Finite Group Databasebeta

← #13054147

Subgroups — #13054147

Order 1344.

300 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 106 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×C4⋊Dic21C211yesmaximal, characteristic
2C2×C7×Dic3⋊C4C211yesmaximal, characteristic
3C2×C3×Dic7⋊C4C211yesmaximal, characteristic
4C2×#66577C211yesmaximal, characteristic
5#10620529C211yesmaximal, characteristic
6#10620529C211yesmaximal, characteristic
7#10620529C211yesmaximal, characteristic
8#89756—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C4⋊Dic21—22no—
17C7×Dic3⋊C4—22no—
18C3×Dic7⋊C4—22no—
19#66577—22no—
20C2×C7×C4⋊C4—31no—
21C2×C4⋊Dic7—31no—
22C2×Dic7⋊C4—31no—
23C2×Dic7⋊C4D311yescharacteristic
24C7⋊#95—31no—
25C7⋊#95—31no—
26C7⋊#95—31no—
27#4982—71nomaximal, Hall
28C23×C3×C7C2311yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7D412yes—
32C2×Dic21—21no—
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21—21no—
36C2×Dic21—21no—
37C2×Dic21D412yes—
38C2×C7×Dic3—21no—
39C2×C7×Dic3—21no—
40C2×C7×Dic3—21no—
41C2×C7×Dic3—21no—
42C2×C7×Dic3—21no—
43C2×C7×Dic3Q812yes—
44C2×C3×Dic7—21no—
45C2×C3×Dic7—21no—
46C2×C3×Dic7—21no—
47C2×C3×Dic7—21no—
48C2×C3×Dic7—21no—
49C2×C3×Dic7Q812yes—
50C22×C4×C7—31no—
51C22×C4×C7—31no—
52C22×C4×C7C2×D311yescharacteristic
53C22×Dic7—31no—
54C22×Dic7—31no—
55C22×Dic7C2×D311yescharacteristic
56C22×Dic7C2×D311yescharacteristic
57C7×C4⋊C4—62no—
58C4⋊Dic7—62no—
59Dic7⋊C4—22no—
60Dic7⋊C4—62no—
61C2×C3×C4⋊C4—71no—
62C2×C4⋊Dic3—71no—
63C2×Dic3⋊C4—71no—
64C2×Dic3⋊C4D711yescharacteristic
65C3⋊#95—71no—
66C3⋊#95—71no—
67C3⋊#95—71no—
68C22×C3×C7C2×D411yescharacteristic
69C22×C3×C7C2×D411yescharacteristic
70C22×C3×C7C2×Q811yescharacteristic
71C22×C3×C7C2×Q811yescharacteristic
72C22×C3×C7#4111yescharacteristic
73C22×C3×C7#4111yescharacteristic
74C22×C3×C7#4111yescharacteristic
75C3×C4×C7—22no—
76Dic21—22no—
77Dic21—41no—
78C3×Dic7—22no—
79C3×Dic7—41no—
80C7×Dic3—22no—
81C7×Dic3—41no—
82#398—211noSylow
83C23×C7C22×D311yescharacteristic
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—32no—
87C2×C4×C7—61no—
88C2×C4×C7—61no—
89C2×C4×C7—61no—
90C2×C4×C7—61no—
91C2×C4×C7—61no—
92C2×C4×C7#6712yes—
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—32no—
99C2×Dic7—61no—
100C2×Dic7—61no—
101C2×Dic7—61no—
102C2×Dic7—61no—
103C2×Dic7—61no—
104C2×Dic7Dic612yes—
105C22×C3×C4—71no—
106C22×C3×C4—71no—
107C22×C3×C4C2×D711yescharacteristic
108C22×Dic3—71no—
109C22×Dic3—71no—
110C22×Dic3C2×D711yescharacteristic
111C22×Dic3C2×D711yescharacteristic
112C3×C4⋊C4—142no—
113C4⋊Dic3—142no—
114Dic3⋊C4—22no—
115Dic3⋊C4—142no—
116C2×C3×C7#12111yescharacteristic
117C2×C3×C7C22⋊Q811yescharacteristic
118C2×C3×C7C22⋊Q811yescharacteristic
119C2×C3×C7#12311yescharacteristic
120C2×C3×C7#12511yescharacteristic
121C2×C3×C7#12511yescharacteristic
122C2×C3×C7C4⋊Q811yescharacteristic
123C2×C4⋊C4—31no—
124C2×C4⋊C4—71no—
125C2×C4⋊C4—211no—
126C2×C4⋊C4—211no—
127#95—211no—
128#95—211no—
129#95—211no—
130C22×C7C2×Dic611yescharacteristic
