Finite Group Databasebeta

← #13513928

Subgroups — #13513928

Order 1792.

2250 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2C7×#2274C211yesmaximal, characteristic
3#11739675C212yesmaximal
4#11739669C212yesmaximal
5#11739589C211yesmaximal, characteristic
6#11731089C218yesmaximal, complemented
7#11731086C218yesmaximal, complemented
8#11730890C214yesmaximal
9#11730882C214yesmaximal
10C23×Dic14C2211yescharacteristic
11C2×C7×#117C2212yes—
12C22×C7×C4⋊C4C2211yescharacteristic
13C2×#6489C2212yes—
14C22×Dic7⋊C4C2212yes—
15C22×C4×Dic7C2211yescharacteristic
16C22×C4×Dic7C2212yes—
17C2×#6450C4116yescomplemented
18C2×#6450C2214yes—
19C2×#6450C2214yes—
20C4×C7×C4⋊C4C2218yes—
21C4×C7×C22⋊C4C2214yes—
22C4×C22⋊Dic7C2214yes—
23#89749C2214yes—
24#89748C2214yes—
25C4⋊C4×Dic7C2218yes—
26#89746C2218yes—
27#89715C2218yes—
28#89714C2214yes—
29#89714C2214yes—
30#89705C22116yes—
31#89704C2218yes—
32C4×Dic7⋊C4C2214yes—
33C4×Dic7⋊C4C2214yes—
34C4×Dic7⋊C4C2218yes—
35#89701C2218yes—
36#89424C22116yes—
37#89423C2218yes—
38#89422C2218yes—
39#34187—71nomaximal, Sylow
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×Dic7C2311yescharacteristic
43C23×Dic7C2312yes—
44C23×Dic7C2311yescharacteristic
45C22×Dic14—28no—
46C22×Dic14C2×C418yes—
47C22×Dic14C2314yes—
48C7×#117—28no—
49C2×C7×C4⋊C4C2×C418yes—
50C2×C7×C4⋊C4C2314yes—
51C2×C7×C4⋊C4C2314yes—
52C2×C7×C22⋊C4C2314yes—
53C2×C42×C7C2314yes—
54C2×C42×C7C2318yes—
55C2×C22⋊Dic7C2314yes—
56#6489—28no—
57C2×C4⋊Dic7C2314yes—
58C2×Dic7⋊C4C2×C4116yes—
59C2×Dic7⋊C4C2318yes—
60C2×Dic7⋊C4C2318yes—
61C2×Dic7⋊C4C2318yes—
62C2×C4×Dic7C2×C4116yes—
63C2×C4×Dic7C2×C418yes—
64C2×C4×Dic7C2314yes—
65C2×C4×Dic7C2318yes—
66C2×C4×Dic7C2314yes—
67C2×C4×Dic7C2314yes—
68C2×C4×Dic7C2312yes—
69C2×C4×Dic7C2312yes—
70#6450C2×C4132yes—
71#6450C2×C4132yes—
72C7×#95C2318yes—
73C7⋊#95C2318yes—
74C7⋊#95C23116yes—
75C7⋊#95C2318yes—
76C7⋊#95C23116yes—
77C7⋊#95C2318yes—
78C22×C4×Q8—71no—
79#2356—78no—
80#2355—78no—
81#2313—74no—
82#2310—72no—
83#2285—72no—
84#2283—74no—
85#2274—71no—
86#2274D711yescharacteristic
87C24×C7C2411yescharacteristic
88C22×C4×C7—44no—
89C22×C4×C7C22×C418yes—
90C22×C4×C7C22×C414yes—
91C22×C4×C7#4118yes—
92C22×C4×C7C2418yes—
93C22×C4×C7C2414yes—
94C22×C4×C7C2412yes—
95C22×Dic7—28no—
96C22×Dic7—24no—
97C22×Dic7—42no—
98C22×Dic7C22×C414yes—
99C22×Dic7C22×C418yes—
100C22×Dic7C22×C414yes—
101C22×Dic7C2418yes—
102C22×Dic7C2411yescharacteristic
103C22×Dic7C2414yes—
