Finite Group Databasebeta

← #13056678

Subgroups — #13056678

Order 1344.

324 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#66728C211yesmaximal, characteristic, complemented
2C3×C7×#121C211yesmaximal, characteristic, complemented
3C2×#66633C211yesmaximal, characteristic, complemented
4C2×#66632C211yesmaximal, characteristic, complemented
5#10620669C211yesmaximal, characteristic, complemented
6#10620649C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89809—31nomaximal, Hall
9C2×C3×C7×D4—21no—
10C2×C3×C7×D4C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C2×#4268—21no—
13#66728—22no—
14#66728—22no—
15C2×D84C2211yescharacteristic
16C2×C4×D21—21no—
17C2×Dic42C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C7×C22⋊C4—21no—
20#66633—22no—
21#66632—22no—
22C2×C3⋊(C7⋊C8)C2211yescharacteristic
23C2×C3⋊(C7⋊C8)C2211yescharacteristic
24C2×#1157—31no—
25C7×#121D311yescharacteristic, complemented
26C2×#1142—31no—
27C2×#1141—31no—
28C22⋊(C7⋊C8)—31no—
29#6384—31no—
30#6383—31no—
31#5037—71nomaximal, Hall
32C23×C3×C7—21no—
33C23×C3×C7D411yescharacteristic
34C22×D21—21no—
35C3×C7×D4—21no—
36C3×C7×D4—21no—
37C3×C7×D4—42no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7D411yescharacteristic
42C2×C3×C4×C7C2311yescharacteristic
43#4268—24no—
44C2×Dic21—21no—
45D84—21no—
46D84D412yescomplemented
47C4×D21—24no—
48Dic42—21no—
49Dic42D412yescomplemented
50C3⋊(C7⋊C8)—21no—
51C3⋊(C7⋊C8)—21no—
52C2×C7×D4—21no—
53C2×C7×D4C2×D311yescharacteristic
54C22×C4×C7C2×D311yescharacteristic
55C2×C22⋊D7—61no—
56#1157—62no—
57#1157—62no—
58C2×D28—31no—
59C2×C4×D7—61no—
60C2×Dic14—31no—
61C7×C4⋊C4C2×D311yescharacteristic
62C7×C22⋊C4—21no—
63#1142—62no—
64#1141—62no—
65C2×C7⋊C8—31no—
66C2×C7⋊C8—31no—
67C2×#235—71no—
68C3×#121D711yescharacteristic, complemented
69C2×#214—71no—
70C2×#213—71no—
71#830—71no—
72#810—71no—
73#809—71no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7—21no—
77C22×C3×C7—41no—
78C22×C3×C7—41no—
79C22×C3×C7—41no—
80C22×C3×C7C2×D411yescharacteristic
81C2×D21—22no—
82C2×D21—41no—
83C2×D21—41no—
84C3×C4×C7—21no—
85C3×C4×C7—41no—
86C3×C4×C7C2×D411yescharacteristic
87C3×C4×C7C2×D411yescharacteristic
88Dic21—22no—
89#449—211noSylow
90C23×C7—21no—
91C23×C7#6711yescharacteristic
92C22×D7—61no—
93C7×D4—21no—
94C7×D4—21no—
95C7×D4—42no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—21no—
99C2×C4×C7#6711yescharacteristic
100C2×C4×C7C22×D311yescharacteristic
101C22⋊D7—64no—
102C2×Dic7—61no—
103D28—32no—
104D28—61no—
105C4×D7—64no—
106Dic14—32no—
107Dic14—61no—
108C7⋊C8—61no—
109C7⋊C8—61no—
110C2×C3×D4—21no—
111C2×C3×D4C2×D711yescharacteristic
112C22×C3×C4C2×D711yescharacteristic
113C2×#67—141no—
114#235—142no—
115#235—142no—
116C2×D12—71no—
117C2×C4×D3—141no—
118C2×Dic6—71no—
119C3×C4⋊C4C2×D711yescharacteristic
120C3×C22⋊C4—21no—
121#214—142no—
122#213—142no—
123C2×C3⋊C8—71no—
124C2×C3⋊C8—71no—
125C2×C3×C7—41no—
126C2×C3×C7—81no—
127C2×C3×C7#12011yescharacteristic
128C2×C3×C7#13511yescharacteristic
129C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
130D21—42no—
