Finite Group Databasebeta

← #13058754

Subgroups — #13058754

Order 1344.

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

normal subgroups only

This group has 432 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#10621370C211yesmaximal, characteristic, complemented
2C7×#895C211yesmaximal, characteristic
3C3×#6451C211yesmaximal, characteristic, complemented
4#10620997C211yesmaximal, characteristic, complemented
5#10620985C212yesmaximal, complemented
6#10620972C212yesmaximal, complemented
7#10620935C211yesmaximal, characteristic, complemented
8#10620933C211yesmaximal, characteristic, complemented
9#10620932C212yesmaximal, complemented
10#10620929C212yesmaximal, complemented
11#10620927C211yesmaximal, characteristic, complemented
12#90294—31nomaximal, Hall
13C2×Dic42C2211yescharacteristic
14C2×C7×D12C2211yescharacteristic
15C2×C7×D12C2212yes—
16C2×C4×D3×C7C2211yescharacteristic
17C2×C3×Dic14C2212yes—
18C2×D3×Dic7C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20#66629C2212yes—
21C4×Dic21C2212yes—
22C4×Dic21C2211yescharacteristic
23C7×(C2×D3)⋊C4C2212yes—
24C3×C4⋊Dic7C2211yescharacteristic
25C3×Dic7⋊C4C2212yes—
26C3×C4×Dic7C2211yescharacteristic
27Dic3⋊Dic7C2211yescharacteristic
28Dic3⋊Dic7C2211yescharacteristic
29Dic7⋊Dic3C2212yes—
30(C2×D3)⋊Dic7C2212yes—
31(C2×D3)⋊Dic7C2212yes—
32(C2×D3)⋊Dic7C2211yescharacteristic
33(C2×D3)⋊Dic7C2212yes—
34(C2×D3)⋊Dic7C2212yes—
35(C2×D3)⋊Dic7C2211yescharacteristic
36C7×#121—31no—
37#6499—32no—
38#6499—31no—
39#6498—32no—
40D4×Dic7—31no—
41#6489—31no—
42#6487—31no—
43#6451D311yescharacteristic, complemented
44#6450—31no—
45#6442—32no—
46#6441—32no—
47#5733—71nomaximal, Hall
48C22×D3×C7C2311yescharacteristic
49C22×D3×C7C2312yes—
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2312yes—
53C2×Dic21C2311yescharacteristic
54C2×Dic21C2311yescharacteristic
55Dic42—22no—
56C2×C7×Dic3C2311yescharacteristic
57C7×D12—22no—
58C7×D12—42no—
59C4×D3×C7—22no—
60C2×C3×Dic7C2311yescharacteristic
61C2×C3×Dic7C2311yescharacteristic
62C2×C3×Dic7C2312yes—
63C3×Dic14—42no—
64D3×Dic7—42no—
65C2×C7×D4—31no—
66C2×C7×D4—32no—
67C22×C4×C7—31no—
68C22×Dic7—32no—
69C2×Dic14—31no—
70C2×Dic14C2×D312yes—
71C7×C4⋊C4C2×D311yescharacteristic
72C7×C22⋊C4—32no—
73C22⋊Dic7—32no—
74C22⋊Dic7—32no—
75C22⋊Dic7—32no—
76C22⋊Dic7—32no—
77C22⋊Dic7—31no—
78C22⋊Dic7—31no—
79C4⋊Dic7—31no—
80C4⋊Dic7—31no—
81C4⋊Dic7C2×D311yescharacteristic
82Dic7⋊C4—32no—
83Dic7⋊C4—32no—
84Dic7⋊C4C2×D312yes—
85C4×Dic7—32no—
86C4×Dic7—31no—
87C4×Dic7C2×D311yescharacteristic
88C3×C4⋊Q8—71no—
89#947—72no—
90#946—72no—
91#898—72no—
92#897—71no—
93#895D711yescharacteristic
94#894—72no—
95#893—71no—
96#887—71no—
97#875—71no—
98#872—71no—
99C22×C3×C7C2411yescharacteristic
100C2×D3×C7—21no—
101C2×D3×C7—22no—
102C2×D3×C7—22no—
103C2×D3×C7—21no—
104C2×D3×C7—21no—
105C2×D3×C7—22no—
106C3×C4×C7—22no—
107C3×C4×C7#4112yes—
108Dic21—22no—
109Dic21—21no—
110Dic21—21no—
111C3×Dic7—22no—
112C3×Dic7—21no—
113C3×Dic7—21no—
114C7×Dic3—21no—
115#542—211noSylow
116C23×C7—31no—
117C23×C7—32no—
118C7×D4—62no—
119C7×D4—122no—
120C2×C4×C7—31no—
121C2×C4×C7—62no—
122C2×C4×C7C22×D312yes—
123C2×C4×C7C22×D311yescharacteristic
124C2×Dic7—31no—
125C2×Dic7—31no—
126C2×Dic7—32no—
127C2×Dic7—122no—
128C2×Dic7C22×D311yescharacteristic
129C2×Dic7C22×D311yescharacteristic
130C2×Dic7C22×D312yes—
131Dic14—42no—
132Dic14—62no—
133C2×C3×Q8—72no—
134C2×D12C2×D711yescharacteristic
135C2×D12C2×D712yes—
136C2×C4×D3—72no—
137C2×C4×D3C2×D711yescharacteristic
138C2×Dic6—71no—
139C3×C4⋊C4—72no—
140C3×C4⋊C4—71no—
141C3×C4⋊C4C2×D711yescharacteristic
142C3×C42—71no—
