Finite Group Databasebeta

← #13062124

Subgroups — #13062124

Order 1344.

280 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C7×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#10621503C211yesmaximal, characteristic, complemented
3#10621502C211yesmaximal, characteristic, complemented
4#10621501C212yesmaximal, complemented
5C2×C7⋊#226C212yesmaximal, complemented
6#90529—31nomaximal, Hall
7#66730—41nomaximal
8C7×#41⋊C3C2211yescharacteristic
9C2×C7×SL(2,𝔽3)C2212yes—
10C7⋊GL(2,𝔽3)C2211yescharacteristic
11C7⋊#226C2211yescharacteristic
12C7⋊#226C2212yes—
13C7×2−1+4D311yescharacteristic, complemented
14#6554—31no—
15#6514—32no—
16#6514—31no—
17#6513—32no—
18#6513—31no—
19#6511—31no—
20C2×#1144—31no—
21C2×#1144—32no—
22#6505—31no—
23#6505—32no—
24#5927—71nomaximal, Hall
25C3×C7×D4—41no—
26#4268—42no—
27C2×Dic21—42no—
28C4×D21—41no—
29Dic42—41no—
30C7×SL(2,𝔽3)C2311yescharacteristic
31C7×#41—31no—
32C7×#41—32no—
33C7×#41C2×D311yescharacteristic
34C2×C7×Q8—31no—
35C2×C7×Q8C2×D312yes—
36Q8×D7—61no—
37#1159—62no—
38#1159—61no—
39#1157—32no—
40#1157—31no—
41C2×Dic14—32no—
42C2×Dic14—31no—
43#1144—32no—
44#1144—32no—
45#1144—32no—
46#1144—32no—
47#1144—31no—
48#1143—32no—
49#1143—31no—
50#1142—32no—
51#1142—31no—
52#1141—31no—
53#1136—32no—
54#1136—31no—
55C2×C7⋊C8—32no—
56C2×C7⋊C8—31no—
572−1+4⋊C3D711yescharacteristic, complemented
58#985—71no—
59#984—71no—
60#983—72no—
61C2×#226—72no—
62C22×C3×C7—42no—
63C2×D21—41no—
64C3×C4×C7—41no—
65Dic21—42no—
66Dic21—41no—
67#578—211noSylow
68C7×Q8—32no—
69C7×Q8—31no—
70C7×Q8C22×D311yescharacteristic
71C7×D4—31no—
72C7×D4—32no—
73C7×D4S411yescharacteristic
74C2×C4×C7—32no—
75C2×C4×C7—31no—
76C2×C4×C7—61no—
77C22⋊D7—61no—
78C22⋊D7—62no—
79C2×Dic7—61no—
80C2×Dic7—62no—
81D28—31no—
82C4×D7—62no—
83C4×D7—61no—
84Dic14—31no—
85Dic14—32no—
86Dic14—62no—
87Dic14—61no—
88C7⋊C8—31no—
89C7⋊C8—32no—
90C7⋊C8—31no—
91#237—281no—
92#41⋊C3C2×D711yescharacteristic
93C2×SL(2,𝔽3)C2×D712yes—
94GL(2,𝔽3)—71no—
95#226—71no—
96#226—72no—
97C2×C3×C7—41no—
98C2×C3×C7—82no—
99D21—81no—
1002−1+4—211no—
1012−1+4D2111yescharacteristic, complemented
102#137—211no—
103#137—212no—
104#137—211no—
105#137—212no—
106#135—212no—
107#135—211no—
108C2×Q16—211no—
109C2×Q16—212no—
110#131—211no—
111C22×C7—31no—
112C22×C7C2×S412yes—
113C2×D7—61no—
114C4×C7—31no—
115C4×C7—32no—
116C4×C7C2×S411yescharacteristic
117Dic7—61no—
118Dic7—62no—
119C3×D4—41no—
120#67—282no—
121C2×Dic3—282no—
122C4×D3—281no—
123Dic6—281no—
124SL(2,𝔽3)C22×D711yescharacteristic
125C3×C7—41noHall
126#41—31no—
127#41—32no—
128#41—212no—
129#41—211no—
130#41—422no—
131#41—421no—
132#41C2×D2111yescharacteristic
133C2×Q8—31no—
134C2×Q8—212no—
135C2×Q8—211no—
136C2×Q8—421no—
137C2×Q8C2×D2112yes—
138Q16—212no—
139Q16—212no—
140Q16—212no—
141Q16—211no—
142Q16—212no—
143SD16—212no—
144SD16—211no—
145SD16—212no—
146SD16—211no—
147D8—211no—
148M16—212no—
149M16—211no—
150C2×C8—212no—
151C2×C8—211no—
152C2×C7—22no—
153C2×C7—61no—
154C2×C7C22×S411yescharacteristic
155D7—121no—
156C22×C3—42no—
157C2×D3—281no—
158C3×C4—41no—
159Dic3—282no—
160Dic3—281no—
161Q8—32no—
162Q8—31no—
163Q8—212no—
164Q8—211no—
165Q8—422no—
166Q8—421no—
167Q8C22×D2111yescharacteristic
168D4—31no—
169D4—32no—
170D4—211no—
171D4—421no—
172D4—422no—
173D4C7⋊S411yescharacteristic
174C2×C4—32no—
175C2×C4—31no—
176C2×C4—61no—
177C2×C4—422no—
178C2×C4—421no—
179C2×C4—422no—
180C2×C4—421no—
181C8—211no—
182C8—212no—
183C8—211no—
184C7#592711yescharacteristic, complemented, Sylow
185C2×C3—41no—
186C2×C3—82no—
187D3—561no—
188C22—31no—
189C22—421no—
190C22C2×C7⋊S412yes—
191C4—31no—
192C4—32no—
193C4—422no—
194C4—421no—
195C4C2×C7⋊S411yescharacteristic
196C3—41noSylow
197C2—22no—
198C2—61no—
199C2—841no—
200C2C22×C7⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C22×C7⋊S4
commutator, cosocle, hypocenterC7×SL(2,𝔽3)C23
FittingC7×2−1+4D3
socleC2×C7C22×S4
hypercenterD4C7⋊S4
LCSG ▷ C7×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C7×2−1+4⋊C3C2semidirect
2. #10621503C2semidirect
3. #10621502C2semidirect
4. #10621501C2semidirect
5. C2×C7⋊#226C2semidirect
13. C7×2−1+4D3semidirect
57. 2−1+4⋊C3D7semidirect
101. 2−1+4D21semidirect
184. C7#5927semidirect

Maximal subgroups

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

subgroupindex
C7×2−1+4⋊C32
#106215032
#106215022
#106215012
C2×C7⋊#2262
#905293
#667304
#59277