Finite Group Databasebeta

← #13058758

Subgroups — #13058758

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#10621378C211yesmaximal, characteristic
2C7×#887C211yesmaximal, characteristic, complemented
3C3×#6451C211yesmaximal, characteristic, complemented
4#10620995C212yesmaximal, complemented
5#10620990C211yesmaximal, characteristic, complemented
6#10620973C212yesmaximal, complemented
7#10620948C211yesmaximal, characteristic, complemented
8#10620945C211yesmaximal, characteristic, complemented
9#10620942C212yesmaximal, complemented
10#10620937C211yesmaximal, characteristic, complemented
11#10620936C212yesmaximal, complemented
12#90273—31nomaximal, Hall
13C2×D84C2211yescharacteristic
14C2×D84C2212yes—
15C2×C4×D21C2211yescharacteristic
16C2×C7×Dic6C2211yescharacteristic
17C2×C3×Dic14C2212yes—
18C2×D21⋊C4C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20#66631C2212yes—
21C7×Dic3⋊C4C2212yes—
22C4×C7×Dic3C2212yes—
23C4×C7×Dic3C2211yescharacteristic
24C3×C4⋊Dic7C2211yescharacteristic
25C3×Dic7⋊C4C2212yes—
26C3×C4×Dic7C2211yescharacteristic
27#66577C2211yescharacteristic
28#66577C2211yescharacteristic
29Dic7⋊Dic3C2212yes—
30#66575C2211yescharacteristic
31#66575C2212yes—
32#66575C2212yes—
33#66575C2212yes—
34#66575C2212yes—
35#66575C2211yescharacteristic
36C4×C7×Q8—31no—
37#6510—31no—
38#6509—31no—
39#6459—32no—
40#6458—31no—
41#6457—32no—
42#6451D311yescharacteristic, complemented
43#6439—32no—
44#6438—32no—
45C4×D28—31no—
46#6435—31no—
47#5733—71nomaximal, Hall
48C22×D21C2311yescharacteristic
49C22×D21C2312yes—
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53D84—22no—
54D84—42no—
55C4×D21—22no—
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2311yescharacteristic
58C2×C7×Dic3C2311yescharacteristic
59C7×Dic6—22no—
60C2×C3×Dic7C2311yescharacteristic
61C2×C3×Dic7C2311yescharacteristic
62C2×C3×Dic7C2312yes—
63C3×Dic14—42no—
64D21⋊C4—42no—
65C2×C7×Q8—31no—
66C2×D28—31no—
67C2×D28—32no—
68C2×C4×D7—31no—
69C2×C4×D7—32no—
70C2×Dic14C2×D312yes—
71C7×C4⋊C4—32no—
72C7×C4⋊C4C2×D311yescharacteristic
73C42×C7—31no—
74C42×C7—32no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—31no—
77(C2×D7)⋊C4—31no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4—32no—
82C4⋊Dic7C2×D311yescharacteristic
83Dic7⋊C4—32no—
84Dic7⋊C4—31no—
85Dic7⋊C4—31no—
86Dic7⋊C4C2×D312yes—
87C4×Dic7C2×D311yescharacteristic
88C3×C4⋊Q8—71no—
89#947—72no—
90#946—72no—
91#898—72no—
92#897—71no—
93#895—71no—
94#894—72no—
95#893—71no—
96#887D711yescharacteristic, complemented
97#875—71no—
98#872—71no—
99C22×C3×C7C2411yescharacteristic
100C2×D21—21no—
101C2×D21—22no—
102C2×D21—22no—
103C2×D21—21no—
104C2×D21—21no—
105C2×D21—22no—
106C3×C4×C7—22no—
107C3×C4×C7#4112yes—
108Dic21—21no—
109C3×Dic7—22no—
110C3×Dic7—21no—
111C3×Dic7—21no—
112C7×Dic3—22no—
113C7×Dic3—21no—
114C7×Dic3—21no—
115#542—211noSylow
116C22×D7—32no—
117C22×D7—31no—
118C7×Q8—62no—
119C2×C4×C7—31no—
120C2×C4×C7—32no—
121C2×C4×C7—31no—
122C2×C4×C7C22×D312yes—
123C2×C4×C7C22×D311yescharacteristic
124C2×Dic7—31no—
125C2×Dic7C22×D311yescharacteristic
126C2×Dic7C22×D311yescharacteristic
127C2×Dic7C22×D312yes—
128D28—62no—
129D28—122no—
130C4×D7—62no—
131C4×D7—122no—
132Dic14—42no—
133C2×C3×Q8—72no—
134C2×D12—71no—
135C2×D12—72no—
136C2×C4×D3—71no—
137C2×C4×D3—72no—
138C2×Dic6C2×D711yescharacteristic
139C3×C4⋊C4—72no—
140C3×C4⋊C4—71no—
141C3×C4⋊C4C2×D711yescharacteristic
142C3×C42—71no—
143(C2×D3)⋊C4—72no—
144(C2×D3)⋊C4—71no—
