Finite Group Databasebeta

← #13056402

Subgroups — #13056402

Order 1344.

204 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 69 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
1D4×Dic21C211yesmaximal, characteristic
2#10621373C211yesmaximal, characteristic, complemented
3C3×C7×#102C211yesmaximal, characteristic
4#10620671C211yesmaximal, characteristic
5#10620657C211yesmaximal, characteristic, complemented
6#10620653C211yesmaximal, characteristic, complemented
7#10620647C211yesmaximal, characteristic, complemented
8#89538—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C22×Dic21—21no—
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13#66636—21no—
14C4⋊Dic21—21no—
15C4⋊Dic21C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20D4×Dic7—31no—
21#6453—31no—
22C7×#102D311yescharacteristic
23#6406—31no—
24#6392—31no—
25Dic7⋊C8—31no—
26#6382—31no—
27#4766—71nomaximal, Hall
28C23×C3×C7—21no—
29C3×C7×D4—21no—
30C3×C7×D4Q812yes—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21—42no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38C3×C7×C8—21no—
39C3⋊(C7⋊C8)—21no—
40C2×C7×D4C2×D311yescharacteristic
41C22×Dic7—61no—
42C2×C7×C8C2×D311yescharacteristic
43C7×C4⋊C4C2×D311yescharacteristic
44C22⋊Dic7—61no—
45C4⋊Dic7—31no—
46C4⋊Dic7—31no—
47C4⋊Dic7—61no—
48Dic7⋊C4—61no—
49C4×Dic7—31no—
50C2×C7⋊C8—31no—
51D4×Dic3—71no—
52#890—71no—
53C3×#102D711yescharacteristic
54#832—71no—
55#818—71no—
56Dic3⋊C8—71no—
57#808—71no—
58C22×C3×C7—22no—
59C22×C3×C7—41no—
60C22×C3×C7—41no—
61C22×C3×C7C2×D411yescharacteristic
62C3×C4×C7—41no—
63C3×C4×C7C2×Q811yescharacteristic
64C3×C4×C7#4111yescharacteristic
65Dic21—21no—
66Dic21—21no—
67Dic21—41no—
68Dic21—41no—
69#478—211noSylow
70C23×C7—21no—
71C7×D4—21no—
72C7×D4Dic612yes—
73C2×C4×C7—21no—
74C2×C4×C7C22×D311yescharacteristic
75C2×Dic7—31no—
76C2×Dic7—31no—
77C2×Dic7—61no—
78C2×Dic7—61no—
79C2×Dic7—122no—
80C7×C8—21no—
81C7⋊C8—61no—
82C2×C3×D4C2×D711yescharacteristic
83C22×Dic3—141no—
84C2×C3×C8C2×D711yescharacteristic
85C3×C4⋊C4C2×D711yescharacteristic
86#217—141no—
87C4⋊Dic3—71no—
88C4⋊Dic3—71no—
89C4⋊Dic3—141no—
90Dic3⋊C4—141no—
91C4×Dic3—71no—
92C2×C3⋊C8—71no—
93C2×C3×C7—42no—
94C2×C3×C7C22⋊Q811yescharacteristic
95C2×C3×C7#13511yescharacteristic
96C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
97#125—211no—
98C4×D4—211no—
99#107—211no—
100#106—211no—
101C4⋊C8—211no—
102#102—211no—
103#102D2111yescharacteristic
104C22×C7—22no—
105C22×C7—41no—
106C22×C7—41no—
107C22×C7D3×D411yescharacteristic
108C4×C7—41no—
109C4×C7C2×Dic611yescharacteristic
110C4×C7#23711yescharacteristic
111Dic7—61no—
112Dic7—61no—
113Dic7—121no—
114Dic7—121no—
115C23×C3—21no—
116C3×D4—21no—
117C3×D4Dic1412yes—
118C2×C3×C4—21no—
119C2×C3×C4C22×D711yescharacteristic
120C2×Dic3—71no—
121C2×Dic3—71no—
122C2×Dic3—141no—
123C2×Dic3—141no—
124C2×Dic3—282no—
125C3×C8—21no—
126C3⋊C8—141no—
127C3×C7#47811yescharacteristic, complemented, Hall
128C2×D4C2×D2111yescharacteristic
129C22×C4—421no—
130C2×C8—211no—
131C2×C8C2×D2111yescharacteristic
132C4⋊C4—211no—
133C4⋊C4—211no—
134C4⋊C4—421no—
135C4⋊C4—421no—
136C4⋊C4C2×D2111yescharacteristic
137C22⋊C4—421no—
138C42—211no—
139C2×C7—42no—
140C2×C7#87811yescharacteristic
141C2×C7#91211yescharacteristic
142C2×C7#91411yescharacteristic
143C22×C3—22no—
144C22×C3—41no—
145C22×C3—41no—
146C22×C3D4×D711yescharacteristic
147C3×C4—41no—
148C3×C4C2×Dic1411yescharacteristic
149C3×C4#115911yescharacteristic
150Dic3—141no—
151Dic3—141no—
152Dic3—281no—
153Dic3—281no—
154C23—21no—
155D4—21no—
156D4Dic4212yes—
157C2×C4—21no—
158C2×C4—211no—
159C2×C4—211no—
160C2×C4—421no—
161C2×C4—421no—
162C2×C4—842no—
163C2×C4C22×D2111yescharacteristic
164C8—21no—
165C8—421no—
166C7#476611yescharacteristic, complemented, Sylow
167C2×C3—42no—
168C2×C3#644111yescharacteristic
169C2×C3#647511yescharacteristic
170C2×C3#647711yescharacteristic
171C22—22no—
172C22—41no—
173C22—41no—
174C22D4×D2111yescharacteristic
175C4—41no—
176C4—421no—
177C4—421no—
178C4—841no—
179C4—841no—
180C4C2×Dic4211yescharacteristic
181C4#6673011yescharacteristic
182C3#8953811yescharacteristic, complemented, Sylow
183C2—42no—
184C2#1062136111yescharacteristic
185C2#1062139511yescharacteristic
186C2#1062139711yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#102C2
socleC22×C3×C7C2×D4
prime radicalC2×C3×C7×D4C22
hypercenter#102D21
hypocenterC3×C7#478
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. #10621373C2semidirect
5. #10620657C2semidirect
6. #10620653C2semidirect
7. #10620647C2semidirect
127. C3×C7#478semidirect
166. C7#4766semidirect
182. C3#89538semidirect

Maximal subgroups

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

subgroupindex
D4×Dic212
#106213732
C3×C7×#1022
#106206712
#106206572
#106206532
#106206472
#895383
#47667