Finite Group Databasebeta

← #13061766

Subgroups — #13061766

Order 1344.

540 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×Q8×D21C211yesmaximal, characteristic, complemented
2C2×#66730C211yesmaximal, characteristic
3C3×C7×#124C211yesmaximal, characteristic, complemented
4#10621427C211yesmaximal, characteristic, complemented
5#10621419C211yesmaximal, characteristic, complemented
6#10621366C214yesmaximal, complemented
7#10621361C214yesmaximal, complemented
8#10621355C211yesmaximal, characteristic, complemented
9#10621352C211yesmaximal, characteristic, complemented
10#90366—31nomaximal, Hall
11C2×C3×C7×Q8C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2212yes—
14C22×Dic21C2212yes—
15Q8×D21—24no—
16#66730—24no—
17C2×C4×D21C2212yes—
18C2×C4×D21C2211yescharacteristic
19C2×Dic42C2211yescharacteristic
20C2×Dic42C2211yescharacteristic
21C2×Dic42C2212yes—
22C3×C7×C22⋊C4C2214yes—
23C3×C42×C7C2211yescharacteristic
24#66636C2214yes—
25#66631C2212yes—
26C4⋊Dic21C2214yes—
27#66629C2212yes—
28#66629C2214yes—
29C4×Dic21C2211yescharacteristic
30C2×Q8×D7—31no—
31C2×#1159—31no—
32C7×#124D311yescharacteristic, complemented
33#6507—31no—
34#6499—31no—
35#6446—34no—
36#6441—34no—
37#6435—31no—
38#6432—31no—
39#5675—71nomaximal, Hall
40C23×C3×C7C2312yes—
41C22×D21C2311yescharacteristic
42C3×C7×Q8—22no—
43C3×C7×D4—22no—
44C2×C3×C4×C7C2312yes—
45C2×C3×C4×C7C2312yes—
46C2×C3×C4×C7C2311yescharacteristic
47#4268—24no—
48C2×Dic21—24no—
49C2×Dic21C2312yes—
50C2×Dic21C2312yes—
51C2×Dic21C2312yes—
52C2×Dic21C2311yescharacteristic
53C4×D21—24no—
54C4×D21D414yes—
55Dic42—22no—
56Dic42—22no—
57Dic42—24no—
58C2×C7×Q8C2×D311yescharacteristic
59C2×C7×D4C2×D311yescharacteristic
60C2×C22⋊D7—32no—
61C22×Dic7—32no—
62Q8×D7—64no—
63#1159—64no—
64C2×C4×D7—31no—
65C2×C4×D7—32no—
66C2×Dic14—31no—
67C2×Dic14—31no—
68C2×Dic14—32no—
69C7×C22⋊C4C2×D314yes—
70C42×C7C2×D311yescharacteristic
71C22⋊Dic7—34no—
72(C2×D7)⋊C4—32no—
73C4⋊Dic7—34no—
74Dic7⋊C4—32no—
75Dic7⋊C4—34no—
76C4×Dic7—31no—
77C2×D3×Q8—71no—
78C2×#237—71no—
79C3×#124D711yescharacteristic, complemented
80#944—71no—
81#936—71no—
82#883—74no—
83#878—74no—
84#872—71no—
85#869—71no—
86C22×C3×C7—22no—
87C22×C3×C7—24no—
88C22×C3×C7C2411yescharacteristic
89C2×D21—22no—
90C2×D21C2×D412yes—
91C3×C4×C7—22no—
92C3×C4×C7—22no—
93C3×C4×C7C2×D412yes—
94Dic21—22no—
95Dic21—22no—
96Dic21—22no—
97Dic21C2×D412yes—
98#536—211noSylow
99C23×C7C22×D312yes—
100C22×D7—31no—
101C7×Q8—22no—
102C7×D4—22no—
103C2×C4×C7C22×D312yes—
104C2×C4×C7C22×D312yes—
105C2×C4×C7C22×D311yescharacteristic
106C22⋊D7—64no—
107C2×Dic7—31no—
108C2×Dic7—32no—
109C2×Dic7—32no—
110C2×Dic7—32no—
111C2×Dic7—64no—
112C4×D7—34no—
113C4×D7—64no—
114Dic14—64no—
115Dic14—62no—
116Dic14—62no—
117C2×C3×Q8C2×D711yescharacteristic
118C2×C3×D4C2×D711yescharacteristic
119C2×#67—72no—
120C22×Dic3—72no—
121D3×Q8—144no—
122#237—144no—
123C2×C4×D3—71no—
124C2×C4×D3—72no—
125C2×Dic6—71no—
126C2×Dic6—71no—
127C2×Dic6—72no—
128C3×C22⋊C4C2×D714yes—
129C3×C42C2×D711yescharacteristic
130#217—74no—
