Finite Group Databasebeta

← #13058536

Subgroups — #13058536

Order 1344.

384 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#10621373C211yesmaximal, characteristic, complemented
2C7×#887C211yesmaximal, characteristic, complemented
3C3×#6460C211yesmaximal, characteristic
4#10620947C211yesmaximal, characteristic, complemented
5#10620943C212yesmaximal, complemented
6#10620925C212yesmaximal
7#10620924C212yesmaximal
8#10620921C211yesmaximal, characteristic
9#10620919C211yesmaximal, characteristic
10#10620901C212yesmaximal, complemented
11Dic3×Dic14C211yesmaximal, characteristic, complemented
12#90274—31nomaximal, Hall
13C2×C7×Dic6C2211yescharacteristic
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×Dic14C2211yescharacteristic
16C3×C7×C4⋊C4C2211yescharacteristic
17C4⋊Dic21C2212yes—
18C4⋊Dic21C2211yescharacteristic
19#66629C2212yes—
20C4×Dic21C2211yescharacteristic
21C7×Dic3⋊C4C2212yes—
22C4×C7×Dic3C2212yes—
23C4×C7×Dic3C2211yescharacteristic
24C3×(C2×D7)⋊C4C2212yes—
25C3×C4⋊Dic7C2212yes—
26#66577C2212yes—
27#66577C2212yes—
28#66577C2212yes—
29Dic7⋊Dic3C2211yescharacteristic
30Dic7⋊Dic3C2212yes—
31Dic7⋊Dic3C2211yescharacteristic
32(C2×D7)⋊Dic3C2211yescharacteristic
33(C2×D7)⋊Dic3C2211yescharacteristic
34(C2×D7)⋊Dic3C2212yes—
35Dic3×Dic7C2212yes—
36C4×C7×Q8—31no—
37#6509—31no—
38#6507—31no—
39#6461—32no—
40#6460D311yescharacteristic
41#6453—31no—
42#6452—32no—
43#6439—32no—
44#6435—31no—
45#6433—32no—
46C4×Dic14—31no—
47#5625—71nomaximal, Hall
48C22×C3×D7C2311yescharacteristic
49C2×C3×C4×C7C2312yes—
50C2×C3×C4×C7C2311yescharacteristic
51C2×Dic21C2312yes—
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2311yescharacteristic
54C2×C7×Dic3C2311yescharacteristic
55C2×C7×Dic3C2311yescharacteristic
56C2×C7×Dic3C2312yes—
57C7×Dic6—22no—
58C2×C3×Dic7C2312yes—
59C2×C3×Dic7C2311yescharacteristic
60C3×C4×D7—22no—
61C3×Dic14—22no—
62C2×C7×Q8—31no—
63C2×C4×D7C2×D311yescharacteristic
64C2×Dic14C2×D311yescharacteristic
65C7×C4⋊C4—32no—
66C7×C4⋊C4C2×D311yescharacteristic
67C42×C7—31no—
68C42×C7—32no—
69(C2×D7)⋊C4—31no—
70(C2×D7)⋊C4—31no—
71(C2×D7)⋊C4—32no—
72(C2×D7)⋊C4C2×D312yes—
73C4⋊Dic7—31no—
74C4⋊Dic7—32no—
75C4⋊Dic7C2×D312yes—
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78Dic7⋊C4—31no—
79Dic7⋊C4—31no—
80Dic7⋊C4—32no—
81Dic7⋊C4—32no—
82Dic7⋊C4—32no—
83C4×Dic7—31no—
84C4×Dic7—32no—
85C3×C22⋊Q8—71no—
86Q8×Dic3—71no—
87#944—71no—
88#926—71no—
89#924—71no—
90#890—71no—
91#890—72no—
92#889—72no—
93#887D711yescharacteristic, complemented
94#879—72no—
95#879—72no—
96C22×C3×C7C2411yescharacteristic
97C2×C3×D7—21no—
98C2×C3×D7—21no—
99C2×C3×D7—21no—
100C3×C4×C7—22no—
101C3×C4×C7#4112yes—
102Dic21—22no—
103Dic21—21no—
104Dic21—21no—
105C3×Dic7—22no—
106C3×Dic7—21no—
107C7×Dic3—22no—
108C7×Dic3—21no—
109C7×Dic3—21no—
110#541—211noSylow
111C22×D7C22×D311yescharacteristic
112C7×Q8—62no—
113C2×C4×C7—31no—
114C2×C4×C7—31no—
115C2×C4×C7—32no—
116C2×C4×C7C22×D312yes—
117C2×C4×C7C22×D311yescharacteristic
118C2×Dic7—31no—
119C2×Dic7—31no—
120C2×Dic7—32no—
121C2×Dic7C22×D312yes—
122C2×Dic7C22×D311yescharacteristic
123C4×D7—22no—
124Dic14—22no—
125C2×C3×Q8—71no—
126C22×C3×C4—71no—
127C2×Dic6C2×D711yescharacteristic
128C3×C4⋊C4—72no—
129C3×C4⋊C4C2×D711yescharacteristic
130C3×C22⋊C4—72no—
131#217—72no—
132#217—71no—
133#217—71no—
