Finite Group Databasebeta

← #13056686

Subgroups — #13056686

Order 1344.

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

normal subgroups only

This group has 240 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
1C3×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#66634C211yesmaximal, characteristic, complemented
3#10621413C211yesmaximal, characteristic
4C2×#66627C211yesmaximal, characteristic, complemented
5#10620674C211yesmaximal, characteristic, complemented
6#10620648C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89821—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C2×#4268—21no—
12C2×D84C2211yescharacteristic
13C3×C7×C4⋊C4—21no—
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C22⋊C4—21no—
16#66634—22no—
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic
19#66627—22no—
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22C7×C22⋊Q8D311yescharacteristic, complemented
23C2×#1143—31no—
24#6493—31no—
25C2×#1136—31no—
26#6409—31no—
27#6383—31no—
28#6381—31no—
29#5049—71nomaximal, Hall
30C23×C3×C7D411yescharacteristic
31C22×D21—21no—
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7D411yescharacteristic
39C2×C3×C4×C7C2311yescharacteristic
40#4268—42no—
41C2×Dic21—21no—
42D84—21no—
43D84—21no—
44C3⋊(C7⋊C8)—21no—
45C3⋊(C7⋊C8)D412yescomplemented
46C2×C7×Q8C2×D311yescharacteristic
47C22×C4×C7C2×D311yescharacteristic
48C2×C22⋊D7—61no—
49C2×D28—31no—
50C7×C4⋊C4—21no—
51C7×C4⋊C4C2×D311yescharacteristic
52C7×C22⋊C4—21no—
53#1143—62no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—31no—
56#1136—62no—
57C2×C7⋊C8—31no—
58C2×C7⋊C8—31no—
59C3×C22⋊Q8D711yescharacteristic, complemented
60C2×#215—71no—
61#930—71no—
62C2×#208—71no—
63#835—71no—
64#809—71no—
65#807—71no—
66C22×C3×C7—21no—
67C22×C3×C7—21no—
68C22×C3×C7—21no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D21—41no—
71C2×D21—41no—
72C2×D21—41no—
73C3×C4×C7—21no—
74C3×C4×C7—41no—
75C3×C4×C7—41no—
76C3×C4×C7C2×D411yescharacteristic
77C3×C4×C7#4111yescharacteristic
78Dic21—41no—
79#468—211noSylow
80C23×C7#6711yescharacteristic
81C22×D7—61no—
82C7×Q8—21no—
83C7×Q8—21no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7#6711yescharacteristic
89C2×C4×C7C22×D311yescharacteristic
90C22⋊D7—122no—
91C2×Dic7—61no—
92D28—61no—
93D28—61no—
94C7⋊C8—32no—
95C7⋊C8—61no—
96C2×C3×Q8C2×D711yescharacteristic
97C22×C3×C4C2×D711yescharacteristic
98C2×#67—141no—
99C2×D12—71no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4C2×D711yescharacteristic
102C3×C22⋊C4—21no—
103#215—142no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—71no—
106#208—142no—
107C2×C3⋊C8—71no—
108C2×C3⋊C8—71no—
109C2×C3×C7—41no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112C2×C3×C7#13711yescharacteristic
113D21—81no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8D2111yescharacteristic, complemented
117#121—211no—
118#107—211no—
119#103—211no—
120#102—211no—
121C22×C7—21no—
122C22×C7—21no—
123C22×C7—21no—
124C22×C7C2×#6711yescharacteristic
125C2×D7—121no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—21no—
129C4×C7—41no—
130C4×C7—41no—
131C4×C7D3×D411yescharacteristic
132C4×C7#23711yescharacteristic
133Dic7—121no—
134C23×C3C22⋊D711yescharacteristic
135C22×D3—141no—
136C3×Q8—21no—
137C3×Q8—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4C22⋊D711yescharacteristic
143C2×C3×C4C22×D711yescharacteristic
144#67—282no—
145C2×Dic3—141no—
146D12—141no—
147D12—141no—
148C3⋊C8—72no—
149C3⋊C8—141no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8C2×D2111yescharacteristic
152C2×D4—211no—
153C2×D4—421no—
154C22×C4C2×D2111yescharacteristic
155SD16—422no—
156M16—422no—
157C2×C8—211no—
158C2×C8—211no—
159C4⋊C4—21no—
160C4⋊C4—211no—
161C4⋊C4C2×D2111yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—421no—
164C2×C7—41no—
165C2×C7#93911yescharacteristic
166C2×C7#94211yescharacteristic
167C2×C7#94911yescharacteristic
168D7—241no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3C2×C22⋊D711yescharacteristic
173C2×D3—281no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—21no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4D4×D711yescharacteristic
180C3×C4#115911yescharacteristic
181Dic3—281no—
182C23—421no—
183C23#426811yescharacteristic
184Q8—21no—
185Q8—21no—
186D4—421no—
187D4—421no—
188D4—842no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—421no—
194C2×C4#426811yescharacteristic
195C2×C4C22×D2111yescharacteristic
196C8—212no—
197C8—421no—
198C7#504911yescharacteristic, complemented, Sylow
199C2×C3—41no—
200C2×C3#650211yescharacteristic
201C2×C3#650511yescharacteristic
202C2×C3#651211yescharacteristic
203D3—561no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—841no—
208C22—841no—
209C22—841no—
210C22C2×#426811yescharacteristic
211C4—21no—
212C4—41no—
213C4—41no—
214C4—841no—
215C4D4×D2111yescharacteristic
216C4#6673011yescharacteristic
217C3#8982111yescharacteristic, complemented, Sylow
218C2—41no—
219C2—1681no—
220C2#1062142211yescharacteristic
221C2#1062142511yescharacteristic
222C2#1062143211yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radical#10621413C2
hypercenterC22⋊Q8D21
hypocenterC3×C7#468
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C7×C22⋊Q8C2semidirect
2. C2×#66634C2semidirect
4. C2×#66627C2semidirect
5. #10620674C2semidirect
6. #10620648C2semidirect
7. #10620646C2semidirect
22. C7×C22⋊Q8D3semidirect
45. C3⋊(C7⋊C8)D4semidirect
59. C3×C22⋊Q8D7semidirect
116. C22⋊Q8D21semidirect
150. C3×C7#468semidirect
198. C7#5049semidirect
217. C3#89821semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
C2×#666342
#106214132
C2×#666272
#106206742
#106206482
#106206462
#898213
#50497