104C22×Dic7C2412yes—
105C22×Dic7C2411yescharacteristic
106C2×Dic14—232no—
107C2×Dic14—28no—
108C2×Dic14—28no—
109C2×Dic14C22×C4116yes—
110C7×C4⋊C4—28no—
111C7×C4⋊C4C22×C4116yes—
112C7×C22⋊C4—28no—
113C42×C7—28no—
114C42×C7—48no—
115C22⋊Dic7—28no—
116C4⋊Dic7—28no—
117Dic7⋊C4—216no—
118Dic7⋊C4—48no—
119Dic7⋊C4C22×C4132yes—
120C4×Dic7—28no—
121C4×Dic7—48no—
122C4×Dic7C22×C418yes—
123C4×Dic7C22×C4116yes—
124C4×Dic7C22×C418yes—
125C4×Dic7C22×C4116yes—
126C23×Q8—71no—
127C2×C4×Q8—716no—
128C2×C4×Q8—74no—
129C2×C4×Q8—74no—
130C2×#117—72no—
131C2×#117C2×D712yes—
132C22×C4⋊C4—72no—
133C22×C4⋊C4C2×D711yescharacteristic
134C22×C42—72no—
135C22×C42—71no—
136#391—78no—
137#391—78no—
138#387—716no—
139#387—716no—
140#384—74no—
141#383—78no—
142#382—74no—
143#382—78no—
144#381—78no—
145#380—74no—
146#380—74no—
147C4×C4⋊C4—78no—
148C4×C4⋊C4—78no—
149C4×C4⋊C4—74no—
150C4×C4⋊C4—74no—
151C4×C4⋊C4C2×D718yes—
152C4×C22⋊C4—74no—
153C4×C22⋊C4C2×D714yes—
154#376—78no—
155C23×C7—22no—
156C23×C7—22no—
157C23×C7C23×C412yes—
158C23×C7C2×#4114yes—
159C23×C7C2511yescharacteristic
160C2×C4×C7—28no—
161C2×C4×C7—28no—
162C2×C4×C7—28no—
163C2×C4×C7—28no—
164C2×C4×C7—48no—
165C2×C4×C7#117116yes—
166C2×C4×C7C23×C418yes—
167C2×C4×C7C23×C414yes—
168C2×C4×C7C2×#4114yes—
169C2×C4×C7C2×#4114yes—
170C2×Dic7—216no—
171C2×Dic7—28no—
172C2×Dic7—22no—
173C2×Dic7—24no—
174C2×Dic7—28no—
175C2×Dic7—28no—
176C2×Dic7—28no—
177C2×Dic7—28no—
178C2×Dic7—44no—
179C2×Dic7C23×C412yes—
180C2×Dic7C23×C414yes—
181C2×Dic7C23×C418yes—
182C2×Dic7C23×C412yes—
183C2×Dic72−1+414yes—
184Dic14—232no—
185C22×Q8—78no—
186C22×Q8—74no—
187C22×Q8—148no—
188C23×C4—72no—
189C23×C4—71no—
190C23×C4—71no—
191C23×C4C22×D712yes—
192C23×C4C22×D711yescharacteristic
193C4×Q8—732no—
194C4×Q8—732no—
195#117—28no—
196#117—148no—
197C2×C4⋊C4—716no—
198C2×C4⋊C4—78no—
199C2×C4⋊C4—74no—
200C2×C4⋊C4—78no—
201C2×C4⋊C4—78no—
202C2×C4⋊C4C4×D718yes—
203C2×C4⋊C4C22×D714yes—
204C2×C4⋊C4C22×D714yes—
205C2×C22⋊C4—74no—
206C2×C22⋊C4C22×D714yes—
207C2×C42—716no—
208C2×C42—78no—
209C2×C42—74no—
210C2×C42—74no—
211C2×C42—72no—
212C2×C42—72no—
213C2×C42—74no—
214C2×C42—78no—
215C2×C42C22×D714yes—
216C2×C42C22×D718yes—
217#95—78no—
218#95—716no—
219#95—78no—
220#95—78no—
221#95—716no—
222#95C22×D718yes—
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—28no—
226C22×C7C2×#11714yes—
227C22×C7C24×C411yescharacteristic
228C22×C7C22×#4112yes—
229C22×C7C2×2−1+412yes—
230C22×C7#58512yes—
