Finite Group Databasebeta

← #13061695

Subgroups — #13061695

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C7×#121C211yesmaximal, characteristic
2#10621419C212yesmaximal, complemented
3#10621419C211yesmaximal, characteristic, complemented
4#10621418C212yesmaximal, complemented
5D4×Dic21C211yesmaximal, characteristic, complemented
6#10621409C211yesmaximal, characteristic, complemented
7#10621407C211yesmaximal, characteristic, complemented
8#10621371C211yesmaximal, characteristic, complemented
9#10621370C211yesmaximal, characteristic, complemented
10#10621362C212yesmaximal, complemented
11#10621361C212yesmaximal, complemented
12#90294—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×C3×C7×D4C2212yes—
15C22×C3×C4×C7C2211yescharacteristic
16C22×Dic21C2212yes—
17C2×Dic42C2211yescharacteristic
18C2×Dic42C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20C3×C7×C22⋊C4C2212yes—
21#66636C2212yes—
22#66636C2212yes—
23#66636C2212yes—
24#66636C2212yes—
25#66636C2211yescharacteristic
26#66636C2211yescharacteristic
27C4⋊Dic21C2211yescharacteristic
28C4⋊Dic21C2211yescharacteristic
29C4⋊Dic21C2211yescharacteristic
30#66629C2212yes—
31#66629C2212yes—
32#66629C2212yes—
33C4×Dic21C2212yes—
34C4×Dic21C2211yescharacteristic
35C4×Dic21C2211yescharacteristic
36C7×#121D311yescharacteristic
37#6499—32no—
38#6499—31no—
39#6498—32no—
40D4×Dic7—31no—
41#6489—31no—
42#6487—31no—
43#6451—31no—
44#6450—31no—
45#6442—32no—
46#6441—32no—
47#5603—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C23×C3×C7C2312yes—
50C3×C7×D4—22no—
51C3×C7×D4—42no—
52C2×C3×C4×C7—22no—
53C2×C3×C4×C7C2312yes—
54C2×C3×C4×C7C2311yescharacteristic
55C2×C3×C4×C7C2311yescharacteristic
56C2×Dic21—42no—
57C2×Dic21C2311yescharacteristic
58C2×Dic21C2311yescharacteristic
59C2×Dic21C2311yescharacteristic
60C2×Dic21C2312yes—
61C2×Dic21C2311yescharacteristic
62C2×Dic21C2312yes—
63Dic42—22no—
64Dic42—42no—
65C2×C7×D4C2×D311yescharacteristic
66C2×C7×D4C2×D312yes—
67C22×C4×C7C2×D311yescharacteristic
68C22×Dic7—32no—
69C2×Dic14—31no—
70C2×Dic14—32no—
71C7×C4⋊C4C2×D311yescharacteristic
72C7×C22⋊C4C2×D312yes—
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⋊Dic7—31no—
82Dic7⋊C4—32no—
83Dic7⋊C4—32no—
84Dic7⋊C4—32no—
85C4×Dic7—32no—
86C4×Dic7—31no—
87C4×Dic7—31no—
88C3×#121D711yescharacteristic
89#936—72no—
90#936—71no—
91#935—72no—
92D4×Dic3—71no—
93#926—71no—
94#924—71no—
95#888—71no—
96#887—71no—
97#879—72no—
98#878—72no—
99C22×C3×C7—21no—
100C22×C3×C7—22no—
101C22×C3×C7—22no—
102C22×C3×C7—21no—
103C22×C3×C7—21no—
104C22×C3×C7—22no—
105C22×C3×C7C2411yescharacteristic
106C3×C4×C7—21no—
107C3×C4×C7—22no—
108C3×C4×C7#4112yes—
109Dic21—22no—
110Dic21—21no—
111Dic21—22no—
112Dic21—21no—
113Dic21—21no—
114Dic21—21no—
115#542—211noSylow
116C23×C7C22×D311yescharacteristic
117C23×C7C22×D312yes—
118C7×D4—22no—
119C7×D4—42no—
120C2×C4×C7—22no—
121C2×C4×C7C22×D312yes—
122C2×C4×C7C22×D311yescharacteristic
123C2×C4×C7C22×D311yescharacteristic
124C2×Dic7—31no—
125C2×Dic7—31no—
126C2×Dic7—31no—
127C2×Dic7—32no—
128C2×Dic7—31no—
129C2×Dic7—32no—
130C2×Dic7—122no—
131Dic14—62no—
132Dic14—122no—
133C2×C3×D4C2×D711yescharacteristic
134C2×C3×D4C2×D712yes—
135C22×C3×C4C2×D711yescharacteristic
136C22×Dic3—72no—
137C2×Dic6—71no—
138C2×Dic6—72no—
139C3×C4⋊C4C2×D711yescharacteristic
140C3×C22⋊C4C2×D712yes—
141#217—72no—
142#217—72no—
143#217—72no—
144#217—72no—
145#217—71no—
146#217—71no—
147C4⋊Dic3—71no—
148C4⋊Dic3—71no—
149C4⋊Dic3—71no—
