Finite Group Databasebeta

← #13054496

Subgroups — #13054496

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
1C7×#932C211yesmaximal, characteristic
2C2×C3×#1142C211yesmaximal, characteristic, complemented
3#10620871C211yesmaximal, characteristic, complemented
4#10620848C211yesmaximal, characteristic, complemented
5#10620672C211yesmaximal, characteristic, complemented
6#10620526C211yesmaximal, characteristic, complemented
7#10620496C211yesmaximal, characteristic, complemented
8#89972—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66627C2211yescharacteristic
13#66627C2211yescharacteristic
14C7×#214—21no—
15C7×#213—21no—
16C7×#208C2211yescharacteristic
17C3×#1142—21no—
18C3×#1142—21no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66569—21no—
21#66564—21no—
22#66558—21no—
23D3×C7⋊C8—21no—
24C7×AGL(1,ℤ/8)—31no—
25#6514—31no—
26#6511—31no—
27C2×#1142D311yescharacteristic, complemented
28#6407—31no—
29#6398—31no—
30#6396—31no—
31#5177—71nomaximal, Hall
32C23×C3×C7—21no—
33C3×C7×D4—21no—
34C3×C7×D4—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×D4C2×D311yescharacteristic
50C2×Dic14C2×D311yescharacteristic
51C7×SD16—61no—
52C7×D8—61no—
53C7×M16—31no—
54#1144—61no—
55#1142—21no—
56#1142—21no—
57#1142—61no—
58#1136—31no—
59#1136—31no—
60#1136—61no—
61C2×C7⋊C8—61no—
62C2×C7⋊C8C2×D311yescharacteristic
63C2×C3×SD16—71no—
64#942—71no—
65#932D711yescharacteristic
66#901—71no—
67#836—71no—
68#833—71no—
69#819—71no—
70C22×C3×C7—41no—
71C22×C3×C7—81no—
72C22×C3×C7#4111yescharacteristic
73C2×D3×C7—21no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76C3×Dic7—41no—
77C7×Dic3—21no—
78#471—211noSylow
79C23×C7—21no—
80C7×Q8—31no—
81C7×D4—21no—
82C7×D4—21no—
83C7×D4—31no—
84C7×D4—61no—
85C2×C4×C7—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—21no—
88Dic14—21no—
89Dic14—21no—
90C7×C8—61no—
91C7⋊C8—61no—
92C7⋊C8—61no—
93C7⋊C8#6711yescharacteristic
94C7⋊C8#6711yescharacteristic
95C2×C3×Q8—71no—
96C2×C3×D4C2×D711yescharacteristic
97#235C2×D711yescharacteristic
98C3×SD16—141no—
99C3×SD16—141no—
100C2×C3×C8—71no—
101#216—141no—
102#215—141no—
103#214—21no—
104#213—21no—
105#208—71no—
106#208—71no—
107#208C2×D711yescharacteristic
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7—81no—
112C2×C3×C7#12111yescharacteristic
113D3×C7—41no—
114#137—211no—
115AGL(1,ℤ/8)—31no—
116C2×SD16—71no—
117#131—211no—
118#108—211no—
119#101—211no—
120#100—211no—
121C22×C7—41no—
122C22×C7—61no—
123C22×C7—81no—
124C22×C7#23711yescharacteristic
125C4×C7—61no—
126C4×C7D3×D411yescharacteristic
127C4×C7C2×#6711yescharacteristic
128Dic7—41no—
129C23×C3—21no—
130C3×Q8—141no—
131C3×Q8—141no—
132C3×D4—21no—
133C3×D4—21no—
134C2×C3×C4—141no—
135C2×C3×C4C22×D711yescharacteristic
136#67—21no—
137D12C22⋊D711yescharacteristic
138C4×D3—21no—
139Dic6C22⋊D711yescharacteristic
140C3×C8—71no—
141C3×C8—71no—
142C3⋊C8—21no—
143C3⋊C8—141no—
144C3⋊C8—141no—
145C3×C7#47111yescharacteristic, complemented, Hall
146#41—31no—
147C2×Q8—71no—
148C2×D4D3×D711yescharacteristic
149Q16—421no—
150SD16—61no—
151SD16—141no—
152SD16—141no—
153SD16—421no—
154D8—61no—
155M16—31no—
156M16—211no—
157M16—211no—
158M16—421no—
159C2×C8—71no—
160C2×C8—421no—
161C2×C7—21no—
162C2×C7—81no—
163C2×C7—121no—
164C2×C7#93811yescharacteristic
165C22×C3—41no—
166C22×C3—81no—
167C22×C3#115911yescharacteristic
168C2×D3—21no—
169C3×C4—281no—
170C3×C4D4×D711yescharacteristic
171C3×C4C2×C22⋊D711yescharacteristic
172Dic3—21no—
173C23—21no—
174Q8—31no—
175Q8—141no—
176Q8—141no—
177D4—21no—
178D4—21no—
179D4—31no—
180D4—61no—
181C2×C4—61no—
182C2×C4—141no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—71no—
186C8—71no—
187C8—421no—
188C8—421no—
189C7#517711yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—81no—
192C2×C3#650211yescharacteristic
193D3—41no—
194C22—41no—
195C22—61no—
196C22—81no—
197C22#6668411yescharacteristic
198C4—61no—
199C4—281no—
200C4D7×#6711yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8997211yescharacteristic, complemented, Sylow
203C2—21no—
204C2—81no—
205C2—121no—
206C2#1062112711yescharacteristic
subgroupG/H
centerC2#10621127
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#121
prime radicalC7×#932C2
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
2. C2×C3×#1142C2semidirect
3. #10620871C2semidirect
4. #10620848C2semidirect
5. #10620672C2semidirect
6. #10620526C2semidirect
7. #10620496C2semidirect
27. C2×#1142D3semidirect
145. C3×C7#471semidirect
189. C7#5177semidirect
202. C3#89972semidirect

Maximal subgroups

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

subgroupindex
C7×#9322
C2×C3×#11422
#106208712
#106208482
#106206722
#106205262
#106204962
#899723
#51777