Finite Group Databasebeta

← #13061735

Subgroups — #13061735

Order 1344.

436 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C3×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10621429C212yesmaximal, complemented
3#10621413C211yesmaximal, characteristic
4C2×#66629C211yesmaximal, characteristic, complemented
5#10621381C212yesmaximal, complemented
6#10621377C212yesmaximal, complemented
7#10621373C212yesmaximal, complemented
8#10621369C212yesmaximal, complemented
9#10621364C212yesmaximal, complemented
10#90336—31nomaximal, Hall
11C2×C3×C7×Q8C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C2×#4268C2212yes—
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C2×D84C2211yescharacteristic
17C2×C4×D21C2212yes—
18C3×C7×C4⋊C4C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20C3×C7×C22⋊C4C2212yes—
21#66631C2212yes—
22#66631C2212yes—
23#66631C2212yes—
24#66631C2212yes—
25C4⋊Dic21C2212yes—
26C4⋊Dic21C2212yes—
27C4⋊Dic21C2211yescharacteristic
28#66629C2212yes—
29#66629C2212yes—
30#66629C2212yes—
31C4×Dic21C2212yes—
32C7×C22⋊Q8D311yescharacteristic, complemented
33#6509—32no—
34#6493—31no—
35C2×Dic7⋊C4—31no—
36#6461—32no—
37#6457—32no—
38#6453—32no—
39#6449—32no—
40(C22⋊D7)⋊C4—32no—
41#5645—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C22×D21C2312yes—
44C3×C7×Q8—41no—
45C2×C3×C4×C7—41no—
46C2×C3×C4×C7C2312yes—
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2311yescharacteristic
49C2×C3×C4×C7C2312yes—
50#4268—24no—
51C2×Dic21—22no—
52C2×Dic21—41no—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56D84—41no—
57C4×D21—42no—
58C2×C7×Q8C2×D311yescharacteristic
59C22×C4×C7C2×D311yescharacteristic
60C2×C22⋊D7—32no—
61C22×Dic7—31no—
62C22×Dic7—31no—
63C2×D28—31no—
64C2×C4×D7—32no—
65C7×C4⋊C4C2×D312yes—
66C7×C4⋊C4C2×D311yescharacteristic
67C7×C22⋊C4C2×D312yes—
68(C2×D7)⋊C4—32no—
69(C2×D7)⋊C4—32no—
70(C2×D7)⋊C4—32no—
71(C2×D7)⋊C4—32no—
72C4⋊Dic7—32no—
73C4⋊Dic7—32no—
74C4⋊Dic7—31no—
75Dic7⋊C4—32no—
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78C4×Dic7—32no—
79C3×C22⋊Q8D711yescharacteristic, complemented
80#946—72no—
81#930—71no—
82C2×Dic3⋊C4—71no—
83#898—72no—
84#894—72no—
85#890—72no—
86#886—72no—
87#67⋊C4—72no—
88C22×C3×C7—21no—
89C22×C3×C7—21no—
90C22×C3×C7#4112yes—
91C22×C3×C7C2411yescharacteristic
92C2×D21—22no—
93C2×D21—22no—
94C2×D21—22no—
95C3×C4×C7—22no—
96C3×C4×C7—21no—
97C3×C4×C7—21no—
98C3×C4×C7—22no—
99Dic21—22no—
100Dic21—22no—
101Dic21—22no—
102#539—211noSylow
103C23×C7C22×D311yescharacteristic
104C22×D7—32no—
105C7×Q8—41no—
106C2×C4×C7—41no—
107C2×C4×C7C22×D312yes—
108C2×C4×C7C22×D311yescharacteristic
109C2×C4×C7C22×D311yescharacteristic
110C2×C4×C7C22×D312yes—
111C22⋊D7—64no—
112C2×Dic7—32no—
113C2×Dic7—32no—
114C2×Dic7—32no—
115C2×Dic7—62no—
116C2×Dic7—121no—
117D28—121no—
118C4×D7—122no—
119C2×C3×Q8C2×D711yescharacteristic
120C22×C3×C4C2×D711yescharacteristic
121C2×#67—72no—
122C22×Dic3—71no—
123C22×Dic3—71no—
124C2×D12—71no—
125C2×C4×D3—72no—
126C3×C4⋊C4C2×D712yes—
127C3×C4⋊C4C2×D711yescharacteristic
128C3×C22⋊C4C2×D712yes—
129(C2×D3)⋊C4—72no—
130(C2×D3)⋊C4—72no—
131(C2×D3)⋊C4—72no—
132(C2×D3)⋊C4—72no—
133C4⋊Dic3—72no—
134C4⋊Dic3—72no—
135C4⋊Dic3—71no—
136Dic3⋊C4—72no—
137Dic3⋊C4—72no—
138Dic3⋊C4—72no—
139C4×Dic3—72no—
140C2×C3×C7—22no—
