Finite Group Databasebeta

← #13054768

Subgroups — #13054768

Order 1344.

200 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 200 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×#216C211yesmaximal, characteristic, complemented
2C3×#6505C211yesmaximal, characteristic
3#10620869C211yesmaximal, characteristic
4#10620845C211yesmaximal, characteristic
5#10620675C211yesmaximal, characteristic, complemented
6#10620528C211yesmaximal, characteristic, complemented
7#10620497C211yesmaximal, characteristic, complemented
8#89964—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12#66627C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#216—21no—
15C7×#216—21no—
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×#1144—21no—
18C3×#1143—21no—
19C3×#1136C2211yescharacteristic
20#66569—21no—
21#66563—21no—
22#66557—21no—
23D7×C3⋊C8—21no—
24C2×C7×Q16—31no—
25#6505—31no—
26#6505D311yescharacteristic
27#6464—31no—
28#6410—31no—
29#6410—31no—
30#6393—31no—
31#5204—71nomaximal, Hall
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—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×Q8C2×D311yescharacteristic
50#1157C2×D311yescharacteristic
51C7×Q16—61no—
52C7×Q16—61no—
53C2×C7×C8—31no—
54#1144—21no—
55#1144—61no—
56#1143—21no—
57#1143—61no—
58#1136—31no—
59#1136—31no—
60#1136C2×D311yescharacteristic
61#1131—61no—
62C8×D7—61no—
63C3×#137—71no—
64#951—71no—
65#948—71no—
66C2×#216D711yescharacteristic, complemented
67#836—71no—
68#824—71no—
69#822—71no—
70C22×C3×C7#4111yescharacteristic
71C2×C3×D7—21no—
72C3×C4×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75C3×Dic7—21no—
76C7×Dic3—41no—
77#473—211noSylow
78C7×Q8—21no—
79C7×Q8—21no—
80C7×Q8—61no—
81C7×Q8—61no—
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C22⋊D7—21no—
86D28#6711yescharacteristic
87C4×D7—21no—
88Dic14#6711yescharacteristic
89C7×C8—31no—
90C7×C8—31no—
91C7⋊C8—21no—
92C7⋊C8—61no—
93C7⋊C8—61no—
94C3×#41—71no—
95C2×C3×Q8C2×D711yescharacteristic
96C2×Dic6C2×D711yescharacteristic
97C3×Q16—141no—
98C3×SD16—141no—
99C3×M16—71no—
100#216—21no—
101#216—21no—
102#216—141no—
103#214—141no—
104#208—71no—
105#208—71no—
106#208—141no—
107C2×C3⋊C8—141no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7—21no—
110C2×C3×C7#12111yescharacteristic
111C3×D7—41no—
112#137—71no—
113#137—211no—
114C2×Q16—31no—
115#131—211no—
116#108—211no—
117#101—211no—
118#101—211no—
119C22×C7#23711yescharacteristic
120C2×D7—21no—
121C4×C7—41no—
122C4×C7—121no—
123C4×C7D3×D411yescharacteristic
124C4×C7C2×#6711yescharacteristic
125Dic7—21no—
126C3×Q8—21no—
127C3×Q8—21no—
128C3×Q8—71no—
129C3×D4—71no—
130C3×D4—141no—
131C2×C3×C4—21no—
132C2×C3×C4—141no—
133C2×C3×C4C22×D711yescharacteristic
134C2×Dic3—21no—
135Dic6—21no—
136Dic6—21no—
137C3×C8—141no—
138C3⋊C8—141no—
139C3⋊C8—141no—
140C3⋊C8C22⋊D711yescharacteristic
141C3⋊C8C22⋊D711yescharacteristic
142C3×C7#47311yescharacteristic, complemented, Hall
143#41—71no—
144C2×Q8—31no—
145C2×Q8D3×D711yescharacteristic
146Q16—61no—
147Q16—61no—
148Q16—141no—
149Q16—421no—
150SD16—141no—
151SD16—421no—
152M16—71no—
153M16—211no—
154M16—211no—
155M16—421no—
156C2×C8—31no—
157C2×C8—421no—
158C2×C7—21no—
159C2×C7#93911yescharacteristic
160D7—41no—
161C22×C3—141no—
162C22×C3#115911yescharacteristic
163C3×C4—41no—
164C3×C4—141no—
165C3×C4D4×D711yescharacteristic
166C3×C4C2×C22⋊D711yescharacteristic
167Dic3—41no—
168Q8—21no—
169Q8—21no—
170Q8—61no—
171Q8—61no—
172Q8—71no—
173D4—71no—
174D4—141no—
175C2×C4—21no—
176C2×C4—61no—
177C2×C4—141no—
178C2×C4C2×D3×D711yescharacteristic
179C8—31no—
180C8—31no—
181C8—141no—
182C8—421no—
183C8—421no—
184C7#520411yescharacteristic, complemented, Sylow
185C2×C3—21no—
186C2×C3—281no—
187C2×C3#650111yescharacteristic
188C22—141no—
189C22#6668411yescharacteristic
190C4—41no—
191C4—121no—
192C4—141no—
193C4D7×#6711yescharacteristic
194C4D3×C22⋊D711yescharacteristic
195C3#8996411yescharacteristic, complemented, Sylow
196C2—21no—
197C2—281no—
198C2#1062111111yescharacteristic
subgroupG/H
centerC2#10621111
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#121
prime radicalC3×#1157C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#473
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. C2×C7×#216C2semidirect
5. #10620675C2semidirect
6. #10620528C2semidirect
7. #10620497C2semidirect
66. C2×#216D7semidirect
142. C3×C7#473semidirect
184. C7#5204semidirect
195. C3#89964semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2162
C3×#65052
#106208692
#106208452
#106206752
#106205282
#106204972
#899643
#52047