Finite Group Databasebeta

← #13054724

Subgroups — #13054724

Order 1344.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#10621425C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3#10620877C211yesmaximal, characteristic
4#10620852C211yesmaximal, characteristic
5C7×#836C211yesmaximal, characteristic, complemented
6#10620526C211yesmaximal, characteristic, complemented
7#10620496C211yesmaximal, characteristic, complemented
8#89527—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66635—21no—
13#66634—21no—
14#66627C2211yescharacteristic
15C7×#208C2211yescharacteristic
16C7×#208C2211yescharacteristic
17C3×#1144—21no—
18C3×#1144—21no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66570—21no—
21#66566—21no—
22#66559—21no—
23D21⋊C8—21no—
24C2×#1144D311yescharacteristic, complemented
25#6505—31no—
26#6472—31no—
27#6470—31no—
28C7×#101—31no—
29#6398—31no—
30#6396—31no—
31#5192—71nomaximal, Hall
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36#4268—21no—
37D84D411yescharacteristic
38C4×D21—21no—
39Dic42D411yescharacteristic
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8D411yescharacteristic
45C3×C7⋊C8D411yescharacteristic
46C7×C3⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×Q8C2×D311yescharacteristic
49#1157—31no—
50C2×Dic14C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16—31no—
53#1144—21no—
54#1144—21no—
55#1144—61no—
56#1143—61no—
57#1136—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59Dic28—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C2×C3×Q16—71no—
64#942—71no—
65#942—71no—
66#901—71no—
67#836—71no—
68#836D711yescharacteristic, complemented
69#819—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C3×C4×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75Dic21—21no—
76C3×Dic7—41no—
77#473—211noSylow
78C7×Q8—21no—
79C7×Q8—21no—
80C2×C4×C7—21no—
81C2×C4×C7C22×D311yescharacteristic
82C22⋊D7—61no—
83C2×Dic7—21no—
84D28—31no—
85C4×D7—61no—
86Dic14—21no—
87Dic14—21no—
88Dic14—31no—
89C7×C8—61no—
90C7×C8—61no—
91C7⋊C8—61no—
92C7⋊C8#6711yescharacteristic
93C7⋊C8#6711yescharacteristic
94C2×C3×Q8—71no—
95C2×C3×Q8C2×D711yescharacteristic
96#235—71no—
97C3×Q16—141no—
98C3×Q16—141no—
99C2×C3×C8—71no—
100#216—141no—
101#216—141no—
102#215—141no—
103#215—141no—
104#208—71no—
105#208C2×D711yescharacteristic
106#208C2×D711yescharacteristic
107ΓL(1,𝔽25)—141no—
108D3×C8—141no—
109C2×C3×C7—21no—
110C2×C3×C7#12111yescharacteristic
111D21—41no—
112#137—211no—
113#137—211no—
114C2×Q16—71no—
115#131—211no—
116#108—211no—
117#101—31no—
118#101—211no—
119C22×C7#23711yescharacteristic
120C2×D7—61no—
121C4×C7—41no—
122C4×C7D3×D411yescharacteristic
123C4×C7C2×#6711yescharacteristic
124Dic7—41no—
125Dic7—61no—
126C3×Q8—21no—
127C3×Q8—21no—
128C3×Q8—141no—
129C3×Q8—141no—
130C2×C3×C4—21no—
131C2×C3×C4—141no—
132C2×C3×C4C22×D711yescharacteristic
133#67—141no—
134D12—71no—
135C4×D3—141no—
136Dic6—71no—
137C3×C8—71no—
138C3×C8—71no—
139C3⋊C8—21no—
140C3⋊C8—21no—
141C3⋊C8—141no—
142C3×C7#47311yescharacteristic, complemented, Hall
143#41—211no—
144C2×Q8—71no—
145C2×Q8D3×D711yescharacteristic
146Q16—141no—
147Q16—141no—
148Q16—421no—
149Q16—421no—
150SD16—421no—
151SD16—421no—
152M16—31no—
153M16—31no—
154M16—211no—
155M16—421no—
156C2×C8—71no—
157C2×C8—421no—
158C2×C7—21no—
159C2×C7#93811yescharacteristic
160D7—121no—
161C22×C3#115711yescharacteristic
162C2×D3—141no—
163C3×C4—41no—
164C3×C4—281no—
165C3×C4D4×D711yescharacteristic
166C3×C4D4×D711yescharacteristic
167Dic3—141no—
168Q8—21no—
169Q8—21no—
170Q8—141no—
171Q8—141no—
172Q8—211no—
173D4—211no—
174D4—421no—
175C2×C4—21no—
176C2×C4—141no—
177C2×C4—421no—
178C2×C4C2×D3×D711yescharacteristic
179C8—61no—
180C8—61no—
181C8—71no—
182C8—71no—
183C8—421no—
184C7#519211yescharacteristic, complemented, Sylow
185C2×C3—21no—
186C2×C3#644711yescharacteristic
187D3—281no—
188C22—421no—
189C22#6668311yescharacteristic
190C4—41no—
191C4—281no—
192C4—421no—
193C4D7×#6711yescharacteristic
194C4#6669411yescharacteristic
195C3#8952711yescharacteristic, complemented, Sylow
196C2—21no—
197C2—841no—
198C2#1062109511yescharacteristic
subgroupG/H
centerC2#10621095
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#121
prime radical#66728C22
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
2. C2×C3×#1144C2semidirect
5. C7×#836C2semidirect
6. #10620526C2semidirect
7. #10620496C2semidirect
24. C2×#1144D3semidirect
68. #836D7semidirect
142. C3×C7#473semidirect
184. C7#5192semidirect
195. C3#89527semidirect

Maximal subgroups

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

subgroupindex
#106214252
C2×C3×#11442
#106208772
#106208522
C7×#8362
#106205262
#106204962
#895273
#51927