131(C2×D3)⋊C4—72no—
132C4⋊Dic3—74no—
133Dic3⋊C4—72no—
134Dic3⋊C4—74no—
135C4×Dic3—71no—
136C2×C3×C7—42no—
137C2×C3×C7C22×D411yescharacteristic
138C2×C3×C72−1+412yes—
139D21—22no—
140C2×#41—211no—
141C22×Q8—211no—
142C4⋊Q8—211no—
143C4⋊Q8—211no—
144#124—211no—
145#124D2111yescharacteristic, complemented
146#123—214no—
147C22⋊Q8—214no—
148#117—211no—
149C22×C7—22no—
150C22×C7—24no—
151C22×C7C23×D311yescharacteristic
152C2×D7—32no—
153C2×D7—62no—
154C4×C7—22no—
155C4×C7—22no—
156C4×C7D3×D412yes—
157Dic7—32no—
158Dic7—62no—
159Dic7—62no—
160Dic7—62no—
161C23×C3C22×D712yes—
162C22×D3—71no—
163C3×Q8—22no—
164C3×D4—22no—
165C2×C3×C4C22×D712yes—
166C2×C3×C4C22×D712yes—
167C2×C3×C4C22×D711yescharacteristic
168#67—144no—
169C2×Dic3—71no—
170C2×Dic3—72no—
171C2×Dic3—72no—
172C2×Dic3—72no—
173C2×Dic3—144no—
174C4×D3—74no—
175C4×D3—144no—
176Dic6—144no—
177Dic6—142no—
178Dic6—142no—
179C3×C7#53611yescharacteristic, complemented, Hall
180#41—424no—
181C2×Q8—212no—
182C2×Q8—211no—
183C2×Q8—211no—
184C2×Q8—424no—
185C2×Q8C2×D2111yescharacteristic
186C2×D4—212no—
187C2×D4C2×D2111yescharacteristic
188C22×C4—211no—
189C22×C4—212no—
190C22×C4—212no—
191C4⋊C4—212no—
192C4⋊C4—214no—
193C4⋊C4—214no—
194C22⋊C4—214no—
195C22⋊C4—212no—
196C22⋊C4C2×D2114yes—
197C42—211no—
198C42C2×D2111yescharacteristic
199C2×C7—42no—
200C2×C7C2×D3×D411yescharacteristic
201C2×C7#101012yes—
202D7—62no—
203C22×C3—22no—
204C22×C3—24no—
205C22×C3C23×D711yescharacteristic
206C2×D3—72no—
207C2×D3—142no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4D4×D712yes—
211Dic3—72no—
212Dic3—142no—
213Dic3—142no—
214Dic3—142no—
215C23—211no—
216C23C22×D2112yes—
217Q8—22no—
218Q8—424no—
219Q8—422no—
220Q8—422no—
221D4—22no—
222D4—424no—
223C2×C4—211no—
224C2×C4—214no—
225C2×C4—212no—
226C2×C4—212no—
227C2×C4—212no—
228C2×C4—424no—
229C2×C4—424no—
230C2×C4C22×D2112yes—
231C2×C4C22×D2112yes—
232C2×C4C22×D2111yescharacteristic
233C7#567511yescharacteristic, complemented, Sylow
234C2×C3—42no—
235C2×C3C2×D4×D711yescharacteristic
236C2×C3#655412yes—
237D3—142no—
238C22—22no—
239C22—24no—
240C22—212no—
241C22—422no—
242C22C23×D2111yescharacteristic
243C4—22no—
244C4—22no—
245C4—212no—
246C4—422no—
247C4—422no—
248C4—422no—
249C4D4×D2112yes—
250C3#9036611yescharacteristic, complemented, Sylow
251C2—42no—
252C2—422no—
253C2C2×D4×D2111yescharacteristic
254C2#1062166312yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
prime radicalC2×#66730C2
hypercenter#124D21
hypocenterC3×C7#536
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×D21C2semidirect
3. C3×C7×#124C2semidirect
4. #10621427C2semidirect
5. #10621419C2semidirect
6. #10621366C2semidirect
7. #10621361C2semidirect
8. #10621355C2semidirect
9. #10621352C2semidirect
32. C7×#124D3semidirect
79. C3×#124D7semidirect
145. #124D21semidirect
179. C3×C7#536semidirect
233. C7#5675semidirect
250. C3#90366semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D212
C2×#667302
C3×C7×#1242
#106214272
#106214192
#106213662
#106213612
#106213552
#106213522
#903663
#56757