Finite Group Databasebeta

← #13059536

Subgroups — #13059536

Order 1344.

588 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66683C212yesmaximal, complemented
2#10621413C211yesmaximal, characteristic, complemented
3C2×C7×Dic3⋊C4C211yesmaximal, characteristic, complemented
4C3×#6493C211yesmaximal, characteristic, complemented
5#10621131C212yesmaximal, complemented
6#10621021C212yesmaximal, complemented
7#10620991C212yesmaximal, complemented
8#10620984C212yesmaximal, complemented
9#10620971C212yesmaximal, complemented
10#90201—31nomaximal, Hall
11C22×C3×C4×C7C2211yescharacteristic
12C2×#4268C2212yes—
13C2×D84C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C7×Dic3C2211yescharacteristic
16C2×C3×C22⋊D7C2212yes—
17C2×C3×D28C2211yescharacteristic
18C2×#4248C2212yes—
19C2×#4246C2212yes—
20C2×#4246C2212yes—
21C2×D21⋊C4C2212yes—
22#66683—28no—
23C2×Dic3×D7C2212yes—
24#66631C2212yes—
25C4⋊Dic21C2211yescharacteristic
26C7×Dic3⋊C4C2212yes—
27C7×Dic3⋊C4C2212yes—
28C3×(C2×D7)⋊C4C2212yes—
29C3×C4⋊Dic7C2211yescharacteristic
30Dic3⋊Dic7C2212yes—
31#66575C2212yes—
32(C2×D7)⋊Dic3C2212yes—
33C2×#1157—32no—
34C2×C7×C4⋊C4—31no—
35#6493—31no—
36#6493—32no—
37#6493D311yescharacteristic, complemented
38#6460—32no—
39#6460—32no—
40#6458—32no—
41#6458—32no—
42#5597—71nomaximal, Hall
43C23×C3×C7C2311yescharacteristic
44C22×D21C2312yes—
45C22×C3×D7C2312yes—
46C2×C3×C4×C7—41no—
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2311yescharacteristic
49#4268—24no—
50C2×Dic21C2312yes—
51D84—41no—
52C2×C7×Dic3—41no—
53C2×C7×Dic3D414yescomplemented
54C2×C7×Dic3C2312yes—
55C2×C7×Dic3C2312yes—
56C3×C22⋊D7—24no—
57C2×C3×Dic7C2312yes—
58C3×D28—41no—
59#4248—24no—
60#4246—24no—
61#4246—42no—
62D21⋊C4—24no—
63Dic3×D7—24no—
64C22×C4×C7—31no—
65C22×C4×C7—31no—
66C22×C4×C7C2×D311yescharacteristic
67C2×C22⋊D7—32no—
68C2×C22⋊D7C2×D312yes—
69#1157—68no—
70C2×D28—32no—
71C2×D28—32no—
72C2×D28—31no—
73C2×D28C2×D311yescharacteristic
74C2×C4×D7—32no—
75C2×C4×D7—32no—
76C2×Dic14—32no—
77C7×C4⋊C4—32no—
78C7×C4⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4—32no—
82(C2×D7)⋊C4C2×D312yes—
83C4⋊Dic7—31no—
84C4⋊Dic7—32no—
85C4⋊Dic7C2×D311yescharacteristic
86C2×#237—72no—
87C3×#121—71no—
88#939—72no—
89#939—72no—
90#930—71no—
91C2×Dic3⋊C4D711yescharacteristic, complemented
92#896—72no—
93#884—72no—
94#878—72no—
95C22×C3×C7—21no—
96C22×C3×C7—21no—
97C22×C3×C7C2×D412yes—
98C22×C3×C7C2411yescharacteristic
99C2×D21—22no—
100C2×D21—22no—
101C2×D21—22no—
102C2×C3×D7—22no—
103C2×C3×D7—22no—
104C2×C3×D7—22no—
105C3×C4×C7—21no—
106C3×C4×C7—21no—
107Dic21—22no—
108C3×Dic7—22no—
109C7×Dic3—22no—
110C7×Dic3C2×D414yes—
111#537—211noSylow
112C23×C7C22×D311yescharacteristic
113C22×D7—32no—
114C22×D7C22×D312yes—
115C2×C4×C7—34no—
116C2×C4×C7—32no—
117C2×C4×C7—32no—
118C2×C4×C7—41no—
119C2×C4×C7—121no—
120C2×C4×C7C22×D311yescharacteristic
121C2×C4×C7C22×D311yescharacteristic
122C22⋊D7—24no—
123C22⋊D7—64no—
124C2×Dic7—32no—
125C2×Dic7C22×D312yes—
126D28—41no—
127D28—64no—
128D28—121no—
129D28—122no—
130C4×D7—64no—
131C4×D7—64no—
132Dic14—64no—
133C2×C3×D4—72no—
134C2×C3×D4—71no—
135C22×C3×C4C2×D711yescharacteristic
136C2×#67—72no—
137C2×#67—72no—
138C2×#67—72no—
139C22×Dic3—72no—
140C22×Dic3C2×D711yescharacteristic
141C22×Dic3C2×D711yescharacteristic
142#237—148no—
143C2×D12—71no—
144C2×C4×D3—72no—
145C2×Dic6—72no—
146C3×C4⋊C4—71no—
147C3×C22⋊C4—72no—
148#217—72no—
149(C2×D3)⋊C4—72no—
150(C2×D3)⋊C4—72no—
151C4⋊Dic3—71no—
152Dic3⋊C4—72no—
153Dic3⋊C4C2×D712yes—
154Dic3⋊C4C2×D712yes—
155C2×C3×C7—22no—
