Finite Group Databasebeta

← #13056456

Subgroups — #13056456

Order 1344.

220 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 79 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
1Q8×Dic21C211yesmaximal, characteristic
2C2×#66635C211yesmaximal, characteristic
3#10621370C211yesmaximal, characteristic
4C3×C7×#103C211yesmaximal, characteristic
5#10620655C211yesmaximal, characteristic
6C8×Dic21C211yesmaximal, characteristic
7#10620646C211yesmaximal, characteristic
8#89562—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13#66635C414yescomplemented
14C4⋊Dic21—21no—
15C4⋊Dic21C2211yescharacteristic
16#66629—21no—
17C4×Dic21—21no—
18C4×Dic21—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21Q8×Dic7—31no—
22C2×#1144—31no—
23#6450—31no—
24C7×#103D311yescharacteristic
25#6390—31no—
26C8×Dic7—31no—
27#6381—31no—
28#4790—71nomaximal, Hall
29C3×C7×Q8—21no—
30C3×C7×Q8C2×C412yes—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38Dic42—21no—
39Dic42C2×C412yes—
40C3×C7×C8—21no—
41C3⋊(C7⋊C8)C2×C412yes—
42C2×C7×Q8C2×D311yescharacteristic
43C2×Dic14—31no—
44C2×C7×C8C2×D311yescharacteristic
45C7×C4⋊C4C2×D311yescharacteristic
46#1144—34no—
47C4⋊Dic7—31no—
48C4⋊Dic7—61no—
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C4×Dic7—61no—
52C4×Dic7—61no—
53C2×C7⋊C8—31no—
54Q8×Dic3—71no—
55C2×#216—71no—
56#887—71no—
57C3×#103D711yescharacteristic
58#816—71no—
59C8×Dic3—71no—
60#807—71no—
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—22no—
63C3×C4×C7—41no—
64C3×C4×C7C22×C411yescharacteristic
65C3×C4×C7#4111yescharacteristic
66Dic21—22no—
67Dic21—21no—
68Dic21—41no—
69Dic21Q1612yescomplemented
70C4×Q16—211noSylow
71C7×Q8—21no—
72C7×Q8C4×D312yes—
73C2×C4×C7—21no—
74C2×C4×C7—21no—
75C2×C4×C7C22×D311yescharacteristic
76C2×Dic7—31no—
77C2×Dic7—31no—
78C2×Dic7—61no—
79C2×Dic7—61no—
80Dic14—32no—
81Dic14—61no—
82C7×C8—21no—
83C7⋊C8—32no—
84C2×C3×Q8C2×D711yescharacteristic
85C2×Dic6—71no—
86C2×C3×C8C2×D711yescharacteristic
87C3×C4⋊C4C2×D711yescharacteristic
88#216—74no—
89C4⋊Dic3—71no—
90C4⋊Dic3—141no—
91Dic3⋊C4—141no—
92C4×Dic3—71no—
93C4×Dic3—141no—
94C4×Dic3—141no—
95C2×C3⋊C8—71no—
96C2×C3×C7C4×D411yescharacteristic
97C2×C3×C7C2×Q1611yescharacteristic
98C2×C3×C7#13511yescharacteristic
99C2×Q16—211no—
100C4×Q8—211no—
101C4×Q8—211no—
102#107—211no—
103#103—211no—
104#103D2111yescharacteristic
105C4×C8—211no—
106C22×C7D3×D411yescharacteristic
107C4×C7—22no—
108C4×C7—41no—
109C4×C7C2×C4×D311yescharacteristic
110C4×C7#23711yescharacteristic
111Dic7—32no—
112Dic7—62no—
113Dic7—61no—
114Dic7—121no—
115C3×Q8—21no—
116C3×Q8C4×D712yes—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4C22×D711yescharacteristic
120C2×Dic3—71no—
121C2×Dic3—71no—
122C2×Dic3—141no—
123C2×Dic3—141no—
124Dic6—72no—
125Dic6—141no—
126C3×C8—21no—
127C3⋊C8—72no—
128C3×C7C4×Q1611yescharacteristic, complemented, Hall
129C2×Q8—211no—
130C2×Q8C2×D2111yescharacteristic
131Q16—214no—
132C2×C8—211no—
133C2×C8C2×D2111yescharacteristic
134C4⋊C4—211no—
135C4⋊C4—421no—
136C4⋊C4—421no—
137C4⋊C4C2×D2111yescharacteristic
138C42—211no—
139C42—421no—
140C42—421no—
141C2×C7#67⋊C411yescharacteristic
142C2×C7#91611yescharacteristic
143C2×C7D3×Q1611yescharacteristic
144C22×C3D4×D711yescharacteristic
145C3×C4—22no—
146C3×C4—41no—
147C3×C4C2×C4×D711yescharacteristic
148C3×C4#115911yescharacteristic
149Dic3—72no—
150Dic3—142no—
151Dic3—141no—
152Dic3—281no—
153Q8—21no—
154Q8—212no—
155Q8—421no—
156Q8C4×D2112yes—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—211no—
160C2×C4—211no—
161C2×C4—421no—
162C2×C4—421no—
163C2×C4C22×D2111yescharacteristic
164C8—21no—
165C8—212no—
166C7#479011yescharacteristic, complemented, Sylow
167C2×C3(C22⋊D7)⋊C411yescharacteristic
168C2×C3#647911yescharacteristic
169C2×C3D7×Q1611yescharacteristic
170C22D4×D2111yescharacteristic
171C4—22no—
172C4—41no—
173C4—212no—
174C4—422no—
175C4—421no—
176C4—841no—
177C4C2×C4×D2111yescharacteristic
178C4#6673011yescharacteristic
179C3#8956211yescharacteristic, complemented, Sylow
180C2#1062136411yescharacteristic
181C2#1062139911yescharacteristic
182C2Q16×D2111yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socle, prime radicalC22×C3×C7C2×D4
cosocleC2×C3×C4×C7C23
hypercenter#103D21
hypocenterC3×C7C4×Q16
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
13. #66635C4semidirect
69. Dic21Q16semidirect
128. C3×C7C4×Q16semidirect
166. C7#4790semidirect
179. C3#89562semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic212
C2×#666352
#106213702
C3×C7×#1032
#106206552
C8×Dic212
#106206462
#895623
#47907