141C2×C3×C7C2×#4111yescharacteristic
142C2×C3×C7H(5,𝔽2)11yescharacteristic
143C2×C3×C72−1+411yescharacteristic
144D21—42no—
145#126—212no—
146#125—212no—
147#123—212no—
148#123—212no—
149C22⋊Q8—212no—
150C22⋊Q8D2111yescharacteristic, complemented
151#121—211no—
152C4×D4—212no—
153C2×C4⋊C4—211no—
154C22×C7—21no—
155C22×C7—21no—
156C22×C7#23912yes—
157C22×C7C23×D311yescharacteristic
158C2×D7—62no—
159C2×D7—62no—
160C2×D7—62no—
161C4×C7—22no—
162C4×C7—21no—
163C4×C7—21no—
164C4×C7—22no—
165Dic7—62no—
166Dic7—62no—
167Dic7—62no—
168C23×C3C22×D711yescharacteristic
169C22×D3—72no—
170C3×Q8—41no—
171C2×C3×C4—41no—
172C2×C3×C4C22×D712yes—
173C2×C3×C4C22×D711yescharacteristic
174C2×C3×C4C22×D711yescharacteristic
175C2×C3×C4C22×D712yes—
176#67—144no—
177C2×Dic3—72no—
178C2×Dic3—72no—
179C2×Dic3—72no—
180C2×Dic3—142no—
181C2×Dic3—281no—
182D12—281no—
183C4×D3—282no—
184C3×C7#53911yescharacteristic, complemented, Hall
185C2×Q8C2×D2111yescharacteristic
186C2×D4—212no—
187C2×D4—211no—
188C22×C4—211no—
189C22×C4—211no—
190C22×C4—212no—
191C22×C4C2×D2111yescharacteristic
192C4⋊C4—212no—
193C4⋊C4—212no—
194C4⋊C4—212no—
195C4⋊C4—211no—
196C4⋊C4—212no—
197C4⋊C4—212no—
198C4⋊C4C2×D2112yes—
199C4⋊C4C2×D2111yescharacteristic
200C22⋊C4—212no—
201C22⋊C4—212no—
202C22⋊C4—212no—
203C22⋊C4—212no—
204C22⋊C4C2×D2112yes—
205C42—212no—
206C2×C7—22no—
207C2×C7#100411yescharacteristic
208C2×C7C2×#23911yescharacteristic
209C2×C7#101011yescharacteristic
210D7—122no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3#116112yes—
214C22×C3C23×D711yescharacteristic
215C2×D3—142no—
216C2×D3—142no—
217C2×D3—142no—
218C3×C4—22no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—22no—
222Dic3—142no—
223Dic3—142no—
224Dic3—142no—
225C23—212no—
226C23C22×D2111yescharacteristic
227Q8—41no—
228D4—424no—
229D4—841no—
230C2×C4—41no—
231C2×C4—212no—
232C2×C4—212no—
233C2×C4—212no—
234C2×C4—422no—
235C2×C4—842no—
236C2×C4—841no—
237C2×C4C22×D2112yes—
238C2×C4C22×D2111yescharacteristic
239C2×C4C22×D2112yes—
240C2×C4C22×D2111yescharacteristic
241C7#564511yescharacteristic, complemented, Sylow
242C2×C3—22no—
243C2×C3#654811yescharacteristic
244C2×C3C2×#116111yescharacteristic
245C2×C3#655411yescharacteristic
246D3—282no—
247C22—21no—
248C22—21no—
249C22—422no—
250C22—422no—
251C22—422no—
252C22#6673212yes—
253C22C23×D2111yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—22no—
258C4—422no—
259C4—422no—
260C4—422no—
261C3#9033611yescharacteristic, complemented, Sylow
262C2—22no—
263C2—842no—
264C2#1062165711yescharacteristic
265C2C2×#6673211yescharacteristic
266C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊Q8C2
prime radical#10621413C2
hypercenterC22⋊Q8D21
hypocenterC3×C7#539
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C7×C22⋊Q8C2semidirect
2. #10621429C2semidirect
4. C2×#66629C2semidirect
5. #10621381C2semidirect
6. #10621377C2semidirect
7. #10621373C2semidirect
8. #10621369C2semidirect
9. #10621364C2semidirect
32. C7×C22⋊Q8D3semidirect
79. C3×C22⋊Q8D7semidirect
150. C22⋊Q8D21semidirect
184. C3×C7#539semidirect
241. C7#5645semidirect
261. C3#90336semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214292
#106214132
C2×#666292
#106213812
#106213772
#106213732
#106213692
#106213642
#903363
#56457