Finite Group Databasebeta

← #13053566

Subgroups — #13053566

Order 1344.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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×#6470C211yesmaximal, characteristic, complemented
2C2×#66568C211yesmaximal, characteristic, complemented
3#10620869C211yesmaximal, characteristic, complemented
4#10620863C211yesmaximal, characteristic
5#10620662C211yesmaximal, characteristic, complemented
6C7×#824C211yesmaximal, characteristic, complemented
7#10620526C211yesmaximal, characteristic, complemented
8#89526—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#208C2211yescharacteristic
15C2×C3×C7⋊C8C2211yescharacteristic
16C3×#1131—21no—
17C3×C8×D7—21no—
18#66569—21no—
19#66568—21no—
20#66568—21no—
21#66566—21no—
22#66563—21no—
23#66561—21no—
24#6505—31no—
25C2×#1143—31no—
26#6471—31no—
27#6470D311yescharacteristic, complemented
28C7×#101—31no—
29#6397—31no—
30#6396—31no—
31#5141—71nomaximal, Hall
32C22×D21—21no—
33C2×C3×C4×C7C2311yescharacteristic
34D84—21no—
35D84—21no—
36C2×C7×Dic3—21no—
37C7×Dic6—21no—
38C7×Dic6—21no—
39C3×C22⋊D7—21no—
40C3×D28D411yescharacteristic
41C3×C4×D7—21no—
42C3×Dic14D411yescharacteristic
43C3×C7×C8—21no—
44C3⋊(C7⋊C8)—21no—
45C3×C7⋊C8D411yescharacteristic
46C3×C7⋊C8D411yescharacteristic
47C7×C3⋊C8—21no—
48C2×C7×Q8—31no—
49#1157C2×D311yescharacteristic
50C2×D28—31no—
51C7×M16—31no—
52C7×M16C2×D311yescharacteristic
53#1144—61no—
54#1143—61no—
55#1143—61no—
56#1143—61no—
57#1136—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59D56—61no—
60#1132—61no—
61#1131—21no—
62C8×D7—21no—
63C3×#131—71no—
64#951—71no—
65#949—71no—
66C2×#204—71no—
67#824D711yescharacteristic, complemented
68#823—71no—
69#819—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—41no—
72C2×D21—81no—
73C2×C3×D7—21no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76C3×Dic7—21no—
77C7×Dic3—41no—
78#471—211noSylow
79C22×D7—61no—
80C7×Q8—61no—
81C7×Q8—61no—
82C2×C4×C7—61no—
83C2×C4×C7C22×D311yescharacteristic
84C22⋊D7—21no—
85D28—61no—
86D28—61no—
87D28D1211yescharacteristic
88C4×D7—21no—
89Dic14D1211yescharacteristic
90C7×C8—21no—
91C7×C8—61no—
92C7⋊C8—61no—
93C7⋊C8#6711yescharacteristic
94C7⋊C8#6711yescharacteristic
95C3×#41—71no—
96C2×D12—71no—
97C2×Dic6C2×D711yescharacteristic
98C3×M16—141no—
99C3×M16C2×D711yescharacteristic
100C2×C3×C8—71no—
101C2×C3×C8—141no—
102#216—141no—
103#215—141no—
104#214—141no—
105#213—141no—
106#208—71no—
107#208C2×D711yescharacteristic
108#204—141no—
109#204—141no—
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112D21—81no—
113C3×D7—41no—
114#137—211no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—211no—
117#131—71no—
118#108—211no—
119#101—31no—
120#100—211no—
121C22×C7#23511yescharacteristic
122C2×D7—21no—
123C2×D7—121no—
124C2×D7—241no—
125C4×C7—121no—
126C4×C7C2×D1211yescharacteristic
127C4×C7C2×#6711yescharacteristic
128Dic7—21no—
129C22×D3—141no—
130C3×Q8—71no—
131C3×D4—71no—
132C3×D4—141no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—21no—
136D12—141no—
137D12—141no—
138Dic6—21no—
139Dic6—21no—
140C3×C8—21no—
141C3×C8—71no—
142C3×C8—71no—
143C3⋊C8—21no—
144C3⋊C8—141no—
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—71no—
147C2×Q8—31no—
148C2×D4—211no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—31no—
156M16—141no—
157M16—211no—
158M16D3×D711yescharacteristic
159C2×C8—71no—
160C2×C8—141no—
161C2×C7—21no—
162C2×C7#93011yescharacteristic
163D7—41no—
164D7—241no—
165C22×C3—141no—
166C22×C3#115711yescharacteristic
167C2×D3—281no—
168C2×D3—561no—
169C3×C4—141no—
170C3×C4D4×D711yescharacteristic
171C3×C4D4×D711yescharacteristic
172Dic3—41no—
173C23—421no—
174Q8—61no—
175Q8—61no—
176Q8—71no—
177D4—71no—
178D4—141no—
179D4—421no—
180D4—421no—
181C2×C4—61no—
182C2×C4—141no—
183C2×C4C2×D3×D711yescharacteristic
184C8—21no—
185C8—61no—
186C8—71no—
187C8—71no—
188C8—421no—
189C7#514111yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—281no—
192C2×C3#644711yescharacteristic
193D3—561no—
194C22—141no—
195C22—841no—
196C22—1681no—
197C22#6667511yescharacteristic
198C4—121no—
199C4—141no—
200C4D7×D1211yescharacteristic
201C4D7×#6711yescharacteristic
202C3#8952611yescharacteristic, complemented, Sylow
203C2—21no—
204C2—281no—
205C2—1681no—
206C2#1062101011yescharacteristic
subgroupG/H
centerC2#10621010
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#121
prime radical#10620863C2
hypercenterM16D3×D7
hypocenterC3×C7#471
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. C3×#6470C2semidirect
2. C2×#66568C2semidirect
3. #10620869C2semidirect
5. #10620662C2semidirect
6. C7×#824C2semidirect
7. #10620526C2semidirect
27. #6470D3semidirect
67. #824D7semidirect
145. C3×C7#471semidirect
189. C7#5141semidirect
202. C3#89526semidirect

Maximal subgroups

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

subgroupindex
C3×#64702
C2×#665682
#106208692
#106208632
#106206622
C7×#8242
#106205262
#895263
#51417