Finite Group Databasebeta

← #125×D21 #13061778

Subgroups — #125×D21 #13061778

Order 1344.

452 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 155 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×C7×#125C211yesmaximal, characteristic, complemented
2C4⋊C4×D21C214yesmaximal
3C4⋊C4×D21C212yesmaximal
4#10621373C212yesmaximal, complemented
5#10621372C214yesmaximal, complemented
6C42×D21C211yesmaximal, characteristic
7#10621353C211yesmaximal, characteristic, complemented
8D7×#125—31nomaximal, Hall
9C2×C4×D21C2212yes—
10C2×C4×D21C2214yes—
11C2×C4×D21C2211yescharacteristic
12C3×C7×C4⋊C4C2214yes—
13C3×C7×C4⋊C4C2212yes—
14C3×C42×C7C2211yescharacteristic
15C4⋊Dic21C2212yes—
16C4⋊Dic21C2214yes—
17#66629C2214yes—
18#66629C2214yes—
19#66629C2214yes—
20C4×Dic21C2212yes—
21C4×Dic21C2211yescharacteristic
22C7×#125D311yescharacteristic, complemented
23D7×C4⋊C4—34no—
24D7×C4⋊C4—32no—
25#6453—32no—
26#6452—34no—
27C42×D7—31no—
28#6433—31no—
29D3×#125—71nomaximal, Hall
30C22×D21C2311yescharacteristic
31C2×C3×C4×C7C2312yes—
32C2×C3×C4×C7C2314yes—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21C2312yes—
35C2×Dic21C2314yes—
36C2×Dic21C2311yescharacteristic
37C4×D21—24no—
38C4×D21—28no—
39C4×D21Q814yes—
40C2×C4×D7—31no—
41C2×C4×D7—32no—
42C2×C4×D7—34no—
43C7×C4⋊C4C2×D314yes—
44C7×C4⋊C4C2×D312yes—
45C42×C7C2×D311yescharacteristic
46C4⋊Dic7—32no—
47C4⋊Dic7—34no—
48Dic7⋊C4—34no—
49Dic7⋊C4—34no—
50Dic7⋊C4—34no—
51C4×Dic7—32no—
52C4×Dic7—31no—
53C3×#125D711yescharacteristic, complemented
54D3×C4⋊C4—74no—
55D3×C4⋊C4—72no—
56#890—72no—
57#889—74no—
58C42×D3—71no—
59#870—71no—
60C22×C3×C7C2411yescharacteristic
61C2×D21C2×Q812yes—
62C2×D21#4114yes—
63C3×C4×C7—22no—
64C3×C4×C7—24no—
65C3×C4×C7C2×Q812yes—
66Dic21—22no—
67Dic21—24no—
68Dic21C2×Q812yes—
69C2×#125—211noSylow
70C22×D7—31no—
71C2×C4×C7C22×D312yes—
72C2×C4×C7C22×D314yes—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—31no—
75C2×Dic7—32no—
76C2×Dic7—34no—
77C4×D7—34no—
78C4×D7—64no—
79C4×D7—68no—
80C2×C4×D3—71no—
81C2×C4×D3—72no—
82C2×C4×D3—74no—
83C3×C4⋊C4C2×D714yes—
84C3×C4⋊C4C2×D712yes—
85C3×C42C2×D711yescharacteristic
86C4⋊Dic3—72no—
87C4⋊Dic3—74no—
88Dic3⋊C4—74no—
89Dic3⋊C4—74no—
90Dic3⋊C4—74no—
91C4×Dic3—72no—
92C4×Dic3—71no—
93C2×C3×C7C22×Q811yescharacteristic
94C2×C3×C7C2×#4112yes—
95D21#12514yesdirect product factor
96#125—214no—
97#125—212no—
98#125—211no—
99#125D2111yescharacteristic, direct product factor
100C2×C4⋊C4—212no—
101C2×C4⋊C4—214no—
102C2×C42—211no—
103C22×C7C23×D311yescharacteristic
104C2×D7—32no—
105C2×D7—34no—
106C4×C7—22no—
107C4×C7—24no—
108C4×C7D3×Q812yes—
109Dic7—32no—
110Dic7—62no—
111Dic7—64no—
112C22×D3—71no—
113C2×C3×C4C22×D712yes—
114C2×C3×C4C22×D714yes—
115C2×C3×C4C22×D711yescharacteristic
116C2×Dic3—71no—
117C2×Dic3—72no—
118C2×Dic3—74no—
119C4×D3—74no—
120C4×D3—144no—
121C4×D3—148no—
122C3×C7C2×#12511yescharacteristic, complemented, Hall
123C22×C4—211no—
124C22×C4—212no—
125C22×C4—214no—
126C4⋊C4—214no—
127C4⋊C4—214no—
128C4⋊C4—214no—
129C4⋊C4—214no—
130C4⋊C4—212no—
131C4⋊C4C2×D2114yes—
132C4⋊C4C2×D2112yes—
133C42—212no—
134C42—211no—
135C42C2×D2111yescharacteristic
136C2×C7C2×D3×Q811yescharacteristic
137C2×C7D3×#4112yes—
138D7—34no—
139C22×C3C23×D711yescharacteristic
140C2×D3—72no—
141C2×D3—74no—
142C3×C4—22no—
143C3×C4—24no—
144C3×C4Q8×D712yes—
145Dic3—72no—
146Dic3—142no—
147Dic3—144no—
148C23—211no—
149C2×C4—211no—
150C2×C4—214no—
151C2×C4—212no—
152C2×C4—214no—
153C2×C4—424no—
154C2×C4—428no—
155C2×C4C22×D2112yes—
156C2×C4C22×D2114yes—
157C2×C4C22×D2111yescharacteristic
158C7D3×#12511yescharacteristic, complemented, Sylow
159C2×C3C2×Q8×D711yescharacteristic
160C2×C3D7×#4112yes—
161D3—74no—
162C22—212no—
163C22—214no—
164C22C23×D2111yescharacteristic
165C4—22no—
166C4—24no—
167C4—212no—
168C4—422no—
169C4—424no—
170C4Q8×D2112yes—
171C3D7×#12511yescharacteristic, complemented, Sylow
172C2—214no—
173C2C2×Q8×D2111yescharacteristic
174C2#41×D2112yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#125C2
prime radicalC22×D21C23
hypercenter#125D21
hypocenterC3×C7C2×#125
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C3×C7×#125C2semidirect
4. #10621373C2semidirect
5. #10621372C2semidirect
7. #10621353C2semidirect
22. C7×#125D3semidirect
53. C3×#125D7semidirect
95. D21#125direct (N × H)
99. #125D21direct (N × H)
122. C3×C7C2×#125semidirect
158. C7D3×#125semidirect
171. C3D7×#125semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1252
C4⋊C4×D212
#106213732
#106213722
C42×D212
#106213532
D7×#1253
D3×#1257