Finite Group Databasebeta

← #13061617

Subgroups — #13061617

Order 1344.

660 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 175 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×#66728C211yesmaximal, characteristic, complemented
2C22×C4×D21C211yesmaximal, characteristic, complemented
3C3×C7×#117C211yesmaximal, characteristic, complemented
4#10621380C212yesmaximal, complemented
5#10621378C212yesmaximal, complemented
6#10621369C212yesmaximal, complemented
7#10621365C212yesmaximal, complemented
8C4×D84C214yesmaximal, complemented
9#90215—31nomaximal, Hall
10C23×D21C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C2×#4268C2212yes—
13C22×Dic21C2211yescharacteristic
14#66728—24no—
15C2×D84C2214yes—
16C2×D84C2211yescharacteristic, complemented
17C2×C4×D21—24no—
18C2×C4×D21C2212yes—
19C2×C4×D21C2212yes—
20C2×C4×D21C2212yescomplemented
21C2×Dic42C2211yescharacteristic, complemented
22C3×C7×C4⋊C4C2212yescomplemented
23C3×C7×C22⋊C4C2212yes—
24C3×C42×C7C2212yescomplemented
25#66631C2214yes—
26#66631C2214yescomplemented
27C4⋊Dic21C2214yescomplemented
28C2×#1157—31no—
29C22×C4×D7—31no—
30C7×#117D311yescharacteristic, complemented
31#6460—32no—
32#6458—32no—
33#6449—32no—
34#6445—32no—
35C4×D28—34no—
36#5524—71nomaximal, Hall
37C23×C3×C7C2311yescharacteristic
38C22×D21—22no—
39C22×D21—24no—
40C22×D21C2312yes—
41C22×D21C2312yes—
42C2×C3×C4×C7D414yescomplemented
43C2×C3×C4×C7C2314yes—
44C2×C3×C4×C7C2311yescharacteristic
45C2×C3×C4×C7C2311yescharacteristic
46#4268—24no—
47C2×Dic21—22no—
48C2×Dic21C2312yes—
49C2×Dic21C2312yes—
50D84—28no—
51D84—22no—
52C4×D21—24no—
53C4×D21—24no—
54C4×D21—24no—
55Dic42—22no—
56C23×D7—31no—
57C22×C4×C7C2×D311yescharacteristic
58C2×C22⋊D7—32no—
59C22×Dic7—31no—
60#1157—64no—
61C2×D28—34no—
62C2×D28—31no—
63C2×C4×D7—32no—
64C2×C4×D7—32no—
65C2×C4×D7—32no—
66C2×C4×D7—64no—
67C2×Dic14—31no—
68C7×C4⋊C4C2×D312yescomplemented
69C7×C22⋊C4C2×D312yes—
70C42×C7C2×D312yescomplemented
71(C2×D7)⋊C4—34no—
72(C2×D7)⋊C4—34no—
73C4⋊Dic7—34no—
74C2×#235—71no—
75C22×C4×D3—71no—
76C3×#117D711yescharacteristic, complemented
77#897—72no—
78#895—72no—
79#886—72no—
80#882—72no—
81C4×D12—74no—
82C22×C3×C7—22no—
83C22×C3×C7C2×D412yes—
84C22×C3×C7C2411yescharacteristic
85C2×D21—28no—
86C2×D21—22no—
87C2×D21—22no—
88C2×D21—22no—
89C2×D21—24no—
90C2×D21#4114yescomplemented
91C3×C4×C7—24no—
92C3×C4×C7C2×D412yes—
93C3×C4×C7C2×D412yes—
94Dic21—22no—
95Dic21—22no—
96#525—211noSylow
97C23×C7C22×D311yescharacteristic
98C22×D7—32no—
99C22×D7—32no—
100C22×D7—64no—
101C22×D7—62no—
102C2×C4×C7D1214yescomplemented
103C2×C4×C7C22×D314yes—
104C2×C4×C7C22×D311yescharacteristic
105C2×C4×C7C22×D311yescharacteristic
106C22⋊D7—64no—
107C2×Dic7—32no—
108C2×Dic7—32no—
109C2×Dic7—62no—
110D28—68no—
111D28—62no—
112C4×D7—64no—
113C4×D7—64no—
114C4×D7—64no—
115Dic14—62no—
116C23×D3—71no—
117C22×C3×C4C2×D711yescharacteristic
118C2×#67—72no—
119C22×Dic3—71no—
120#235—144no—
121C2×D12—74no—
122C2×D12—71no—
123C2×C4×D3—72no—
124C2×C4×D3—72no—
125C2×C4×D3—72no—
126C2×C4×D3—144no—
127C2×Dic6—71no—
128C3×C4⋊C4C2×D712yescomplemented
129C3×C22⋊C4C2×D712yes—
130C3×C42C2×D712yescomplemented
131(C2×D3)⋊C4—74no—
132(C2×D3)⋊C4—74no—
133C4⋊Dic3—74no—
134C2×C3×C7—22no—
135C2×C3×C7C22×D411yescharacteristic
136C2×C3×C7C2×#4112yes—
137D21—24no—
138D21—42no—
139C2×#41—211no—
140C23×C4—211no—
141#123—212no—
142C22⋊Q8—212no—
143#121—212no—
144#120—212no—
145C4×D4—214no—
146#117D2111yescharacteristic, complemented