150Dic3⋊C4—72no—
151Dic3⋊C4—72no—
152Dic3⋊C4—72no—
153C4×Dic3—72no—
154C4×Dic3—71no—
155C4×Dic3—71no—
156C2×C3×C7—41no—
157C2×C3×C7—42no—
158C2×C3×C7C2×#4111yescharacteristic
159C2×C3×C7H(5,𝔽2)11yescharacteristic
160C2×C3×C72−1+411yescharacteristic
161C4⋊Q8—211no—
162#126—212no—
163#124—211no—
164#124—212no—
165#123—212no—
166C22⋊Q8—211no—
167C22⋊Q8—212no—
168#121D2111yescharacteristic
169C4×Q8—211no—
170C4×D4—211no—
171#117—211no—
172C22×C7—21no—
173C22×C7—22no—
174C22×C7—22no—
175C22×C7—21no—
176C22×C7—21no—
177C22×C7—22no—
178C22×C7C23×D311yescharacteristic
179C4×C7—21no—
180C4×C7—22no—
181C4×C7#23712yes—
182Dic7—62no—
183Dic7—61no—
184Dic7—62no—
185Dic7—61no—
186Dic7—61no—
187Dic7—61no—
188C23×C3C22×D711yescharacteristic
189C23×C3C22×D712yes—
190C3×D4—22no—
191C3×D4—42no—
192C2×C3×C4—22no—
193C2×C3×C4C22×D712yes—
194C2×C3×C4C22×D711yescharacteristic
195C2×C3×C4C22×D711yescharacteristic
196C2×Dic3—71no—
197C2×Dic3—71no—
198C2×Dic3—71no—
199C2×Dic3—72no—
200C2×Dic3—71no—
201C2×Dic3—72no—
202C2×Dic3—282no—
203Dic6—142no—
204Dic6—282no—
205C3×C7#54211yescharacteristic, complemented, Hall
206C2×Q8—211no—
207C2×Q8—212no—
208C2×D4C2×D2111yescharacteristic
209C2×D4C2×D2112yes—
210C22×C4—212no—
211C22×C4C2×D2111yescharacteristic
212C4⋊C4—212no—
213C4⋊C4—212no—
214C4⋊C4—212no—
215C4⋊C4—211no—
216C4⋊C4—211no—
217C4⋊C4—211no—
218C4⋊C4C2×D2111yescharacteristic
219C22⋊C4—211no—
220C22⋊C4—211no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4C2×D2112yes—
226C42—211no—
227C42—211no—
228C42—212no—
229C2×C7—41no—
230C2×C7—42no—
231C2×C7C2×#23711yescharacteristic
232C2×C7#100411yescharacteristic
233C2×C7#101011yescharacteristic
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—22no—
240C22×C3C23×D711yescharacteristic
241C3×C4—21no—
242C3×C4—22no—
243C3×C4#115912yes—
244Dic3—142no—
245Dic3—141no—
246Dic3—142no—
247Dic3—141no—
248Dic3—141no—
249Dic3—141no—
250C23C22×D2111yescharacteristic
251C23C22×D2112yes—
252Q8—422no—
253Q8—842no—
254D4—22no—
255D4—42no—
256C2×C4—22no—
257C2×C4—211no—
258C2×C4—211no—
259C2×C4—212no—
260C2×C4—211no—
261C2×C4—211no—
262C2×C4—212no—
263C2×C4—842no—
264C2×C4C22×D2112yes—
265C2×C4C22×D2111yescharacteristic
266C2×C4C22×D2111yescharacteristic
267C7#560311yescharacteristic, complemented, Sylow
268C2×C3—41no—
269C2×C3—42no—
270C2×C3C2×#115911yescharacteristic
271C2×C3#654811yescharacteristic
272C2×C3#655411yescharacteristic
273C22—21no—
274C22—22no—
275C22—22no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22C23×D2111yescharacteristic
280C4—21no—
281C4—22no—
282C4—422no—
283C4—421no—
284C4—422no—
285C4—421no—
286C4—421no—
287C4—421no—
288C4#6673012yes—
289C3#9029411yescharacteristic, complemented, Sylow
290C2—41no—
291C2—42no—
292C2C2×#6673011yescharacteristic
293C2#1062165711yescharacteristic
294C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
Fitting, prime radicalC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#542
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #10621419C2semidirect
3. #10621419C2semidirect
4. #10621418C2semidirect
5. D4×Dic21C2semidirect
6. #10621409C2semidirect
7. #10621407C2semidirect
8. #10621371C2semidirect
9. #10621370C2semidirect
10. #10621362C2semidirect
11. #10621361C2semidirect
205. C3×C7#542semidirect
267. C7#5603semidirect
289. C3#90294semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
#106214192
#106214182
D4×Dic212
#106214092
#106214072
#106213712
#106213702
#106213622
#106213612
#902943
#56037