Finite Group Databasebeta

← #13056688

Subgroups — #13056688

Order 1344.

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

normal subgroups only

This group has 304 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
1C2×#66728C211yesmaximal, characteristic
2C3×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#66635C211yesmaximal, characteristic, complemented
4C2×#66634C211yesmaximal, characteristic, complemented
5#10620669C211yesmaximal, characteristic, complemented
6#10620649C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89819—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C2×#4268—21no—
12#66728—22no—
13#66728—22no—
14C2×D84C2211yescharacteristic
15C2×C4×D21—21no—
16C2×Dic42C2211yescharacteristic
17C3×C7×C4⋊C4—21no—
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C7×C22⋊C4—21no—
20#66635—22no—
21#66634—22no—
22C2×C3⋊(C7⋊C8)C2211yescharacteristic
23C2×C3⋊(C7⋊C8)C2211yescharacteristic
24C2×#1157—31no—
25C7×C22⋊Q8D311yescharacteristic, complemented
26C2×#1144—31no—
27C2×#1143—31no—
28C22⋊(C7⋊C8)—31no—
29#6384—31no—
30#6383—31no—
31#5047—71nomaximal, Hall
32C23×C3×C7D411yescharacteristic
33C22×D21—21no—
34C3×C7×Q8—21no—
35C3×C7×Q8—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7D411yescharacteristic
41C2×C3×C4×C7C2311yescharacteristic
42#4268—24no—
43C2×Dic21—21no—
44D84—21no—
45D84D412yes—
46C4×D21—24no—
47Dic42—21no—
48Dic42D412yes—
49C3⋊(C7⋊C8)—21no—
50C3⋊(C7⋊C8)—21no—
51C2×C7×Q8C2×D311yescharacteristic
52C22×C4×C7C2×D311yescharacteristic
53C2×C22⋊D7—61no—
54#1157—62no—
55#1157—62no—
56C2×D28—31no—
57C2×C4×D7—61no—
58C2×Dic14—31no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4C2×D311yescharacteristic
61C7×C22⋊C4—21no—
62#1144—62no—
63#1143—62no—
64C2×C7⋊C8—31no—
65C2×C7⋊C8—31no—
66C2×#235—71no—
67C3×C22⋊Q8D711yescharacteristic, complemented
68C2×#216—71no—
69C2×#215—71no—
70#830—71no—
71#810—71no—
72#809—71no—
73C22×C3×C7—21no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7C2×D411yescharacteristic
77C2×D21—22no—
78C2×D21—41no—
79C2×D21—41no—
80C3×C4×C7—21no—
81C3×C4×C7—41no—
82C3×C4×C7—41no—
83C3×C4×C7C2×D411yescharacteristic
84C3×C4×C7C2×D411yescharacteristic
85Dic21—22no—
86#452—211noSylow
87C23×C7#6711yescharacteristic
88C22×D7—61no—
89C7×Q8—21no—
90C7×Q8—21no—
91C2×C4×C7—21no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7—21no—
95C2×C4×C7#6711yescharacteristic
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—64no—
98C2×Dic7—61no—
99D28—32no—
100D28—61no—
101C4×D7—64no—
102Dic14—32no—
103Dic14—61no—
104C7⋊C8—61no—
105C7⋊C8—61no—
106C2×C3×Q8C2×D711yescharacteristic
107C22×C3×C4C2×D711yescharacteristic
108C2×#67—141no—
109#235—142no—
110#235—142no—
111C2×D12—71no—
112C2×C4×D3—141no—
113C2×Dic6—71no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4C2×D711yescharacteristic
116C3×C22⋊C4—21no—
117#216—142no—
118#215—142no—
119C2×C3⋊C8—71no—
120C2×C3⋊C8—71no—
121C2×C3×C7—41no—
122C2×C3×C7#12011yescharacteristic
123C2×C3×C7#13511yescharacteristic
124C2×C3×C7#13711yescharacteristic
125D21—42no—
126C2×#41—211no—
127C2×Q16—211no—
128C2×SD16—211no—
129C22⋊Q8D2111yescharacteristic, complemented
130#103—211no—
131#102—211no—
132C22⋊C8—211no—
133C22×C7—21no—
134C22×C7—21no—
135C22×C7—21no—
136C22×C7C2×#6711yescharacteristic
137C2×D7—62no—
138C2×D7—121no—
139C2×D7—121no—
140C4×C7—21no—
141C4×C7—41no—
142C4×C7—41no—
143C4×C7D3×D411yescharacteristic
144C4×C7D3×D411yescharacteristic
145Dic7—62no—
146C23×C3C22⋊D711yescharacteristic
147C22×D3—141no—
148C3×Q8—21no—
149C3×Q8—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4C22⋊D711yescharacteristic
155C2×C3×C4C22×D711yescharacteristic
156#67—144no—
157C2×Dic3—141no—
158D12—72no—
159D12—141no—
160C4×D3—144no—
161Dic6—72no—
162Dic6—141no—
163C3⋊C8—141no—
164C3⋊C8—141no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—422no—
167#41—422no—
168C2×Q8—211no—
169C2×Q8C2×D2111yescharacteristic
170C2×D4—211no—
171C2×D4—421no—
172C22×C4—421no—
173C22×C4C2×D2111yescharacteristic
174Q16—422no—
175SD16—422no—
176C2×C8—211no—
177C2×C8—211no—
178C4⋊C4—21no—
179C4⋊C4C2×D2111yescharacteristic
180C22⋊C4—21no—
181C2×C7—41no—
182C2×C7#93711yescharacteristic
183C2×C7#94211yescharacteristic
184C2×C7#95011yescharacteristic
185D7—122no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3C2×C22⋊D711yescharacteristic
190C2×D3—142no—
191C2×D3—281no—
192C2×D3—281no—
193C3×C4—21no—
194C3×C4—41no—
195C3×C4—41no—
196C3×C4D4×D711yescharacteristic
197C3×C4D4×D711yescharacteristic
198Dic3—142no—
199C23—421no—
200C23#426811yescharacteristic
201Q8—21no—
202Q8—21no—
203Q8—212no—
204Q8—421no—
205D4—212no—
206D4—424no—
207D4—421no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—424no—
213C2×C4—421no—
214C2×C4#426811yescharacteristic
215C2×C4C22×D2111yescharacteristic
216C8—421no—
217C8—421no—
218C7#504711yescharacteristic, complemented, Sylow
219C2×C3—41no—
220C2×C3#650011yescharacteristic
221C2×C3#650511yescharacteristic
222C2×C3#651311yescharacteristic
223D3—282no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—422no—
228C22—841no—
229C22—841no—
230C22C2×#426811yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—422no—
235C4D4×D2111yescharacteristic
236C4D4×D2111yescharacteristic
237C3#8981911yescharacteristic, complemented, Sylow
238C2—41no—
239C2—842no—
240C2#1062142011yescharacteristic
241C2#1062142511yescharacteristic
242C2#1062143311yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radicalC2×#66728C2
hypercenterC22⋊Q8D21
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C3×C7×C22⋊Q8C2semidirect
3. C2×#66635C2semidirect
4. C2×#66634C2semidirect
5. #10620669C2semidirect
6. #10620649C2semidirect
7. #10620648C2semidirect
25. C7×C22⋊Q8D3semidirect
67. C3×C22⋊Q8D7semidirect
129. C22⋊Q8D21semidirect
165. C3×C7#452semidirect
218. C7#5047semidirect
237. C3#89819semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
C3×C7×C22⋊Q82
C2×#666352
C2×#666342
#106206692
#106206492
#106206482
#898193
#50477