Finite Group Databasebeta

← #13056370

Subgroups — #13056370

Order 1344.

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

normal subgroups only

This group has 304 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
2#10621407C211yesmaximal, characteristic, complemented
3C2×Dic84C211yesmaximal, characteristic, complemented
4C2×#66623C211yesmaximal, characteristic, complemented
5C3×C7×C22⋊C8C211yesmaximal, characteristic, complemented
6#10620660C211yesmaximal, characteristic, complemented
7#10620655C211yesmaximal, characteristic, complemented
8#89505—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×#4268—21no—
11#66728—22no—
12#66728—22no—
13C2×D84C2211yescharacteristic
14C2×C4×D21—21no—
15C2×Dic42C2211yescharacteristic
16C2×Dic42C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C2×C3×C7×C8C2211yescharacteristic
19#66636—21no—
20C4⋊Dic21C2211yescharacteristic
21#66629—21no—
22Dic84—22no—
23#66623—22no—
24C2×#1157—31no—
25#6487—31no—
26C2×Dic28—31no—
27C2×#1132—31no—
28C7×C22⋊C8D311yescharacteristic, complemented
29#6395—31no—
30#6390—31no—
31#4733—71nomaximal, Hall
32C23×C3×C7D411yescharacteristic
33C22×D21—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic
37C2×C3×C4×C7C2311yescharacteristic
38#4268—24no—
39C2×Dic21—21no—
40C2×Dic21—21no—
41C2×Dic21—21no—
42D84—21no—
43D84D412yes—
44C4×D21—24no—
45Dic42—21no—
46Dic42—21no—
47Dic42—21no—
48Dic42D412yes—
49C3×C7×C8—21no—
50C3×C7×C8—21no—
51C22×C4×C7C2×D311yescharacteristic
52C2×C22⋊D7—61no—
53#1157—62no—
54#1157—62no—
55C2×D28—31no—
56C2×C4×D7—61no—
57C2×Dic14—31no—
58C2×Dic14—31no—
59C2×C7×C8C2×D311yescharacteristic
60C2×C7×C8C2×D311yescharacteristic
61C22⋊Dic7—61no—
62C4⋊Dic7—31no—
63Dic7⋊C4—61no—
64Dic28—62no—
65#1132—62no—
66C2×#235—71no—
67#924—71no—
68C2×Dic12—71no—
69C2×#204—71no—
70C3×C22⋊C8D711yescharacteristic, complemented
71#821—71no—
72#816—71no—
73C22×C3×C7—21no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7C2×D411yescharacteristic
77C2×D21—22no—
78C2×D21—41no—
79C2×D21—41no—
80C3×C4×C7—21no—
81C3×C4×C7C2×D411yescharacteristic
82C3×C4×C7C2×D411yescharacteristic
83Dic21—22no—
84Dic21—41no—
85Dic21—41no—
86#452—211noSylow
87C23×C7D1211yescharacteristic
88C22×D7—61no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7D1211yescharacteristic
92C2×C4×C7C22×D311yescharacteristic
93C22⋊D7—64no—
94C2×Dic7—61no—
95C2×Dic7—61no—
96C2×Dic7—61no—
97D28—32no—
98D28—61no—
99C4×D7—64no—
100Dic14—32no—
101Dic14—61no—
102Dic14—61no—
103Dic14—61no—
104C7×C8—21no—
105C7×C8—21no—
106C22×C3×C4C2×D711yescharacteristic
107C2×#67—141no—
108#235—142no—
109#235—142no—
110C2×D12—71no—
111C2×C4×D3—141no—
112C2×Dic6—71no—
113C2×Dic6—71no—
114C2×C3×C8C2×D711yescharacteristic
115C2×C3×C8C2×D711yescharacteristic
116#217—141no—
117C4⋊Dic3—71no—
118Dic3⋊C4—141no—
119Dic12—142no—
120#204—142no—
121C2×C3×C7—41no—
122C2×C3×C7#12011yescharacteristic
123C2×C3×C7#13511yescharacteristic
124C2×C3×C7#13711yescharacteristic
125D21—42no—
126C2×#41—211no—
127C2×Q16—211no—
128C2×SD16—211no—
129C22⋊Q8—211no—
