Finite Group Databasebeta

← #13054679

Subgroups — #13054679

Order 1344.

180 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 58 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
2#10620946C211yesmaximal, characteristic
3C7×#835C211yesmaximal, characteristic
4C3×#6409C211yesmaximal, characteristic
5#10620510C211yesmaximal, characteristic
6#10620509C211yesmaximal, characteristic
7#10620489C211yesmaximal, characteristic
8#89569—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11#66629—21no—
12#66629—21no—
13C4×Dic21C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×C4⋊Dic7C2211yescharacteristic
17C2×C3×C7⋊C8C2211yescharacteristic
18Dic3⋊Dic7—21no—
19Dic7⋊Dic3—21no—
20#6507—31no—
21#6453—31no—
22C7×#103—31no—
23#6409D311yescharacteristic
24#6390—31no—
25#6389—31no—
26#6384—31no—
27#4797—71nomaximal, Hall
28C3×C7×Q8—21no—
29C3×C7×Q8—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35Dic42—21no—
36Dic42—21no—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39C3×C7⋊C8—21no—
40C7×C3⋊C8—21no—
41C2×C7×Q8C2×D311yescharacteristic
42C2×Dic14—31no—
43C2×C7×C8—31no—
44C7×C4⋊C4—31no—
45C4⋊Dic7—61no—
46C4⋊Dic7C2×D311yescharacteristic
47Dic7⋊C4—61no—
48Dic7⋊C4—61no—
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C2×C7⋊C8C2×D311yescharacteristic
52#944—71no—
53#890—71no—
54C3×#103—71no—
55#835D711yescharacteristic
56#816—71no—
57#815—71no—
58#810—71no—
59C22×C3×C7C2×D411yescharacteristic
60C3×C4×C7—41no—
61C3×C4×C7#4111yescharacteristic
62C3×C4×C7#4111yescharacteristic
63Dic21—21no—
64Dic21—21no—
65Dic21—41no—
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—31no—
76C2×Dic7—31no—
77C2×Dic7—61no—
78Dic14—61no—
79Dic14—61no—
80C7×C8—61no—
81C7⋊C8—21no—
82C2×C3×Q8C2×D711yescharacteristic
83C2×Dic6—71no—
84C2×C3×C8—71no—
85C3×C4⋊C4—71no—
86C4⋊Dic3—141no—
87C4⋊Dic3C2×D711yescharacteristic
88Dic3⋊C4—141no—
89Dic3⋊C4—141no—
90Dic3⋊C4—141no—
91C4×Dic3—71no—
92C2×C3⋊C8C2×D711yescharacteristic
93C2×C3×C7#12411yescharacteristic
94C2×C3×C7#13711yescharacteristic
95C2×C3×C7#13711yescharacteristic
96C4⋊Q8—211no—
97#125—211no—
98#103—31no—
99#103—71no—
100#103—211no—
101#103—211no—
102#97—211no—
103C22×C7D3×D411yescharacteristic
104C4×C7—41no—
105C4×C7—121no—
106C4×C7#23511yescharacteristic
107C4×C7#23711yescharacteristic
108Dic7—41no—
109Dic7—61no—
110Dic7—61no—
111Dic7—121no—
112C3×Q8—21no—
113C3×Q8—21no—
114C2×C3×C4—21no—
115C2×C3×C4—141no—
116C2×C3×C4C22×D711yescharacteristic
117C2×Dic3—21no—
118C2×Dic3—71no—
119C2×Dic3—71no—
120C2×Dic3—141no—
121Dic6—141no—
122Dic6—141no—
123C3×C8—141no—
124C3⋊C8—21no—
125C3×C7#49111yescharacteristic, complemented, Hall
126C2×Q8—211no—
127C2×Q8D3×D711yescharacteristic
128C2×C8—31no—
129C2×C8—71no—
130C4⋊C4—31no—
131C4⋊C4—71no—
132C4⋊C4—421no—
133C4⋊C4—421no—
134C4⋊C4—421no—
135C4⋊C4—421no—
136C42—211no—
137C2×C7#88511yescharacteristic
138C2×C7#91511yescharacteristic
139C2×C7#91811yescharacteristic
140C22×C3D4×D711yescharacteristic
141C3×C4—41no—
142C3×C4—281no—
143C3×C4#115711yescharacteristic
144C3×C4#115911yescharacteristic
145Dic3—41no—
146Dic3—141no—
147Dic3—141no—
148Dic3—281no—
149Q8—21no—
150Q8—21no—
151Q8—421no—
152Q8—421no—
153C2×C4—21no—
154C2×C4—61no—
155C2×C4—141no—
156C2×C4—211no—
157C2×C4—211no—
158C2×C4—421no—
159C2×C4C2×D3×D711yescharacteristic
160C8—61no—
161C8—141no—
162C7#479711yescharacteristic, complemented, Sylow
163C2×C3#644811yescharacteristic
164C2×C3#647811yescharacteristic
165C2×C3#648111yescharacteristic
166C22#6669411yescharacteristic
167C4—41no—
168C4—121no—
169C4—281no—
170C4—421no—
171C4—421no—
172C4—841no—
173C4#6668311yescharacteristic
174C4#6668611yescharacteristic
175C3#8956911yescharacteristic, complemented, Sylow
176C2#1062108411yescharacteristic
177C2#1062108611yescharacteristic
178C2#1062108811yescharacteristic
subgroupG/H
centerC22#66694
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#4797semidirect
175. C3#89569semidirect

Maximal subgroups

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

subgroupindex
#106214272
#106209462
C7×#8352
C3×#64092
#106205102
#106205092
#106204892
#895693
#47977