156C2×C3×C7C22×D411yescharacteristic
157C2×C3×C7H(5,𝔽2)11yescharacteristic
158C2×C3×C72−1+411yescharacteristic
159D21—42no—
160C3×D7—42no—
161C2×#41—212no—
162C22⋊Q8—212no—
163C22⋊Q8—212no—
164#121—71no—
165#121—212no—
166#121—212no—
167#121—212no—
168#121—211no—
169C2×C4⋊C4—31no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7D3×D412yes—
173C22×C7C23×D311yescharacteristic
174C2×D7—22no—
175C2×D7—22no—
176C2×D7—22no—
177C2×D7—62no—
178C2×D7—62no—
179C2×D7—62no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—34no—
183C4×C7—62no—
184Dic7—22no—
185Dic7—62no—
186C23×C3—72no—
187C23×C3C22×D711yescharacteristic
188C22×D3—72no—
189C3×D4—144no—
190C3×D4—281no—
191C2×C3×C4—41no—
192C2×C3×C4—72no—
193C2×C3×C4C22×D711yescharacteristic
194C2×C3×C4C22×D711yescharacteristic
195#67—144no—
196#67—144no—
197#67—282no—
198C2×Dic3—41no—
199C2×Dic3—72no—
200C2×Dic3—144no—
201C2×Dic3D2814yescomplemented
202C2×Dic3C22×D712yes—
203C2×Dic3C22×D712yes—
204D12—281no—
205C4×D3—144no—
206Dic6—144no—
207C3×C7#53711yescharacteristic, complemented, Hall
208#41—428no—
209C2×Q8—212no—
210C2×D4—72no—
211C2×D4—71no—
212C2×D4—212no—
213C2×D4—212no—
214C2×D4—212no—
215C2×D4—211no—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—212no—
219C22×C4—212no—
220C22×C4D3×D711yescharacteristic
221C4⋊C4—32no—
222C4⋊C4—32no—
223C4⋊C4—71no—
224C4⋊C4—212no—
225C4⋊C4—211no—
226C22⋊C4—72no—
227C22⋊C4—212no—
228C22⋊C4—212no—
229C22⋊C4—212no—
230C2×C7—22no—
231C2×C7C2×D3×D411yescharacteristic
232C2×C7#100411yescharacteristic
233C2×C7#101011yescharacteristic
234D7—42no—
235D7—122no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—142no—
239C22×C3—142no—
240C22×C3—142no—
241C22×C3C2×D2812yes—
242C22×C3C23×D711yescharacteristic
243C2×D3—142no—
244C2×D3—142no—
245C2×D3—142no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—142no—
249Dic3—22no—
250Dic3—142no—
251Dic3C2×D2814yes—
252C23—72no—
253C23—212no—
254C23C2×D3×D711yescharacteristic
255Q8—424no—
256D4—144no—
257D4—281no—
258D4—424no—
259D4—424no—
260D4—841no—
261D4—842no—
262C2×C4—34no—
263C2×C4—32no—
264C2×C4—32no—
265C2×C4—41no—
266C2×C4—72no—
267C2×C4—121no—
268C2×C4—212no—
269C2×C4—424no—
270C2×C4—424no—
271C2×C4C2×D3×D711yescharacteristic
272C2×C4C2×D3×D711yescharacteristic
273C7#559711yescharacteristic, complemented, Sylow
274C2×C3—22no—
275C2×C3—282no—
276C2×C3C22×D2811yescharacteristic
277C2×C3#654811yescharacteristic
278C2×C3#655111yescharacteristic
279D3—282no—
280C22—21no—
281C22—21no—
282C22—142no—
283C22—142no—
284C22—142no—
285C22—422no—
286C22—422no—
287C22—422no—
288C22D3×D2812yes—
289C22C22×D3×D711yescharacteristic
290C4—21no—
291C4—21no—
292C4—34no—
293C4—62no—
294C4—142no—
295C4—422no—
296C3#9020111yescharacteristic, complemented, Sylow
297C2—22no—
298C2—282no—
299C2—842no—
300C2#1062154711yescharacteristic
301C2C2×D3×D2811yescharacteristic
302C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#537
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#66683C2semidirect
2. #10621413C2semidirect
3. C2×C7×Dic3⋊C4C2semidirect
4. C3×#6493C2semidirect
5. #10621131C2semidirect
6. #10621021C2semidirect
7. #10620991C2semidirect
8. #10620984C2semidirect
9. #10620971C2semidirect
37. #6493D3semidirect
53. C2×C7×Dic3D4semidirect
91. C2×Dic3⋊C4D7semidirect
201. C2×Dic3D28semidirect
207. C3×C7#537semidirect
273. C7#5597semidirect
296. C3#90201semidirect

Maximal subgroups

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

subgroupindex
C2×#666832
#106214132
C2×C7×Dic3⋊C42
C3×#64932
#106211312
#106210212
#106209912
#106209842
#106209712
#902013
#55977