Finite Group Databasebeta

← #13052465

Subgroups — #13052465

Order 1344.

216 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C4×Dic42C211yesmaximal, characteristic
2C3×C4×Dic14C211yesmaximal, characteristic
3C2×#66570C211yesmaximal, characteristic
4C7×#803C211yesmaximal, characteristic
5#10620522C211yesmaximal, characteristic
6#10620509C211yesmaximal, characteristic
7#10620506C211yesmaximal, characteristic
8#89488—31nomaximal, Hall
9C2×Dic42C2211yescharacteristic
10C2×C3×Dic14C2211yescharacteristic
11C3×C42×C7C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13#66629—21no—
14C4×Dic21—21no—
15C2×C7×C3⋊C8C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C3×Dic7⋊C4—21no—
19C3×C4×Dic7—21no—
20#66570C414yescomplemented
21C2×Dic28—31no—
22C4×Dic14—31no—
23C4×Dic14D311yescharacteristic
24C7×#97—31no—
25#6391—31no—
26#6390—31no—
27#6390—31no—
28#5030—71nomaximal, Hall
29C2×C3×C4×C7D411yescharacteristic
30C2×C3×C4×C7D411yescharacteristic
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21—21no—
34Dic42—21no—
35Dic42C2×C412yes—
36C2×C3×Dic7—21no—
37C2×C3×Dic7—21no—
38C3×Dic14—21no—
39C3×Dic14C2×C412yes—
40C7×C3⋊C8—21no—
41C7×C3⋊C8C2×C412yes—
42C2×Dic14—31no—
43C2×Dic14C2×D311yescharacteristic
44C2×C7×C8—31no—
45C2×C7×C8—31no—
46C42×C7C2×D311yescharacteristic
47C4⋊Dic7—31no—
48C4⋊Dic7C2×D311yescharacteristic
49Dic7⋊C4—21no—
50Dic7⋊C4—61no—
51C4×Dic7—21no—
52C4×Dic7—61no—
53Dic28—34no—
54C3×C4×Q8—71no—
55C2×#216—71no—
56C4×Dic6—71no—
57#835—71no—
58#810—71no—
59#808—71no—
60#803D711yescharacteristic
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—21no—
63C3×C4×C7—21no—
64C3×C4×C7C22×C411yescharacteristic
65C3×C4×C7#4111yescharacteristic
66Dic21—22no—
67Dic21—41no—
68C3×Dic7—22no—
69C3×Dic7—41no—
70#441—211noSylow
71C2×C4×C7#6711yescharacteristic
72C2×C4×C7#6711yescharacteristic
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75C2×Dic7—21no—
76C2×Dic7—61no—
77C2×Dic7—61no—
78Dic14—21no—
79Dic14—32no—
80Dic14—61no—
81Dic14C4×D312yes—
82C7×C8—32no—
83C7×C8—61no—
84C2×C3×Q8—71no—
85C2×Dic6—71no—
86C3×C4⋊C4—71no—
87C3×C4⋊C4—141no—
88C3×C42—141no—
89C3×C42C2×D711yescharacteristic
90#216—74no—
91C4⋊Dic3—71no—
92Dic3⋊C4—141no—
93C4×Dic3—141no—
94C2×C3⋊C8C2×D711yescharacteristic
95C2×C3⋊C8C2×D711yescharacteristic
96C2×C3×C7C4×D411yescharacteristic
97C2×C3×C7#13711yescharacteristic
98C2×C3×C7#13711yescharacteristic
99C2×Q16—211no—
100C4×Q8—71no—
101C4×Q8—211no—
102#106—211no—
103#103—211no—
104#103—211no—
105#97—31no—
106C22×C7C2×#6711yescharacteristic
107C4×C7—21no—
108C4×C7—21no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23511yescharacteristic
111Dic7—22no—
112Dic7—41no—
113Dic7—62no—
114Dic7—121no—
115C3×Q8—72no—
116C3×Q8—141no—
117C2×C3×C4—141no—
118C2×C3×C4—141no—
119C2×C3×C4D2811yescharacteristic
120C2×C3×C4D2811yescharacteristic
121C2×C3×C4C22×D711yescharacteristic
122C2×Dic3—141no—
123C2×Dic3—141no—
124Dic6—72no—
125Dic6—141no—
126C3⋊C8—21no—
127C3⋊C8C4×D712yes—
128C3×C7#44111yescharacteristic, complemented, Hall
129C2×Q8—71no—
130C2×Q8—211no—
131Q16—214no—
132C2×C8—31no—
133C2×C8—31no—
134C4⋊C4—71no—
135C4⋊C4—141no—
136C4⋊C4—211no—
137C4⋊C4—421no—
138C42—141no—
139C42—421no—
140C42D3×D711yescharacteristic
141C2×C7C4×#6711yescharacteristic
142C2×C7#94211yescharacteristic
143C2×C7#95111yescharacteristic
144C22×C3C2×D2811yescharacteristic
145C3×C4—21no—
146C3×C4—21no—
147C3×C4—142no—
148C3×C4—281no—
149C3×C4C2×C4×D711yescharacteristic
150C3×C4#115711yescharacteristic
151Dic3—142no—
152Dic3—281no—
153Q8—72no—
154Q8—141no—
155Q8—212no—
156Q8—421no—
157C2×C4—141no—
158C2×C4—141no—
159C2×C4—421no—
160C2×C4—421no—
161C2×C4#424611yescharacteristic
162C2×C4#424611yescharacteristic
163C2×C4C2×D3×D711yescharacteristic
164C8—32no—
165C8—61no—
166C7#503011yescharacteristic, complemented, Sylow
167C2×C3C4×D2811yescharacteristic
168C2×C3#647211yescharacteristic
169C2×C3#647211yescharacteristic
170C22C2×#424611yescharacteristic
171C4—21no—
172C4—21no—
173C4—142no—
174C4—281no—
175C4—422no—
176C4—841no—
177C4#6667511yescharacteristic
178C4C4×D3×D711yescharacteristic
179C3#8948811yescharacteristic, complemented, Sylow
180C2#1062087511yescharacteristic
181C2#1062087711yescharacteristic
182C2C4×#424611yescharacteristic
subgroupG/H
centerC22C2×#4246
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C42×C7C22
socle, prime radicalC22×C3×C7C2×D4
cosocleC2×C3×C4×C7C23
hypercenterC42D3×D7
hypocenterC3×C7#441
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
20. #66570C4semidirect
128. C3×C7#441semidirect
166. C7#5030semidirect
179. C3#89488semidirect

Maximal subgroups

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

subgroupindex
C4×Dic422
C3×C4×Dic142
C2×#665702
C7×#8032
#106205222
#106205092
#106205062
#894883
#50307