Finite Group Databasebeta

← #13058752

Subgroups — #13058752

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×#888C211yesmaximal, characteristic, complemented
3C3×#6458C211yesmaximal, characteristic
4#10620996C211yesmaximal, characteristic, complemented
5#10620983C212yesmaximal, complemented
6#10620969C212yesmaximal, complemented
7#10620930C212yesmaximal, complemented
8#10620928C212yesmaximal, complemented
9#10620926C211yesmaximal, characteristic, complemented
10#10620922C211yesmaximal, characteristic, complemented
11#10620919C211yesmaximal, characteristic, complemented
12#90424—31nomaximal, Hall
13C2×Dic42C2211yescharacteristic
14C2×C7×Dic6C2212yes—
15C2×C3×D28C2211yescharacteristic
16C2×C3×D28C2212yes—
17C2×C3×C4×D7C2211yescharacteristic
18C2×Dic3×D7C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20#66629C2212yes—
21C4×Dic21C2212yes—
22C4×Dic21C2211yescharacteristic
23C7×C4⋊Dic3C2211yescharacteristic
24C7×Dic3⋊C4C2212yes—
25C4×C7×Dic3C2211yescharacteristic
26C3×(C2×D7)⋊C4C2212yes—
27Dic3⋊Dic7C2212yes—
28Dic7⋊Dic3C2211yescharacteristic
29Dic7⋊Dic3C2211yescharacteristic
30(C2×D7)⋊Dic3C2212yes—
31(C2×D7)⋊Dic3C2212yes—
32(C2×D7)⋊Dic3C2211yescharacteristic
33(C2×D7)⋊Dic3C2212yes—
34(C2×D7)⋊Dic3C2212yes—
35(C2×D7)⋊Dic3C2211yescharacteristic
36C7×C4⋊Q8—31no—
37#6510—32no—
38#6509—32no—
39#6461—32no—
40#6460—31no—
41#6458D311yescharacteristic
42#6457—32no—
43#6456—31no—
44#6450—31no—
45#6438—31no—
46#6435—31no—
47#5603—71nomaximal, Hall
48C22×C3×D7C2311yescharacteristic
49C22×C3×D7C2312yes—
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2312yes—
54C2×Dic21C2311yescharacteristic
55Dic42—22no—
56C2×C7×Dic3C2311yescharacteristic
57C2×C7×Dic3C2311yescharacteristic
58C2×C7×Dic3C2312yes—
59C7×Dic6—42no—
60C2×C3×Dic7C2311yescharacteristic
61C3×D28—22no—
62C3×D28—42no—
63C3×C4×D7—22no—
64Dic3×D7—42no—
65C2×C7×Q8—32no—
66C2×D28C2×D311yescharacteristic
67C2×D28C2×D312yes—
68C2×C4×D7—32no—
69C2×C4×D7C2×D311yescharacteristic
70C2×Dic14—31no—
71C7×C4⋊C4—32no—
72C7×C4⋊C4—31no—
73C7×C4⋊C4C2×D311yescharacteristic
74C42×C7—31no—
75(C2×D7)⋊C4—31no—
76(C2×D7)⋊C4—31no—
77(C2×D7)⋊C4—32no—
78(C2×D7)⋊C4—32no—
79(C2×D7)⋊C4—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4C2×D312yes—
82C4⋊Dic7—32no—
83Dic7⋊C4—32no—
84Dic7⋊C4—31no—
85Dic7⋊C4—31no—
86C4×Dic7—31no—
87C4×Dic7—32no—
88C3×#121—71no—
89#936—72no—
90#936—71no—
91#935—72no—
92D4×Dic3—71no—
93#926—71no—
94#924—71no—
95#888D711yescharacteristic, complemented
96#887—71no—
97#879—72no—
98#878—72no—
99C22×C3×C7C2411yescharacteristic
100C2×C3×D7—21no—
101C2×C3×D7—22no—
102C2×C3×D7—22no—
103C2×C3×D7—21no—
104C2×C3×D7—21no—
105C2×C3×D7—22no—
106C3×C4×C7—22no—
107C3×C4×C7#4112yes—
108Dic21—22no—
109Dic21—21no—
110Dic21—21no—
111C3×Dic7—21no—
112C7×Dic3—22no—
113C7×Dic3—21no—
114C7×Dic3—21no—
115#542—211noSylow
116C22×D7C22×D311yescharacteristic
117C22×D7C22×D312yes—
118C7×Q8—122no—
119C2×C4×C7—31no—
120C2×C4×C7—31no—
121C2×C4×C7—32no—
122C2×C4×C7C22×D312yes—
123C2×C4×C7C22×D311yescharacteristic
124C2×Dic7—31no—
125C2×Dic7—31no—
126C2×Dic7—32no—
127C2×Dic7C22×D311yescharacteristic
128D28—22no—
129D28—42no—
130C4×D7—22no—
131C4×D7—122no—
132Dic14—62no—
133C2×C3×D4—71no—
134C2×C3×D4—72no—
135C22×C3×C4—71no—
136C22×Dic3—72no—
137C2×Dic6—71no—
138C2×Dic6C2×D712yes—
139C3×C4⋊C4C2×D711yescharacteristic
140C3×C22⋊C4—72no—
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⋊Dic3C2×D711yescharacteristic
