Finite Group Databasebeta

← #13058723

Subgroups — #13058723

Order 1344.

456 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 140 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#10621376C211yesmaximal, characteristic
2C7×#888C211yesmaximal, characteristic, complemented
3C3×#6453C211yesmaximal, characteristic, complemented
4C4×#4248C211yesmaximal, characteristic, complemented
5#10620991C212yesmaximal, complemented
6#10620990C211yesmaximal, characteristic
7#10620986C212yesmaximal, complemented
8#10620980C211yesmaximal, characteristic
9#10620966C212yesmaximal, complemented
10#10620946C211yesmaximal, characteristic, complemented
11#10620945C212yesmaximal, complemented
12#90421—31nomaximal, Hall
13C2×D84C2211yescharacteristic
14C2×C4×D21C2212yes—
15C2×C7×Dic6C2212yes—
16C2×#4248C2211yescharacteristic
17C2×D21⋊C4C2212yes—
18C2×D21⋊C4C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20#66631C2212yes—
21C4×Dic21C2211yescharacteristic
22C7×C4⋊Dic3C2211yescharacteristic
23C7×Dic3⋊C4C2212yes—
24C4×C7×Dic3C2211yescharacteristic
25C3×C4⋊Dic7C2212yes—
26C3×C4⋊Dic7C2211yescharacteristic
27C3×Dic7⋊C4C2212yes—
28C3×C4×Dic7C2211yescharacteristic
29Dic3⋊Dic7C2212yes—
30Dic3⋊Dic7C2211yescharacteristic
31#66577C2211yescharacteristic
32Dic7⋊Dic3C2211yescharacteristic
33Dic7⋊Dic3C2212yes—
34#66575C2212yes—
35#66575C2212yes—
36C7×C4⋊Q8—31no—
37#6509—32no—
38#6509—32no—
39#6460—32no—
40#6456—31no—
41#6456—31no—
42D7×C4⋊C4—32no—
43#6453—31no—
44#6453D311yescharacteristic, complemented
45C4×D28—31no—
46C4×Dic14—31no—
47#5690—71nomaximal, Hall
48C22×D21C2312yes—
49C2×C3×C4×C7C2312yes—
50C2×C3×C4×C7C2311yescharacteristic
51C2×Dic21C2311yescharacteristic
52C2×Dic21C2311yescharacteristic
53D84Q814yescomplemented
54C4×D21—42no—
55C2×C7×Dic3C2311yescharacteristic
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2311yescharacteristic
58C7×Dic6—42no—
59C2×C3×Dic7C2311yescharacteristic
60C2×C3×Dic7C2312yes—
61C2×C3×Dic7C2311yescharacteristic
62#4248—22no—
63D21⋊C4—24no—
64D21⋊C4—42no—
65C2×C7×Q8—32no—
66C2×D28—31no—
67C2×C4×D7—32no—
68C2×C4×D7—32no—
69C2×C4×D7—32no—
70C2×Dic14—31no—
71C7×C4⋊C4—32no—
72C7×C4⋊C4—31no—
73C7×C4⋊C4C2×D311yescharacteristic
74C42×C7—31no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—32no—
78C4⋊Dic7—31no—
79C4⋊Dic7—32no—
80C4⋊Dic7C2×D311yescharacteristic
81C4⋊Dic7C2×D312yes—
82Dic7⋊C4—31no—
83Dic7⋊C4—32no—
84Dic7⋊C4—31no—
85Dic7⋊C4C2×D312yes—
86C4×Dic7—31no—
87C4×Dic7C2×D311yescharacteristic
88C3×#125—71no—
89#897—72no—
90#896—72no—
91#896—72no—
92#893—71no—
93#893—71no—
94D3×C4⋊C4—72no—
95#890—71no—
96#888D711yescharacteristic, complemented
97C4×D12—71no—
98C4×Dic6—71no—
99C22×C3×C7C2411yescharacteristic
100C2×D21—22no—
101C2×D21—22no—
102C2×D21C2×Q814yes—
103C3×C4×C7—22no—
104C3×C4×C7C2×Q812yes—
105Dic21—21no—
106Dic21—21no—
107C3×Dic7—22no—
108C3×Dic7—21no—
109C3×Dic7—21no—
110C7×Dic3—22no—
111C7×Dic3—21no—
112C7×Dic3#4112yes—
113#554—211noSylow
114C22×D7—32no—
115C7×Q8—122no—
116C2×C4×C7—31no—
117C2×C4×C7—31no—
118C2×C4×C7—32no—
119C2×C4×C7C22×D312yes—
120C2×C4×C7C22×D311yescharacteristic
121C2×Dic7—31no—
122C2×Dic7—31no—
123C2×Dic7C22×D311yescharacteristic
124C2×Dic7C22×D311yescharacteristic
125C2×Dic7C22×D312yes—
126D28—34no—
127C4×D7—64no—
128C4×D7—122no—
129C4×D7—122no—
130Dic14—62no—
131C2×D12—71no—
132C2×C4×D3—72no—
133C2×C4×D3—72no—
134C2×C4×D3—72no—
135C2×Dic6—71no—
136C2×Dic6C2×D712yes—
137C3×C4⋊C4—72no—
138C3×C4⋊C4—71no—
139C3×C4⋊C4—72no—
140C3×C4⋊C4C2×D711yescharacteristic
141C3×C42—71no—
