Finite Group Databasebeta

← #13058534

Subgroups — #13058534

Order 1344.

384 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#10621380C211yesmaximal, characteristic
2C7×#890C211yesmaximal, characteristic, complemented
3C3×#6450C211yesmaximal, characteristic, complemented
4#10620952C211yesmaximal, characteristic
5#10620951C211yesmaximal, characteristic, complemented
6#10620950C212yesmaximal, complemented
7#10620949C212yesmaximal
8#10620948C211yesmaximal, characteristic
9#10620905C212yesmaximal, complemented
10#10620899C212yesmaximal
11#10620895C211yesmaximal, characteristic, complemented
12#90388—31nomaximal, Hall
13C2×C4×D21C2211yescharacteristic
14C2×Dic42C2211yescharacteristic
15C2×C3×Dic14C2211yescharacteristic
16C3×C7×C4⋊C4C2211yescharacteristic
17#66631C2212yes—
18C4⋊Dic21C2212yes—
19C7×C4⋊Dic3C2212yes—
20C7×C4⋊Dic3C2211yescharacteristic
21C7×Dic3⋊C4C2212yes—
22C4×C7×Dic3C2211yescharacteristic
23C3×Dic7⋊C4C2212yes—
24C3×C4×Dic7C2211yescharacteristic
25C3×C4×Dic7C2212yes—
26Dic3⋊Dic7C2212yes—
27Dic3⋊Dic7C2212yes—
28Dic3⋊Dic7C2212yes—
29#66577C2211yescharacteristic
30#66577C2211yescharacteristic
31#66577C2212yes—
32#66575C2211yescharacteristic
33#66575C2211yescharacteristic
34#66575C2212yes—
35Dic3×Dic7C2212yes—
36C7×#125—31no—
37#6461—32no—
38#6461—32no—
39#6460—31no—
40#6460—31no—
41#6453—32no—
42#6453—32no—
43#6450—31no—
44#6450D311yescharacteristic, complemented
45#6435—31no—
46#6432—31no—
47#5583—71nomaximal, Hall
48C22×D21C2311yescharacteristic
49C2×C3×C4×C7C2312yes—
50C2×C3×C4×C7C2311yescharacteristic
51C2×Dic21C2312yes—
52C2×Dic21C2311yescharacteristic
53C4×D21—22no—
54Dic42—22no—
55C2×C7×Dic3C2312yes—
56C2×C7×Dic3C2311yescharacteristic
57C2×C7×Dic3C2311yescharacteristic
58C2×C3×Dic7C2311yescharacteristic
59C2×C3×Dic7C2311yescharacteristic
60C2×C3×Dic7C2312yes—
61C3×Dic14—22no—
62C2×C4×D7—31no—
63C2×Dic14—31no—
64C2×Dic14C2×D311yescharacteristic
65C7×C4⋊C4—31no—
66C7×C4⋊C4—32no—
67C7×C4⋊C4—32no—
68C7×C4⋊C4C2×D311yescharacteristic
69C42×C7—31no—
70(C2×D7)⋊C4—32no—
71(C2×D7)⋊C4—31no—
72(C2×D7)⋊C4—31no—
73(C2×D7)⋊C4—32no—
74C4⋊Dic7—32no—
75C4⋊Dic7—32no—
76C4⋊Dic7—32no—
77C4⋊Dic7—32no—
78Dic7⋊C4—31no—
79Dic7⋊C4—31no—
80Dic7⋊C4—32no—
81Dic7⋊C4C2×D312yes—
82C4×Dic7—32no—
83C4×Dic7C2×D311yescharacteristic
84C4×Dic7C2×D312yes—
85C3×C4×Q8—71no—
86#946—71no—
87#944—71no—
88#898—72no—
89#897—71no—
90#890D711yescharacteristic, complemented
91#889—72no—
92#876—72no—
93#872—71no—
94#870—72no—
95C4×Dic6—71no—
96C22×C3×C7C2411yescharacteristic
97C2×D21—21no—
98C2×D21—21no—
99C2×D21—21no—
100C3×C4×C7—22no—
101C3×C4×C7#4112yes—
102Dic21—22no—
103Dic21—21no—
104C3×Dic7—22no—
105C3×Dic7—21no—
106C3×Dic7—21no—
107C7×Dic3—22no—
108C7×Dic3—21no—
109C7×Dic3—21no—
110#541—211noSylow
111C22×D7—31no—
112C2×C4×C7—32no—
113C2×C4×C7—31no—
114C2×C4×C7—31no—
115C2×C4×C7C22×D312yes—
116C2×C4×C7C22×D311yescharacteristic
117C2×Dic7—32no—
118C2×Dic7—31no—
119C2×Dic7C22×D311yescharacteristic
120C2×Dic7C22×D311yescharacteristic
121C2×Dic7C22×D312yes—
122C4×D7—62no—
123Dic14—22no—
124Dic14—62no—
125C2×C3×Q8—71no—
126C2×C4×D3—71no—
127C2×Dic6—71no—
128C3×C4⋊C4—72no—
129C3×C4⋊C4C2×D711yescharacteristic
130C3×C42—71no—
131C3×C42—72no—
132(C2×D3)⋊C4—72no—
133(C2×D3)⋊C4—71no—
134(C2×D3)⋊C4—71no—
135(C2×D3)⋊C4—72no—