231C4×C7—28no—
232C4×C7—48no—
233C4×C7C2×#11718yes—
234Dic7—28no—
235Dic7—22no—
236Dic7—24no—
237Dic7—48no—
238Dic7#51914yes—
239C24C23×D711yescharacteristic
240C2×Q8—716no—
241C2×Q8—1432no—
242C2×Q8—148no—
243C2×Q8—148no—
244C22×C4—44no—
245C22×C4—71no—
246C22×C4—74no—
247C22×C4—78no—
248C22×C4—71no—
249C22×C4—74no—
250C22×C4—72no—
251C22×C4—74no—
252C22×C4—78no—
253C22×C4—144no—
254C22×C4—148no—
255C22×C4—282no—
256C22×C4C2×C4×D718yes—
257C22×C4C2×C4×D714yes—
258C22×C4#115718yes—
259C22×C4C23×D718yes—
260C22×C4C23×D714yes—
261C22×C4C23×D712yes—
262C4⋊C4—28no—
263C4⋊C4—732no—
264C4⋊C4—1416no—
265C4⋊C4—148no—
266C4⋊C4—288no—
267C4⋊C4C2×C4×D7116yes—
268C22⋊C4—28no—
269C22⋊C4—148no—
270C42—28no—
271C42—48no—
272C42—78no—
273C42—716no—
274C42—78no—
275C42—716no—
276C42—148no—
277C42—288no—
278C2×C7—24no—
279C2×C7C22×#11711yescharacteristic
280C2×C7C2×#51911yescharacteristic
281C2×C7#342811yescharacteristic
282C2×C7#350714yes—
283C23—22no—
284C23—22no—
285C23C22×C4×D712yes—
286C23C2×#115714yes—
287C23C24×D711yescharacteristic
288Q8—1432no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—48no—
294C2×C4—72no—
295C2×C4—74no—
296C2×C4—72no—
297C2×C4—74no—
298C2×C4—78no—
299C2×C4—148no—
300C2×C4—1416no—
301C2×C4—148no—
302C2×C4—144no—
303C2×C4—142no—
304C2×C4—148no—
305C2×C4—148no—
306C2×C4—148no—
307C2×C4—284no—
308C2×C4#6435116yes—
309C2×C4C22×C4×D718yes—
310C2×C4C22×C4×D714yes—
311C2×C4C2×#115714yes—
312C2×C4C2×#115714yes—
313C7#3418711yescharacteristic, complemented, Sylow
314C22—22no—
315C22—22no—
316C22—28no—
317C22C2×#643514yes—
318C22C23×C4×D711yescharacteristic
319C22C22×#115712yes—
320C22#9061812yes—
321C22D7×2−1+412yes—
322C4—28no—
323C4—48no—
324C4—74no—
325C4—148no—
326C4—144no—
327C4—142no—
328C4—288no—
329C4C2×#643518yes—
330C2—24no—
331C2C22×#643511yescharacteristic
332C2#1174292411yescharacteristic
333C2D7×#51911yescharacteristic
334C2#1174296014yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2274C2
socle, cosocleC23×C7C25
prime radicalC24×C7C24
hypercenter#2274D7
hypocenterC7#34187
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2274

Complements

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

NH ≅ G/Ntype
6. #11731089C2semidirect
7. #11731086C2semidirect
17. C2×#6450C4semidirect
313. C7#34187semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C7×#22742
#117396752
#117396692
#117395892
#117310892
#117310862
#117308902
#117308822
#341877