Dic14—24no—
40C22×Dic14C2×C414yes—
41C2×C7×C4⋊C4—24no—
42C2×C7×C4⋊C4—24no—
43C2×C7×C4⋊C4—24no—
44C2×C42×C7C2×C414yes—
45C2×C42×C7D414yes—
46C2×C42×C7D414yes—
47C2×(C2×D7)⋊C4—28no—
48C2×(C2×D7)⋊C4—28no—
49C2×(C2×D7)⋊C4—216no—
50C2×(C2×D7)⋊C4—232no—
51C2×(C2×D7)⋊C4C2×C4116yes—
52C2×C4×Dic7—28no—
53C2×C4×Dic7—24no—
54C2×C4×Dic7—24no—
55C2×C4×Dic7—216no—
56C2×C4×Dic7D418yes—
57#6438—232no—
58C7×#95—28no—
59C7⋊#95—216no—
60C22×#124—71no—
61C2×#391—72no—
62C2×#390—72no—
63C2×#382D711yescharacteristic, complemented
64C2×C43—71no—
65#1437—78no—
66C24×C7C2411yescharacteristic
67C23×D7—22no—
68C23×D7—22no—
69C23×D7—24no—
70C23×D7—216no—
71C23×D7—24no—
72C23×D7C22×C414yes—
73C22×C4×C7—28no—
74C22×C4×C7—24no—
75C22×C4×C7—24no—
76C22×C4×C7—24no—
77C22×C4×C7—216no—
78C22×C4×C7C22×C414yes—
79C22×C4×C7C22×C412yes—
80C22×C4×C7C2×D414yes—
81C22×C4×C7C2×D414yes—
82C22×C4×C7C2×D412yes—
83C22×C4×C7C2×D412yes—
84C22×C4×C7C2×D414yes—
85C22×C4×C7#4114yes—
86C22×Dic7—24no—
87C22×Dic7—24no—
88C22×Dic7—28no—
89C22×Dic7—24no—
90C22×Dic7—22no—
91C22×Dic7—22no—
92C22×Dic7—24no—
93C22×Dic7C22×C414yes—
94C22×Dic7C2×D418yes—
95C22×Dic7#41116yes—
96C2×D28—28no—
97C2×D28—216no—
98C2×D28—48no—
99C2×D28—44no—
100C2×D28—44no—
101C2×Dic14—28no—
102C2×Dic14—216no—
103C2×Dic14—48no—
104C2×Dic14—44no—
105C2×Dic14—44no—
106C7×C4⋊C4—48no—
107C42×C7C22⋊C4116yescomplemented
108(C2×D7)⋊C4—232no—
109(C2×D7)⋊C4—432no—
110C4×Dic7—232no—
111C4×Dic7—216no—
112C23×Q8—71no—
113C23×D4—71no—
114C2×#124—78no—
115C2×#124—144no—
116C2×#124—144no—
117C22×C4⋊C4C2×D712yes—
118C22×C22⋊C4—74no—
119C22×C22⋊C4—78no—
120C22×C42—72no—
121C22×C42—74no—
122C22×C42C2×D711yescharacteristic
123#391—148no—
124#390—148no—
125#382—24no—
126C2×#95—78no—
127C2×#95C2×D714yes—
128C43—144no—
129C23×C7C23×C411yescharacteristic
130C23×C7C22×D411yescharacteristic
131C23×C7C22×D412yes—
132C23×C7C22×D411yescharacteristic
133C23×C7C2×#4112yes—
134C23×C7C2×#4118yes—
135C22×D7—24no—
136C22×D7—28no—
137C22×D7—216no—
138C22×D7—44no—
139C22×D7—48no—
140C22×D7—48no—
141C22×D7—42no—
142C22×D7—416no—
143C22×D7—42no—
144C22×D7—48no—
145C22×D7—48no—
146C2×C4×C7—28no—
147C2×C4×C7—28no—
148C2×C4×C7—28no—
149C2×C4×C7—48no—
150C2×C4×C7—44no—
151C2×C4×C7—48no—
152C2×C4×C7—48no—
153C2×C4×C7C2×C22⋊C418yes—
154C2×C4×C7C2×C22⋊C414yes—
155C2×C4×C7C4×D418yes—
156C2×C4×C7#12118yes—
157C2×C4×C7#12414yes—
158C2×C4×C7#12714yes—
159C2×Dic7—28no—
160C2×Dic7—216no—
161C2×Dic7—216no—
162C2×Dic7—28no—
163C2×Dic7—24no—
164C2×Dic7—28no—
165C2×Dic7—44no—
166C2×Dic7—42no—
167C2×Dic7—42no—
168C2×Dic7#124116yescomplemented
169D28—416no—
170Dic14—416no—
171C25—72no—
172C22×Q8—74no—
173C22×Q8—142no—
174C22×Q8—144no—
175C22×Q8—144no—
176C22×Q8—142no—
177C22×D4—74no—
178C22×D4—142no—
179C22×D4—144no—
180C22×D4—144no—
181C22×D4—142no—
182C23×C4—74no—
183C23×C4—72no—
184C23×C4C22×D714yes—
185C23×C4C22×D712yes—
186C23×C4C22×D711yescharacteristic
187#124—1432no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C22⋊C4—716no—
192C2×C22⋊C4—148no—
193C2×C22⋊C4—148no—
194C2×C22⋊C4—1432no—
195C2×C22⋊C4—1416no—
196C2×C42—78no—
197C2×C42—1416no—
198C2×C42—148no—
199C2×C42—144no—
200C2×C42—144no—
201C2×C42C4×D714yes—
202C2×C42D2814yes—
203C2×C42C22⋊D714yes—
204#95—28no—
205#95—1416no—
206C22×C7C22×C22⋊C411yescharacteristic
207C22×C7C2×C4×D412yes—
