Finite Group Databasebeta

← #13059842

Subgroups — #13059842

Order 1344.

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

normal subgroups only

This group has 412 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
1#10621407C211yesmaximal, characteristic
2C7×#924C211yesmaximal, characteristic
3C2×C3×Dic7⋊C4C211yesmaximal, characteristic
4#10621136C212yesmaximal
5#10621090C212yesmaximal
6#10620897C212yesmaximal, complemented
7#10620896C212yesmaximal, complemented
8#10620891C212yesmaximal, complemented
9#10620890C212yesmaximal, complemented
10#90314—31nomaximal, Hall
11C22×C3×C4×C7C2211yescharacteristic
12C2×Dic42C2211yescharacteristic
13C2×C7×Dic6C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C2×#4248C2212yes—
17#66636C2212yes—
18C4⋊Dic21C2211yescharacteristic
19#66629C2212yes—
20C7×#217C2212yes—
21C7×C4⋊Dic3C2211yescharacteristic
22C7×Dic3⋊C4C2212yes—
23C3×Dic7⋊C4C2212yes—
24C3×Dic7⋊C4C2212yes—
25Dic3⋊Dic7C2212yes—
26#66577C2212yes—
27Dic7⋊Dic3C2212yes—
28Dic7⋊Dic3C2212yes—
29Dic7⋊Dic3C2212yes—
30Dic3×Dic7C2212yes—
31Dic3×Dic7C2212yes—
32C7×C22⋊Q8—31no—
33#6507—32no—
34#6487—31no—
35C2×Dic7⋊C4D311yescharacteristic
36#6452—32no—
37#6452—32no—
38#6451—32no—
39#6441—32no—
40#6440—32no—
41#5500—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C2×C3×C4×C7—41no—
44C2×C3×C4×C7C2311yescharacteristic
45C2×C3×C4×C7C2311yescharacteristic
46C2×Dic21C2312yes—
47C2×Dic21C2312yes—
48Dic42—41no—
49C2×C7×Dic3C2312yes—
50C2×C7×Dic3C2312yes—
51C7×Dic6—41no—
52C2×C3×Dic7—41no—
53C2×C3×Dic7Q814yescomplemented
54C2×C3×Dic7C2312yes—
55C2×C3×Dic7C2312yes—
56#4248—42no—
57C2×C7×Q8—31no—
58C22×C4×C7C2×D311yescharacteristic
59C22×Dic7C2×D311yescharacteristic
60C22×Dic7C2×D311yescharacteristic
61C2×Dic14—31no—
62C2×Dic14—32no—
63C7×C4⋊C4—32no—
64C7×C4⋊C4—31no—
65C7×C22⋊C4—32no—
66C22⋊Dic7—32no—
67C4⋊Dic7—31no—
68C4⋊Dic7—32no—
69Dic7⋊C4—32no—
70Dic7⋊C4—32no—
71Dic7⋊C4—32no—
72Dic7⋊C4—32no—
73Dic7⋊C4—32no—
74Dic7⋊C4C2×D312yes—
75Dic7⋊C4C2×D312yes—
76C4×Dic7—32no—
77C4×Dic7—32no—
78C2×C3×C4⋊C4—71no—
79#926—72no—
80#924—71no—
81#924—72no—
82#924D711yescharacteristic
83#890—72no—
84#890—72no—
85#888—72no—
86#888—72no—
87C22×C3×C7—21no—
88C22×C3×C7—21no—
89C22×C3×C7C2×Q812yes—
90C22×C3×C7C2411yescharacteristic
91C3×C4×C7—21no—
92C3×C4×C7—21no—
93Dic21—22no—
94Dic21—22no—
95C3×Dic7—22no—
96C3×Dic7C2×Q814yes—
97C7×Dic3—22no—
98C7×Dic3—22no—
99#544—211noSylow
100C23×C7C22×D311yescharacteristic
101C7×Q8—121no—
102C2×C4×C7—32no—
103C2×C4×C7—32no—
104C2×C4×C7—41no—
105C2×C4×C7C22×D311yescharacteristic
106C2×C4×C7C22×D311yescharacteristic
107C2×Dic7—32no—
108C2×Dic7—32no—
109C2×Dic7—41no—
110C2×Dic7Dic614yescomplemented
111C2×Dic7C22×D312yes—
112C2×Dic7C22×D312yes—
113Dic14—122no—
114Dic14—121no—
115C22×C3×C4—71no—
116C22×C3×C4—71no—
117C22×C3×C4C2×D711yescharacteristic
118C2×Dic6—71no—
119C2×Dic6—72no—
120C2×Dic6C2×D711yescharacteristic
121C3×C4⋊C4—72no—
122C3×C4⋊C4—72no—
123#217—72no—
124#217C2×D712yes—
125C4⋊Dic3—71no—
126C4⋊Dic3—72no—
127C4⋊Dic3—72no—
128C4⋊Dic3—72no—
129C4⋊Dic3C2×D711yescharacteristic
