Finite Group Databasebeta

← #13054813

Subgroups — #13054813

Order 1344.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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#10621433C211yesmaximal, characteristic
2C7×#948C211yesmaximal, characteristic, complemented
3#10620853C211yesmaximal, characteristic, complemented
4C3×#6411C211yesmaximal, characteristic, complemented
5#10620527C211yesmaximal, characteristic, complemented
6#10620516C211yesmaximal, characteristic, complemented
7C3⋊C8×Dic7C211yesmaximal, characteristic, complemented
8#89916—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10#66728C2211yescharacteristic
11#66635C411yescharacteristic, complemented
12#66634C411yescharacteristic, complemented
13#66633C411yescharacteristic, complemented
14#66632C411yescharacteristic, complemented
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C7×#208—21no—
17C7×#208C2211yescharacteristic
18C2×C7×C3⋊C8—21no—
19C2×C7×C3⋊C8C2211yescharacteristic
20C3×C4×Dic7C2211yescharacteristic
21C3×#1136C2211yescharacteristic
22#66559—21no—
23D21⋊C8—21no—
24C7×#131—31no—
25#6513—31no—
26#6464—31no—
27#6411D311yescharacteristic, complemented
28#6399—31no—
29#6396—31no—
30C8×Dic7—31no—
31#4824—71nomaximal, Hall
32C3×C7×Q8C2×C411yescharacteristic
33C3×C7×D4—21no—
34C3×C7×D4C2×C411yescharacteristic
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37#4268—21no—
38D84C2×C411yescharacteristic
39C4×D21—21no—
40Dic42C2×C411yescharacteristic
41C2×C3×Dic7—21no—
42C3⋊(C7⋊C8)C2×C411yescharacteristic
43C3⋊(C7⋊C8)C2×C411yescharacteristic
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C7×C3⋊C8D411yescharacteristic
47C7×C3⋊C8D411yescharacteristic
48C7×#41C2×D311yescharacteristic
49#1157—31no—
50C7×M16—31no—
51C7×M16—61no—
52C2×C7×C8—31no—
53C2×C7×C8—61no—
54#1144—31no—
55#1143—31no—
56#1142—31no—
57#1141—31no—
58C4×Dic7C2×D311yescharacteristic
59#1136C2×D311yescharacteristic
60C2×C7⋊C8—31no—
61#1131—61no—
62C8×D7—61no—
63#950—71no—
64#948D711yescharacteristic, complemented
65#907—71no—
66C3×#104—71no—
67#822—71no—
68#805—71no—
69C4×C3⋊C8—71no—
70C22×C3×C7—21no—
71C22×C3×C7#4111yescharacteristic
72C2×D21—21no—
73C3×C4×C7—21no—
74C3×C4×C7C22×C411yescharacteristic
75C3×C4×C7C2×D411yescharacteristic
76Dic21—21no—
77C3×Dic7—22no—
78#443—211noSylow
79C7×Q8C4×D311yescharacteristic
80C7×D4—21no—
81C7×D4C4×D311yescharacteristic
82C2×C4×C7—21no—
83C2×C4×C7C22×D311yescharacteristic
84C22⋊D7—61no—
85C2×Dic7—21no—
86D28—31no—
87C4×D7—61no—
88Dic14—31no—
89C7×C8—31no—
90C7×C8—31no—
91C7×C8—61no—
92C7⋊C8—21no—
93C7⋊C8—31no—
94C7⋊C8—31no—
95C3×#41C2×D711yescharacteristic
96#235—71no—
97C3×M16—71no—
98C3×C42—71no—
99#216—71no—
100#215—71no—
101#214—71no—
102#213—71no—
103#208—21no—
104#208C2×D711yescharacteristic
105C2×C3⋊C8—21no—
106C2×C3⋊C8—71no—
107C2×C3⋊C8C2×D711yescharacteristic
108ΓL(1,𝔽25)—141no—
109D3×C8—141no—
110C2×C3×C7—21no—
111C2×C3×C7—41no—
112C2×C3×C7C4×D411yescharacteristic
113D21—41no—
114#135—211no—
115#131—31no—
116#131—211no—
117#108—211no—
118#104—71no—
119#104—211no—
120C4×C8—211no—
121C22×C7—21no—
122C22×C7#23711yescharacteristic
123C2×D7—61no—
124C4×C7—21no—
125C4×C7C2×C4×D311yescharacteristic
126C4×C7D3×D411yescharacteristic
127Dic7—22no—
128Dic7—61no—
129C3×Q8C4×D711yescharacteristic
130C3×D4—21no—
131C3×D4C4×D711yescharacteristic
132C2×C3×C4—21no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135#67—141no—
136D12—71no—
137C4×D3—141no—
138Dic6—71no—
139C3×C8—141no—
140C3⋊C8—21no—
141C3⋊C8—71no—
142C3⋊C8—71no—
143C3⋊C8C22⋊D711yescharacteristic
144C3⋊C8C22⋊D711yescharacteristic
145C3×C7#44311yescharacteristic, complemented, Hall
146#41—211no—
147#41D3×D711yescharacteristic
148Q16—211no—
149SD16—211no—
150SD16—211no—
151D8—211no—
152M16—31no—
153M16—61no—
154M16—71no—
155M16—421no—
156C2×C8—31no—
157C2×C8—61no—
158C2×C8—211no—
159C2×C8—421no—
160C42—71no—
161C2×C7—21no—
162C2×C7—41no—
163C2×C7#67⋊C411yescharacteristic
164D7—121no—
165C22×C3—21no—
166C22×C3#115711yescharacteristic
167C2×D3—141no—
168C3×C4—21no—
169C3×C4—142no—
170C3×C4C2×C4×D711yescharacteristic
171C3×C4C2×C22⋊D711yescharacteristic
172Dic3—141no—
173Q8—211no—
174Q8D21⋊C411yescharacteristic
175D4—21no—
176D4—211no—
177D4—421no—
178D4D21⋊C411yescharacteristic
179C2×C4—21no—
180C2×C4—141no—
181C2×C4—421no—
182C2×C4C2×D3×D711yescharacteristic
183C8—31no—
184C8—31no—
185C8—61no—
186C8—141no—
187C8—211no—
188C8—211no—
189C7#482411yescharacteristic, complemented, Sylow
190C2×C3—21no—
191C2×C3—41no—
192C2×C3C4×C22⋊D711yescharacteristic
193D3—281no—
194C22—21no—
195C22—421no—
196C22#6668311yescharacteristic
197C4—21no—
198C4—142no—
199C4—421no—
200C4C2×D21⋊C411yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8991611yescharacteristic, complemented, Sylow
203C2—21no—
204C2—41no—
205C2—841no—
206C2#1062111811yescharacteristic
subgroupG/H
centerC4D3×C22⋊D7
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7C4×D4
prime radical#10621433C2
cosocleC2×C3×C4×C7C23
hypercenter#41D3×D7
hypocenterC3×C7#443
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C7×#948C2semidirect
3. #10620853C2semidirect
4. C3×#6411C2semidirect
5. #10620527C2semidirect
6. #10620516C2semidirect
7. C3⋊C8×Dic7C2semidirect
11. #66635C4semidirect
12. #66634C4semidirect
13. #66633C4semidirect
14. #66632C4semidirect
27. #6411D3semidirect
64. #948D7semidirect
145. C3×C7#443semidirect
189. C7#4824semidirect
202. C3#89916semidirect

Maximal subgroups

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

subgroupindex
#106214332
C7×#9482
#106208532
C3×#64112
#106205272
#106205162
C3⋊C8×Dic72
#899163
#48247