208C22×C7C4×#4114yes—
209C22×C7C2×#12112yes—
210C22×C7#52514yes—
211C22×C7C2×#12411yescharacteristic
212C22×C7C2×#12414yes—
213C22×C7#52918yes—
214C22×C7C2×#12711yescharacteristic
215C22×C7#53214yes—
216C22×C7#53214yes—
217C2×D7—44no—
218C2×D7—48no—
219C2×D7—416no—
220C2×D7—88no—
221C2×D7—84no—
222C2×D7—84no—
223C4×C7—28no—
224C4×C7—84no—
225C4×C7#39018yes—
226Dic7—216no—
227Dic7—44no—
228C24—74no—
229C24—142no—
230C24—144no—
231C24—144no—
232C24—1416no—
233C24—142no—
234C24C23×D711yescharacteristic
235C2×Q8—1416no—
236C2×Q8—148no—
237C2×Q8—288no—
238C2×Q8—284no—
239C2×Q8—284no—
240C2×D4—1416no—
241C2×D4—148no—
242C2×D4—288no—
243C2×D4—284no—
244C2×D4—284no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—24no—
250C22×C4—78no—
251C22×C4—716no—
252C22×C4—74no—
253C22×C4—144no—
254C22×C4—144no—
255C22×C4—148no—
256C22×C4—142no—
257C22×C4—144no—
258C22×C4—144no—
259C22×C4—142no—
260C22×C4C2×C4×D714yes—
261C22×C4C2×C4×D712yes—
262C22×C4C2×D2814yes—
263C22×C4C2×D2812yes—
264C22×C4D4×D714yes—
265C22×C4#116114yes—
266C22×C4C2×C22⋊D714yes—
267C22×C4C2×C22⋊D712yes—
268C4⋊C4—48no—
269C22⋊C4—1432no—
270C22⋊C4—2832no—
271C42—1432no—
272C42—1416no—
273C42(C2×D7)⋊C4116yescomplemented
274C2×C7C2×#39011yescharacteristic
275C2×C7#229712yes—
276C2×C7C4×#12414yes—
277C2×C7#285014yes—
278C2×C7#285812yes—
279C2×C7#286112yes—
280D7—84no—
281C23—148no—
282C23—1416no—
283C23—144no—
284C23—2816no—
285C23—284no—
286C23—288no—
287C23—282no—
288C23—282no—
289C23—288no—
290C23—288no—
291C23—288no—
292C23C22×C4×D711yescharacteristic
293C23C22×D2811yescharacteristic
294C23C2×D4×D712yes—
295C23C2×#116112yes—
296C23D7×#4118yes—
297C23C22×C22⋊D711yescharacteristic
298Q8—2816no—
299D4—2816no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—48no—
307C2×C4—716no—
308C2×C4—1416no—
309C2×C4—148no—
310C2×C4—1416no—
311C2×C4—148no—
312C2×C4—148no—
313C2×C4—144no—
314C2×C4—284no—
315C2×C4—282no—
316C2×C4—282no—
317C2×C4#645618yes—
318C2×C4#645818yes—
319C2×C4C2×(C2×D7)⋊C418yes—
320C2×C4C2×(C2×D7)⋊C414yes—
321C2×C4#650314yes—
322C2×C4#651014yes—
323C7#1470811yescharacteristic, complemented, Sylow
324C22—288no—
325C22—2816no—
326C22—284no—
327C22—564no—
328C22—564no—
329C22—568no—
330C22C2×#645612yes—
331C22C2×#645812yes—
332C22#9021314yes—
333C22#9021514yes—
334C22#9035914yes—
335C22D7×#12414yes—
336C22#9038218yes—
337C22C22×(C2×D7)⋊C411yescharacteristic
338C22C2×#650311yescharacteristic
339C22C2×#651011yescharacteristic
340C22#9052314yes—
341C4—28no—
342C4—84no—
343C4—1416no—
344C4—284no—
345C4#8976018yes—
346C2—564no—
347C2#1173159414yes—
348C2#1173170114yes—
349C2C2×#8976011yescharacteristic
350C2#1173967312yes—
351C2#1174043812yes—
352C2#1174050212yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#382C2
socle, cosocleC24×C7C24
prime radicalC23×D28C22
hypercenterC2×#382D7
hypocenterC7#14708
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×C7×#382C2semidirect
3. C2×#89746C2semidirect
5. C2×C42×Dic7C2semidirect
6. #11727162C2semidirect
17. C2×#6438C4semidirect
63. C2×#382D7semidirect
107. C42×C7C22⋊C4semidirect
168. C2×Dic7#124semidirect
273. C42(C2×D7)⋊C4semidirect
323. C7#14708semidirect

Maximal subgroups

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

subgroupindex
C22×#64382
C2×C7×#3822
C2×#897462
C2×#897602
C2×C42×Dic72
#117271622
#147087