145(C2×D3)⋊C4—71no—
146(C2×D3)⋊C4—72no—
147(C2×D3)⋊C4—72no—
148(C2×D3)⋊C4—72no—
149(C2×D3)⋊C4—72no—
150C4⋊Dic3—72no—
151Dic3⋊C4—71no—
152Dic3⋊C4—71no—
153Dic3⋊C4C2×D712yes—
154C4×Dic3C2×D711yescharacteristic
155C4×Dic3C2×D712yes—
156C2×C3×C7C2×#4111yescharacteristic
157C2×C3×C7H(5,𝔽2)11yescharacteristic
158C2×C3×C72−1+411yescharacteristic
159D21—41no—
160D21—42no—
161C4⋊Q8—71no—
162#126—212no—
163#124—212no—
164#124—211no—
165#123—212no—
166C22⋊Q8—212no—
167C22⋊Q8—211no—
168#121—211no—
169C4×Q8—31no—
170C4×D4—211no—
171#117—211no—
172C22×C7C23×D311yescharacteristic
173C2×D7—62no—
174C2×D7—61no—
175C2×D7—61no—
176C2×D7—62no—
177C2×D7—62no—
178C2×D7—61no—
179C4×C7—22no—
180C4×C7—61no—
181C4×C7—62no—
182C4×C7—61no—
183C4×C7#23912yes—
184Dic7—21no—
185Dic7—22no—
186Dic7—21no—
187Dic7—61no—
188C22×D3—71no—
189C22×D3—72no—
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×Dic3C22×D711yescharacteristic
198C2×Dic3C22×D711yescharacteristic
199C2×Dic3C22×D712yes—
200D12—142no—
201D12—282no—
202C4×D3—142no—
203C4×D3—282no—
204Dic6—22no—
205C3×C7#54211yescharacteristic, complemented, Hall
206C2×Q8—31no—
207C2×Q8—72no—
208C2×D4—211no—
209C2×D4—212no—
210C22×C4—212no—
211C22×C4—211no—
212C4⋊C4—32no—
213C4⋊C4—72no—
214C4⋊C4—71no—
215C4⋊C4—211no—
216C4⋊C4—211no—
217C4⋊C4—212no—
218C4⋊C4D3×D711yescharacteristic
219C22⋊C4—211no—
220C22⋊C4—211no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C42—31no—
227C42—32no—
228C42—71no—
229C2×C7#100411yescharacteristic
230C2×C7C2×#23911yescharacteristic
231C2×C7#100711yescharacteristic
232D7—122no—
233D7—121no—
234C22×C3C23×D711yescharacteristic
235C2×D3—141no—
236C2×D3—142no—
237C2×D3—141no—
238C2×D3—142no—
239C2×D3—142no—
240C2×D3—141no—
241C3×C4—22no—
242C3×C4—141no—
243C3×C4—142no—
244C3×C4—141no—
245C3×C4#115712yes—
246Dic3—22no—
247Dic3—21no—
248Dic3—21no—
249Dic3—141no—
250C23—211no—
251C23—212no—
252Q8—62no—
253Q8—282no—
254D4—422no—
255D4—842no—
256C2×C4—31no—
257C2×C4—32no—
258C2×C4—31no—
259C2×C4—71no—
260C2×C4—71no—
261C2×C4—72no—
262C2×C4—211no—
263C2×C4—422no—
264C2×C4—842no—
265C2×C4C2×D3×D712yes—
266C2×C4C2×D3×D711yescharacteristic
267C7#573311yescharacteristic, complemented, Sylow
268C2×C3C2×#115711yescharacteristic
269C2×C3#655111yescharacteristic
270C2×C3#655311yescharacteristic
271D3—281no—
272D3—282no—
273C22—421no—
274C22—422no—
275C22—422no—
276C22—421no—
277C22—421no—
278C22—422no—
279C22C22×D3×D711yescharacteristic
280C4—22no—
281C4—62no—
282C4—61no—
283C4—61no—
284C4—142no—
285C4—141no—
286C4—141no—
287C4—421no—
288C4#6667312yes—
289C3#9027311yescharacteristic, complemented, Sylow
290C2—841no—
291C2—842no—
292C2C2×#6667311yescharacteristic
293C2#1062157311yescharacteristic
294C2#1062157611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radical#10621378C2
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
2. C7×#887C2semidirect
3. C3×#6451C2semidirect
4. #10620995C2semidirect
5. #10620990C2semidirect
6. #10620973C2semidirect
7. #10620948C2semidirect
8. #10620945C2semidirect
9. #10620942C2semidirect
10. #10620937C2semidirect
11. #10620936C2semidirect
42. #6451D3semidirect
96. #887D7semidirect
205. C3×C7#542semidirect
267. C7#5733semidirect
289. C3#90273semidirect

Maximal subgroups

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

subgroupindex
#106213782
C7×#8872
C3×#64512
#106209952
#106209902
#106209732
#106209482
#106209452
#106209422
#106209372
#106209362
#902733
#57337