150Dic3⋊C4—72no—
151Dic3⋊C4—72no—
152Dic3⋊C4C2×D712yes—
153C4×Dic3—72no—
154C4×Dic3—71no—
155C4×Dic3C2×D711yescharacteristic
156C2×C3×C7C2×#4111yescharacteristic
157C2×C3×C7H(5,𝔽2)11yescharacteristic
158C2×C3×C72−1+411yescharacteristic
159C3×D7—41no—
160C3×D7—42no—
161C4⋊Q8—31no—
162#126—212no—
163#124—211no—
164#124—212no—
165#123—212no—
166C22⋊Q8—212no—
167C22⋊Q8—211no—
168#121—71no—
169C4×Q8—211no—
170C4×D4—211no—
171#117—211no—
172C22×C7C23×D311yescharacteristic
173C2×D7—21no—
174C2×D7—22no—
175C2×D7—22no—
176C2×D7—21no—
177C2×D7—21no—
178C2×D7—22no—
179C4×C7—22no—
180C4×C7—62no—
181C4×C7—61no—
182C4×C7—61no—
183C4×C7#23712yes—
184Dic7—21no—
185Dic7—62no—
186Dic7—61no—
187Dic7—61no—
188C23×C3—71no—
189C23×C3—72no—
190C3×D4—142no—
191C3×D4—282no—
192C2×C3×C4—71no—
193C2×C3×C4—142no—
194C2×C3×C4C22×D712yes—
195C2×C3×C4C22×D711yescharacteristic
196C2×Dic3—71no—
197C2×Dic3—71no—
198C2×Dic3—72no—
199C2×Dic3—282no—
200C2×Dic3C22×D711yescharacteristic
201C2×Dic3C22×D711yescharacteristic
202C2×Dic3C22×D712yes—
203Dic6—42no—
204Dic6—142no—
205C3×C7#54211yescharacteristic, complemented, Hall
206C2×Q8—32no—
207C2×Q8—211no—
208C2×D4—71no—
209C2×D4—72no—
210C22×C4—71no—
211C22×C4—212no—
212C4⋊C4—31no—
213C4⋊C4—32no—
214C4⋊C4—211no—
215C4⋊C4—211no—
216C4⋊C4—212no—
217C4⋊C4—212no—
218C4⋊C4D3×D711yescharacteristic
219C22⋊C4—72no—
220C22⋊C4—211no—
221C22⋊C4—211no—
222C22⋊C4—212no—
223C22⋊C4—212no—
224C22⋊C4—212no—
225C22⋊C4—212no—
226C42—31no—
227C42—211no—
228C42—212no—
229C2×C7C2×#23711yescharacteristic
230C2×C7#100411yescharacteristic
231C2×C7#101011yescharacteristic
232D7—41no—
233D7—42no—
234C22×C3—142no—
235C22×C3—141no—
236C22×C3—141no—
237C22×C3—142no—
238C22×C3—141no—
239C22×C3—142no—
240C22×C3C23×D711yescharacteristic
241C3×C4—22no—
242C3×C4—141no—
243C3×C4#116112yes—
244Dic3—21no—
245Dic3—22no—
246Dic3—21no—
247Dic3—141no—
248Dic3—142no—
249Dic3—141no—
250C23—72no—
251C23—71no—
252Q8—122no—
253Q8—422no—
254D4—142no—
255D4—282no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—32no—
259C2×C4—71no—
260C2×C4—142no—
261C2×C4—211no—
262C2×C4—212no—
263C2×C4—211no—
264C2×C4—842no—
265C2×C4C2×D3×D712yes—
266C2×C4C2×D3×D711yescharacteristic
267C7#560311yescharacteristic, complemented, Sylow
268C2×C3—282no—
269C2×C3—281no—
270C2×C3#654811yescharacteristic
271C2×C3C2×#116111yescharacteristic
272C2×C3#655111yescharacteristic
273C22—141no—
274C22—142no—
275C22—142no—
276C22—141no—
277C22—142no—
278C22—141no—
279C22C22×D3×D711yescharacteristic
280C4—22no—
281C4—62no—
282C4—61no—
283C4—61no—
284C4—141no—
285C4—422no—
286C4—421no—
287C4—421no—
288C4#6667012yes—
289C3#9042411yescharacteristic, complemented, Sylow
290C2—281no—
291C2—282no—
292C2C2×#6667011yescharacteristic
293C2#1062157211yescharacteristic
294C2#1062157511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC3×#6458C2
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
2. C7×#888C2semidirect
4. #10620996C2semidirect
5. #10620983C2semidirect
6. #10620969C2semidirect
7. #10620930C2semidirect
8. #10620928C2semidirect
9. #10620926C2semidirect
10. #10620922C2semidirect
11. #10620919C2semidirect
95. #888D7semidirect
205. C3×C7#542semidirect
267. C7#5603semidirect
289. C3#90424semidirect

Maximal subgroups

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

subgroupindex
#106213702
C7×#8882
C3×#64582
#106209962
#106209832
#106209692
#106209302
#106209282
#106209262
#106209222
#106209192
#904243
#56037