Finite Group Databasebeta

← #13054591

Subgroups — #13054591

Order 1344.

240 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66634C211yesmaximal, characteristic
2C3×#6507C211yesmaximal, characteristic, complemented
3#10620980C211yesmaximal, characteristic
4C2×#66566C211yesmaximal, characteristic
5C7×#835C211yesmaximal, characteristic, complemented
6Dic7⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#10620505C211yesmaximal, characteristic, complemented
8#89590—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C2×D21⋊C4—21no—
13#66634—22no—
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×Dic7⋊C4—21no—
18C3×Dic7⋊C4—21no—
19C3×C4×Dic7C2211yescharacteristic
20#66575—21no—
21#66566—22no—
22#6507D311yescharacteristic, complemented
23C2×#1143—31no—
24C2×#1132—31no—
25#6456—31no—
26C7×#103—31no—
27Dic7⋊C8—31no—
28#6384—31no—
29#5083—71nomaximal, Hall
30C22×D21—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35D84—21no—
36D84D412yes—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7D411yescharacteristic
40C2×C3×Dic7D411yescharacteristic
41C3×Dic14—21no—
42C3×Dic14—21no—
43D21⋊C4—42no—
44C3⋊(C7⋊C8)—21no—
45C7×C3⋊C8—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28—31no—
48C2×C4×D7—61no—
49C2×Dic14C2×D311yescharacteristic
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52#1143—62no—
53(C2×D7)⋊C4—61no—
54Dic7⋊C4—21no—
55Dic7⋊C4—21no—
56C4×Dic7C2×D311yescharacteristic
57C2×C7⋊C8—31no—
58#1132—62no—
59C3×C4⋊Q8—71no—
60C2×#215—71no—
61C2×#215—71no—
62C4×D12—71no—
63#835—71no—
64#835D711yescharacteristic, complemented
65C4⋊(C3⋊C8)—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—22no—
68C2×D21—41no—
69C2×D21—41no—
70C3×C4×C7—41no—
71C3×C4×C7C2×D411yescharacteristic
72C3×C4×C7#4111yescharacteristic
73C3×Dic7—21no—
74C3×Dic7—41no—
75C3×Dic7SD1612yescomplemented
76C7×Dic3—41no—
77#461—211noSylow
78C22×D7—61no—
79C7×Q8—21no—
80C7×Q8—21no—
81C2×C4×C7—21no—
82C2×C4×C7—61no—
83C2×C4×C7C22×D311yescharacteristic
84C2×Dic7—21no—
85C2×Dic7#6711yescharacteristic
86C2×Dic7#6711yescharacteristic
87D28—32no—
88D28—61no—
89C4×D7—122no—
90Dic14—21no—
91Dic14—21no—
92C7×C8—61no—
93C7⋊C8—61no—
94C2×C3×Q8—71no—
95C2×C3×Q8C2×D711yescharacteristic
96C2×D12—71no—
97C2×C4×D3—141no—
98C3×C4⋊C4—141no—
99C3×C4⋊C4—141no—
100C3×C42—71no—
101#215—142no—
102#215—142no—
103(C2×D3)⋊C4—141no—
104C4⋊Dic3C2×D711yescharacteristic
105C2×C3⋊C8—71no—
106C2×C3⋊C8C2×D711yescharacteristic
107C2×C3×C7#12111yescharacteristic
108C2×C3×C7C2×SD1611yescharacteristic
109C2×C3×C7#13711yescharacteristic
110D21—42no—
111C2×SD16—211no—
112C2×SD16—211no—
113C4⋊Q8—71no—
114C4×D4—211no—
115C4⋊C8—211no—
116#103—31no—
117#103—211no—
118C22×C7C2×#6711yescharacteristic
119C2×D7—62no—
120C2×D7—121no—
121C2×D7—121no—
122C4×C7—41no—
123C4×C7—121no—
124C4×C7D3×D411yescharacteristic
125C4×C7#23711yescharacteristic
126Dic7—21no—
127Dic7—41no—
128Dic7#21512yescomplemented
129C22×D3—141no—
130C3×Q8—21no—
131C3×Q8—21no—
132C3×Q8—141no—
133C3×Q8—141no—
134C2×C3×C4—21no—
135C2×C3×C4—71no—
136C2×C3×C4—71no—
137C2×C3×C4—141no—
138C2×C3×C4C22×D711yescharacteristic
139C2×Dic3—21no—
140D12—72no—
141D12—141no—
142C4×D3—282no—
143C3⋊C8—21no—
144C3⋊C8—141no—
145C3×C7#46111yescharacteristic, complemented, Hall
146C2×Q8—71no—
147C2×Q8D3×D711yescharacteristic
148C2×D4—211no—
149C22×C4—421no—
150SD16—422no—
151SD16—422no—
152C2×C8—31no—
153C2×C8—211no—
154C4⋊C4—31no—
155C4⋊C4—141no—
156C4⋊C4—141no—
157C22⋊C4—421no—
158C42—71no—
159C2×C7#93811yescharacteristic
160C2×C7C2×#21511yescharacteristic
161C2×C7#94211yescharacteristic
162D7—122no—
163C22×C3D4×D711yescharacteristic
164C2×D3—142no—
165C2×D3—281no—
166C2×D3—281no—
167C3×C4—41no—
168C3×C4—72no—
169C3×C4—141no—
170C3×C4—281no—
171C3×C4#115711yescharacteristic
172C3×C4D4×D711yescharacteristic
173Dic3—41no—
174C23—421no—
175Q8—21no—
176Q8—21no—
177Q8—141no—
178Q8—141no—
179D4—212no—
180D4—421no—
181C2×C4—21no—
182C2×C4—61no—
183C2×C4—71no—
184C2×C4—71no—
185C2×C4—141no—
186C2×C4—842no—
187C2×C4C2×D3×D711yescharacteristic
188C8—61no—
189C8—421no—
190C7#508311yescharacteristic, complemented, Sylow
191C2×C3#644711yescharacteristic
192C2×C3D7×SD1611yescharacteristic
193C2×C3#648111yescharacteristic
194D3—282no—
195C22—422no—
196C22—841no—
197C22—841no—
198C22D7×#6711yescharacteristic
199C4—41no—
200C4—72no—
201C4—121no—
202C4—141no—
203C4—281no—
204C4#6668311yescharacteristic
205C4#6669411yescharacteristic
206C3#8959011yescharacteristic, complemented, Sylow
207C2—842no—
208C2D7×#21511yescharacteristic
209C2#1062107711yescharacteristic
210C2#1062109511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#461
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#6507C2semidirect
5. C7×#835C2semidirect
6. Dic7⋊(C3⋊C8)C2semidirect
7. #10620505C2semidirect
22. #6507D3semidirect
64. #835D7semidirect
75. C3×Dic7SD16semidirect
128. Dic7#215semidirect
145. C3×C7#461semidirect
190. C7#5083semidirect
206. C3#89590semidirect

Maximal subgroups

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

subgroupindex
C2×#666342
C3×#65072
#106209802
C2×#665662
C7×#8352
Dic7⋊(C3⋊C8)2
#106205052
#895903
#50837