Finite Group Databasebeta

← #13053584

Subgroups — #13053584

Order 1344.

212 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 64 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#10621390C211yesmaximal, characteristic, complemented
2#10620873C211yesmaximal, characteristic
3#10620846C211yesmaximal, characteristic, complemented
4C7×#822C211yesmaximal, characteristic, complemented
5C3×#6399C211yesmaximal, characteristic, complemented
6#10620516C211yesmaximal, characteristic, complemented
7C3⋊C8×Dic7C211yesmaximal, characteristic, complemented
8#89667—31nomaximal, Hall
9#66728C2211yescharacteristic
10C3×#1157C2211yescharacteristic
11C3×C7×M16C2211yescharacteristic
12C2×C3⋊(C7⋊C8)C2211yescharacteristic
13#66622—21no—
14C8×D21—21no—
15C7×#208C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4×Dic7C2211yescharacteristic
18#66570C411yescharacteristic, complemented
19#66566C411yescharacteristic, complemented
20#66565C411yescharacteristic, complemented
21#66561C411yescharacteristic, complemented
22#66557—21no—
23D7×C3⋊C8—21no—
24#6470—31no—
25#6470—31no—
26#6467—31no—
27C7×#108—31no—
28#6399—31no—
29#6399D311yescharacteristic, complemented
30C8×Dic7—31no—
31#4824—71nomaximal, Hall
32C2×C3×C4×C7C2311yescharacteristic
33#4268—21no—
34D84C2×C411yescharacteristic
35C4×D21—21no—
36Dic42C2×C411yescharacteristic
37C3×C22⋊D7—21no—
38C2×C3×Dic7—21no—
39C3×D28C2×C411yescharacteristic
40C3×C4×D7—21no—
41C3×Dic14C2×C411yescharacteristic
42C3×C7×C8—21no—
43C3⋊(C7⋊C8)D411yescharacteristic
44C3⋊(C7⋊C8)D411yescharacteristic
45C7×C3⋊C8—21no—
46C7×C3⋊C8C2×C411yescharacteristic
47C7×C3⋊C8C2×C411yescharacteristic
48#1157—31no—
49#1157C2×D311yescharacteristic
50C7×M16—31no—
51C7×M16C2×D311yescharacteristic
52C2×C7×C8—31no—
53C4×Dic7C2×D311yescharacteristic
54C2×C7⋊C8—31no—
55Dic28—31no—
56D56—31no—
57#1132—31no—
58#1132—31no—
59#1131—61no—
60#1131—61no—
61C8×D7—61no—
62C8×D7—61no—
63#950—71no—
64#948—71no—
65#907—71no—
66C3×#104—71no—
67#822D711yescharacteristic, complemented
68#805—71no—
69C4×C3⋊C8—71no—
70C22×C3×C7#4111yescharacteristic
71C2×D21—21no—
72C2×C3×D7—21no—
73C3×C4×C7C22×C411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75Dic21—21no—
76C3×Dic7—22no—
77C3×Dic7—21no—
78#443—211noSylow
79C2×C4×C7C22×D311yescharacteristic
80C22⋊D7—21no—
81C22⋊D7—61no—
82C2×Dic7—21no—
83D28—31no—
84D28C4×D311yescharacteristic
85C4×D7—21no—
86C4×D7—61no—
87Dic14—31no—
88Dic14C4×D311yescharacteristic
89C7×C8—21no—
90C7×C8—31no—
91C7×C8—31no—
92C7×C8—61no—
93C7⋊C8—31no—
94C7⋊C8—31no—
95C3×#41—71no—
96#235—71no—
97C3×M16C2×D711yescharacteristic
98C3×C42—71no—
99#216—71no—
100#215—71no—
101#214—71no—
102#213—71no—
103#208—141no—
104#208C2×D711yescharacteristic
105C2×C3⋊C8—71no—
106C2×C3⋊C8—141no—
107C2×C3⋊C8C2×D711yescharacteristic
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7C4×D411yescharacteristic
112D21—41no—
113C3×D7—41no—
114#135—211no—
115#131—211no—
116#131—211no—
117#108—31no—
118#104—71no—
119#104—211no—
120C4×C8—211no—
121C22×C7#23711yescharacteristic
122C2×D7—21no—
123C2×D7—61no—
124C4×C7C2×C4×D311yescharacteristic
125C4×C7D3×D411yescharacteristic
126Dic7—22no—
127Dic7—21no—
128Dic7—61no—
129C3×Q8—71no—
130C3×D4—71no—
131C3×D4—141no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135#67—141no—
136D12—71no—
137C4×D3—141no—
138Dic6—71no—
139C3×C8—21no—
140C3⋊C8—21no—
141C3⋊C8—71no—
142C3⋊C8—71no—
143C3⋊C8C4×D711yescharacteristic
144C3⋊C8C4×D711yescharacteristic
145C3×C7#44311yescharacteristic, complemented, Hall
146#41—71no—
147#41—211no—
148Q16—211no—
149SD16—211no—
150SD16—211no—
151D8—211no—
152M16—31no—
153M16—421no—
154M16—421no—
155M16D3×D711yescharacteristic
156C2×C8—31no—
157C2×C8—211no—
158C2×C8—421no—
159C2×C8—421no—
160C42—71no—
161C2×C7—21no—
162C2×C7#67⋊C411yescharacteristic
163D7—41no—
164D7—121no—
165C22×C3—141no—
166C22×C3#116111yescharacteristic
167C2×D3—141no—
168C3×C4—142no—
169C3×C4—141no—
170C3×C4C2×C4×D711yescharacteristic
171C3×C4D4×D711yescharacteristic
172Dic3—141no—
173Q8—71no—
174Q8—211no—
175D4—71no—
176D4—141no—
177D4—211no—
178D4—421no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—421no—
182C2×C4C2×D3×D711yescharacteristic
183C8—21no—
184C8—31no—
185C8—31no—
186C8—61no—
187C8—211no—
188C8—211no—
189C7#482411yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—281no—
192C2×C3#645611yescharacteristic
193D3—281no—
194C22—141no—
195C22—421no—
196C22#6667011yescharacteristic
197C4—142no—
198C4—141no—
199C4—421no—
200C4C4×D3×D711yescharacteristic
201C4#6669411yescharacteristic
202C3#8966711yescharacteristic, complemented, Sylow
203C2—21no—
204C2—281no—
205C2—841no—
206C2#1062095811yescharacteristic
subgroupG/H
centerC4#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7C4×D4
prime radical#10620873C2
cosocleC2×C3×C4×C7C23
hypercenterM16D3×D7
hypocenterC3×C7#443
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10621390C2semidirect
3. #10620846C2semidirect
4. C7×#822C2semidirect
5. C3×#6399C2semidirect
6. #10620516C2semidirect
7. C3⋊C8×Dic7C2semidirect
18. #66570C4semidirect
19. #66566C4semidirect
20. #66565C4semidirect
21. #66561C4semidirect
29. #6399D3semidirect
67. #822D7semidirect
145. C3×C7#443semidirect
189. C7#4824semidirect
202. C3#89667semidirect

Maximal subgroups

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

subgroupindex
#106213902
#106208732
#106208462
C7×#8222
C3×#63992
#106205162
C3⋊C8×Dic72
#896673
#48247