143(C2×D3)⋊C4—71no—
144(C2×D3)⋊C4—71no—
145(C2×D3)⋊C4—72no—
146(C2×D3)⋊C4—72no—
147(C2×D3)⋊C4—72no—
148(C2×D3)⋊C4—72no—
149(C2×D3)⋊C4C2×D712yes—
150C4⋊Dic3—72no—
151Dic3⋊C4—72no—
152Dic3⋊C4—71no—
153Dic3⋊C4—71no—
154C4×Dic3—71no—
155C4×Dic3—72no—
156C2×C3×C7C2×#4111yescharacteristic
157C2×C3×C7H(5,𝔽2)11yescharacteristic
158C2×C3×C72−1+411yescharacteristic
159D3×C7—41no—
160D3×C7—42no—
161C4⋊Q8—71no—
162#126—212no—
163#124—211no—
164#124—212no—
165#123—212no—
166C22⋊Q8—212no—
167C22⋊Q8—211no—
168#121—31no—
169C4×Q8—211no—
170C4×D4—211no—
171#117—211no—
172C22×C7—62no—
173C22×C7—61no—
174C22×C7—61no—
175C22×C7—62no—
176C22×C7—61no—
177C22×C7—62no—
178C22×C7C23×D311yescharacteristic
179C4×C7—22no—
180C4×C7—61no—
181C4×C7#23912yes—
182Dic7—21no—
183Dic7—22no—
184Dic7—21no—
185Dic7—61no—
186Dic7—62no—
187Dic7—61no—
188C22×D3C22×D711yescharacteristic
189C22×D3C22×D712yes—
190C3×Q8—282no—
191C2×C3×C4—71no—
192C2×C3×C4—71no—
193C2×C3×C4—72no—
194C2×C3×C4C22×D712yes—
195C2×C3×C4C22×D711yescharacteristic
196C2×Dic3—71no—
197C2×Dic3—71no—
198C2×Dic3—72no—
199C2×Dic3C22×D711yescharacteristic
200D12—22no—
201D12—42no—
202C4×D3—22no—
203C4×D3—282no—
204Dic6—142no—
205C3×C7#54211yescharacteristic, complemented, Hall
206C2×Q8—72no—
207C2×Q8—211no—
208C2×D4—31no—
209C2×D4—32no—
210C22×C4—31no—
211C22×C4—212no—
212C4⋊C4—72no—
213C4⋊C4—71no—
214C4⋊C4—212no—
215C4⋊C4—211no—
216C4⋊C4—211no—
217C4⋊C4—212no—
218C4⋊C4D3×D711yescharacteristic
219C22⋊C4—32no—
220C22⋊C4—211no—
221C22⋊C4—211no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C42—71no—
227C42—211no—
228C42—212no—
229C2×C7—122no—
230C2×C7—121no—
231C2×C7#100411yescharacteristic
232C2×C7C2×#23911yescharacteristic
233C2×C7#100711yescharacteristic
234C22×C3C23×D711yescharacteristic
235C2×D3—21no—
236C2×D3—22no—
237C2×D3—22no—
238C2×D3—21no—
239C2×D3—21no—
240C2×D3—22no—
241C3×C4—22no—
242C3×C4—142no—
243C3×C4—141no—
244C3×C4—141no—
245C3×C4#115912yes—
246Dic3—21no—
247Dic3—142no—
248Dic3—141no—
249Dic3—141no—
250C23—32no—
251C23—31no—
252Q8—282no—
253Q8—422no—
254D4—62no—
255D4—122no—
256C2×C4—31no—
257C2×C4—62no—
258C2×C4—71no—
259C2×C4—71no—
260C2×C4—72no—
261C2×C4—211no—
262C2×C4—212no—
263C2×C4—211no—
264C2×C4—842no—
265C2×C4C2×D3×D712yes—
266C2×C4C2×D3×D711yescharacteristic
267C7#573311yescharacteristic, complemented, Sylow
268C2×C3C2×#115911yescharacteristic
269C2×C3#654811yescharacteristic
270C2×C3#655411yescharacteristic
271D3—41no—
272D3—42no—
273C22—61no—
274C22—62no—
275C22—62no—
276C22—61no—
277C22—62no—
278C22—61no—
279C22C22×D3×D711yescharacteristic
280C4—22no—
281C4—61no—
282C4—142no—
283C4—141no—
284C4—141no—
285C4—422no—
286C4—421no—
287C4—421no—
288C4#6667212yes—
289C3#9029411yescharacteristic, complemented, Sylow
290C2—121no—
291C2—122no—
292C2C2×#6667211yescharacteristic
293C2#1062157211yescharacteristic
294C2#1062157711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC7×#895C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#542
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. #10621370C2semidirect
3. C3×#6451C2semidirect
4. #10620997C2semidirect
5. #10620985C2semidirect
6. #10620972C2semidirect
7. #10620935C2semidirect
8. #10620933C2semidirect
9. #10620932C2semidirect
10. #10620929C2semidirect
11. #10620927C2semidirect
43. #6451D3semidirect
205. C3×C7#542semidirect
267. C7#5733semidirect
289. C3#90294semidirect

Maximal subgroups

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

subgroupindex
#106213702
C7×#8952
C3×#64512
#106209972
#106209852
#106209722
#106209352
#106209332
#106209322
#106209292
#106209272
#902943
#57337