Finite Group Databasebeta

← #13054748

Subgroups — #13054748

Order 1344.

240 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 64 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#10621427C211yesmaximal, characteristic, complemented
2C2×C7×#215C211yesmaximal, characteristic
3#10620997C211yesmaximal, characteristic
4C2×#66567C211yesmaximal, characteristic
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620518C211yesmaximal, characteristic, complemented
7#10620509C211yesmaximal, characteristic, complemented
8#89934—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×D3×Dic7—21no—
13#66629—21no—
14#66629—21no—
15C4×Dic21C2211yescharacteristic
16C7×#215—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21#66567—22no—
22C2×C7×SD16—31no—
23#6507—31no—
24D4×Dic7—31no—
25C2×#1142—31no—
26#6409D311yescharacteristic, complemented
27#6390—31no—
28Dic7⋊C8—31no—
29#4818—71nomaximal, Hall
30C22×D3×C7—21no—
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38Dic42—21no—
39Dic42—21no—
40C7×D12—21no—
41C7×D12D412yes—
42C2×C3×Dic7—21no—
43D3×Dic7—42no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×C7×D4—31no—
48C22×Dic7—61no—
49C2×Dic14—31no—
50C7×SD16—62no—
51C2×C7×C8—31no—
52C22⋊Dic7—61no—
53#1142—62no—
54C4⋊Dic7C2×D311yescharacteristic
55Dic7⋊C4—61no—
56Dic7⋊C4—61no—
57C4×Dic7—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59#944—71no—
60C2×#215D711yescharacteristic
61C2×#204—71no—
62#893—71no—
63C3×#103—71no—
64Dic3⋊C8—71no—
65#810—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D3×C7—22no—
68C2×D3×C7—41no—
69C2×D3×C7—41no—
70C3×C4×C7—41no—
71C3×C4×C7C2×D411yescharacteristic
72C3×C4×C7#4111yescharacteristic
73Dic21—21no—
74Dic21—41no—
75Dic21SD1612yescomplemented
76C3×Dic7—41no—
77#461—211noSylow
78C23×C7—61no—
79C7×Q8—21no—
80C7×Q8—21no—
81C7×D4—32no—
82C7×D4—61no—
83C2×C4×C7—21no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7—31no—
87C2×Dic7—31no—
88C2×Dic7—61no—
89C2×Dic7—122no—
90Dic14—61no—
91Dic14—61no—
92C7×C8—61no—
93C7⋊C8—21no—
94C2×C3×Q8C2×D711yescharacteristic
95C2×D12C2×D711yescharacteristic
96C2×C4×D3—141no—
97C2×Dic6—71no—
98C2×C3×C8—71no—
99C3×C4⋊C4—71no—
100#215—22no—
101(C2×D3)⋊C4—141no—
102Dic3⋊C4—141no—
103Dic3⋊C4—141no—
104C4×Dic3—71no—
105C2×C3⋊C8C2×D711yescharacteristic
106#204—142no—
107C2×C3×C7#12111yescharacteristic
108C2×C3×C7C2×SD1611yescharacteristic
109C2×C3×C7#13711yescharacteristic
110D3×C7—42no—
111C2×SD16—31no—
112C2×SD16—211no—
113C4⋊Q8—211no—
114C4×D4—211no—
115C4⋊C8—211no—
116#103—71no—
117#103—211no—
118C22×C7—62no—
119C22×C7—121no—
120C22×C7—121no—
121C22×C7D3×D411yescharacteristic
122C4×C7—41no—
123C4×C7#23511yescharacteristic
124C4×C7D3×D411yescharacteristic
125Dic7—32no—
126Dic7—41no—
127Dic7—61no—
128Dic7—121no—
129C22×D3—21no—
130C3×Q8—21no—
131C3×Q8—21no—
132C2×C3×C4—21no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C2×Dic3—71no—
136C2×Dic3—71no—
137C2×Dic3—141no—
138D12—21no—
139D12C22⋊D712yes—
140C4×D3—282no—
141Dic6—141no—
142Dic6—141no—
143C3×C8—141no—
144C3⋊C8—21no—
145C3×C7#46111yescharacteristic, complemented, Hall
146C2×Q8—211no—
147C2×Q8D3×D711yescharacteristic
148C2×D4—31no—
149C22×C4—421no—
150SD16—62no—
151SD16—422no—
152C2×C8—31no—
153C2×C8—71no—
154C4⋊C4—71no—
155C4⋊C4—421no—
156C4⋊C4—421no—
157C22⋊C4—421no—
158C42—211no—
159C2×C7—122no—
160C2×C7#88411yescharacteristic
161C2×C7D3×SD1611yescharacteristic
162C2×C7#91811yescharacteristic
163C22×C3D4×D711yescharacteristic
164C2×D3—22no—
165C2×D3—41no—
166C2×D3—41no—
167C3×C4—41no—
168C3×C4—281no—
169C3×C4#115911yescharacteristic
170C3×C4C2×C22⋊D711yescharacteristic
171Dic3—72no—
172Dic3—141no—
173Dic3—281no—
174C23—61no—
175Q8—21no—
176Q8—21no—
177Q8—421no—
178Q8—421no—
179D4—32no—
180D4—61no—
181C2×C4—21no—
182C2×C4—141no—
183C2×C4—211no—
184C2×C4—211no—
185C2×C4—421no—
186C2×C4—842no—
187C2×C4C2×D3×D711yescharacteristic
188C8—61no—
189C8—141no—
190C7#481811yescharacteristic, complemented, Sylow
191C2×C3D7×SD1611yescharacteristic
192C2×C3#647811yescharacteristic
193C2×C3#650211yescharacteristic
194D3—42no—
195C22—62no—
196C22—121no—
197C22—121no—
198C22#6669411yescharacteristic
199C4—41no—
200C4—212no—
201C4—281no—
202C4—421no—
203C4—841no—
204C4#6668611yescharacteristic
205C4D3×C22⋊D711yescharacteristic
206C3#8993411yescharacteristic, complemented, Sylow
207C2—122no—
208C2#1062106711yescharacteristic
209C2#1062108611yescharacteristic
210C2#1062112011yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×D12C22
hypercenterC2×Q8D3×D7
hypocenterC3×C7#461
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #10621427C2semidirect
5. C3×#6409C2semidirect
6. #10620518C2semidirect
7. #10620509C2semidirect
26. #6409D3semidirect
75. Dic21SD16semidirect
145. C3×C7#461semidirect
190. C7#4818semidirect
206. C3#89934semidirect

Maximal subgroups

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

subgroupindex
#106214272
C2×C7×#2152
#106209972
C2×#665672
C3×#64092
#106205182
#106205092
#899343
#48187