Finite Group Databasebeta

← #13054474

Subgroups — #13054474

Order 1344.

240 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D4×Dic21C211yesmaximal, characteristic
2C2×C7×#214C211yesmaximal, characteristic
3C2×C3×#1142C211yesmaximal, characteristic
4#10620947C211yesmaximal, characteristic, complemented
5#10620518C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620506C211yesmaximal, characteristic, complemented
8#89934—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C22×Dic21—21no—
11C2×C7×Dic6C2211yescharacteristic
12C2×C3×Dic14C2211yescharacteristic
13#66636—21no—
14C4⋊Dic21C2211yescharacteristic
15C4×Dic21C2211yescharacteristic
16C7×#214—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1142—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20#66577—21no—
21#66577—21no—
22C2×C7×SD16—31no—
23#6507—31no—
24D4×Dic7—31no—
25C2×#1142D311yescharacteristic
26#6409—31no—
27#6390—31no—
28Dic7⋊C8—31no—
29#5162—71nomaximal, Hall
30C23×C3×C7—21no—
31C3×C7×D4—21no—
32C3×C7×D4D412yes—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21—42no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38C2×C7×Dic3—21no—
39C7×Dic6—21no—
40C7×Dic6—21no—
41C2×C3×Dic7—21no—
42C3×Dic14—21no—
43C3×Dic14—21no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C2×C7×Q8—31no—
47C2×C7×D4C2×D311yescharacteristic
48C22×Dic7—61no—
49C2×Dic14C2×D311yescharacteristic
50C7×SD16—62no—
51C2×C7×C8—31no—
52C22⋊Dic7—61no—
53#1142—22no—
54C4⋊Dic7—31no—
55Dic7⋊C4—61no—
56Dic7⋊C4—61no—
57C4×Dic7—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59C2×C3×SD16—71no—
60#944—71no—
61D4×Dic3—71no—
62C2×#214D711yescharacteristic
63#835—71no—
64#816—71no—
65Dic3⋊C8—71no—
66C22×C3×C7—22no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—21no—
73Dic21—41no—
74Dic21SD1612yescomplemented
75C3×Dic7—41no—
76C7×Dic3—41no—
77#461—211noSylow
78C23×C7—21no—
79C7×Q8—61no—
80C7×Q8—61no—
81C7×D4—21no—
82C7×D4#6712yes—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7—31no—
87C2×Dic7—31no—
88C2×Dic7—61no—
89C2×Dic7—122no—
90Dic14—21no—
91Dic14—21no—
92C7×C8—61no—
93C7⋊C8—21no—
94C2×C3×Q8—71no—
95C2×C3×D4C2×D711yescharacteristic
96C22×Dic3—141no—
97C2×Dic6C2×D711yescharacteristic
98C3×SD16—142no—
99C2×C3×C8—71no—
100#217—141no—
101#214—22no—
102C4⋊Dic3—71no—
103Dic3⋊C4—141no—
104Dic3⋊C4—141no—
105C4×Dic3—71no—
106C2×C3⋊C8C2×D711yescharacteristic
107C2×C3×C7—42no—
108C2×C3×C7#12111yescharacteristic
109C2×C3×C7C2×SD1611yescharacteristic
110C2×C3×C7#13711yescharacteristic
111C2×SD16—31no—
112C2×SD16—71no—
113C4⋊Q8—211no—
114C4×D4—211no—
115C4⋊C8—211no—
116#103—211no—
117#103—211no—
118C22×C7—22no—
119C22×C7—41no—
120C22×C7—41no—
121C22×C7D3×D411yescharacteristic
122C4×C7—121no—
123C4×C7#23711yescharacteristic
124C4×C7C2×#6711yescharacteristic
125Dic7—32no—
126Dic7—41no—
127Dic7—61no—
128Dic7—121no—
129C23×C3—21no—
130C3×Q8—141no—
131C3×Q8—141no—
132C3×D4—21no—
133C3×D4C22⋊D712yes—
134C2×C3×C4—141no—
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—71no—
138C2×Dic3—71no—
139C2×Dic3—141no—
140C2×Dic3—282no—
141Dic6—21no—
142Dic6—21no—
143C3×C8—141no—
144C3⋊C8—21no—
145C3×C7#46111yescharacteristic, complemented, Hall
146C2×Q8—31no—
147C2×Q8—71no—
148C2×D4D3×D711yescharacteristic
149C22×C4—421no—
150SD16—62no—
151SD16—142no—
152C2×C8—31no—
153C2×C8—71no—
154C4⋊C4—211no—
155C4⋊C4—421no—
156C4⋊C4—421no—
157C22⋊C4—421no—
158C42—211no—
159C2×C7—42no—
160C2×C7D3×SD1611yescharacteristic
161C2×C7#91511yescharacteristic
162C2×C7#93911yescharacteristic
163C22×C3—22no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3D4×D711yescharacteristic
167C3×C4—281no—
168C3×C4#115911yescharacteristic
169C3×C4C2×C22⋊D711yescharacteristic
170Dic3—41no—
171Dic3—72no—
172Dic3—141no—
173Dic3—281no—
174C23—21no—
175Q8—61no—
176Q8—61no—
177Q8—141no—
178Q8—141no—
179D4—21no—
180D4#424512yes—
181C2×C4—61no—
182C2×C4—141no—
183C2×C4—211no—
184C2×C4—211no—
185C2×C4—421no—
186C2×C4—842no—
187C2×C4C2×D3×D711yescharacteristic
188C8—61no—
189C8—141no—
190C7#516211yescharacteristic, complemented, Sylow
191C2×C3—42no—
192C2×C3D7×SD1611yescharacteristic
193C2×C3#647811yescharacteristic
194C2×C3#650211yescharacteristic
195C22—22no—
196C22—41no—
197C22—41no—
198C22#6669411yescharacteristic
199C4—121no—
200C4—212no—
201C4—281no—
202C4—421no—
203C4—841no—
204C4#6668411yescharacteristic
205C4C2×#424511yescharacteristic
206C3#8993411yescharacteristic, complemented, Sylow
207C2—42no—
208C2#1062104911yescharacteristic
209C2#1062105011yescharacteristic
210C2#1062113311yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
Fitting, prime radicalC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#461
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #10620947C2semidirect
5. #10620518C2semidirect
6. #10620508C2semidirect
7. #10620506C2semidirect
74. Dic21SD16semidirect
145. C3×C7#461semidirect
190. C7#5162semidirect
206. C3#89934semidirect

Maximal subgroups

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

subgroupindex
D4×Dic212
C2×C7×#2142
C2×C3×#11422
#106209472
#106205182
#106205082
#106205062
#899343
#51627