131C22×C7#23511yescharacteristic
132C22×C7D3×D411yescharacteristic
133C22×C7#23711yescharacteristic
134C22×C7D3×Q811yescharacteristic
135C22×C7#23911yescharacteristic
136C22×C7C2×#6711yescharacteristic
137C4×C7—22no—
138C4×C7—62no—
139C4×C7—121no—
140Dic7—22no—
141Dic7—41no—
142Dic7—62no—
143Dic7—121no—
144C23×C3C22×D711yescharacteristic
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—72no—
148C2×C3×C4—141no—
149C2×C3×C4—141no—
150C2×C3×C4—141no—
151C2×C3×C4—141no—
152C2×C3×C4—141no—
153C2×C3×C4C22⋊D712yes—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—72no—
160C2×Dic3—141no—
161C2×Dic3—141no—
162C2×Dic3—141no—
163C2×Dic3—141no—
164C2×Dic3—141no—
165C2×Dic3Dic1412yes—
166C3×C7#39811yescharacteristic, complemented, Hall
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—71no—
170C22×C4—71no—
171C22×C4—211no—
172C22×C4—211no—
173C22×C4D3×D711yescharacteristic
174C4⋊C4—62no—
175C4⋊C4—142no—
176C4⋊C4—422no—
177C4⋊C4—422no—
178C2×C7#88811yescharacteristic
179C2×C7#88911yescharacteristic
180C2×C7#89011yescharacteristic
181C2×C7#89411yescharacteristic
182C2×C7#92411yescharacteristic
183C2×C7#93911yescharacteristic
184C2×C7#94611yescharacteristic
185C22×C3C2×Dic1411yescharacteristic
186C22×C3#115711yescharacteristic
187C22×C3D4×D711yescharacteristic
188C22×C3#115911yescharacteristic
189C22×C3Q8×D711yescharacteristic
190C22×C3#116111yescharacteristic
191C22×C3C2×C22⋊D711yescharacteristic
192C3×C4—22no—
193C3×C4—142no—
194C3×C4—281no—
195Dic3—22no—
196Dic3—41no—
197Dic3—142no—
198Dic3—281no—
199C23C2×D3×D711yescharacteristic
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—32no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—72no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—212no—
215C2×C4—421no—
216C2×C4—421no—
217C2×C4—421no—
218C2×C4—421no—
219C2×C4—421no—
220C2×C4#424512yes—
221C7#498211yescharacteristic, complemented, Sylow
222C2×C3#645111yescharacteristic
223C2×C3#645211yescharacteristic
224C2×C3#645311yescharacteristic
225C2×C3#645711yescharacteristic
226C2×C3#648711yescharacteristic
227C2×C3#650211yescharacteristic
228C2×C3#650911yescharacteristic
229C22D7×Dic611yescharacteristic
230C22D3×Dic1411yescharacteristic
231C22#6667311yescharacteristic
232C22#6667711yescharacteristic
233C22#6668411yescharacteristic
234C22C2×#424511yescharacteristic
235C22#6669411yescharacteristic
236C4—22no—
237C4—62no—
238C4—121no—
239C4—142no—
240C4—281no—
241C4—422no—
242C4—841no—
243C3#8975611yescharacteristic, complemented, Sylow
244C2#1062089011yescharacteristic
245C2#1062089611yescharacteristic
246C2#1062090511yescharacteristic
247C2#1062092111yescharacteristic
248C2#1062093111yescharacteristic
249C2#1062099511yescharacteristic
250C2#1062113311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#398
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
166. C3×C7#398semidirect
221. C7#4982semidirect
243. C3#89756semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊Dic212
C2×C7×Dic3⋊C42
C2×C3×Dic7⋊C42
C2×#665772
#106205292
#897563
#49827