130#103—211no—
131#102—211no—
132C22⋊C8D2111yescharacteristic, complemented
133C22×C7—21no—
134C22×C7—21no—
135C22×C7—21no—
136C22×C7C2×D1211yescharacteristic
137C2×D7—62no—
138C2×D7—121no—
139C2×D7—121no—
140C4×C7—21no—
141C4×C7D3×D411yescharacteristic
142C4×C7D3×D411yescharacteristic
143Dic7—62no—
144Dic7—121no—
145Dic7—121no—
146C23×C3D2811yescharacteristic
147C22×D3—141no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4D2811yescharacteristic
151C2×C3×C4C22×D711yescharacteristic
152#67—144no—
153C2×Dic3—141no—
154C2×Dic3—141no—
155C2×Dic3—141no—
156D12—72no—
157D12—141no—
158C4×D3—144no—
159Dic6—72no—
160Dic6—141no—
161Dic6—141no—
162Dic6—141no—
163C3×C8—21no—
164C3×C8—21no—
165C3×C7#45211yescharacteristic, complemented, Hall
166#41—422no—
167#41—422no—
168C2×Q8—211no—
169C2×Q8—211no—
170C2×D4—211no—
171C2×D4—421no—
172C22×C4—421no—
173C22×C4C2×D2111yescharacteristic
174Q16—422no—
175SD16—422no—
176C2×C8C2×D2111yescharacteristic
177C2×C8C2×D2111yescharacteristic
178C4⋊C4—211no—
179C4⋊C4—421no—
180C22⋊C4—421no—
181C2×C7—41no—
182C2×C7#88211yescharacteristic
183C2×C7#90411yescharacteristic
184C2×C7#90911yescharacteristic
185D7—122no—
186C22×C3—21no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3C2×D2811yescharacteristic
190C2×D3—142no—
191C2×D3—281no—
192C2×D3—281no—
193C3×C4—21no—
194C3×C4D4×D711yescharacteristic
195C3×C4D4×D711yescharacteristic
196Dic3—142no—
197Dic3—281no—
198Dic3—281no—
199C23—421no—
200C23D8411yescharacteristic
201Q8—212no—
202Q8—421no—
203Q8—421no—
204Q8—421no—
205D4—212no—
206D4—424no—
207D4—421no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—424no—
211C2×C4—421no—
212C2×C4—421no—
213C2×C4—421no—
214C2×C4D8411yescharacteristic
215C2×C4C22×D2111yescharacteristic
216C8—21no—
217C8—21no—
218C7#473311yescharacteristic, complemented, Sylow
219C2×C3—41no—
220C2×C3#644511yescharacteristic
221C2×C3#646711yescharacteristic
222C2×C3#647211yescharacteristic
223D3—282no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—422no—
228C22—841no—
229C22—841no—
230C22C2×D8411yescharacteristic
231C4—21no—
232C4—422no—
233C4—841no—
234C4—841no—
235C4D4×D2111yescharacteristic
236C4D4×D2111yescharacteristic
237C3#8950511yescharacteristic, complemented, Sylow
238C2—41no—
239C2—842no—
240C2#1062136511yescharacteristic
241C2#1062138711yescharacteristic
242C2#1062139211yescharacteristic
subgroupG/H
centerC22C2×D84
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊C8C2
socleC22×C3×C7C2×D4
prime radicalC2×#66728C2
hypercenterC22⋊C8D21
hypocenterC3×C7#452
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #10621407C2semidirect
3. C2×Dic84C2semidirect
4. C2×#66623C2semidirect
5. C3×C7×C22⋊C8C2semidirect
6. #10620660C2semidirect
7. #10620655C2semidirect
28. C7×C22⋊C8D3semidirect
70. C3×C22⋊C8D7semidirect
132. C22⋊C8D21semidirect
165. C3×C7#452semidirect
218. C7#4733semidirect
237. C3#89505semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
#106214072
C2×Dic842
C2×#666232
C3×C7×C22⋊C82
#106206602
#106206552
#895053
#47337