147C22×C7—22no—
148C22×C7C2×D1212yes—
149C22×C7C23×D311yescharacteristic
150C2×D7—34no—
151C2×D7—68no—
152C2×D7—62no—
153C2×D7—62no—
154C2×D7—62no—
155C2×D7—64no—
156C4×C7—24no—
157C4×C7C2×D1212yes—
158C4×C7C2×D1212yes—
159Dic7—62no—
160Dic7—62no—
161C23×C3C22×D711yescharacteristic
162C22×D3—72no—
163C22×D3—72no—
164C22×D3—144no—
165C22×D3—142no—
166C2×C3×C4D2814yescomplemented
167C2×C3×C4C22×D714yes—
168C2×C3×C4C22×D711yescharacteristic
169C2×C3×C4C22×D711yescharacteristic
170#67—144no—
171C2×Dic3—72no—
172C2×Dic3—72no—
173C2×Dic3—142no—
174D12—148no—
175D12—142no—
176C4×D3—144no—
177C4×D3—144no—
178C4×D3—144no—
179Dic6—142no—
180C3×C7#52511yescharacteristic, complemented, Hall
181C24—211no—
182#41—424no—
183C2×Q8—211no—
184C2×D4—212no—
185C2×D4—214no—
186C2×D4—211no—
187C22×C4—211no—
188C22×C4—212no—
189C22×C4—212no—
190C22×C4—212no—
191C22×C4—424no—
192C22×C4C2×D2111yescharacteristic
193C4⋊C4—214no—
194C4⋊C4C2×D2112yescomplemented
195C22⋊C4—214no—
196C22⋊C4—214no—
197C22⋊C4C2×D2112yes—
198C42C2×D2112yescomplemented
199C2×C7—22no—
200C2×C7C22×D1211yescharacteristic
201C2×C7D3×#4112yes—
202D7—64no—
203D7—122no—
204C22×C3—22no—
205C22×C3C2×D2812yes—
206C22×C3C23×D711yescharacteristic
207C2×D3—74no—
208C2×D3—148no—
209C2×D3—142no—
210C2×D3—142no—
211C2×D3—142no—
212C2×D3—144no—
213C3×C4—24no—
214C3×C4C2×D2812yes—
215C3×C4C2×D2812yes—
216Dic3—142no—
217Dic3—142no—
218C23—212no—
219C23—212no—
220C23—424no—
221C23—422no—
222C23C22×D2111yescharacteristic
223Q8—422no—
224D4—428no—
225D4—424no—
226D4—422no—
227C2×C4—212no—
228C2×C4—212no—
229C2×C4—422no—
230C2×C4—424no—
231C2×C4—424no—
232C2×C4—424no—
233C2×C4D8414yescomplemented
234C2×C4C22×D2114yes—
235C2×C4C22×D2111yescharacteristic
236C2×C4C22×D2111yescharacteristic
237C7#552411yescharacteristic, complemented, Sylow
238C2×C3—22no—
239C2×C3C22×D2811yescharacteristic
240C2×C3D7×#4112yes—
241D3—144no—
242D3—282no—
243C22—22no—
244C22—214no—
245C22—428no—
246C22—422no—
247C22—422no—
248C22—422no—
249C22—424no—
250C22C2×D8412yes—
251C22C23×D2111yescharacteristic
252C4—24no—
253C4—422no—
254C4—422no—
255C4C2×D8412yes—
256C4C2×D8412yes—
257C3#9021511yescharacteristic, complemented, Sylow
258C2—22no—
259C2—424no—
260C2—842no—
261C2C22×D8411yescharacteristic
262C2#41×D2112yes—
subgroupG/H
centerC2×C4C22×D21
commutator, socle, cosocleC22×C3×C7C24
FrattiniC22C23×D21
FittingC3×C7×#117C2
hypercenter#117D21
hypocenterC3×C7#525
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#66728C2semidirect
2. C22×C4×D21C2semidirect
3. C3×C7×#117C2semidirect
4. #10621380C2semidirect
5. #10621378C2semidirect
6. #10621369C2semidirect
7. #10621365C2semidirect
8. C4×D84C2semidirect
16. C2×D84C22semidirect
20. C2×C4×D21C22semidirect
21. C2×Dic42C22semidirect
22. C3×C7×C4⋊C4C22semidirect
24. C3×C42×C7C22semidirect
26. #66631C22semidirect
27. C4⋊Dic21C22semidirect
30. C7×#117D3semidirect
42. C2×C3×C4×C7D4semidirect
68. C7×C4⋊C4C2×D3semidirect
70. C42×C7C2×D3semidirect
76. C3×#117D7semidirect
90. C2×D21#41semidirect
102. C2×C4×C7D12semidirect
128. C3×C4⋊C4C2×D7semidirect
130. C3×C42C2×D7semidirect
146. #117D21semidirect
166. C2×C3×C4D28semidirect
180. C3×C7#525semidirect
194. C4⋊C4C2×D21semidirect
198. C42C2×D21semidirect
233. C2×C4D84semidirect
237. C7#5524semidirect
257. C3#90215semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
C22×C4×D212
C3×C7×#1172
#106213802
#106213782
#106213692
#106213652
C4×D842
#902153
#55247