Finite Group Databasebeta

← #13053894

Subgroups — #13053894

Order 1344.

308 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 112 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×#66629C211yesmaximal, characteristic
2C2×C3×C4⋊Dic7C211yesmaximal, characteristic
3C2×Dic3×Dic7C211yesmaximal, characteristic
4C7×C3⋊#95C211yesmaximal, characteristic
5#10620529C211yesmaximal, characteristic
6#10620529C211yesmaximal, characteristic
7#10620529C211yesmaximal, characteristic
8#89427—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16#66629C414yescomplemented
17C3×C4⋊Dic7—22no—
18Dic3×Dic7—22no—
19C2×C4⋊Dic7D311yescharacteristic
20C2×Dic7⋊C4—31no—
21C2×C4×Dic7—31no—
22C7×#95—31no—
23C7⋊#95—31no—
24C7⋊#95—31no—
25C7⋊#95—31no—
26#4978—71nomaximal, Hall
27C23×C3×C7C2311yescharacteristic
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7C2×C412yes—
31C2×Dic21—21no—
32C2×Dic21—21no—
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21C2×C412yes—
36C2×Dic21C2×C412yes—
37C2×C7×Dic3—21no—
38C2×C7×Dic3—21no—
39C2×C7×Dic3—21no—
40C2×C7×Dic3—21no—
41C2×C7×Dic3—21no—
42C2×C7×Dic3Q812yes—
43C2×C3×Dic7—21no—
44C2×C3×Dic7—21no—
45C2×C3×Dic7—21no—
46C2×C3×Dic7—21no—
47C2×C3×Dic7—21no—
48C2×C3×Dic7D412yes—
49C22×C4×C7—31no—
50C22×C4×C7—31no—
51C22×C4×C7C2×D311yescharacteristic
52C22×Dic7—31no—
53C22×Dic7—31no—
54C22×Dic7C2×D311yescharacteristic
55C22×Dic7C2×D311yescharacteristic
56C4⋊Dic7—22no—
57Dic7⋊C4—34no—
58C4×Dic7—62no—
59C2×C3×C4⋊C4—71no—
60C2×Dic3⋊C4—71no—
61C2×C4×Dic3—71no—
62C3⋊#95—71no—
63C3⋊#95—71no—
64C3⋊#95—71no—
65C3⋊#95D711yescharacteristic
66C22×C3×C7C22×C411yescharacteristic
67C22×C3×C7C2×D411yescharacteristic
68C22×C3×C7C2×Q811yescharacteristic
69C22×C3×C7#4111yescharacteristic
70C22×C3×C7#4111yescharacteristic
71C22×C3×C7#4111yescharacteristic
72C22×C3×C7#4111yescharacteristic
73C3×C4×C7—22no—
74Dic21—22no—
75Dic21—22no—
76C3×Dic7—22no—
77C3×Dic7—41no—
78C7×Dic3—22no—
79C7×Dic3—41no—
80#387—211noSylow
81C23×C7C22×D311yescharacteristic
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7—32no—
85C2×C4×C7—61no—
86C2×C4×C7—61no—
87C2×C4×C7—61no—
88C2×C4×C7—61no—
89C2×C4×C7—61no—
90C2×C4×C7C4×D312yes—
91C2×Dic7—21no—
92C2×Dic7—21no—
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—32no—
97C2×Dic7—32no—
98C2×Dic7—61no—
99C2×Dic7—61no—
100C2×Dic7—61no—
101C2×Dic7—61no—
102C2×Dic7#6712yes—
103C22×C3×C4—71no—
104C22×C3×C4—71no—
105C22×C3×C4C2×D711yescharacteristic
106C22×Dic3—71no—
107C22×Dic3—71no—
108C22×Dic3C2×D711yescharacteristic
109C22×Dic3C2×D711yescharacteristic
110C3×C4⋊C4—142no—
111Dic3⋊C4—74no—
112C4×Dic3—142no—
113C2×C3×C7#11711yescharacteristic
114C2×C3×C7C4×D411yescharacteristic
115C2×C3×C7C4×Q811yescharacteristic
116C2×C3×C7C22⋊Q811yescharacteristic
117C2×C3×C7#12311yescharacteristic
118C2×C3×C7#12511yescharacteristic
119C2×C3×C7#12611yescharacteristic
120C2×C4⋊C4—71no—
121C2×C4⋊C4—211no—
122C2×C42—211no—
123#95—31no—
124#95—211no—
125#95—211no—
126#95—211no—
127C22×C7C2×C4×D311yescharacteristic
128C22×C7#23511yescharacteristic
129C22×C7#23711yescharacteristic
130C22×C7#23711yescharacteristic
131C22×C7D3×Q811yescharacteristic
