Finite Group Databasebeta

← #13058539

Subgroups — #13058539

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#10621370C211yesmaximal, characteristic, complemented
2C7×#897C211yesmaximal, characteristic
3C3×#6453C211yesmaximal, characteristic, complemented
4#10620946C212yesmaximal, complemented
5#10620938C211yesmaximal, characteristic, complemented
6#10620933C211yesmaximal, characteristic
7#10620929C212yesmaximal
8#10620927C211yesmaximal, characteristic
9#10620906C211yesmaximal, characteristic, complemented
10#10620903C212yesmaximal
11#10620897C212yesmaximal, complemented
12#90316—31nomaximal, Hall
13C2×Dic42C2211yescharacteristic
14C2×C4×D3×C7C2211yescharacteristic
15C2×C7×Dic6C2211yescharacteristic
16C3×C7×C4⋊C4C2211yescharacteristic
17#66629C2212yes—
18C4×Dic21C2212yes—
19C4×Dic21C2211yescharacteristic
20C7×(C2×D3)⋊C4C2212yes—
21C7×C4⋊Dic3C2212yes—
22C3×C4⋊Dic7C2212yes—
23C3×C4⋊Dic7C2211yescharacteristic
24C3×Dic7⋊C4C2212yes—
25C3×C4×Dic7C2211yescharacteristic
26Dic3⋊Dic7C2212yes—
27Dic3⋊Dic7C2211yescharacteristic
28Dic3⋊Dic7C2211yescharacteristic
29Dic7⋊Dic3C2212yes—
30Dic7⋊Dic3C2212yes—
31Dic7⋊Dic3C2212yes—
32(C2×D3)⋊Dic7C2211yescharacteristic
33(C2×D3)⋊Dic7C2212yes—
34(C2×D3)⋊Dic7C2211yescharacteristic
35Dic3×Dic7C2212yes—
36C7×C22⋊Q8—31no—
37Q8×Dic7—31no—
38#6507—31no—
39#6489—31no—
40#6487—31no—
41#6453—32no—
42#6453D311yescharacteristic, complemented
43#6452—32no—
44#6450—31no—
45#6442—32no—
46#6442—32no—
47#5698—71nomaximal, Hall
48C22×D3×C7C2311yescharacteristic
49C2×C3×C4×C7C2312yes—
50C2×C3×C4×C7C2311yescharacteristic
51C2×Dic21C2311yescharacteristic
52C2×Dic21C2312yes—
53C2×Dic21C2311yescharacteristic
54Dic42—22no—
55C2×C7×Dic3C2312yes—
56C2×C7×Dic3C2311yescharacteristic
57C4×D3×C7—22no—
58C7×Dic6—22no—
59C2×C3×Dic7C2312yes—
60C2×C3×Dic7C2311yescharacteristic
61C2×C3×Dic7C2311yescharacteristic
62C2×C7×Q8—31no—
63C22×C4×C7—31no—
64C2×Dic14—31no—
65C7×C4⋊C4—32no—
66C7×C4⋊C4C2×D311yescharacteristic
67C7×C22⋊C4—32no—
68C22⋊Dic7—32no—
69C22⋊Dic7—31no—
70C22⋊Dic7—31no—
71C4⋊Dic7—32no—
72C4⋊Dic7—31no—
73C4⋊Dic7—31no—
74C4⋊Dic7C2×D312yes—
75C4⋊Dic7C2×D311yescharacteristic
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78Dic7⋊C4—32no—
79Dic7⋊C4—32no—
80Dic7⋊C4C2×D312yes—
81C4×Dic7—32no—
82C4×Dic7—31no—
83C4×Dic7—32no—
84C4×Dic7C2×D311yescharacteristic
85C3×#125—71no—
86#898—72no—
87#898—72no—
88#897—71no—
89#897D711yescharacteristic
90#890—72no—
91#890—72no—
92#887—71no—
93#887—71no—
94#872—71no—
95#869—71no—
96C22×C3×C7C2411yescharacteristic
97C2×D3×C7—21no—
98C2×D3×C7—21no—
99C2×D3×C7—21no—
100C3×C4×C7—22no—
101C3×C4×C7#4112yes—
102Dic21—22no—
103Dic21—21no—
104Dic21—21no—
105C3×Dic7—22no—
106C3×Dic7—21no—
107C3×Dic7—21no—
108C7×Dic3—22no—
109C7×Dic3—21no—
110#541—211noSylow
111C23×C7—31no—
112C7×Q8—62no—
113C2×C4×C7—32no—
114C2×C4×C7—31no—
115C2×C4×C7—62no—
116C2×C4×C7C22×D312yes—
117C2×C4×C7C22×D311yescharacteristic
118C2×Dic7—31no—
119C2×Dic7—31no—
120C2×Dic7—32no—
121C2×Dic7C22×D312yes—
122C2×Dic7C22×D311yescharacteristic
123C2×Dic7C22×D311yescharacteristic
124Dic14—62no—
125C2×C4×D3C2×D711yescharacteristic
126C2×Dic6—71no—
127C2×Dic6C2×D711yescharacteristic
128C3×C4⋊C4—72no—
129C3×C4⋊C4—71no—
130C3×C4⋊C4—72no—
131C3×C4⋊C4C2×D711yescharacteristic
