Finite Group Databasebeta

← #13054789

Subgroups — #13054789

Order 1344.

304 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 58 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#10621565C211yesmaximal, characteristic
2#10621425C211yesmaximal, characteristic, complemented
3C7×#947C211yesmaximal, characteristic, complemented
4C3×#6505C211yesmaximal, characteristic, complemented
5#10620515C211yesmaximal, characteristic, complemented
6#10620512C211yesmaximal, characteristic, complemented
7#10620494C211yesmaximal, characteristic, complemented
8#89833—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic, complemented
10#66728C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C3×#1157C2211yescharacteristic
13D3×C22⋊D7—22no—
14#66681—21no—
15D7×D12—21no—
16#66676—21no—
17#66672—21no—
18#66635—21no—
19#66634—21no—
20#66627C2211yescharacteristic, complemented
21C7×(C2×D3)⋊C4—22no—
22C4×C7×Dic3C2211yescharacteristic, complemented
23C3×#1144—21no—
24C3×#1143—21no—
25C3×#1136C2211yescharacteristic, complemented
26#6548—31no—
27C7×#124—31no—
28#6505—31no—
29#6505D311yescharacteristic, complemented
30#6407—31no—
31#6379—31no—
32#6379—31no—
33#5202—71nomaximal, Hall
34C22×D3×C7D412yes—
35C2×D3×D7—22no—
36C3×C7×Q8—21no—
37C3×C7×Q8—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7C2311yescharacteristic
40#4268—21no—
41D84D411yescharacteristic
42C4×D21—21no—
43Dic42D411yescharacteristic
44C2×C7×Dic3—21no—
45C7×D12—21no—
46C7×D12—21no—
47C3×C22⋊D7—21no—
48C3×D28D411yescharacteristic
49C3×C4×D7—21no—
50C3×Dic14D411yescharacteristic
51#4247—22no—
52#4245—22no—
53D3×Dic7—22no—
54C3⋊(C7⋊C8)—21no—
55C3×C7⋊C8—21no—
56C2×C7×Q8C2×D311yescharacteristic, complemented
57C2×C7×D4—31no—
58C2×C22⋊D7—62no—
59#1159—61no—
60#1159—61no—
61D4×D7—61no—
62D4×D7—61no—
63#1157—31no—
64#1157C2×D311yescharacteristic
65C7×C22⋊C4—62no—
66C42×C7—31no—
67#1144—21no—
68#1144—61no—
69#1143—21no—
70#1143—61no—
71#1136—31no—
72#1136C2×D311yescharacteristic, complemented
73#1009—71no—
74C3×#137—71no—
75#947D711yescharacteristic, complemented
76#942—71no—
77#837—71no—
78#825—71no—
79#823—71no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—21no—
82C2×D3×C7—22no—
83C2×D3×C7—42no—
84C2×C3×D7—21no—
85D3×D7—82no—
86C3×C4×C7—41no—
87C3×C4×C7C2×D411yescharacteristic
88C3×C4×C7C2×D411yescharacteristic
89Dic21—21no—
90C3×Dic7—21no—
91C7×Dic3—41no—
92#455—211noSylow
93C23×C7—32no—
94C22×D7—62no—
95C7×Q8—21no—
96C7×Q8—21no—
97C7×D4—61no—
98C7×D4—61no—
99C2×C4×C7—21no—
100C2×C4×C7—61no—
101C2×C4×C7C22×D311yescharacteristic
102C22⋊D7—21no—
103C22⋊D7—62no—
104C22⋊D7—62no—
105C22⋊D7—61no—
106C2×Dic7—62no—
107D28—31no—
108D28#6711yescharacteristic
109C4×D7—21no—
110C4×D7—61no—
111Dic14—31no—
112Dic14#6711yescharacteristic
113C7⋊C8—21no—
114C7⋊C8—61no—
115C3×#41—71no—
116C2×C3×Q8C2×D711yescharacteristic, complemented
117#239—141no—
118D3×D4—141no—
119D3×D4—142no—
120#235—71no—
121#235—141no—
122C2×D12—141no—
123C2×D12C2×D711yescharacteristic
124C3×Q16—141no—
125C3×SD16—141no—
126C3×M16—71no—
127#216—141no—
128#215—141no—
129(C2×D3)⋊C4—22no—
130C4×Dic3C2×D711yescharacteristic, complemented
131#208—71no—