142(C2×D3)⋊C4—72no—
143(C2×D3)⋊C4—72no—
144(C2×D3)⋊C4—72no—
145C4⋊Dic3—71no—
146C4⋊Dic3—72no—
147C4⋊Dic3C2×D711yescharacteristic
148Dic3⋊C4—71no—
149Dic3⋊C4—71no—
150Dic3⋊C4—72no—
151Dic3⋊C4C2×D712yes—
152C4×Dic3—71no—
153C4×Dic3C2×D711yescharacteristic
154C2×C3×C7C22×Q811yescharacteristic
155C2×C3×C7C2×#4111yescharacteristic
156C2×C3×C7H(5,𝔽2)11yescharacteristic
157D21—24no—
158C4⋊Q8—31no—
159#125—71no—
160#125—211no—
161C22⋊Q8—212no—
162C22⋊Q8—212no—
163C22⋊Q8—212no—
164C4×Q8—211no—
165C4×D4—211no—
166C4×D4—211no—
167C4×D4—211no—
168C2×C4⋊C4—212no—
169C22×C7C23×D311yescharacteristic
170C2×D7—34no—
171C2×D7—62no—
172C2×D7—62no—
173C4×C7—22no—
174C4×C7—32no—
175C4×C7—62no—
176C4×C7—61no—
177C4×C7D3×Q812yes—
178Dic7—21no—
179Dic7—22no—
180Dic7—21no—
181Dic7—61no—
182Dic7—61no—
183C22×D3—72no—
184C2×C3×C4—71no—
185C2×C3×C4—72no—
186C2×C3×C4—71no—
187C2×C3×C4C22×D712yes—
188C2×C3×C4C22×D711yescharacteristic
189C2×Dic3—71no—
190C2×Dic3—71no—
191C2×Dic3C22×D711yescharacteristic
192C2×Dic3C22×D712yes—
193C2×Dic3C22×D711yescharacteristic
194D12—74no—
195C4×D3—144no—
196C4×D3—282no—
197C4×D3—282no—
198Dic6—42no—
199Dic6—142no—
200C3×C7#55411yescharacteristic, complemented, Hall
201C2×Q8—32no—
202C2×Q8—211no—
203C2×D4—211no—
204C22×C4—212no—
205C22×C4—212no—
206C22×C4—212no—
207C4⋊C4—32no—
208C4⋊C4—31no—
209C4⋊C4—72no—
210C4⋊C4—71no—
211C4⋊C4—72no—
212C4⋊C4—212no—
213C4⋊C4—211no—
214C4⋊C4—211no—
215C4⋊C4—211no—
216C4⋊C4—212no—
217C4⋊C4D3×D711yescharacteristic
218C22⋊C4—212no—
219C22⋊C4—212no—
220C22⋊C4—212no—
221C42—31no—
222C42—71no—
223C42—211no—
224C2×C7C2×D3×Q811yescharacteristic
225C2×C7D3×#4111yescharacteristic
226C2×C7#100911yescharacteristic
227D7—64no—
228C22×C3C23×D711yescharacteristic
229C2×D3—74no—
230C2×D3—142no—
231C2×D3—142no—
232C3×C4—22no—
233C3×C4—141no—
234C3×C4—141no—
235C3×C4—142no—
236C3×C4Q8×D712yes—
237Dic3—21no—
238Dic3—22no—
239Dic3—141no—
240Dic3—141no—
241Dic3#116112yes—
242C23—212no—
243Q8—122no—
244Q8—422no—
245D4—214no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—32no—
249C2×C4—71no—
250C2×C4—71no—
251C2×C4—72no—
252C2×C4—211no—
253C2×C4—211no—
254C2×C4—424no—
255C2×C4—842no—
256C2×C4—842no—
257C2×C4C2×D3×D712yes—
258C2×C4C2×D3×D711yescharacteristic
259C7#569011yescharacteristic, complemented, Sylow
260C2×C3#654811yescharacteristic
261C2×C3C2×Q8×D711yescharacteristic
262C2×C3C2×#116111yescharacteristic
263D3—144no—
264C22—214no—
265C22—422no—
266C22—422no—
267C22C22×D3×D711yescharacteristic
268C4—22no—
269C4—32no—
270C4—62no—
271C4—61no—
272C4—141no—
273C4—142no—
274C4—141no—
275C4—421no—
276C4—421no—
277C4D21⋊Q812yes—
278C3#9042111yescharacteristic, complemented, Sylow
279C2—424no—
280C2C2×D21⋊Q811yescharacteristic
281C2#1062157411yescharacteristic
282C2D3×#116111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
prime radicalC2×D84C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#554
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×#888C2semidirect
3. C3×#6453C2semidirect
4. C4×#4248C2semidirect
5. #10620991C2semidirect
7. #10620986C2semidirect
9. #10620966C2semidirect
10. #10620946C2semidirect
11. #10620945C2semidirect
44. #6453D3semidirect
53. D84Q8semidirect
96. #888D7semidirect
200. C3×C7#554semidirect
259. C7#5690semidirect
278. C3#90421semidirect

Maximal subgroups

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

subgroupindex
#106213762
C7×#8882
C3×#64532
C4×#42482
#106209912
#106209902
#106209862
#106209802
#106209662
#106209462
#106209452
#904213
#56907