Finite Group Databasebeta

← #13054669

Subgroups — #13054669

Order 1344.

180 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×#6507C211yesmaximal, characteristic
2#10620950C211yesmaximal, characteristic
3#10620674C211yesmaximal, characteristic
4C7×#835C211yesmaximal, characteristic
5#10620506C211yesmaximal, characteristic
6#10620505C211yesmaximal, characteristic
7#10620487C211yesmaximal, characteristic
8#89569—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C3×Dic14C2211yescharacteristic
11C4⋊Dic21C2211yescharacteristic
12C2×C3⋊(C7⋊C8)C2211yescharacteristic
13C7×C4⋊Dic3C2211yescharacteristic
14C2×C7×C3⋊C8C2211yescharacteristic
15C3×Dic7⋊C4—21no—
16C3×Dic7⋊C4—21no—
17C3×C4×Dic7C2211yescharacteristic
18Dic3⋊Dic7—21no—
19#66577—21no—
20#6507D311yescharacteristic
21#6453—31no—
22C7×#103—31no—
23#6409—31no—
24#6390—31no—
25#6389—31no—
26#6384—31no—
27#5082—71nomaximal, Hall
28C3×C7×Q8—21no—
29C3×C7×Q8—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×C7×Dic3—21no—
34C2×C3×Dic7—21no—
35C2×C3×Dic7D411yescharacteristic
36C2×C3×Dic7D411yescharacteristic
37C3×Dic14—21no—
38C3×Dic14—21no—
39C3⋊(C7⋊C8)—21no—
40C7×C3⋊C8—21no—
41C2×C7×Q8C2×D311yescharacteristic
42C2×Dic14C2×D311yescharacteristic
43C2×C7×C8—31no—
44C7×C4⋊C4—31no—
45C4⋊Dic7—31no—
46C4⋊Dic7—61no—
47Dic7⋊C4—21no—
48Dic7⋊C4—21no—
49Dic7⋊C4—61no—
50C4×Dic7C2×D311yescharacteristic
51C2×C7⋊C8—31no—
52C3×C4⋊Q8—71no—
53#870—71no—
54#835—71no—
55#835—71no—
56#835—71no—
57#835D711yescharacteristic
58#803—71no—
59C22×C3×C7C2×D411yescharacteristic
60C3×C4×C7—41no—
61C3×C4×C7#4111yescharacteristic
62C3×C4×C7#4111yescharacteristic
63Dic21—41no—
64C3×Dic7—21no—
65C3×Dic7—21no—
66C3×Dic7—41no—
67C7×Dic3—41no—
68#491—211noSylow
69C7×Q8—21no—
70C7×Q8—21no—
71C2×C4×C7—21no—
72C2×C4×C7—61no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—21no—
75C2×Dic7—61no—
76C2×Dic7#6711yescharacteristic
77C2×Dic7#6711yescharacteristic
78Dic14—21no—
79Dic14—21no—
80C7×C8—61no—
81C7⋊C8—61no—
82C2×C3×Q8—71no—
83C2×C3×Q8C2×D711yescharacteristic
84C3×C4⋊C4—141no—
85C3×C4⋊C4—141no—
86C3×C42—71no—
87C4⋊Dic3—71no—
88C4⋊Dic3C2×D711yescharacteristic
89Dic3⋊C4—141no—
90Dic3⋊C4—141no—
91C2×C3⋊C8—71no—
92C2×C3⋊C8C2×D711yescharacteristic
93C2×C3×C7#12411yescharacteristic
94C2×C3×C7#13711yescharacteristic
95C2×C3×C7#13711yescharacteristic
96C4⋊Q8—71no—
97#125—211no—
98#103—31no—
99#103—211no—
100#103—211no—
101#103—211no—
102#97—211no—
103C22×C7C2×#6711yescharacteristic
104C4×C7—41no—
105C4×C7—121no—
106C4×C7#23711yescharacteristic
107C4×C7#23711yescharacteristic
108Dic7—21no—
109Dic7—21no—
110Dic7—41no—
111Dic7—121no—
112C3×Q8—21no—
113C3×Q8—21no—
114C3×Q8—141no—
115C3×Q8—141no—
116C2×C3×C4—21no—
117C2×C3×C4—71no—
118C2×C3×C4—71no—
119C2×C3×C4—141no—
120C2×C3×C4C22×D711yescharacteristic
121C2×Dic3—21no—
122C2×Dic3—141no—
123C3⋊C8—21no—
124C3⋊C8—141no—
125C3×C7#49111yescharacteristic, complemented, Hall
126C2×Q8—71no—
127C2×Q8D3×D711yescharacteristic
128C2×C8—31no—
129C2×C8—211no—
130C4⋊C4—31no—
131C4⋊C4—141no—
132C4⋊C4—141no—
133C4⋊C4—211no—
134C4⋊C4—421no—
135C4⋊C4—421no—
136C42—71no—
137C2×C7#93611yescharacteristic
138C2×C7#94211yescharacteristic
139C2×C7#94211yescharacteristic
140C22×C3D4×D711yescharacteristic
141C3×C4—41no—
142C3×C4—141no—
143C3×C4—141no—
144C3×C4—281no—
145C3×C4#115711yescharacteristic
146C3×C4#115911yescharacteristic
147Dic3—41no—
148Dic3—281no—
149Q8—21no—
150Q8—21no—
151Q8—141no—
152Q8—141no—
153C2×C4—21no—
154C2×C4—61no—
155C2×C4—71no—
156C2×C4—71no—
157C2×C4—141no—
158C2×C4—421no—
159C2×C4C2×D3×D711yescharacteristic
160C8—61no—
161C8—421no—
162C7#508211yescharacteristic, complemented, Sylow
163C2×C3#644811yescharacteristic
164C2×C3#647811yescharacteristic
165C2×C3#648111yescharacteristic
166C22D7×#6711yescharacteristic
167C4—41no—
168C4—121no—
169C4—141no—
170C4—141no—
171C4—281no—
172C4—841no—
173C4#6668311yescharacteristic
174C4#6668411yescharacteristic
175C3#8956911yescharacteristic, complemented, Sylow
176C2#1062107511yescharacteristic
177C2#1062107711yescharacteristic
178C2#1062109611yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socle, prime radicalC22×C3×C7C2×D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#491
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
125. C3×C7#491semidirect
162. C7#5082semidirect
175. C3#89569semidirect

Maximal subgroups

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

subgroupindex
C3×#65072
#106209502
#106206742
C7×#8352
#106205062
#106205052
#106204872
#895693
#50827