132C22×C7#23911yescharacteristic
133C22×C7C2×#6711yescharacteristic
134C4×C7—22no—
135C4×C7—62no—
136C4×C7—121no—
137Dic7—22no—
138Dic7—41no—
139Dic7—62no—
140Dic7—62no—
141C23×C3C22×D711yescharacteristic
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—72no—
145C2×C3×C4—141no—
146C2×C3×C4—141no—
147C2×C3×C4—141no—
148C2×C3×C4—141no—
149C2×C3×C4—141no—
150C2×C3×C4C4×D712yes—
151C2×Dic3—21no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—72no—
157C2×Dic3—72no—
158C2×Dic3—141no—
159C2×Dic3—141no—
160C2×Dic3—141no—
161C2×Dic3—141no—
162C2×Dic3Dic1412yes—
163C3×C7#38711yescharacteristic, complemented, Hall
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—71no—
167C22×C4—71no—
168C22×C4—211no—
169C22×C4—211no—
170C22×C4D3×D711yescharacteristic
171C4⋊C4—142no—
172C4⋊C4—214no—
173C42—422no—
174C2×C7#88711yescharacteristic
175C2×C7#88911yescharacteristic
176C2×C7#89211yescharacteristic
177C2×C7#89811yescharacteristic
178C2×C7C4×#6711yescharacteristic
179C2×C7#93511yescharacteristic
180C2×C7#94611yescharacteristic
181C22×C3C2×Dic1411yescharacteristic
182C22×C3C2×C4×D711yescharacteristic
183C22×C3#115711yescharacteristic
184C22×C3D4×D711yescharacteristic
185C22×C3#115911yescharacteristic
186C22×C3#115911yescharacteristic
187C22×C3#116111yescharacteristic
188C3×C4—22no—
189C3×C4—142no—
190C3×C4—281no—
191Dic3—22no—
192Dic3—41no—
193Dic3—142no—
194Dic3—142no—
195C23C2×D3×D711yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—32no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—72no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4—141no—
208C2×C4—141no—
209C2×C4—141no—
210C2×C4—212no—
211C2×C4—212no—
212C2×C4—421no—
213C2×C4—421no—
214C2×C4—421no—
215C2×C4—421no—
216C2×C4D21⋊C412yes—
217C7#497811yescharacteristic, complemented, Sylow
218C2×C3C4×Dic1411yescharacteristic
219C2×C3#644111yescharacteristic
220C2×C3#644211yescharacteristic
221C2×C3(C22⋊D7)⋊C411yescharacteristic
222C2×C3#645311yescharacteristic
223C2×C3#645511yescharacteristic
224C2×C3#645711yescharacteristic
225C22#6666911yescharacteristic
226C22#6667011yescharacteristic
227C22D3×Dic1411yescharacteristic
228C22#6667211yescharacteristic
229C22#6668611yescharacteristic
230C22C2×D21⋊C411yescharacteristic
231C22D7×#6711yescharacteristic
232C4—22no—
233C4—62no—
234C4—121no—
235C4—142no—
236C4—281no—
237C4—422no—
238C4—422no—
239C3#8942711yescharacteristic, complemented, Sylow
240C2#1062090211yescharacteristic
241C2#1062090311yescharacteristic
242C2#1062090811yescharacteristic
243C2#1062090911yescharacteristic
244C2#1062096911yescharacteristic
245C2#1062097211yescharacteristic
246C2#1062111411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutatorC22×C3×C7C22×C4
FittingC22×C3×C4×C7C22
socle, prime radical, cosocleC23×C3×C7C23
hypercenterC22×C4D3×D7
hypocenterC3×C7#387
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
16. #66629C4semidirect
163. C3×C7#387semidirect
217. C7#4978semidirect
239. C3#89427semidirect

Maximal subgroups

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

subgroupindex
C2×#666292
C2×C3×C4⋊Dic72
C2×Dic3×Dic72
C7×C3⋊#952
#106205292
#894273
#49787