Finite Group Databasebeta

← #13061745

Subgroups — #13061745

Order 1344.

392 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×C7×#123C211yesmaximal, characteristic
2#10621419C211yesmaximal, characteristic
3#10621418C211yesmaximal, characteristic
4#10621407C212yesmaximal, complemented
5#10621372C212yesmaximal, complemented
6#10621371C212yesmaximal, complemented
7#10621362C212yesmaximal, complemented
8#10621362C212yesmaximal, complemented
9#10621361C212yesmaximal, complemented
10#90342—31nomaximal, Hall
11C2×C3×C7×D4C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C2×Dic42C2212yes—
15C2×Dic42C2211yescharacteristic
16C3×C7×C4⋊C4C2212yes—
17C3×C7×C22⋊C4C2212yes—
18C3×C7×C22⋊C4C2211yescharacteristic
19#66636C2212yes—
20#66636C2211yescharacteristic
21#66636C2212yes—
22#66636C2212yes—
23C4⋊Dic21C2212yes—
24C4⋊Dic21C2212yes—
25#66629C2212yes—
26#66629C2212yes—
27#66629C2212yes—
28#66629C2212yes—
29#66629C2212yes—
30C4×Dic21C2212yes—
31C4×Dic21C2211yescharacteristic
32C7×#123D311yescharacteristic
33#6499—31no—
34#6498—31no—
35#6487—32no—
36#6452—32no—
37#6451—32no—
38#6442—32no—
39#6442—32no—
40#6441—32no—
41#5651—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C23×C3×C7C2311yescharacteristic
44C3×C7×D4—41no—
45C2×C3×C4×C7—41no—
46C2×C3×C4×C7C2312yes—
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2312yes—
49C2×Dic21—41no—
50C2×Dic21C2312yes—
51C2×Dic21C2312yes—
52C2×Dic21C2312yes—
53C2×Dic21C2312yes—
54Dic42—41no—
55Dic42—42no—
56C2×C7×D4C2×D311yescharacteristic
57C22×C4×C7C2×D311yescharacteristic
58C22×Dic7—31no—
59C2×Dic14—32no—
60C2×Dic14—31no—
61C7×C4⋊C4C2×D312yes—
62C7×C22⋊C4C2×D312yes—
63C7×C22⋊C4C2×D311yescharacteristic
64C22⋊Dic7—32no—
65C22⋊Dic7—31no—
66C22⋊Dic7—32no—
67C22⋊Dic7—32no—
68C4⋊Dic7—32no—
69C4⋊Dic7—32no—
70Dic7⋊C4—32no—
71Dic7⋊C4—32no—
72Dic7⋊C4—32no—
73Dic7⋊C4—32no—
74Dic7⋊C4—32no—
75C4×Dic7—32no—
76C4×Dic7—31no—
77C3×#123D711yescharacteristic
78#936—71no—
79#935—71no—
80#924—72no—
81#889—72no—
82#888—72no—
83#879—72no—
84#879—72no—
85#878—72no—
86C22×C3×C7—21no—
87C22×C3×C7—21no—
88C22×C3×C7—22no—
89C22×C3×C7—22no—
90C22×C3×C7C2411yescharacteristic
91C3×C4×C7—22no—
92C3×C4×C7—21no—
93C3×C4×C7—22no—
94Dic21—22no—
95Dic21—22no—
96Dic21—22no—
97Dic21—22no—
98#563—211noSylow
99C23×C7C22×D311yescharacteristic
100C23×C7C22×D311yescharacteristic
101C7×D4—41no—
102C2×C4×C7—41no—
103C2×C4×C7C22×D312yes—
104C2×C4×C7C22×D311yescharacteristic
105C2×C4×C7C22×D312yes—
106C2×Dic7—32no—
107C2×Dic7—32no—
108C2×Dic7—32no—
109C2×Dic7—32no—
110C2×Dic7—121no—
111Dic14—121no—
112Dic14—122no—
113C2×C3×D4C2×D711yescharacteristic
114C22×C3×C4C2×D711yescharacteristic
115C22×Dic3—71no—
116C2×Dic6—72no—
117C2×Dic6—71no—
118C3×C4⋊C4C2×D712yes—
119C3×C22⋊C4C2×D712yes—
120C3×C22⋊C4C2×D711yescharacteristic
121#217—72no—
122#217—71no—
123#217—72no—
124#217—72no—
125C4⋊Dic3—72no—
126C4⋊Dic3—72no—