130Dic3⋊C4—72no—
131Dic3⋊C4—72no—
132Dic3⋊C4—72no—
133Dic3⋊C4C2×D712yes—
134C4×Dic3—72no—
135C4×Dic3—72no—
136C2×C3×C7—22no—
137C2×C3×C7C22×Q811yescharacteristic
138C2×C3×C7H(5,𝔽2)11yescharacteristic
139C2×C3×C72−1+411yescharacteristic
140C4⋊Q8—212no—
141C4⋊Q8—212no—
142#125—212no—
143#125—212no—
144C22⋊Q8—31no—
145C22⋊Q8—212no—
146C22⋊Q8—211no—
147#117—212no—
148C2×C4⋊C4—71no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7C2×Dic612yes—
152C22×C7C23×D311yescharacteristic
153C4×C7—21no—
154C4×C7—21no—
155C4×C7—62no—
156C4×C7—62no—
157Dic7—22no—
158Dic7—62no—
159Dic7—62no—
160Dic7C2×Dic614yes—
161C23×C3C22×D711yescharacteristic
162C2×C3×C4—41no—
163C2×C3×C4—74no—
164C2×C3×C4—72no—
165C2×C3×C4—72no—
166C2×C3×C4—281no—
167C2×C3×C4C22×D711yescharacteristic
168C2×C3×C4C22×D711yescharacteristic
169C2×Dic3—72no—
170C2×Dic3—72no—
171C2×Dic3C22×D712yes—
172C2×Dic3C22×D712yes—
173Dic6—41no—
174Dic6—282no—
175Dic6—281no—
176C3×C7#54411yescharacteristic, complemented, Hall
177C2×Q8—31no—
178C2×Q8—212no—
179C2×Q8—211no—
180C22×C4—71no—
181C22×C4—71no—
182C22×C4D3×D711yescharacteristic
183C4⋊C4—31no—
184C4⋊C4—32no—
185C4⋊C4—72no—
186C4⋊C4—72no—
187C4⋊C4—212no—
188C4⋊C4—212no—
189C4⋊C4—212no—
190C4⋊C4—212no—
191C4⋊C4—212no—
192C4⋊C4—212no—
193C4⋊C4—211no—
194C22⋊C4—32no—
195C22⋊C4—212no—
196C42—212no—
197C42—212no—
198C2×C7—22no—
199C2×C7C22×Dic611yescharacteristic
200C2×C7#100411yescharacteristic
201C2×C7#100711yescharacteristic
202C22×C3—21no—
203C22×C3—21no—
204C22×C3Q8×D712yes—
205C22×C3C23×D711yescharacteristic
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—74no—
209C3×C4—142no—
210Dic3—22no—
211Dic3—22no—
212Dic3—142no—
213Dic3—142no—
214C23C2×D3×D711yescharacteristic
215Q8—121no—
216Q8—841no—
217Q8—842no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—41no—
221C2×C4—74no—
222C2×C4—72no—
223C2×C4—72no—
224C2×C4—212no—
225C2×C4—212no—
226C2×C4—281no—
227C2×C4C2×D3×D711yescharacteristic
228C2×C4C2×D3×D711yescharacteristic
229C7#550011yescharacteristic, complemented, Sylow
230C2×C3—22no—
231C2×C3#654811yescharacteristic
232C2×C3C2×Q8×D711yescharacteristic
233C2×C3#655411yescharacteristic
234C22—21no—
235C22—21no—
236C22D7×Dic612yes—
237C22C22×D3×D711yescharacteristic
238C4—21no—
239C4—21no—
240C4—62no—
241C4—62no—
242C4—74no—
243C4—142no—
244C4—422no—
245C4—422no—
246C3#9031411yescharacteristic, complemented, Sylow
247C2—22no—
248C2C2×D7×Dic611yescharacteristic
249C2#1062155111yescharacteristic
250C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC23×C3×C7C23
hypercenterC22×C4D3×D7
hypocenterC3×C7#544
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. #10620897C2semidirect
7. #10620896C2semidirect
8. #10620891C2semidirect
9. #10620890C2semidirect
53. C2×C3×Dic7Q8semidirect
110. C2×Dic7Dic6semidirect
176. C3×C7#544semidirect
229. C7#5500semidirect
246. C3#90314semidirect

Maximal subgroups

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

subgroupindex
#106214072
C7×#9242
C2×C3×Dic7⋊C42
#106211362
#106210902
#106208972
#106208962
#106208912
#106208902
#903143
#55007