132C3×C42—71no—
133(C2×D3)⋊C4—71no—
134(C2×D3)⋊C4—71no—
135(C2×D3)⋊C4—72no—
136(C2×D3)⋊C4C2×D712yes—
137C4⋊Dic3—72no—
138C4⋊Dic3—72no—
139C4⋊Dic3—72no—
140C4⋊Dic3C2×D712yes—
141Dic3⋊C4—72no—
142Dic3⋊C4—71no—
143Dic3⋊C4—71no—
144Dic3⋊C4—72no—
145C4×Dic3—71no—
146C4×Dic3—72no—
147C4×Dic3—72no—
148C2×C3×C7C2×#4111yescharacteristic
149C2×C3×C72−1+411yescharacteristic
150C2×C3×C72−1+411yescharacteristic
151D3×C7—41no—
152C4⋊Q8—211no—
153#126—212no—
154#126—212no—
155#125—71no—
156#125—212no—
157#125—212no—
158C22⋊Q8—31no—
159C22⋊Q8—211no—
160C4×Q8—211no—
161C4×Q8—211no—
162#117—211no—
163C22×C7—61no—
164C22×C7—61no—
165C22×C7—61no—
166C22×C7C23×D311yescharacteristic
167C4×C7—22no—
168C4×C7—61no—
169C4×C7—62no—
170C4×C7#23912yes—
171Dic7—21no—
172Dic7—22no—
173Dic7—21no—
174Dic7—62no—
175Dic7—61no—
176Dic7—61no—
177C22×D3C22×D711yescharacteristic
178C2×C3×C4—72no—
179C2×C3×C4—71no—
180C2×C3×C4—71no—
181C2×C3×C4C22×D712yes—
182C2×C3×C4C22×D711yescharacteristic
183C2×Dic3—71no—
184C2×Dic3—71no—
185C2×Dic3—72no—
186C2×Dic3C22×D712yes—
187C2×Dic3C22×D711yescharacteristic
188C4×D3—22no—
189Dic6—22no—
190Dic6—142no—
191C3×C7#54111yescharacteristic, complemented, Hall
192C2×Q8—31no—
193C2×Q8—211no—
194C22×C4—31no—
195C4⋊C4—32no—
196C4⋊C4—72no—
197C4⋊C4—71no—
198C4⋊C4—72no—
199C4⋊C4—212no—
200C4⋊C4—212no—
201C4⋊C4—212no—
202C4⋊C4—211no—
203C4⋊C4—211no—
204C4⋊C4—212no—
205C4⋊C4—212no—
206C4⋊C4D3×D711yescharacteristic
207C22⋊C4—32no—
208C22⋊C4—211no—
209C22⋊C4—211no—
210C22⋊C4—212no—
211C42—71no—
212C42—212no—
213C42—211no—
214C42—212no—
215C2×C7—121no—
216C2×C7C2×#23911yescharacteristic
217C2×C7#101011yescharacteristic
218C2×C7#101011yescharacteristic
219C22×C3C23×D711yescharacteristic
220C2×D3—21no—
221C2×D3—21no—
222C2×D3—21no—
223C3×C4—22no—
224C3×C4—142no—
225C3×C4—141no—
226C3×C4—141no—
227C3×C4#115912yes—
228Dic3—22no—
229Dic3—21no—
230Dic3—142no—
231Dic3—141no—
232Dic3—141no—
233C23—31no—
234Q8—62no—
235Q8—422no—
236C2×C4—32no—
237C2×C4—31no—
238C2×C4—62no—
239C2×C4—71no—
240C2×C4—71no—
241C2×C4—72no—
242C2×C4—211no—
243C2×C4—211no—
244C2×C4—212no—
245C2×C4C2×D3×D712yes—
246C2×C4C2×D3×D711yescharacteristic
247C7#569811yescharacteristic, complemented, Sylow
248C2×C3C2×#115911yescharacteristic
249C2×C3#655111yescharacteristic
250C2×C3#655411yescharacteristic
251D3—41no—
252C22—61no—
253C22—61no—
254C22—61no—
255C22C22×D3×D711yescharacteristic
256C4—22no—
257C4—61no—
258C4—62no—
259C4—141no—
260C4—141no—
261C4—142no—
262C4—422no—
263C4—421no—
264C4—421no—
265C4#6667212yes—
266C3#9031611yescharacteristic, complemented, Sylow
267C2—121no—
268C2C2×#6667211yescharacteristic
269C2#1062156711yescharacteristic
270C2#1062157511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC22×D3×C7C23
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
1. #10621370C2semidirect
3. C3×#6453C2semidirect
4. #10620946C2semidirect
5. #10620938C2semidirect
9. #10620906C2semidirect
11. #10620897C2semidirect
42. #6453D3semidirect
191. C3×C7#541semidirect
247. C7#5698semidirect
266. C3#90316semidirect

Maximal subgroups

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

subgroupindex
#106213702
C7×#8972
C3×#64532
#106209462
#106209382
#106209332
#106209292
#106209272
#106209062
#106209032
#106208972
#903163
#56987