131C2×#41—211no—
132C2×SD16—211no—
133C2×D8—211no—
134#121D2111yescharacteristic, complemented
135#103—211no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7—21no—
141C22×C7—41no—
142C22×C7—41no—
143C22×C7—41no—
144C22×C7C2×#6711yescharacteristic
145C2×D7—62no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—21no—
149C4×C7—41no—
150C4×C7D3×D411yescharacteristic
151C4×C7D3×D411yescharacteristic
152Dic7—62no—
153C23×C3—21no—
154C23×C3C22⋊D711yescharacteristic
155C22×D3—141no—
156C3×D4—21no—
157C3×D4—21no—
158C3×D4—42no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4C22⋊D711yescharacteristic
163C2×C3×C4C22×D711yescharacteristic
164#67—144no—
165C2×Dic3—141no—
166D12—72no—
167D12—141no—
168C4×D3—144no—
169Dic6—72no—
170Dic6—141no—
171C3⋊C8—141no—
172C3⋊C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—211no—
177C2×D4—21no—
178C2×D4—211no—
179C2×D4—421no—
180C2×D4C2×D2111yescharacteristic
181C22×C4—421no—
182C22×C4C2×D2111yescharacteristic
183SD16—422no—
184D8—422no—
185C2×C8—211no—
186C2×C8—211no—
187C4⋊C4C2×D2111yescharacteristic
188C22⋊C4—21no—
189C2×C7—41no—
190C2×C7—81no—
191C2×C7#93211yescharacteristic
192C2×C7#93711yescharacteristic
193C2×C7#95011yescharacteristic
194D7—122no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3C2×C22⋊D711yescharacteristic
202C2×D3—142no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—21no—
206C3×C4—41no—
207C3×C4D4×D711yescharacteristic
208C3×C4D4×D711yescharacteristic
209Dic3—142no—
210C23—21no—
211C23—421no—
212C23#426811yescharacteristic
213Q8—212no—
214Q8—421no—
215D4—21no—
216D4—21no—
217D4—42no—
218D4—212no—
219D4—424no—
220D4—421no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—424no—
225C2×C4—421no—
226C2×C4#426811yescharacteristic
227C2×C4C22×D2111yescharacteristic
228C8—421no—
229C8—421no—
230C7#503711yescharacteristic, complemented, Sylow
231C2×C3—41no—
232C2×C3—81no—
233C2×C3#649511yescharacteristic
234C2×C3#650011yescharacteristic
235C2×C3#651311yescharacteristic
236D3—282no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—422no—
244C22—841no—
245C22—841no—
246C22C2×#426811yescharacteristic
247C4—21no—
248C4—41no—
249C4—422no—
250C4D4×D2111yescharacteristic
251C4D4×D2111yescharacteristic
252C3#8980911yescharacteristic, complemented, Sylow
253C2—41no—
254C2—81no—
255C2—842no—
256C2#1062141511yescharacteristic
257C2#1062142011yescharacteristic
258C2#1062143311yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#121C2
socleC22×C3×C7C2×D4
hypercenter#121D21
hypocenterC3×C7#449
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. C2×#66728C2semidirect
2. C3×C7×#121C2semidirect
3. C2×#66633C2semidirect
4. C2×#66632C2semidirect
5. #10620669C2semidirect
6. #10620649C2semidirect
7. #10620648C2semidirect
25. C7×#121D3semidirect
46. D84D4semidirect
49. Dic42D4semidirect
68. C3×#121D7semidirect
134. #121D21semidirect
173. C3×C7#449semidirect
230. C7#5037semidirect
252. C3#89809semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
C3×C7×#1212
C2×#666332
C2×#666322
#106206692
#106206492
#106206482
#898093
#50377