Finite Group Databasebeta

← #13054497

Subgroups — #13054497

Order 1344.

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

normal subgroups only

This group has 216 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#10621415C211yesmaximal, characteristic, complemented
2C2×C3×#1141C211yesmaximal, characteristic, complemented
3#10620874C211yesmaximal, characteristic, complemented
4#10620852C211yesmaximal, characteristic, complemented
5C7×#833C211yesmaximal, characteristic, complemented
6#10620526C211yesmaximal, characteristic, complemented
7#10620493C211yesmaximal, characteristic, complemented
8#89521—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10#66728C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12#66633—21no—
13#66632—21no—
14#66627C2211yescharacteristic
15C7×#208C2211yescharacteristic
16C7×#208C2211yescharacteristic
17C3×#1141—21no—
18C3×#1141—21no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66565—21no—
21#66561—21no—
22#66559—21no—
23D21⋊C8—21no—
24#6495—31no—
25C2×#1141D311yescharacteristic, complemented
26#6471—31no—
27#6470—31no—
28C7×#100—31no—
29#6397—31no—
30#6396—31no—
31#5160—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×C3×D7—21no—
34C3×C7×D4—21no—
35C3×C7×D4—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—21no—
38D84D411yescharacteristic
39C4×D21—21no—
40Dic42D411yescharacteristic
41C3×D28—21no—
42C3×D28—21no—
43C3⋊(C7⋊C8)—21no—
44C3×C7⋊C8D411yescharacteristic
45C3×C7⋊C8D411yescharacteristic
46C7×C3⋊C8—21no—
47C7×C3⋊C8—21no—
48C2×C7×D4C2×D311yescharacteristic
49#1157—31no—
50C2×D28C2×D311yescharacteristic
51C7×M16—31no—
52C7×M16—31no—
53#1142—61no—
54#1141—21no—
55#1141—21no—
56#1141—61no—
57#1136—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59D56—61no—
60#1132—61no—
61#1131—61no—
62C8×D7—61no—
63C2×C3×D8—71no—
64#932—71no—
65#932—71no—
66#901—71no—
67#833—71no—
68#833D711yescharacteristic, complemented
69#819—71no—
70C22×C3×C7—41no—
71C22×C3×C7—81no—
72C22×C3×C7#4111yescharacteristic
73C2×D21—21no—
74C2×C3×D7—41no—
75C2×C3×D7—81no—
76C3×C4×C7C2×D411yescharacteristic
77C3×C4×C7C2×D411yescharacteristic
78Dic21—21no—
79#472—211noSylow
80C23×C7—21no—
81C22×D7—21no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7C22×D311yescharacteristic
85C22⋊D7—61no—
86D28—21no—
87D28—21no—
88D28—31no—
89C4×D7—61no—
90Dic14—31no—
91C7×C8—61no—
92C7×C8—61no—
93C7⋊C8—61no—
94C7⋊C8#6711yescharacteristic
95C7⋊C8#6711yescharacteristic
96C2×C3×D4—71no—
97C2×C3×D4C2×D711yescharacteristic
98#235—71no—
99C3×D8—141no—
100C3×D8—141no—
101C2×C3×C8—71no—
102#214—141no—
103#214—141no—
104#213—141no—
105#213—141no—
106#208—71no—
107#208C2×D711yescharacteristic
108#208C2×D711yescharacteristic
109ΓL(1,𝔽25)—141no—
110D3×C8—141no—
111C2×C3×C7—21no—
112C2×C3×C7—81no—
113C2×C3×C7#12111yescharacteristic
114D21—41no—
115C3×D7—81no—
116AGL(1,ℤ/8)—211no—
117AGL(1,ℤ/8)—211no—
118C2×D8—71no—
119#131—211no—
120#108—211no—
121#100—31no—
122#100—211no—
123C22×C7—41no—
124C22×C7—81no—
125C22×C7#23711yescharacteristic
126C2×D7—41no—
127C2×D7—61no—
128C2×D7—81no—
129C4×C7D3×D411yescharacteristic
130C4×C7C2×#6711yescharacteristic
131Dic7—61no—
132C23×C3—21no—
133C23×C3—141no—
134C3×D4—21no—
135C3×D4—21no—
136C3×D4—141no—
137C3×D4—141no—
138C2×C3×C4C22×D711yescharacteristic
139#67—141no—
140D12—71no—
141C4×D3—141no—
142Dic6—71no—
143C3×C8—71no—
144C3×C8—71no—
145C3⋊C8—21no—
146C3⋊C8—21no—
147C3⋊C8—141no—
148C3×C7#47211yescharacteristic, complemented, Hall
149#41—211no—
150C2×D4—71no—
151C2×D4D3×D711yescharacteristic
152SD16—421no—
153SD16—421no—
154D8—141no—
155D8—141no—
156D8—421no—
157D8—421no—
158M16—31no—
159M16—31no—
160M16—211no—
161M16—421no—
162C2×C8—71no—
163C2×C8—421no—
164C2×C7—21no—
165C2×C7—81no—
166C2×C7#93811yescharacteristic
167D7—81no—
168D7—121no—
169C22×C3—41no—
170C22×C3—81no—
171C22×C3—281no—
172C22×C3—561no—
173C22×C3#115711yescharacteristic
174C2×D3—141no—
175C3×C4D4×D711yescharacteristic
176C3×C4D4×D711yescharacteristic
177Dic3—141no—
178C23—21no—
179C23—141no—
180Q8—211no—
181D4—21no—
182D4—21no—
183D4—141no—
184D4—141no—
185D4—211no—
186D4—421no—
187C2×C4—421no—
188C2×C4C2×D3×D711yescharacteristic
189C8—61no—
190C8—61no—
191C8—71no—
192C8—71no—
193C8—421no—
194C7#516011yescharacteristic, complemented, Sylow
195C2×C3—21no—
196C2×C3—81no—
197C2×C3—561no—
198C2×C3#644711yescharacteristic
199D3—281no—
200C22—41no—
201C22—81no—
202C22—281no—
203C22—421no—
204C22—561no—
205C22#6668311yescharacteristic
206C4—421no—
207C4D7×#6711yescharacteristic
208C4#6669411yescharacteristic
209C3#8952111yescharacteristic, complemented, Sylow
210C2—21no—
211C2—81no—
212C2—561no—
213C2—841no—
214C2#1062109511yescharacteristic
subgroupG/H
centerC2#10621095
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC2×C3×C7#121
hypercenterC2×D4D3×D7
hypocenterC3×C7#472
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
1. #10621415C2semidirect
2. C2×C3×#1141C2semidirect
3. #10620874C2semidirect
4. #10620852C2semidirect
5. C7×#833C2semidirect
6. #10620526C2semidirect
7. #10620493C2semidirect
25. C2×#1141D3semidirect
68. #833D7semidirect
148. C3×C7#472semidirect
194. C7#5160semidirect
209. C3#89521semidirect

Maximal subgroups

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

subgroupindex
#106214152
C2×C3×#11412
#106208742
#106208522
C7×#8332
#106205262
#106204932
#895213
#51607