Finite Group Databasebeta

← #13054722

Subgroups — #13054722

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
1C7×#942C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic, complemented
3#10620871C211yesmaximal, characteristic
4#10620848C211yesmaximal, characteristic
5#10620675C211yesmaximal, characteristic, complemented
6#10620526C211yesmaximal, characteristic, complemented
7#10620496C211yesmaximal, characteristic, complemented
8#89976—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66627C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#216—21no—
15C7×#215—21no—
16C7×#208C2211yescharacteristic
17C3×#1144—21no—
18C3×#1144—21no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66569—21no—
21#66564—21no—
22#66558—21no—
23D3×C7⋊C8—21no—
24C7×#137—31no—
25#6514—31no—
26#6511—31no—
27C2×#1144D311yescharacteristic, complemented
28#6410—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
36C7×#67—21no—
37C7×D12D411yescharacteristic
38C4×D3×C7—21no—
39C7×Dic6D411yescharacteristic
40C2×C3×Dic7—21no—
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)—21no—
44C3⋊(C7⋊C8)—21no—
45C3×C7⋊C8D411yescharacteristic
46C3×C7⋊C8D411yescharacteristic
47C7×C3⋊C8—21no—
48C7×#41—31no—
49C2×C7×Q8C2×D311yescharacteristic
50C2×Dic14C2×D311yescharacteristic
51C7×Q16—61no—
52C7×SD16—61no—
53C7×M16—31no—
54#1144—21no—
55#1144—21no—
56#1144—61no—
57#1142—61no—
58#1136—31no—
59#1136—31no—
60#1136—61no—
61C2×C7⋊C8—61no—
62C2×C7⋊C8C2×D311yescharacteristic
63C2×C3×Q16—71no—
64#942—71no—
65#942D711yescharacteristic
66#901—71no—
67#836—71no—
68#836—71no—
69#819—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D3×C7—21no—
72C3×C4×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75C3×Dic7—41no—
76C7×Dic3—21no—
77#473—211noSylow
78C7×Q8—21no—
79C7×Q8—21no—
80C7×Q8—31no—
81C7×D4—31no—
82C7×D4—61no—
83C2×C4×C7—21no—
84C2×C4×C7—61no—
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—21no—
87Dic14—21no—
88Dic14—21no—
89C7×C8—61no—
90C7⋊C8—61no—
91C7⋊C8—61no—
92C7⋊C8#6711yescharacteristic
93C7⋊C8#6711yescharacteristic
94C2×C3×Q8—71no—
95C2×C3×Q8C2×D711yescharacteristic
96#235C2×D711yescharacteristic
97C3×Q16—141no—
98C3×Q16—141no—
99C2×C3×C8—71no—
100#216—21no—
101#216—141no—
102#215—21no—
103#215—141no—
104#208—71no—
105#208—71no—
106#208C2×D711yescharacteristic
107ΓL(1,𝔽25)—141no—
108D3×C8—141no—
109C2×C3×C7—21no—
110C2×C3×C7#12111yescharacteristic
111D3×C7—41no—
112#137—31no—
113#137—211no—
114C2×Q16—71no—
115#131—211no—
116#108—211no—
117#101—211no—
118#101—211no—
119C22×C7—61no—
120C22×C7#23711yescharacteristic
121C4×C7—41no—
122C4×C7—61no—
123C4×C7D3×D411yescharacteristic
124C4×C7C2×#6711yescharacteristic
125Dic7—41no—
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—21no—
134D12C22⋊D711yescharacteristic
135C4×D3—21no—
136Dic6C22⋊D711yescharacteristic
137C3×C8—71no—
138C3×C8—71no—
139C3⋊C8—21no—
140C3⋊C8—141no—
141C3⋊C8—141no—
142C3×C7#47311yescharacteristic, complemented, Hall
143#41—31no—
144C2×Q8—71no—
145C2×Q8D3×D711yescharacteristic
146Q16—61no—
147Q16—141no—
148Q16—141no—
149Q16—421no—
150SD16—61no—
151SD16—421no—
152M16—31no—
153M16—211no—
154M16—211no—
155M16—421no—
156C2×C8—71no—
157C2×C8—421no—
158C2×C7—21no—
159C2×C7—121no—
160C2×C7#93811yescharacteristic
161C22×C3#115911yescharacteristic
162C2×D3—21no—
163C3×C4—41no—
164C3×C4—281no—
165C3×C4D4×D711yescharacteristic
166C3×C4C2×C22⋊D711yescharacteristic
167Dic3—21no—
168Q8—21no—
169Q8—21no—
170Q8—31no—
171Q8—141no—
172Q8—141no—
173D4—31no—
174D4—61no—
175C2×C4—21no—
176C2×C4—61no—
177C2×C4—141no—
178C2×C4C2×D3×D711yescharacteristic
179C8—61no—
180C8—71no—
181C8—71no—
182C8—421no—
183C8—421no—
184C7#519211yescharacteristic, complemented, Sylow
185C2×C3—21no—
186C2×C3#650211yescharacteristic
187D3—41no—
188C22—61no—
189C22#6668411yescharacteristic
190C4—41no—
191C4—61no—
192C4—281no—
193C4D7×#6711yescharacteristic
194C4D3×C22⋊D711yescharacteristic
195C3#8997611yescharacteristic, complemented, Sylow
196C2—21no—
197C2—121no—
198C2#1062112711yescharacteristic
subgroupG/H
centerC2#10621127
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#121
prime radicalC7×#235C22
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. #10620675C2semidirect
6. #10620526C2semidirect
7. #10620496C2semidirect
27. C2×#1144D3semidirect
142. C3×C7#473semidirect
184. C7#5192semidirect
195. C3#89976semidirect

Maximal subgroups

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

subgroupindex
C7×#9422
C2×C3×#11442
#106208712
#106208482
#106206752
#106205262
#106204962
#899763
#51927