Finite Group Databasebeta

← #13059360

Subgroups — #13059360

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#10621409C211yesmaximal, characteristic
2C7×#924C211yesmaximal, characteristic
3C3×#6487C211yesmaximal, characteristic
4#10621093C212yesmaximal
5#10621092C212yesmaximal
6#10620947C211yesmaximal, characteristic, complemented
7#10620946C211yesmaximal, characteristic, complemented
8#10620906C211yesmaximal, characteristic, complemented
9#10620905C212yesmaximal, complemented
10#10620896C212yesmaximal, complemented
11#10620893C211yesmaximal, characteristic, complemented
12#90316—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×C7×Dic6C2211yescharacteristic
15C2×C3×Dic14C2211yescharacteristic
16#66636C2211yescharacteristic
17#66636C2211yescharacteristic
18C4⋊Dic21C2211yescharacteristic
19C4⋊Dic21C2211yescharacteristic
20C4×Dic21C2211yescharacteristic
21C4×Dic21C2211yescharacteristic
22C7×#217C2212yes—
23C7×C4⋊Dic3C2211yescharacteristic
24C7×Dic3⋊C4C2212yes—
25C3×C22⋊Dic7C2212yes—
26C3×C4⋊Dic7C2211yescharacteristic
27C3×Dic7⋊C4C2212yes—
28Dic3⋊Dic7C2212yes—
29Dic3⋊Dic7C2212yes—
30#66577C2212yes—
31#66577C2212yes—
32Dic7⋊Dic3C2212yes—
33Dic7⋊Dic3C2212yes—
34Dic3×Dic7C2212yes—
35Dic3×Dic7C2212yes—
36C7×C22⋊Q8—31no—
37Q8×Dic7—31no—
38#6507—31no—
39#6489—31no—
40#6487D311yescharacteristic
41#6453—31no—
42#6453—32no—
43#6452—32no—
44#6450—31no—
45#6442—32no—
46#6442—32no—
47#5625—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C2×C3×C4×C7—22no—
50C2×C3×C4×C7C2311yescharacteristic
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2311yescharacteristic
54C2×Dic21C2311yescharacteristic
55C2×Dic21C2311yescharacteristic
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2312yes—
58C7×Dic6—22no—
59C2×C3×Dic7C2312yes—
60C2×C3×Dic7C2312yes—
61C3×Dic14—22no—
62C2×C7×Q8—31no—
63C22×C4×C7C2×D311yescharacteristic
64C2×Dic14C2×D311yescharacteristic
65C7×C4⋊C4—31no—
66C7×C4⋊C4—32no—
67C7×C22⋊C4—32no—
68C22⋊Dic7—31no—
69C22⋊Dic7—31no—
70C22⋊Dic7C2×D312yes—
71C4⋊Dic7—31no—
72C4⋊Dic7—31no—
73C4⋊Dic7—32no—
74C4⋊Dic7—32no—
75C4⋊Dic7C2×D311yescharacteristic
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78Dic7⋊C4—32no—
79Dic7⋊C4—32no—
80Dic7⋊C4C2×D312yes—
81C4×Dic7—31no—
82C4×Dic7—31no—
83C4×Dic7—32no—
84C4×Dic7—32no—
85C3×C22⋊Q8—71no—
86Q8×Dic3—71no—
87#944—71no—
88#926—71no—
89#924D711yescharacteristic
90#890—71no—
91#890—72no—
92#889—72no—
93#887—71no—
94#879—72no—
95#879—72no—
96C22×C3×C7—21no—
97C22×C3×C7—21no—
98C22×C3×C7—21no—
99C22×C3×C7C2411yescharacteristic
100C3×C4×C7—21no—
101C3×C4×C7#4112yes—
102Dic21—21no—
103Dic21—21no—
104Dic21—21no—
105Dic21—21no—
106C3×Dic7—22no—
107C3×Dic7—22no—
108C7×Dic3—22no—
109C7×Dic3—22no—
110#541—211noSylow
111C23×C7C22×D311yescharacteristic
112C7×Q8—62no—
113C2×C4×C7—22no—
114C2×C4×C7—32no—
115C2×C4×C7—32no—
116C2×C4×C7C22×D311yescharacteristic
117C2×C4×C7C22×D311yescharacteristic
118C2×Dic7—31no—
119C2×Dic7—31no—
120C2×Dic7—31no—
121C2×Dic7—31no—
122C2×Dic7C22×D312yes—
123C2×Dic7C22×D312yes—
124Dic14—22no—
125C2×C3×Q8—71no—
126C22×C3×C4C2×D711yescharacteristic
127C2×Dic6C2×D711yescharacteristic
128C3×C4⋊C4—72no—
129C3×C4⋊C4—71no—
130C3×C22⋊C4—72no—
131#217—71no—
132#217—71no—
133#217C2×D712yes—
134C4⋊Dic3—71no—
135C4⋊Dic3—71no—
