Finite Group Databasebeta

← #13054143

Subgroups — #13054143

Order 1344.

300 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 100 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×C7×Dic3⋊C4C211yesmaximal, characteristic
3C2×#66577C211yesmaximal, characteristic
4C2×Dic7⋊Dic3C211yesmaximal, characteristic
5C3×C7⋊#95C211yesmaximal, characteristic
6#10620529C211yesmaximal, characteristic
7#10620529C211yesmaximal, characteristic
8#89752—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16#66629—22no—
17C7×Dic3⋊C4—22no—
18#66577—22no—
19Dic7⋊Dic3—22no—
20C2×C7×C4⋊C4—31no—
21C2×Dic7⋊C4—31no—
22C2×Dic7⋊C4—31no—
23C2×Dic7⋊C4—31no—
24C7⋊#95—31no—
25C7⋊#95—31no—
26C7⋊#95D311yescharacteristic
27#4657—71nomaximal, Hall
28C23×C3×C7C2311yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×Dic21—21no—
33C2×Dic21—21no—
34C2×Dic21—21no—
35C2×Dic21—21no—
36C2×Dic21D412yes—
37C2×Dic21Q812yes—
38C2×C7×Dic3—21no—
39C2×C7×Dic3—21no—
40C2×C7×Dic3—21no—
41C2×C7×Dic3—21no—
42C2×C7×Dic3—21no—
43C2×C7×Dic3D412yes—
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—
58Dic7⋊C4—62no—
59Dic7⋊C4—62no—
60Dic7⋊C4—62no—
61C2×C4⋊Dic3—71no—
62C2×Dic3⋊C4—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—41no—
76Dic21—22no—
77Dic21—22no—
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—21no—
87C2×C4×C7—32no—
88C2×C4×C7—61no—
89C2×C4×C7—61no—
90C2×C4×C7—61no—
91C2×C4×C7—61no—
92C2×C4×C7—61no—
93C2×Dic7—21no—
94C2×Dic7—21no—
95C2×Dic7—21no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—32no—
99C2×Dic7—32no—
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
112C4⋊Dic3—142no—
113Dic3⋊C4—22no—
114Dic3⋊C4—142no—
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—211no—
125C2×C4⋊C4—211no—
126C2×C4⋊C4—211no—
127#95—71no—
128#95—211no—
129#95—211no—
130C22×C7C2×Dic611yescharacteristic
131C22×C7#23511yescharacteristic
132C22×C7#23511yescharacteristic
133C22×C7D3×D411yescharacteristic
134C22×C7D3×D411yescharacteristic
135C22×C7#23711yescharacteristic
136C22×C7D3×Q811yescharacteristic
137C4×C7—41no—
138C4×C7—62no—
139C4×C7—121no—
140Dic7—22no—
141Dic7—41no—
142Dic7—62no—
143Dic7—62no—
144C23×C3C22×D711yescharacteristic
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—72no—
149C2×C3×C4—141no—
150C2×C3×C4—141no—
151C2×C3×C4—141no—
152C2×C3×C4—141no—
153C2×C3×C4—141no—
154C2×Dic3—21no—
155C2×Dic3—21no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—72no—
160C2×Dic3—72no—
161C2×Dic3—141no—
162C2×Dic3—141no—
163C2×Dic3—141no—
164C2×Dic3—141no—
165C2×Dic3C22⋊D712yes—
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—422no—
176C4⋊C4—422no—
177C4⋊C4—422no—
178C2×C7#87011yescharacteristic
179C2×C7#87811yescharacteristic
180C2×C7#88311yescharacteristic
181C2×C7#88411yescharacteristic
182C2×C7#88811yescharacteristic
183C2×C7#88911yescharacteristic
184C2×C7#89611yescharacteristic
185C22×C3#115711yescharacteristic
186C22×C3#115711yescharacteristic
187C22×C3D4×D711yescharacteristic
188C22×C3#115911yescharacteristic
189C22×C3Q8×D711yescharacteristic
190C22×C3Q8×D711yescharacteristic
191C22×C3C2×C22⋊D711yescharacteristic
192C3×C4—41no—
193C3×C4—142no—
194C3×C4—281no—
195Dic3—22no—
196Dic3—41no—
197Dic3—142no—
198Dic3—142no—
199C23C2×D3×D711yescharacteristic
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—32no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—72no—
210C2×C4—141no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4—212no—
216C2×C4—212no—
217C2×C4—421no—
218C2×C4—421no—
219C2×C4—421no—
220C2×C4—421no—
221C7#465711yescharacteristic, complemented, Sylow
222C2×C3#645211yescharacteristic
223C2×C3#645211yescharacteristic
224C2×C3#645911yescharacteristic
225C2×C3#645911yescharacteristic
226C2×C3#649211yescharacteristic
227C2×C3#650111yescharacteristic
228C2×C3#650711yescharacteristic
229C22D7×Dic611yescharacteristic
230C22#6666911yescharacteristic
231C22D21⋊Q811yescharacteristic
232C22#6667511yescharacteristic
233C22#6668611yescharacteristic
234C22D3×C22⋊D711yescharacteristic
235C22#6669411yescharacteristic
236C4—41no—
237C4—62no—
238C4—121no—
239C4—142no—
240C4—281no—
241C4—422no—
242C4—422no—
243C3#8975211yescharacteristic, complemented, Sylow
244C2#1062089111yescharacteristic
245C2#1062090111yescharacteristic
246C2#1062093911yescharacteristic
247C2#1062094011yescharacteristic
248C2#1062099311yescharacteristic
249C2#1062101411yescharacteristic
250C2#1062111911yescharacteristic
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#4657semidirect
243. C3#89752semidirect

Maximal subgroups

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

subgroupindex
C2×#666292
C2×C7×Dic3⋊C42
C2×#665772
C2×Dic7⋊Dic32
C3×C7⋊#952
#106205292
#897523
#46577