Finite Group Databasebeta

← #13052823

Subgroups — #13052823

Order 1344.

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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#10621370C211yesmaximal, characteristic, complemented
2#10620996C211yesmaximal, characteristic
3C2×#66565C211yesmaximal, characteristic
4C7×#807C211yesmaximal, characteristic, complemented
5C3×#6383C211yesmaximal, characteristic
6#10620510C211yesmaximal, characteristic, complemented
7#10620489C211yesmaximal, characteristic, complemented
8#89661—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×D28C2211yescharacteristic
11C2×Dic3×D7—21no—
12C3×C7×C4⋊C4C2211yescharacteristic
13#66629—21no—
14C4×Dic21—21no—
15C4×Dic21C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D7)⋊Dic3—21no—
20#66565C414yescomplemented
21C2×#1132—31no—
22#6456—31no—
23#6450—31no—
24C7×#107—31no—
25#6389—31no—
26#6384—31no—
27#6383D311yescharacteristic
28#4757—71nomaximal, Hall
29C22×C3×D7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35Dic42—21no—
36Dic42C2×C412yescomplemented
37C2×C7×Dic3—21no—
38C3×D28—21no—
39C3×D28C2×C412yes—
40Dic3×D7—42no—
41C3×C7⋊C8—21no—
42C7×C3⋊C8C2×C412yescomplemented
43C2×D28C2×D311yescharacteristic
44C2×C4×D7—61no—
45C2×Dic14—31no—
46C2×C7×C8—31no—
47C7×C4⋊C4—31no—
48C7×C4⋊C4C2×D311yescharacteristic
49(C2×D7)⋊C4—61no—
50Dic7⋊C4—61no—
51C4×Dic7—31no—
52C4×Dic7—61no—
53C2×C7⋊C8C2×D311yescharacteristic
54#1132—34no—
55D4×Dic3—71no—
56C2×#214—71no—
57#887—71no—
58C3×#102—71no—
59#816—71no—
60#815—71no—
61#807D711yescharacteristic, complemented
62C22×C3×C7C2×D411yescharacteristic
63C2×C3×D7—22no—
64C2×C3×D7—41no—
65C2×C3×D7—41no—
66C3×C4×C7—41no—
67C3×C4×C7C22×C411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—22no—
70Dic21—21no—
71Dic21—21no—
72C7×Dic3—41no—
73#440—211noSylow
74C22×D7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—61no—
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—31no—
79C2×Dic7—31no—
80C2×Dic7—61no—
81D28—21no—
82D28C4×D312yes—
83C4×D7—122no—
84Dic14—32no—
85Dic14—61no—
86C7×C8—32no—
87C7⋊C8—21no—
88C2×C3×D4—71no—
89C22×Dic3—141no—
90C2×Dic6—71no—
91C2×C3×C8—71no—
92C3×C4⋊C4C2×D711yescharacteristic
93#217—141no—
94#214—74no—
95C4⋊Dic3C2×D711yescharacteristic
96Dic3⋊C4—141no—
97C4×Dic3—71no—
98C4×Dic3—141no—
99C2×C3⋊C8C2×D711yescharacteristic
100C2×C3×C7C4×D411yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102C2×C3×C7#13711yescharacteristic
103C3×D7—42no—
104C2×SD16—211no—
105C4×Q8—211no—
106C4×D4—211no—
107#107—31no—
108#103—211no—
109#102—71no—
110#97—211no—
111C22×C7D3×D411yescharacteristic
112C2×D7—22no—
113C2×D7—41no—
114C2×D7—41no—
115C4×C7—41no—
116C4×C7—121no—
117C4×C7C2×C4×D311yescharacteristic
118C4×C7#23711yescharacteristic
119Dic7—62no—
120Dic7—61no—
121Dic7—61no—
122C23×C3—141no—
123C3×D4—72no—
124C3×D4—141no—
125C2×C3×C4—21no—
126C2×C3×C4C22×D711yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—71no—
129C2×Dic3—71no—
130C2×Dic3—141no—
131C2×Dic3—282no—
132Dic6—72no—
133Dic6—141no—
134C3×C8—141no—
135C3⋊C8C4×D712yescomplemented
136C3×C7#44011yescharacteristic, complemented, Hall
137C2×Q8—211no—
138C2×D4—71no—
139C22×C4—421no—
140SD16—214no—
141C2×C8—31no—
142C2×C8—71no—
143C4⋊C4—31no—
144C4⋊C4—421no—
145C4⋊C4D3×D711yescharacteristic
146C22⋊C4—421no—
147C42—211no—
148C42—421no—
149C2×C7#67⋊C411yescharacteristic
150C2×C7#91111yescharacteristic
151C2×C7#91511yescharacteristic
152D7—42no—
153C22×C3—142no—
154C22×C3—281no—
155C22×C3—281no—
156C22×C3D4×D711yescharacteristic
157C3×C4—41no—
158C3×C4C2×C4×D711yescharacteristic
159C3×C4#116111yescharacteristic
160Dic3—41no—
161Dic3—142no—
162Dic3—141no—
163Dic3—141no—
164C23—141no—
165Q8—212no—
166Q8—421no—
167D4—72no—
168D4—141no—
169C2×C4—21no—
170C2×C4—61no—
171C2×C4—211no—
172C2×C4—211no—
173C2×C4—421no—
174C2×C4—842no—
175C2×C4C2×D3×D711yescharacteristic
176C8—32no—
177C8—141no—
178C7#475711yescharacteristic, complemented, Sylow
179C2×C3—282no—
180C2×C3#645611yescharacteristic
181C2×C3#647411yescharacteristic
182C2×C3#648111yescharacteristic
183C22—142no—
184C22—281no—
185C22—281no—
186C22#6669411yescharacteristic
187C4—41no—
188C4—121no—
189C4—422no—
190C4—421no—
191C4—421no—
192C4#6667011yescharacteristic
193C4C4×D3×D711yescharacteristic
194C3#8966111yescharacteristic, complemented, Sylow
195C2—282no—
196C2#1062095811yescharacteristic
197C2#1062104411yescharacteristic
198C2#1062108411yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×D28C22
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#440
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621370C2semidirect
4. C7×#807C2semidirect
6. #10620510C2semidirect
7. #10620489C2semidirect
20. #66565C4semidirect
36. Dic42C2×C4semidirect
42. C7×C3⋊C8C2×C4semidirect
61. #807D7semidirect
135. C3⋊C8C4×D7semidirect
136. C3×C7#440semidirect
178. C7#4757semidirect
194. C3#89661semidirect

Maximal subgroups

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

subgroupindex
#106213702
#106209962
C2×#665652
C7×#8072
C3×#63832
#106205102
#106204892
#896613
#47577