136C4⋊Dic3—72no—
137C4⋊Dic3—72no—
138C4⋊Dic3C2×D711yescharacteristic
139Dic3⋊C4—72no—
140Dic3⋊C4—72no—
141Dic3⋊C4—72no—
142Dic3⋊C4—72no—
143Dic3⋊C4C2×D712yes—
144C4×Dic3—71no—
145C4×Dic3—71no—
146C4×Dic3—72no—
147C4×Dic3—72no—
148C2×C3×C7—41no—
149C2×C3×C7C2×#4111yescharacteristic
150C2×C3×C72−1+411yescharacteristic
151C2×C3×C72−1+411yescharacteristic
152C4⋊Q8—211no—
153#126—212no—
154#126—212no—
155#125—212no—
156#125—212no—
157#125—211no—
158C22⋊Q8—31no—
159C22⋊Q8—71no—
160C4×Q8—211no—
161C4×Q8—211no—
162#117—211no—
163C22×C7—21no—
164C22×C7—21no—
165C22×C7—21no—
166C22×C7C23×D311yescharacteristic
167C4×C7—21no—
168C4×C7—62no—
169C4×C7—62no—
170C4×C7#23712yes—
171Dic7—22no—
172Dic7—22no—
173Dic7—61no—
174Dic7—61no—
175Dic7—61no—
176Dic7—61no—
177C23×C3C22×D711yescharacteristic
178C3×Q8—142no—
179C2×C3×C4—22no—
180C2×C3×C4—72no—
181C2×C3×C4—72no—
182C2×C3×C4C22×D711yescharacteristic
183C2×C3×C4C22×D711yescharacteristic
184C2×Dic3—71no—
185C2×Dic3—71no—
186C2×Dic3—71no—
187C2×Dic3—71no—
188C2×Dic3C22×D712yes—
189C2×Dic3C22×D712yes—
190Dic6—22no—
191C3×C7#54111yescharacteristic, complemented, Hall
192C2×Q8—31no—
193C2×Q8—71no—
194C22×C4D3×D711yescharacteristic
195C4⋊C4—31no—
196C4⋊C4—32no—
197C4⋊C4—72no—
198C4⋊C4—71no—
199C4⋊C4—212no—
200C4⋊C4—212no—
201C4⋊C4—212no—
202C4⋊C4—212no—
203C4⋊C4—212no—
204C4⋊C4—212no—
205C4⋊C4—211no—
206C4⋊C4—211no—
207C22⋊C4—32no—
208C22⋊C4—72no—
209C22⋊C4—211no—
210C22⋊C4—211no—
211C42—212no—
212C42—212no—
213C42—211no—
214C42—211no—
215C2×C7—41no—
216C2×C7C2×#23711yescharacteristic
217C2×C7#100711yescharacteristic
218C2×C7#101011yescharacteristic
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3C23×D711yescharacteristic
223C3×C4—21no—
224C3×C4—142no—
225C3×C4—142no—
226C3×C4#115912yes—
227Dic3—22no—
228Dic3—22no—
229Dic3—141no—
230Dic3—141no—
231Dic3—141no—
232Dic3—141no—
233C23C2×D3×D711yescharacteristic
234Q8—62no—
235Q8—142no—
236C2×C4—22no—
237C2×C4—32no—
238C2×C4—32no—
239C2×C4—72no—
240C2×C4—72no—
241C2×C4—211no—
242C2×C4—211no—
243C2×C4—211no—
244C2×C4—211no—
245C2×C4C2×D3×D711yescharacteristic
246C2×C4C2×D3×D711yescharacteristic
247C7#562511yescharacteristic, complemented, Sylow
248C2×C3—41no—
249C2×C3C2×#115911yescharacteristic
250C2×C3#655111yescharacteristic
251C2×C3#655411yescharacteristic
252C22—21no—
253C22—21no—
254C22—21no—
255C22C22×D3×D711yescharacteristic
256C4—21no—
257C4—62no—
258C4—62no—
259C4—142no—
260C4—142no—
261C4—421no—
262C4—421no—
263C4—421no—
264C4—421no—
265C4#6668412yes—
266C3#9031611yescharacteristic, complemented, Sylow
267C2—41no—
268C2#1062154711yescharacteristic
269C2#1062155111yescharacteristic
270C2C2×#6668411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC23×C3×C7C23
hypercenterC22×C4D3×D7
hypocenterC3×C7#541
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #10620947C2semidirect
7. #10620946C2semidirect
8. #10620906C2semidirect
9. #10620905C2semidirect
10. #10620896C2semidirect
11. #10620893C2semidirect
191. C3×C7#541semidirect
247. C7#5625semidirect
266. C3#90316semidirect

Maximal subgroups

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

subgroupindex
#106214092
C7×#9242
C3×#64872
#106210932
#106210922
#106209472
#106209462
#106209062
#106209052
#106208962
#106208932
#903163
#56257