Finite Group Databasebeta

← #13054725

Subgroups — #13054725

Order 1344.

208 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 54 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#10621425C211yesmaximal, characteristic, complemented
2C2×C3×#1143C211yesmaximal, characteristic, complemented
3#10620874C211yesmaximal, characteristic
4#10620852C211yesmaximal, characteristic, complemented
5C7×#836C211yesmaximal, characteristic, complemented
6#10620526C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89526—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12#66635—21no—
13#66634—21no—
14#66627C2211yescharacteristic
15C7×#208C2211yescharacteristic
16C7×#208C2211yescharacteristic
17C3×#1143—21no—
18C3×#1143—21no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66565—21no—
21#66561—21no—
22#66559—21no—
23D21⋊C8—21no—
24#6505—31no—
25C2×#1143D311yescharacteristic, complemented
26#6471—31no—
27#6470—31no—
28C7×#101—31no—
29#6397—31no—
30#6396—31no—
31#5177—71nomaximal, Hall
32C22×C3×D7—21no—
33C3×C7×Q8—21no—
34C3×C7×Q8—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—21no—
38D84D411yescharacteristic
39C4×D21—21no—
40Dic42D411yescharacteristic
41C3×D28—21no—
42C3×D28—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8D411yescharacteristic
45C3×C7⋊C8D411yescharacteristic
46C7×C3⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×Q8C2×D311yescharacteristic
49#1157—31no—
50C2×D28C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16—31no—
53#1144—61no—
54#1143—21no—
55#1143—21no—
56#1143—61no—
57#1136—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59D56—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C2×C3×SD16—71no—
64#942—71no—
65#932—71no—
66#901—71no—
67#836D711yescharacteristic, complemented
68#833—71no—
69#819—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C2×C3×D7—41no—
73C2×C3×D7—81no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7C2×D411yescharacteristic
77Dic21—21no—
78#471—211noSylow
79C22×D7—21no—
80C7×Q8—21no—
81C7×Q8—21no—
82C2×C4×C7—21no—
83C2×C4×C7C22×D311yescharacteristic
84C22⋊D7—61no—
85D28—21no—
86D28—21no—
87D28—31no—
88C4×D7—61no—
89Dic14—31no—
90C7×C8—61no—
91C7×C8—61no—
92C7⋊C8—61no—
93C7⋊C8#6711yescharacteristic
94C7⋊C8#6711yescharacteristic
95C2×C3×Q8C2×D711yescharacteristic
96C2×C3×D4—71no—
97#235—71no—
98C3×SD16—141no—
99C3×SD16—141no—
100C2×C3×C8—71no—
101#216—141no—
102#215—141no—
103#214—141no—
104#213—141no—
105#208—71no—
106#208C2×D711yescharacteristic
107#208C2×D711yescharacteristic
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112D21—41no—
113C3×D7—81no—
114#137—211no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—71no—
117#131—211no—
118#108—211no—
119#101—31no—
120#100—211no—
121C22×C7#23711yescharacteristic
122C2×D7—41no—
123C2×D7—61no—
124C2×D7—81no—
125C4×C7—41no—
126C4×C7D3×D411yescharacteristic
127C4×C7C2×#6711yescharacteristic
128Dic7—61no—
129C23×C3—141no—
130C3×Q8—21no—
131C3×Q8—21no—
132C3×D4—141no—
133C3×D4—141no—
134C2×C3×C4—21no—
135C2×C3×C4C22×D711yescharacteristic
136#67—141no—
137D12—71no—
138C4×D3—141no—
139Dic6—71no—
140C3×C8—71no—
141C3×C8—71no—
142C3⋊C8—21no—
143C3⋊C8—21no—
144C3⋊C8—141no—
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—211no—
147C2×Q8D3×D711yescharacteristic
148C2×D4—71no—
149Q16—421no—
150SD16—141no—
151SD16—141no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—31no—
156M16—31no—
157M16—211no—
158M16—421no—
159C2×C8—71no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7#93811yescharacteristic
163D7—81no—
164D7—121no—
165C22×C3—281no—
166C22×C3—561no—
167C22×C3#115711yescharacteristic
168C2×D3—141no—
169C3×C4—41no—
170C3×C4D4×D711yescharacteristic
171C3×C4D4×D711yescharacteristic
172Dic3—141no—
173C23—141no—
174Q8—21no—
175Q8—21no—
176Q8—211no—
177D4—141no—
178D4—141no—
179D4—211no—
180D4—421no—
181C2×C4—21no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—61no—
186C8—71no—
187C8—71no—
188C8—421no—
189C7#517711yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—561no—
192C2×C3#644711yescharacteristic
193D3—281no—
194C22—281no—
195C22—421no—
196C22—561no—
197C22#6668311yescharacteristic
198C4—41no—
199C4—421no—
200C4D7×#6711yescharacteristic
201C4#6669411yescharacteristic
202C3#8952611yescharacteristic, complemented, Sylow
203C2—21no—
204C2—561no—
205C2—841no—
206C2#1062109511yescharacteristic
subgroupG/H
centerC2#10621095
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#121
prime radical#10620874C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#471
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #10621425C2semidirect
2. C2×C3×#1143C2semidirect
4. #10620852C2semidirect
5. C7×#836C2semidirect
6. #10620526C2semidirect
7. #10620493C2semidirect
25. C2×#1143D3semidirect
67. #836D7semidirect
145. C3×C7#471semidirect
189. C7#5177semidirect
202. C3#89526semidirect

Maximal subgroups

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

subgroupindex
#106214252
C2×C3×#11432
#106208742
#106208522
C7×#8362
#106205262
#106204932
#895263
#51777