Finite Group Databasebeta

← #13054824

Subgroups — #13054824

Order 1344.

208 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 60 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#10621434C211yesmaximal, characteristic, complemented
2C7×#948C211yesmaximal, characteristic, complemented
3C3×#6512C211yesmaximal, characteristic
4C2×#66565C211yesmaximal, characteristic, complemented
5#10620528C211yesmaximal, characteristic, complemented
6#10620498C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89913—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12#66635—21no—
13#66633—21no—
14#66627C2211yescharacteristic
15C7×#208—21no—
16C7×#208C2211yescharacteristic
17C2×C7×C3⋊C8—21no—
18C2×C7×C3⋊C8C2211yescharacteristic
19C3×#1143—21no—
20C3×#1141—21no—
21C3×#1136C2211yescharacteristic
22#66565—21no—
23#66565—21no—
24C7×#131—31no—
25#6514—31no—
26#6512D311yescharacteristic
27C2×#1132—31no—
28#6398—31no—
29#6397—31no—
30#6393—31no—
31#5200—71nomaximal, Hall
32C22×C3×D7—21no—
33C3×C7×Q8D411yescharacteristic
34C3×C7×D4—21no—
35C3×C7×D4D411yescharacteristic
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39Dic42—21no—
40Dic42—21no—
41C3×D28—21no—
42C3×D28—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C7×#41C2×D311yescharacteristic
49C2×D28C2×D311yescharacteristic
50C2×Dic14—31no—
51C7×M16—31no—
52C7×M16—61no—
53C2×C7×C8—31no—
54C2×C7×C8—61no—
55#1144—61no—
56#1143—21no—
57#1142—61no—
58#1141—21no—
59#1136—31no—
60#1136C2×D311yescharacteristic
61#1132—61no—
62#1132—61no—
63C3×AGL(1,ℤ/8)—71no—
64#951—71no—
65#948D711yescharacteristic, complemented
66C2×#214—71no—
67#833—71no—
68#824—71no—
69#822—71no—
70C22×C3×C7—21no—
71C22×C3×C7#4111yescharacteristic
72C2×C3×D7—41no—
73C2×C3×D7—81no—
74C3×C4×C7—21no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7C2×D411yescharacteristic
77Dic21—41no—
78#471—211noSylow
79C22×D7—21no—
80C7×Q8#6711yescharacteristic
81C7×D4—21no—
82C7×D4#6711yescharacteristic
83C2×C4×C7—21no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—61no—
86D28—21no—
87D28—21no—
88Dic14—61no—
89Dic14—61no—
90C7×C8—31no—
91C7×C8—31no—
92C7×C8—61no—
93C7⋊C8—21no—
94C7⋊C8—61no—
95C3×#41C2×D711yescharacteristic
96C2×C3×D4—71no—
97C2×Dic6—71no—
98C3×SD16—141no—
99C3×D8—141no—
100C3×M16—71no—
101#216—141no—
102#214—141no—
103#214—141no—
104#214—141no—
105#208—21no—
106#208—71no—
107#208C2×D711yescharacteristic
108C2×C3⋊C8—21no—
109C2×C3⋊C8C2×D711yescharacteristic
110C2×C3×C7—21no—
111C2×C3×C7—41no—
112C2×C3×C7#12111yescharacteristic
113C3×D7—81no—
114#137—211no—
115AGL(1,ℤ/8)—71no—
116C2×SD16—211no—
117#131—31no—
118#108—211no—
119#101—211no—
120#100—211no—
121C22×C7—21no—
122C22×C7#23711yescharacteristic
123C2×D7—41no—
124C2×D7—81no—
125C4×C7—21no—
126C4×C7D3×D411yescharacteristic
127C4×C7C2×#6711yescharacteristic
128Dic7—121no—
129C23×C3—141no—
130C3×Q8D2811yescharacteristic
131C3×D4—21no—
132C3×D4—141no—
133C3×D4—141no—
134C3×D4D2811yescharacteristic
135C2×C3×C4—21no—
136C2×C3×C4C22×D711yescharacteristic
137C2×Dic3—141no—
138Dic6—141no—
139Dic6—141no—
140C3×C8—141no—
141C3⋊C8—21no—
142C3⋊C8—141no—
143C3⋊C8C22⋊D711yescharacteristic
144C3⋊C8C22⋊D711yescharacteristic
145C3×C7#47111yescharacteristic, complemented, Hall
146#41D3×D711yescharacteristic
147C2×Q8—211no—
148C2×D4—71no—
149Q16—421no—
150SD16—141no—
151SD16—421no—
152SD16—421no—
153SD16—421no—
154D8—141no—
155M16—31no—
156M16—61no—
157M16—71no—
158M16—211no—
159C2×C8—31no—
160C2×C8—61no—
161C2×C7—21no—
162C2×C7—41no—
163C2×C7#93911yescharacteristic
164D7—81no—
165C22×C3—21no—
166C22×C3—281no—
167C22×C3—561no—
168C22×C3#115711yescharacteristic
169C3×C4—21no—
170C3×C4C2×D2811yescharacteristic
171C3×C4C2×C22⋊D711yescharacteristic
172Dic3—281no—
173C23—141no—
174Q8—421no—
175Q8—421no—
176Q8#424611yescharacteristic
177D4—21no—
178D4—141no—
179D4—141no—
180D4#424611yescharacteristic
181C2×C4—21no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—31no—
185C8—31no—
186C8—61no—
187C8—141no—
188C8—421no—
189C7#520011yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—41no—
192C2×C3—561no—
193C2×C3#649311yescharacteristic
194C22—21no—
195C22—281no—
196C22—561no—
197C22#6668311yescharacteristic
198C4—21no—
199C4—841no—
200C4C2×#424611yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8991311yescharacteristic, complemented, Sylow
203C2—21no—
204C2—41no—
205C2—561no—
206C2#1062113111yescharacteristic
subgroupG/H
centerC2#10621131
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7#121
prime radicalC3×#6512C2
hypercenter#41D3×D7
hypocenterC3×C7#471
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #10621434C2semidirect
2. C7×#948C2semidirect
4. C2×#66565C2semidirect
5. #10620528C2semidirect
6. #10620498C2semidirect
7. #10620493C2semidirect
65. #948D7semidirect
145. C3×C7#471semidirect
189. C7#5200semidirect
202. C3#89913semidirect

Maximal subgroups

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

subgroupindex
#106214342
C7×#9482
C3×#65122
C2×#665652
#106205282
#106204982
#106204932
#899133
#52007