Finite Group Databasebeta

← #13054727

Subgroups — #13054727

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×C7×#215C211yesmaximal, characteristic, complemented
3#10620866C211yesmaximal, characteristic
4#10620853C211yesmaximal, characteristic, complemented
5C3×#6410C211yesmaximal, characteristic, complemented
6#10620528C211yesmaximal, characteristic, complemented
7#10620494C211yesmaximal, characteristic, complemented
8#89949—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12#66635—21no—
13#66634—21no—
14#66627C2211yescharacteristic
15C7×#215—21no—
16C7×#215—21no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1136C2211yescharacteristic
19C3×#1136C2211yescharacteristic
20#66567—21no—
21#66562—21no—
22#66559—21no—
23D21⋊C8—21no—
24C2×C7×SD16—31no—
25#6505—31no—
26#6495—31no—
27#6464—31no—
28#6410D311yescharacteristic, complemented
29#6407—31no—
30#6393—31no—
31#4754—71nomaximal, Hall
32C22×D3×C7—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
41C7×D12—21no—
42C7×D12—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8—21no—
45C3×C7⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C2×C7×Q8C2×D311yescharacteristic
49C2×C7×D4—31no—
50#1157—31no—
51C7×SD16—61no—
52C7×SD16—61no—
53C2×C7×C8—31no—
54#1144—61no—
55#1143—61no—
56#1142—61no—
57#1141—61no—
58#1136—31no—
59#1136C2×D311yescharacteristic
60#1136C2×D311yescharacteristic
61#1131—61no—
62C8×D7—61no—
63#942—71no—
64C2×#215D711yescharacteristic, complemented
65#908—71no—
66#907—71no—
67C3×#101—71no—
68#823—71no—
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C2×D3×C7—41no—
73C2×D3×C7—81no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7C2×D411yescharacteristic
77Dic21—21no—
78#471—211noSylow
79C23×C7—61no—
80C7×Q8—21no—
81C7×Q8—21no—
82C7×D4—61no—
83C7×D4—61no—
84C2×C4×C7—21no—
85C2×C4×C7C22×D311yescharacteristic
86C22⋊D7—61no—
87D28—31no—
88C4×D7—61no—
89Dic14—31no—
90C7×C8—31no—
91C7×C8—31no—
92C7⋊C8—21no—
93C7⋊C8—21no—
94C7⋊C8—61no—
95C2×C3×Q8C2×D711yescharacteristic
96#235—71no—
97C2×D12C2×D711yescharacteristic
98C3×M16—71no—
99C3×M16—71no—
100#216—141no—
101#215—21no—
102#215—21no—
103#215—141no—
104#208—71no—
105C2×C3⋊C8C2×D711yescharacteristic
106D24—141no—
107#204—141no—
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112D21—41no—
113D3×C7—81no—
114#137—211no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—31no—
117#131—211no—
118#108—211no—
119#101—71no—
120#100—211no—
121C22×C7—121no—
122C22×C7—241no—
123C22×C7#23511yescharacteristic
124C2×D7—61no—
125C4×C7—41no—
126C4×C7D3×D411yescharacteristic
127C4×C7D3×D411yescharacteristic
128Dic7—61no—
129C22×D3—21no—
130C3×Q8—21no—
131C3×Q8—21no—
132C2×C3×C4—21no—
133C2×C3×C4C22×D711yescharacteristic
134#67—141no—
135D12—21no—
136D12—21no—
137D12—71no—
138C4×D3—141no—
139Dic6—71no—
140C3×C8—141no—
141C3×C8—141no—
142C3⋊C8—141no—
143C3⋊C8C22⋊D711yescharacteristic
144C3⋊C8C22⋊D711yescharacteristic
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—211no—
147C2×Q8D3×D711yescharacteristic
148C2×D4—31no—
149Q16—421no—
150SD16—61no—
151SD16—61no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—71no—
156M16—71no—
157M16—211no—
158M16—421no—
159C2×C8—31no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7—241no—
163C2×C7#88411yescharacteristic
164D7—121no—
165C22×C3#115911yescharacteristic
166C2×D3—41no—
167C2×D3—81no—
168C2×D3—141no—
169C3×C4—41no—
170C3×C4D4×D711yescharacteristic
171C3×C4C2×C22⋊D711yescharacteristic
172Dic3—141no—
173C23—61no—
174Q8—21no—
175Q8—21no—
176Q8—211no—
177D4—61no—
178D4—61no—
179D4—211no—
180D4—421no—
181C2×C4—21no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—31no—
185C8—31no—
186C8—141no—
187C8—141no—
188C8—421no—
189C7#475411yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3#650111yescharacteristic
192D3—81no—
193D3—281no—
194C22—121no—
195C22—241no—
196C22—421no—
197C22#6668611yescharacteristic
198C4—41no—
199C4—421no—
200C4D3×C22⋊D711yescharacteristic
201C4#6669411yescharacteristic
202C3#8994911yescharacteristic, complemented, Sylow
203C2—21no—
204C2—241no—
205C2—841no—
206C2#1062111911yescharacteristic
subgroupG/H
centerC2#10621119
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#121
prime radical#10620866C2
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×C7×#215C2semidirect
4. #10620853C2semidirect
5. C3×#6410C2semidirect
6. #10620528C2semidirect
7. #10620494C2semidirect
28. #6410D3semidirect
64. C2×#215D7semidirect
145. C3×C7#471semidirect
189. C7#4754semidirect
202. C3#89949semidirect

Maximal subgroups

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

subgroupindex
#106214252
C2×C7×#2152
#106208662
#106208532
C3×#64102
#106205282
#106204942
#899493
#47547