Finite Group Databasebeta

← #13056511

Subgroups — #13056511

Order 1344.

204 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621376C211yesmaximal, characteristic
2#10621373C211yesmaximal, characteristic, complemented
3C3×C7×#107C211yesmaximal, characteristic, complemented
4#10620660C211yesmaximal, characteristic
5#10620653C211yesmaximal, characteristic, complemented
6#10620648C211yesmaximal, characteristic
7#10620647C211yesmaximal, characteristic, complemented
8#89663—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C4×D21—21no—
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14#66631—21no—
15C4⋊Dic21—21no—
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6456—31no—
21#6453—31no—
22C7×#107D311yescharacteristic, complemented
23#6395—31no—
24Dic7⋊C8—31no—
25#6383—31no—
26#6382—31no—
27#4891—71nomaximal, Hall
28C22×D21—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C2×Dic21D411yescharacteristic
35D84—21no—
36D84Q812yes—
37C4×D21—42no—
38C3×C7×C8—21no—
39C3⋊(C7⋊C8)—21no—
40C2×D28—31no—
41C2×C4×D7—61no—
42C2×C7×C8C2×D311yescharacteristic
43C7×C4⋊C4C2×D311yescharacteristic
44C7×C4⋊C4C2×D311yescharacteristic
45(C2×D7)⋊C4—61no—
46C4⋊Dic7—31no—
47C4⋊Dic7—61no—
48Dic7⋊C4—61no—
49C4×Dic7—31no—
50C2×C7⋊C8—31no—
51#893—71no—
52#890—71no—
53C3×#107D711yescharacteristic, complemented
54#821—71no—
55Dic3⋊C8—71no—
56#809—71no—
57#808—71no—
58C22×C3×C7C2×D411yescharacteristic
59C2×D21—22no—
60C2×D21—41no—
61C2×D21—41no—
62C3×C4×C7—41no—
63C3×C4×C7—41no—
64C3×C4×C7C2×Q811yescharacteristic
65C3×C4×C7#4111yescharacteristic
66Dic21—21no—
67Dic21—21no—
68Dic21—41no—
69#478—211noSylow
70C22×D7—61no—
71C2×C4×C7—21no—
72C2×C4×C7—21no—
73C2×C4×C7C22×D311yescharacteristic
74C2×Dic7—31no—
75C2×Dic7—31no—
76C2×Dic7—61no—
77D28—32no—
78D28—61no—
79C4×D7—122no—
80C7×C8—21no—
81C7⋊C8—61no—
82C2×D12—71no—
83C2×C4×D3—141no—
84C2×C3×C8C2×D711yescharacteristic
85C3×C4⋊C4C2×D711yescharacteristic
86C3×C4⋊C4C2×D711yescharacteristic
87(C2×D3)⋊C4—141no—
88C4⋊Dic3—71no—
89C4⋊Dic3—141no—
90Dic3⋊C4—141no—
91C4×Dic3—71no—
92C2×C3⋊C8—71no—
93C2×C3×C7C22⋊Q811yescharacteristic
94C2×C3×C7#13511yescharacteristic
95C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
96D21—42no—
97#125—211no—
98C4×D4—211no—
99#107D2111yescharacteristic, complemented
100#106—211no—
101C4⋊C8—211no—
102#102—211no—
103#102—211no—
104C22×C7D3×D411yescharacteristic
105C2×D7—62no—
106C2×D7—121no—
107C2×D7—121no—
108C4×C7—41no—
109C4×C7—41no—
110C4×C7#23511yescharacteristic
111C4×C7D3×Q811yescharacteristic
112Dic7—61no—
113Dic7—61no—
114Dic7—121no—
115C22×D3—141no—
116C2×C3×C4—21no—
117C2×C3×C4—21no—
118C2×C3×C4C22×D711yescharacteristic
119C2×Dic3—71no—
120C2×Dic3—71no—
121C2×Dic3—141no—
122D12—72no—
123D12—141no—
124C4×D3—282no—
125C3×C8—21no—
126C3⋊C8—141no—
127C3×C7#47811yescharacteristic, complemented, Hall
128C2×D4—211no—
129C22×C4—421no—
130C2×C8—211no—
131C2×C8C2×D2111yescharacteristic
132C4⋊C4—211no—
133C4⋊C4—421no—
134C4⋊C4—421no—
135C4⋊C4C2×D2111yescharacteristic
136C4⋊C4C2×D2111yescharacteristic
137C22⋊C4—421no—
138C42—211no—
139C2×C7#89611yescharacteristic
140C2×C7#91111yescharacteristic
141C2×C7#91911yescharacteristic
142D7—122no—
143C22×C3D4×D711yescharacteristic
144C2×D3—142no—
145C2×D3—281no—
146C2×D3—281no—
147C3×C4—41no—
148C3×C4—41no—
149C3×C4#115711yescharacteristic
150C3×C4Q8×D711yescharacteristic
151Dic3—141no—
152Dic3—141no—
153Dic3—281no—
154C23—421no—
155D4—212no—
156D4—421no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—211no—
160C2×C4—211no—
161C2×C4—421no—
162C2×C4—842no—
163C2×C4C22×D2111yescharacteristic
164C8—21no—
165C8—421no—
166C7#489111yescharacteristic, complemented, Sylow
167C2×C3#645911yescharacteristic
168C2×C3#647411yescharacteristic
169C2×C3#648211yescharacteristic
170D3—282no—
171C22—422no—
172C22—841no—
173C22—841no—
174C22D4×D2111yescharacteristic
175C4—41no—
176C4—41no—
177C4—421no—
178C4—421no—
179C4—841no—
180C4#6672811yescharacteristic
181C4Q8×D2111yescharacteristic
182C3#8966311yescharacteristic, complemented, Sylow
183C2—842no—
184C2#1062137911yescharacteristic
185C2#1062139411yescharacteristic
186C2#1062140211yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#107C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenter#107D21
hypocenterC3×C7#478
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #10621373C2semidirect
3. C3×C7×#107C2semidirect
5. #10620653C2semidirect
7. #10620647C2semidirect
22. C7×#107D3semidirect
53. C3×#107D7semidirect
99. #107D21semidirect
127. C3×C7#478semidirect
166. C7#4891semidirect
182. C3#89663semidirect

Maximal subgroups

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

subgroupindex
#106213762
#106213732
C3×C7×#1072
#106206602
#106206532
#106206482
#106206472
#896633
#48917