127Dic3⋊C4—72no—
128Dic3⋊C4—72no—
129Dic3⋊C4—72no—
130Dic3⋊C4—72no—
131Dic3⋊C4—72no—
132C4×Dic3—72no—
133C4×Dic3—71no—
134C2×C3×C7—41no—
135C2×C3×C7—41no—
136C2×C3×C7H(5,𝔽2)11yescharacteristic
137C2×C3×C72−1+412yes—
138C4⋊Q8—212no—
139#126—212no—
140#126—212no—
141#125—212no—
142#124—211no—
143#123—211no—
144#123D2111yescharacteristic
145C22⋊Q8—212no—
146C22⋊Q8—212no—
147C22×C7—21no—
148C22×C7—21no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7C23×D311yescharacteristic
152C4×C7—22no—
153C4×C7—21no—
154C4×C7—22no—
155Dic7—62no—
156Dic7—62no—
157Dic7—62no—
158Dic7—62no—
159C23×C3C22×D711yescharacteristic
160C23×C3C22×D711yescharacteristic
161C3×D4—41no—
162C2×C3×C4—41no—
163C2×C3×C4C22×D712yes—
164C2×C3×C4C22×D711yescharacteristic
165C2×C3×C4C22×D712yes—
166C2×Dic3—72no—
167C2×Dic3—72no—
168C2×Dic3—72no—
169C2×Dic3—72no—
170C2×Dic3—281no—
171Dic6—281no—
172Dic6—282no—
173C3×C7#56311yescharacteristic, complemented, Hall
174C2×Q8—212no—
175C2×Q8—211no—
176C2×D4C2×D2111yescharacteristic
177C22×C4—211no—
178C22×C4C2×D2111yescharacteristic
179C4⋊C4—212no—
180C4⋊C4—212no—
181C4⋊C4—212no—
182C4⋊C4—212no—
183C4⋊C4—212no—
184C4⋊C4—212no—
185C4⋊C4—212no—
186C4⋊C4C2×D2112yes—
187C22⋊C4—212no—
188C22⋊C4—212no—
189C22⋊C4—212no—
190C22⋊C4—211no—
191C22⋊C4C2×D2112yes—
192C22⋊C4C2×D2111yescharacteristic
193C42—211no—
194C42—212no—
195C2×C7—41no—
196C2×C7—41no—
197C2×C7#100411yescharacteristic
198C2×C7#101012yes—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3C23×D711yescharacteristic
204C3×C4—22no—
205C3×C4—21no—
206C3×C4—22no—
207Dic3—142no—
208Dic3—142no—
209Dic3—142no—
210Dic3—142no—
211C23C22×D2111yescharacteristic
212C23C22×D2111yescharacteristic
213Q8—842no—
214Q8—841no—
215D4—41no—
216C2×C4—41no—
217C2×C4—212no—
218C2×C4—212no—
219C2×C4—212no—
220C2×C4—212no—
221C2×C4—841no—
222C2×C4C22×D2112yes—
223C2×C4C22×D2111yescharacteristic
224C2×C4C22×D2112yes—
225C7#565111yescharacteristic, complemented, Sylow
226C2×C3—41no—
227C2×C3—41no—
228C2×C3#654811yescharacteristic
229C2×C3#655412yes—
230C22—21no—
231C22—21no—
232C22—22no—
233C22—22no—
234C22C23×D2111yescharacteristic
235C4—22no—
236C4—21no—
237C4—22no—
238C4—422no—
239C4—422no—
240C4—422no—
241C4—422no—
242C3#9034211yescharacteristic, complemented, Sylow
243C2—41no—
244C2—41no—
245C2#1062165711yescharacteristic
246C2#1062166312yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#123C2
prime radicalC2×C3×C7×D4C22
hypercenter#123D21
hypocenterC3×C7#563
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #10621407C2semidirect
5. #10621372C2semidirect
6. #10621371C2semidirect
7. #10621362C2semidirect
8. #10621362C2semidirect
9. #10621361C2semidirect
173. C3×C7#563semidirect
225. C7#5651semidirect
242. C3#90342semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1232
#106214192
#106214182
#106214072
#106213722
#106213712
#106213622
#106213612
#903423
#56517