132C2×C3×C7—21no—
133C2×C3×C7#12011yescharacteristic
134D21—41no—
135C3×D7—41no—
136D3×C7—42no—
137H(5,𝔽2)—211no—
138#137—71no—
139#137—211no—
140#124—31no—
141#104—211no—
142#104—211no—
143#100—211no—
144C22×C7—62no—
145C22×C7—122no—
146C22×C7D3×D411yescharacteristic
147C2×D7—21no—
148C2×D7—61no—
149C2×D7—242no—
150C4×C7—41no—
151C4×C7—121no—
152C4×C7D3×D411yescharacteristic
153C4×C7C2×#6711yescharacteristic
154Dic7—21no—
155Dic7—61no—
156C22×D3—142no—
157C22×D3C22⋊D712yes—
158C3×Q8—21no—
159C3×Q8—21no—
160C3×Q8—71no—
161C3×D4—71no—
162C3×D4—141no—
163C2×C3×C4—21no—
164C2×C3×C4—141no—
165C2×C3×C4C22×D711yescharacteristic
166#67—142no—
167#67—141no—
168C2×Dic3—21no—
169D12—21no—
170D12—21no—
171D12—71no—
172D12—142no—
173C4×D3—142no—
174C4×D3—141no—
175Dic6—71no—
176C3×C8—141no—
177C3⋊C8—141no—
178C3×C7#45511yescharacteristic, complemented, Hall
179#41—71no—
180#41—211no—
181#41—421no—
182#41—421no—
183C2×Q8D3×D711yescharacteristic, complemented
184C2×D4—31no—
185C2×D4—421no—
186C2×D4—421no—
187C2×D4—422no—
188Q16—141no—
189Q16—421no—
190SD16—141no—
191SD16—421no—
192M16—71no—
193M16—211no—
194C22⋊C4—62no—
195C42—31no—
196C2×C7—21no—
197C2×C7—122no—
198C2×C7#93711yescharacteristic
199D7—41no—
200D7—121no—
201C22×C3—141no—
202C22×C3C2×C22⋊D711yescharacteristic
203C2×D3—22no—
204C2×D3—42no—
205C2×D3—141no—
206C2×D3—562no—
207C3×C4—41no—
208C3×C4—141no—
209C3×C4D4×D711yescharacteristic
210C3×C4D4×D711yescharacteristic
211Dic3—41no—
212Dic3—141no—
213C23—32no—
214C23—422no—
215Q8—21no—
216Q8—21no—
217Q8—71no—
218Q8—211no—
219D4—61no—
220D4—61no—
221D4—71no—
222D4—141no—
223D4—211no—
224D4—422no—
225D4—422no—
226D4—421no—
227C2×C4—21no—
228C2×C4—61no—
229C2×C4—141no—
230C2×C4—422no—
231C2×C4—421no—
232C2×C4C2×D3×D711yescharacteristic
233C8—141no—
234C8—421no—
235C7#520211yescharacteristic, complemented, Sylow
236C2×C3—21no—
237C2×C3—281no—
238C2×C3#650011yescharacteristic
239D3—42no—
240D3—281no—
241C22—62no—
242C22—122no—
243C22—141no—
244C22—421no—
245C22—1682no—
246C22D3×C22⋊D711yescharacteristic
247C4—41no—
248C4—121no—
249C4—141no—
250C4—421no—
251C4D7×#6711yescharacteristic
252C4#6669411yescharacteristic
253C3#8983311yescharacteristic, complemented, Sylow
254C2—21no—
255C2—122no—
256C2—281no—
257C2—841no—
258C2#1062113911yescharacteristic
subgroupG/H
centerC2#10621139
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC2×C3×C7#120
prime radical#10621565C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #10621425C2semidirect
3. C7×#947C2semidirect
4. C3×#6505C2semidirect
5. #10620515C2semidirect
6. #10620512C2semidirect
7. #10620494C2semidirect
9. C2×C3×C7×Q8C22semidirect
20. #66627C22semidirect
22. C4×C7×Dic3C22semidirect
25. C3×#1136C22semidirect
29. #6505D3semidirect
56. C2×C7×Q8C2×D3semidirect
72. #1136C2×D3semidirect
75. #947D7semidirect
116. C2×C3×Q8C2×D7semidirect
130. C4×Dic3C2×D7semidirect
178. C3×C7#455semidirect
183. C2×Q8D3×D7semidirect
235. C7#5202semidirect
253. C3#89833semidirect

Maximal subgroups

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

subgroupindex
#106215652
#106214252
C7×#9472
C3×#65052
#106205152
#106205122
#106204942
#898333
#52027