136C4⋊Dic3—72no—
137C4⋊Dic3C2×D711yescharacteristic
138C4⋊Dic3C2×D712yes—
139Dic3⋊C4—72no—
140Dic3⋊C4—71no—
141Dic3⋊C4—71no—
142Dic3⋊C4—72no—
143Dic3⋊C4—72no—
144Dic3⋊C4—72no—
145Dic3⋊C4C2×D712yes—
146C4×Dic3—72no—
147C4×Dic3C2×D711yescharacteristic
148C2×C3×C7C2×#4111yescharacteristic
149C2×C3×C72−1+411yescharacteristic
150C2×C3×C72−1+411yescharacteristic
151D21—41no—
152C4⋊Q8—211no—
153#126—212no—
154#126—212no—
155#125—31no—
156#125—212no—
157#125—212no—
158C22⋊Q8—211no—
159C22⋊Q8—211no—
160C4×Q8—71no—
161C4×Q8—211no—
162#117—211no—
163C22×C7C23×D311yescharacteristic
164C2×D7—61no—
165C2×D7—61no—
166C2×D7—61no—
167C4×C7—22no—
168C4×C7—62no—
169C4×C7—61no—
170C4×C7—61no—
171C4×C7#23512yes—
172Dic7—22no—
173Dic7—21no—
174Dic7—21no—
175Dic7—62no—
176Dic7—61no—
177C22×D3—71no—
178C3×Q8—142no—
179C2×C3×C4—71no—
180C2×C3×C4—71no—
181C2×C3×C4—72no—
182C2×C3×C4C22×D712yes—
183C2×C3×C4C22×D711yescharacteristic
184C2×Dic3—72no—
185C2×Dic3—71no—
186C2×Dic3C22×D712yes—
187C2×Dic3C22×D711yescharacteristic
188C2×Dic3C22×D711yescharacteristic
189C4×D3—142no—
190Dic6—142no—
191C3×C7#54111yescharacteristic, complemented, Hall
192C2×Q8—71no—
193C2×Q8—211no—
194C22×C4—211no—
195C4⋊C4—31no—
196C4⋊C4—32no—
197C4⋊C4—32no—
198C4⋊C4—72no—
199C4⋊C4—211no—
200C4⋊C4—211no—
201C4⋊C4—212no—
202C4⋊C4—212no—
203C4⋊C4—212no—
204C4⋊C4—212no—
205C4⋊C4—212no—
206C4⋊C4D3×D711yescharacteristic
207C22⋊C4—211no—
208C22⋊C4—211no—
209C22⋊C4—212no—
210C22⋊C4—212no—
211C42—31no—
212C42—71no—
213C42—72no—
214C42—212no—
215C2×C7C2×#23511yescharacteristic
216C2×C7#100711yescharacteristic
217C2×C7#101011yescharacteristic
218D7—121no—
219C22×C3C23×D711yescharacteristic
220C2×D3—141no—
221C2×D3—141no—
222C2×D3—141no—
223C3×C4—22no—
224C3×C4—142no—
225C3×C4—141no—
226C3×C4—141no—
227C3×C4#116112yes—
228Dic3—21no—
229Dic3—22no—
230Dic3—21no—
231Dic3—142no—
232Dic3—141no—
233C23—211no—
234Q8—142no—
235Q8—422no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—71no—
240C2×C4—71no—
241C2×C4—72no—
242C2×C4—212no—
243C2×C4—211no—
244C2×C4—422no—
245C2×C4C2×D3×D712yes—
246C2×C4C2×D3×D711yescharacteristic
247C7#558311yescharacteristic, complemented, Sylow
248C2×C3C2×#116111yescharacteristic
249C2×C3#655411yescharacteristic
250C2×C3#655411yescharacteristic
251D3—281no—
252C22—421no—
253C22—421no—
254C22—421no—
255C22C22×D3×D711yescharacteristic
256C4—22no—
257C4—61no—
258C4—61no—
259C4—62no—
260C4—142no—
261C4—141no—
262C4—141no—
263C4—421no—
264C4—422no—
265C4#6667712yes—
266C3#9038811yescharacteristic, complemented, Sylow
267C2—841no—
268C2C2×#6667711yescharacteristic
269C2#1062156711yescharacteristic
270C2#1062157711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC22×D21C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#541
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C7×#890C2semidirect
3. C3×#6450C2semidirect
5. #10620951C2semidirect
6. #10620950C2semidirect
9. #10620905C2semidirect
11. #10620895C2semidirect
44. #6450D3semidirect
90. #890D7semidirect
191. C3×C7#541semidirect
247. C7#5583semidirect
266. C3#90388semidirect

Maximal subgroups

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

subgroupindex
#106213802
C7×#8902
C3×#64502
#106209522
#106209512
#106209502
#106209492
#106209482
#106209052
#106208992
#106208952
#903883
#55837