Finite Group Databasebeta

← #13056693

Subgroups — #13056693

Order 1344.

260 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C3×C7×#124C211yesmaximal, characteristic, complemented
2C2×#66635C211yesmaximal, characteristic, complemented
3C2×#66634C211yesmaximal, characteristic, complemented
4C2×#66633C211yesmaximal, characteristic, complemented
5C2×#66632C211yesmaximal, characteristic
6#10621358C211yesmaximal, characteristic, complemented
7C4×C3⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
8#89831—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C3×C7×D4C2211yescharacteristic
11C2×D84C2211yescharacteristic
12C2×Dic42C2211yescharacteristic
13C3×C7×C22⋊C4—22no—
14C3×C42×C7C2211yescharacteristic
15#66635—22no—
16#66634—22no—
17#66633—22no—
18#66632—22no—
19#66631—22no—
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22C7×#124D311yescharacteristic, complemented
23C2×#1144—31no—
24C2×#1143—31no—
25C2×#1142—31no—
26C2×#1141—31no—
27#6438—31no—
28C4×C7⋊C8—31no—
29#5059—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D21—21no—
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C3×C7×D4—21no—
35C3×C7×D4—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7D412yes—
38C2×C3×C4×C7C2311yescharacteristic
39C2×Dic21—21no—
40D84—21no—
41D84—21no—
42Dic42—21no—
43Dic42—21no—
44C3⋊(C7⋊C8)D412yes—
45C3⋊(C7⋊C8)D412yes—
46C2×C7×Q8C2×D311yescharacteristic
47C2×C7×D4C2×D311yescharacteristic
48C2×D28—31no—
49C2×Dic14—31no—
50C7×C22⋊C4—22no—
51C42×C7C2×D311yescharacteristic
52#1144—62no—
53#1143—62no—
54#1142—62no—
55#1141—62no—
56(C2×D7)⋊C4—62no—
57C2×C7⋊C8—31no—
58C2×C7⋊C8—31no—
59C3×#124D711yescharacteristic, complemented
60C2×#216—71no—
61C2×#215—71no—
62C2×#214—71no—
63C2×#213—71no—
64#875—71no—
65C4×C3⋊C8—71no—
66C22×C3×C7—41no—
67C22×C3×C7—42no—
68C22×C3×C7C2×D411yescharacteristic
69C2×D21—41no—
70C2×D21—42no—
71C3×C4×C7—22no—
72C3×C4×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7C2×D411yescharacteristic
75Dic21—41no—
76#495—211noSylow
77C23×C7—21no—
78C22×D7—61no—
79C7×Q8—21no—
80C7×Q8—21no—
81C7×D4—21no—
82C7×D4—21no—
83C2×C4×C7—21no—
84C2×C4×C7#6712yes—
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—61no—
87D28—61no—
88D28—61no—
89Dic14—61no—
90Dic14—61no—
91C7⋊C8—32no—
92C7⋊C8—32no—
93C2×C3×Q8C2×D711yescharacteristic
94C2×C3×D4C2×D711yescharacteristic
95C2×D12—71no—
96C2×Dic6—71no—
97C3×C22⋊C4—22no—
98C3×C42C2×D711yescharacteristic
99#216—142no—
100#215—142no—
101#214—142no—
102#213—142no—
103(C2×D3)⋊C4—142no—
104C2×C3⋊C8—71no—
105C2×C3⋊C8—71no—
106C2×C3×C7—81no—
107C2×C3×C7#12711yescharacteristic
108C2×C3×C7#13512yes—
109D21—81no—
110C2×Q16—211no—
111C2×SD16—211no—
112C2×SD16—211no—
113C2×D8—211no—
114#124—211no—
115#124D2111yescharacteristic, complemented
116C4×C8—211no—
117C22×C7—41no—
118C22×C7—42no—
119C22×C7C2×#6711yescharacteristic
120C2×D7—121no—
121C2×D7—122no—
122C4×C7—22no—
123C4×C7—41no—
124C4×C7D3×D411yescharacteristic
125C4×C7D3×D411yescharacteristic
126Dic7—121no—
127C23×C3—21no—
128C22×D3—141no—
129C3×Q8—21no—
130C3×Q8—21no—
131C3×D4—21no—
132C3×D4—21no—
133C2×C3×C4—21no—
134C2×C3×C4C22⋊D712yes—
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—141no—
137D12—141no—
138D12—141no—
139Dic6—141no—
140Dic6—141no—
141C3⋊C8—72no—
142C3⋊C8—72no—
143C3×C7#49511yescharacteristic, complemented, Hall
144C2×Q8—211no—
145C2×Q8C2×D2111yescharacteristic
146C2×D4—211no—
147C2×D4C2×D2111yescharacteristic
148Q16—422no—
149SD16—422no—
150SD16—422no—
151D8—422no—
152C2×C8—211no—
153C2×C8—211no—
154C22⋊C4—22no—
155C22⋊C4—422no—
156C42C2×D2111yescharacteristic
157C2×C7—81no—
158C2×C7#94011yescharacteristic
159C2×C7#95012yes—
160D7—241no—
161C22×C3—41no—
162C22×C3—42no—
163C22×C3C2×C22⋊D711yescharacteristic
164C2×D3—281no—
165C2×D3—282no—
166C3×C4—22no—
167C3×C4—41no—
168C3×C4D4×D711yescharacteristic
169C3×C4D4×D711yescharacteristic
170Dic3—281no—
171C23—21no—
172C23—421no—
173Q8—21no—
174Q8—21no—
175Q8—421no—
176Q8—421no—
177D4—21no—
178D4—21no—
179D4—421no—
180D4—421no—
181C2×C4—21no—
182C2×C4—421no—
183C2×C4#426812yes—
184C2×C4C22×D2111yescharacteristic
185C8—212no—
186C8—212no—
187C7#505911yescharacteristic, complemented, Sylow
188C2×C3—81no—
189C2×C3#650311yescharacteristic
190C2×C3#651312yes—
191D3—561no—
192C22—41no—
193C22—42no—
194C22—841no—
195C22—842no—
196C22C2×#426811yescharacteristic
197C4—22no—
198C4—41no—
199C4—841no—
200C4D4×D2111yescharacteristic
201C4D4×D2111yescharacteristic
202C3#8983111yescharacteristic, complemented, Sylow
203C2—81no—
204C2—1681no—
205C2#1062142311yescharacteristic
206C2#1062143312yes—
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#124C2
socleC22×C3×C7C2×D4
prime radicalC2×#66632C2
hypercenter#124D21
hypocenterC3×C7#495
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×C7×#124C2semidirect
2. C2×#66635C2semidirect
3. C2×#66634C2semidirect
4. C2×#66633C2semidirect
6. #10621358C2semidirect
7. C4×C3⋊(C7⋊C8)C2semidirect
22. C7×#124D3semidirect
59. C3×#124D7semidirect
115. #124D21semidirect
143. C3×C7#495semidirect
187. C7#5059semidirect
202. C3#89831semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1242
C2×#666352
C2×#666342
C2×#666332
C2×#666322
#106213582
C4×C3⋊(C7⋊C8)2
#898313
#50597