Finite Group Databasebeta

← #13056674

Subgroups — #13056674

Order 1344.

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

normal subgroups only

This group has 260 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×#121C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3#10621413C211yesmaximal, characteristic, complemented
4C2×#66627C211yesmaximal, characteristic, complemented
5#10620671C211yesmaximal, characteristic, complemented
6#10620648C211yesmaximal, characteristic, complemented
7#10620647C211yesmaximal, characteristic, complemented
8#89811—31nomaximal, Hall
9C2×C3×C7×D4—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C2×#4268—21no—
13C2×D84C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C22⋊C4—21no—
16#66632—22no—
17#66631—21no—
18C4⋊Dic21C2211yescharacteristic
19#66627—22no—
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22C7×#121D311yescharacteristic, complemented
23C2×#1141—31no—
24#6493—31no—
25C2×#1136—31no—
26#6406—31no—
27#6383—31no—
28#6382—31no—
29#5039—71nomaximal, Hall
30C23×C3×C7—21no—
31C23×C3×C7D411yescharacteristic
32C22×D21—21no—
33C3×C7×D4—21no—
34C3×C7×D4—21no—
35C3×C7×D4—42no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7D411yescharacteristic
40C2×C3×C4×C7C2311yescharacteristic
41#4268—42no—
42C2×Dic21—21no—
43D84—21no—
44D84—21no—
45C3⋊(C7⋊C8)—21no—
46C3⋊(C7⋊C8)D412yescomplemented
47C2×C7×D4—21no—
48C2×C7×D4C2×D311yescharacteristic
49C22×C4×C7C2×D311yescharacteristic
50C2×C22⋊D7—61no—
51C2×D28—31no—
52C7×C4⋊C4C2×D311yescharacteristic
53C7×C22⋊C4—21no—
54#1141—62no—
55(C2×D7)⋊C4—61no—
56C4⋊Dic7—31no—
57#1136—62no—
58C2×C7⋊C8—31no—
59C2×C7⋊C8—31no—
60C3×#121D711yescharacteristic, complemented
61C2×#213—71no—
62#930—71no—
63C2×#208—71no—
64#832—71no—
65#809—71no—
66#808—71no—
67C22×C3×C7—21no—
68C22×C3×C7—21no—
69C22×C3×C7—21no—
70C22×C3×C7—41no—
71C22×C3×C7—41no—
72C22×C3×C7—41no—
73C22×C3×C7C2×D411yescharacteristic
74C2×D21—41no—
75C2×D21—41no—
76C2×D21—41no—
77C3×C4×C7—21no—
78C3×C4×C7—41no—
79C3×C4×C7C2×D411yescharacteristic
80C3×C4×C7#4111yescharacteristic
81Dic21—41no—
82#469—211noSylow
83C23×C7—21no—
84C23×C7#6711yescharacteristic
85C22×D7—61no—
86C7×D4—21no—
87C7×D4—21no—
88C7×D4—42no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7#6711yescharacteristic
93C2×C4×C7C22×D311yescharacteristic
94C22⋊D7—122no—
95C2×Dic7—61no—
96D28—61no—
97D28—61no—
98C7⋊C8—32no—
99C7⋊C8—61no—
100C2×C3×D4—21no—
101C2×C3×D4C2×D711yescharacteristic
102C22×C3×C4C2×D711yescharacteristic
103C2×#67—141no—
104C2×D12—71no—
105C3×C4⋊C4C2×D711yescharacteristic
106C3×C22⋊C4—21no—
107#213—142no—
108(C2×D3)⋊C4—141no—
109C4⋊Dic3—71no—
110#208—142no—
111C2×C3⋊C8—71no—
112C2×C3⋊C8—71no—
113C2×C3×C7—41no—
114C2×C3×C7—81no—
115C2×C3×C7#12111yescharacteristic
116C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118D21—81no—
119C2×D8—211no—
120C2×M16—211no—
121#121—211no—
122#121D2111yescharacteristic, complemented
123#106—211no—
124#102—211no—
125#102—211no—
126C22×C7—21no—
127C22×C7—21no—
128C22×C7—21no—
129C22×C7—41no—
130C22×C7—41no—
131C22×C7—41no—
132C22×C7C2×#6711yescharacteristic
133C2×D7—121no—
134C2×D7—121no—
135C2×D7—121no—
136C4×C7—21no—
137C4×C7—41no—
138C4×C7D3×D411yescharacteristic
139C4×C7#23711yescharacteristic
140Dic7—121no—
141C23×C3—21no—
142C23×C3C22⋊D711yescharacteristic
143C22×D3—141no—
144C3×D4—21no—
145C3×D4—21no—
146C3×D4—42no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4C22⋊D711yescharacteristic
151C2×C3×C4C22×D711yescharacteristic
152#67—282no—
153C2×Dic3—141no—
154D12—141no—
155D12—141no—
156C3⋊C8—72no—
157C3⋊C8—141no—
158C3×C7#46911yescharacteristic, complemented, Hall
159C2×D4—21no—
160C2×D4—211no—
161C2×D4—421no—
162C2×D4C2×D2111yescharacteristic
163C22×C4C2×D2111yescharacteristic
164D8—422no—
165M16—422no—
166C2×C8—211no—
167C2×C8—211no—
168C4⋊C4—211no—
169C4⋊C4C2×D2111yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—421no—
172C2×C7—41no—
173C2×C7—81no—
174C2×C7#93211yescharacteristic
175C2×C7#93911yescharacteristic
176C2×C7#94911yescharacteristic
177D7—241no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3C2×C22⋊D711yescharacteristic
185C2×D3—281no—
186C2×D3—281no—
187C2×D3—281no—
188C3×C4—21no—
189C3×C4—41no—
190C3×C4D4×D711yescharacteristic
191C3×C4#115911yescharacteristic
192Dic3—281no—
193C23—21no—
194C23—421no—
195C23#426811yescharacteristic
196D4—21no—
197D4—21no—
198D4—42no—
199D4—421no—
200D4—421no—
201D4—842no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—421no—
206C2×C4#426811yescharacteristic
207C2×C4C22×D2111yescharacteristic
208C8—212no—
209C8—421no—
210C7#503911yescharacteristic, complemented, Sylow
211C2×C3—41no—
212C2×C3—81no—
213C2×C3#649511yescharacteristic
214C2×C3#650211yescharacteristic
215C2×C3#651211yescharacteristic
216D3—561no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—841no—
224C22—841no—
225C22—841no—
226C22C2×#426811yescharacteristic
227C4—21no—
228C4—41no—
229C4—841no—
230C4D4×D2111yescharacteristic
231C4#6673011yescharacteristic
232C3#8981111yescharacteristic, complemented, Sylow
233C2—41no—
234C2—81no—
235C2—1681no—
236C2#1062141511yescharacteristic
237C2#1062142211yescharacteristic
238C2#1062143211yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenter#121D21
hypocenterC3×C7#469
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C3×C7×#121C2semidirect
2. C2×#66632C2semidirect
3. #10621413C2semidirect
4. C2×#66627C2semidirect
5. #10620671C2semidirect
6. #10620648C2semidirect
7. #10620647C2semidirect
22. C7×#121D3semidirect
46. C3⋊(C7⋊C8)D4semidirect
60. C3×#121D7semidirect
122. #121D21semidirect
158. C3×C7#469semidirect
210. C7#5039semidirect
232. C3#89811semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
C2×#666322
#106214132
C2×#666272
#106206712
#106206482
#106206472
#898113
#50397