Finite Group Databasebeta

← #13054540

Subgroups — #13054540

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
1C2×C7×#214C211yesmaximal, characteristic, complemented
2C3×#6495C211yesmaximal, characteristic
3#10620869C211yesmaximal, characteristic, complemented
4#10620845C211yesmaximal, characteristic, complemented
5#10620672C211yesmaximal, characteristic, complemented
6#10620528C211yesmaximal, characteristic, complemented
7#10620497C211yesmaximal, characteristic, complemented
8#89949—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12#66627C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#214—21no—
15C7×#214—21no—
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×#1142—21no—
18C3×#1141—21no—
19C3×#1136C2211yescharacteristic
20#66569—21no—
21#66563—21no—
22#66557—21no—
23D7×C3⋊C8—21no—
24C2×C7×SD16—31no—
25#6505—31no—
26#6495D311yescharacteristic
27#6464—31no—
28#6410—31no—
29#6407—31no—
30#6393—31no—
31#5200—71nomaximal, Hall
32C23×C3×C7—21no—
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C2×C3×C4×C7C2311yescharacteristic
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⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C2×C7×Q8—31no—
49C2×C7×D4C2×D311yescharacteristic
50#1157C2×D311yescharacteristic
51C7×SD16—61no—
52C7×SD16—61no—
53C2×C7×C8—31no—
54#1144—61no—
55#1143—61no—
56#1142—21no—
57#1141—21no—
58#1136—31no—
59#1136—31no—
60#1136C2×D311yescharacteristic
61#1131—61no—
62C8×D7—61no—
63C3×AGL(1,ℤ/8)—71no—
64#951—71no—
65#948—71no—
66C2×#214D711yescharacteristic, complemented
67#833—71no—
68#824—71no—
69#822—71no—
70C22×C3×C7—41no—
71C22×C3×C7—81no—
72C22×C3×C7#4111yescharacteristic
73C2×C3×D7—21no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76C3×Dic7—21no—
77C7×Dic3—41no—
78#471—211noSylow
79C23×C7—21no—
80C7×Q8—61no—
81C7×Q8—61no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7—61no—
85C2×C4×C7C22×D311yescharacteristic
86C22⋊D7—21no—
87D28#6711yescharacteristic
88C4×D7—21no—
89Dic14#6711yescharacteristic
90C7×C8—31no—
91C7×C8—31no—
92C7⋊C8—21no—
93C7⋊C8—61no—
94C7⋊C8—61no—
95C3×#41—71no—
96C2×C3×D4C2×D711yescharacteristic
97C2×Dic6C2×D711yescharacteristic
98C3×SD16—141no—
99C3×D8—141no—
100C3×M16—71no—
101#216—141no—
102#214—21no—
103#214—21no—
104#214—141no—
105#208—71no—
106#208—71no—
107#208—141no—
108C2×C3⋊C8—141no—
109C2×C3⋊C8C2×D711yescharacteristic
110C2×C3×C7—21no—
111C2×C3×C7—81no—
112C2×C3×C7#12111yescharacteristic
113C3×D7—41no—
114#137—211no—
115AGL(1,ℤ/8)—71no—
116C2×SD16—31no—
117#131—211no—
118#108—211no—
119#101—211no—
120#100—211no—
121C22×C7—41no—
122C22×C7—81no—
123C22×C7#23711yescharacteristic
124C2×D7—21no—
125C4×C7—121no—
126C4×C7D3×D411yescharacteristic
127C4×C7C2×#6711yescharacteristic
128Dic7—21no—
129C23×C3—21no—
130C3×Q8—71no—
131C3×D4—21no—
132C3×D4—21no—
133C3×D4—71no—
134C3×D4—141no—
135C2×C3×C4—141no—
136C2×C3×C4C22×D711yescharacteristic
137C2×Dic3—21no—
138Dic6—21no—
139Dic6—21no—
140C3×C8—141no—
141C3⋊C8—141no—
142C3⋊C8—141no—
143C3⋊C8C22⋊D711yescharacteristic
144C3⋊C8C22⋊D711yescharacteristic
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—71no—
147C2×Q8—31no—
148C2×D4D3×D711yescharacteristic
149Q16—421no—
150SD16—61no—
151SD16—61no—
152SD16—141no—
153SD16—421no—
154D8—141no—
155M16—71no—
156M16—211no—
157M16—211no—
158M16—421no—
159C2×C8—31no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7—81no—
163C2×C7#93911yescharacteristic
164D7—41no—
165C22×C3—41no—
166C22×C3—81no—
167C22×C3—141no—
168C22×C3#115911yescharacteristic
169C3×C4—141no—
170C3×C4D4×D711yescharacteristic
171C3×C4C2×C22⋊D711yescharacteristic
172Dic3—41no—
173C23—21no—
174Q8—61no—
175Q8—61no—
176Q8—71no—
177D4—21no—
178D4—21no—
179D4—71no—
180D4—141no—
181C2×C4—61no—
182C2×C4—141no—
183C2×C4C2×D3×D711yescharacteristic
184C8—31no—
185C8—31no—
186C8—141no—
187C8—421no—
188C8—421no—
189C7#520011yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—81no—
192C2×C3—281no—
193C2×C3#650111yescharacteristic
194C22—41no—
195C22—81no—
196C22—141no—
197C22#6668411yescharacteristic
198C4—121no—
199C4—141no—
200C4D7×#6711yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8994911yescharacteristic, complemented, Sylow
203C2—21no—
204C2—81no—
205C2—281no—
206C2#1062111111yescharacteristic
subgroupG/H
centerC2#10621111
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#121
prime radicalC3×#6495C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#471
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C7×#214C2semidirect
3. #10620869C2semidirect
4. #10620845C2semidirect
5. #10620672C2semidirect
6. #10620528C2semidirect
7. #10620497C2semidirect
66. C2×#214D7semidirect
145. C3×C7#471semidirect
189. C7#5200semidirect
202. C3#89949semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2142
C3×#64952
#106208692
#106208452
#106206722
#106205282
#106204972
#899493
#52007