Finite Group Databasebeta

← #13053567

Subgroups — #13053567

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
1C7×#907C211yesmaximal, characteristic, complemented
2#10620884C211yesmaximal, characteristic, complemented
3C2×#66565C211yesmaximal, characteristic, complemented
4#10620860C211yesmaximal, characteristic
5#10620663C211yesmaximal, characteristic, complemented
6C3×#6397C211yesmaximal, characteristic, complemented
7#10620528C211yesmaximal, characteristic, complemented
8#89913—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C3×C7×M16C2211yescharacteristic
13#66627C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C7×ΓL(1,𝔽25)—21no—
16D3×C7×C8—21no—
17C3×#1136C2211yescharacteristic
18#66571—21no—
19#66567—21no—
20#66565—21no—
21#66565—21no—
22#66563—21no—
23#66560—21no—
24C7×#131—31no—
25#6514—31no—
26#6512—31no—
27C2×#1132—31no—
28#6398—31no—
29#6397D311yescharacteristic, complemented
30#6393—31no—
31#4748—71nomaximal, Hall
32C22×C3×D7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35Dic42—21no—
36Dic42—21no—
37C7×#67—21no—
38C7×D12D411yescharacteristic
39C4×D3×C7—21no—
40C7×Dic6D411yescharacteristic
41C3×D28—21no—
42C3×D28—21no—
43C3×C7×C8—21no—
44C3⋊(C7⋊C8)—21no—
45C3×C7⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C7×#41—31no—
49C2×D28C2×D311yescharacteristic
50C2×Dic14—31no—
51C7×M16—61no—
52C7×M16C2×D311yescharacteristic
53C2×C7×C8—31no—
54C2×C7×C8—61no—
55#1144—61no—
56#1143—61no—
57#1142—61no—
58#1141—61no—
59#1136—31no—
60#1136C2×D311yescharacteristic
61#1132—61no—
62#1132—61no—
63C2×#214—71no—
64#932—71no—
65#909—71no—
66#907D711yescharacteristic, complemented
67C3×#100—71no—
68#824—71no—
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D3×C7—21no—
72C2×C3×D7—41no—
73C2×C3×D7—81no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—41no—
77C7×Dic3—21no—
78#471—211noSylow
79C22×D7—21no—
80C7×Q8—31no—
81C7×D4—31no—
82C7×D4—61no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—61no—
86D28—21no—
87D28—21no—
88Dic14—61no—
89Dic14—61no—
90C7×C8—21no—
91C7×C8—31no—
92C7×C8—31no—
93C7⋊C8—21no—
94C7⋊C8—61no—
95C2×C3×D4—71no—
96#235C2×D711yescharacteristic
97C2×Dic6—71no—
98C3×M16—71no—
99C3×M16C2×D711yescharacteristic
100#214—141no—
101#214—141no—
102#214—141no—
103#213—141no—
104#208—71no—
105C2×C3⋊C8C2×D711yescharacteristic
106Dic12—141no—
107#204—141no—
108ΓL(1,𝔽25)—21no—
109D3×C8—21no—
110C2×C3×C7—21no—
111C2×C3×C7#12111yescharacteristic
112C3×D7—81no—
113D3×C7—41no—
114#137—211no—
115AGL(1,ℤ/8)—211no—
116C2×SD16—211no—
117#131—31no—
118#108—211no—
119#101—211no—
120#100—71no—
121C22×C7—61no—
122C22×C7#23511yescharacteristic
123C2×D7—41no—
124C2×D7—81no—
125C4×C7—61no—
126C4×C7D3×D411yescharacteristic
127C4×C7D3×D411yescharacteristic
128Dic7—121no—
129C23×C3—141no—
130C3×D4—141no—
131C3×D4—141no—
132C2×C3×C4C22×D711yescharacteristic
133#67—21no—
134C2×Dic3—141no—
135D12D2811yescharacteristic
136C4×D3—21no—
137Dic6—141no—
138Dic6—141no—
139Dic6D2811yescharacteristic
140C3×C8—21no—
141C3×C8—141no—
142C3⋊C8—141no—
143C3⋊C8C22⋊D711yescharacteristic
144C3⋊C8C22⋊D711yescharacteristic
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—31no—
147C2×Q8—211no—
148C2×D4—71no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—421no—
155M16—61no—
156M16—71no—
157M16—211no—
158M16D3×D711yescharacteristic
159C2×C8—31no—
160C2×C8—61no—
161C2×C7—21no—
162C2×C7—121no—
163C2×C7#88411yescharacteristic
164D7—81no—
165C22×C3—281no—
166C22×C3—561no—
167C22×C3#115711yescharacteristic
168C2×D3—21no—
169C3×C4C2×D2811yescharacteristic
170C3×C4C2×C22⋊D711yescharacteristic
171Dic3—21no—
172Dic3—281no—
173C23—141no—
174Q8—31no—
175Q8—421no—
176Q8—421no—
177D4—31no—
178D4—61no—
179D4—141no—
180D4—141no—
181C2×C4—61no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—21no—
185C8—31no—
186C8—31no—
187C8—141no—
188C8—421no—
189C7#474811yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—561no—
192C2×C3#649311yescharacteristic
193D3—41no—
194C22—61no—
195C22—281no—
196C22—561no—
197C22#6667511yescharacteristic
198C4—61no—
199C4—841no—
200C4D3×D2811yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8991311yescharacteristic, complemented, Sylow
203C2—21no—
204C2—121no—
205C2—561no—
206C2#1062101611yescharacteristic
subgroupG/H
centerC2#10621016
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7#121
prime radical#10620860C2
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. C7×#907C2semidirect
2. #10620884C2semidirect
3. C2×#66565C2semidirect
5. #10620663C2semidirect
6. C3×#6397C2semidirect
7. #10620528C2semidirect
29. #6397D3semidirect
66. #907D7semidirect
145. C3×C7#471semidirect
189. C7#4748semidirect
202. C3#89913semidirect

Maximal subgroups

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

subgroupindex
C7×#9072
#106208842
C2×#665652
#106208602
#106206632
C3×#63972
#106205282
#899133
#47487