134C4⋊Dic3—72no—
135C4⋊Dic3—71no—
136C4⋊Dic3—72no—
137C4⋊Dic3—71no—
138C4⋊Dic3—71no—
139Dic3⋊C4—72no—
140Dic3⋊C4—72no—
141Dic3⋊C4—72no—
142Dic3⋊C4—72no—
143Dic3⋊C4C2×D712yes—
144C4×Dic3—71no—
145C4×Dic3—72no—
146C4×Dic3C2×D712yes—
147C4×Dic3C2×D711yescharacteristic
148C2×C3×C7C2×#4111yescharacteristic
149C2×C3×C72−1+411yescharacteristic
150C2×C3×C72−1+411yescharacteristic
151C3×D7—41no—
152C4⋊Q8—211no—
153#126—212no—
154#126—212no—
155#125—212no—
156#125—212no—
157#125—211no—
158C22⋊Q8—71no—
159C22⋊Q8—211no—
160C4×Q8—31no—
161C4×Q8—211no—
162#117—211no—
163C22×C7C23×D311yescharacteristic
164C2×D7—21no—
165C2×D7—21no—
166C2×D7—21no—
167C4×C7—22no—
168C4×C7—62no—
169C4×C7—61no—
170C4×C7—61no—
171C4×C7#23712yes—
172Dic7—22no—
173Dic7—21no—
174Dic7—61no—
175Dic7—62no—
176Dic7—61no—
177C23×C3—71no—
178C3×Q8—142no—
179C2×C3×C4—72no—
180C2×C3×C4—71no—
181C2×C3×C4—142no—
182C2×C3×C4C22×D712yes—
183C2×C3×C4C22×D711yescharacteristic
184C2×Dic3—71no—
185C2×Dic3—72no—
186C2×Dic3—71no—
187C2×Dic3C22×D711yescharacteristic
188C2×Dic3C22×D711yescharacteristic
189C2×Dic3C22×D712yes—
190Dic6—22no—
191C3×C7#54111yescharacteristic, complemented, Hall
192C2×Q8—31no—
193C2×Q8—71no—
194C22×C4—71no—
195C4⋊C4—32no—
196C4⋊C4—72no—
197C4⋊C4—211no—
198C4⋊C4—211no—
199C4⋊C4—212no—
200C4⋊C4—212no—
201C4⋊C4—212no—
202C4⋊C4—212no—
203C4⋊C4—212no—
204C4⋊C4—211no—
205C4⋊C4—212no—
206C4⋊C4D3×D711yescharacteristic
207C22⋊C4—72no—
208C22⋊C4—211no—
209C22⋊C4—211no—
210C22⋊C4—212no—
211C42—31no—
212C42—32no—
213C42—212no—
214C42—211no—
215C2×C7C2×#23711yescharacteristic
216C2×C7#100711yescharacteristic
217C2×C7#101011yescharacteristic
218D7—41no—
219C22×C3—141no—
220C22×C3—141no—
221C22×C3—141no—
222C22×C3C23×D711yescharacteristic
223C3×C4—22no—
224C3×C4—141no—
225C3×C4—142no—
226C3×C4#115712yes—
227Dic3—22no—
228Dic3—21no—
229Dic3—21no—
230Dic3—142no—
231Dic3—141no—
232Dic3—141no—
233C23—71no—
234Q8—62no—
235Q8—142no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—32no—
239C2×C4—72no—
240C2×C4—71no—
241C2×C4—142no—
242C2×C4—211no—
243C2×C4—212no—
244C2×C4—211no—
245C2×C4C2×D3×D712yes—
246C2×C4C2×D3×D711yescharacteristic
247C7#562511yescharacteristic, complemented, Sylow
248C2×C3—281no—
249C2×C3C2×#115711yescharacteristic
250C2×C3#655111yescharacteristic
251C2×C3#655411yescharacteristic
252C22—141no—
253C22—141no—
254C22—141no—
255C22C22×D3×D711yescharacteristic
256C4—22no—
257C4—62no—
258C4—61no—
259C4—61no—
260C4—141no—
261C4—142no—
262C4—421no—
263C4—422no—
264C4—421no—
265C4#6666912yes—
266C3#9027411yescharacteristic, complemented, Sylow
267C2—281no—
268C2C2×#6666911yescharacteristic
269C2#1062156711yescharacteristic
270C2#1062157611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC22×C3×D7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#541
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621373C2semidirect
2. C7×#887C2semidirect
4. #10620947C2semidirect
5. #10620943C2semidirect
10. #10620901C2semidirect
11. Dic3×Dic14C2semidirect
93. #887D7semidirect
191. C3×C7#541semidirect
247. C7#5625semidirect
266. C3#90274semidirect

Maximal subgroups

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

subgroupindex
#106213732
C7×#8872
C3×#64602
#106209472
#106209432
#106209252
#106209242
#106209212
#106209192
#106209012
Dic3×Dic142
#902743
#56257