Finite Group Databasebeta

← #13054311

Subgroups — #13054311

Order 1344.

364 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×#4248C211yesmaximal, characteristic
2C2×#66629C211yesmaximal, characteristic
3C2×C7×Dic3⋊C4C211yesmaximal, characteristic
4C2×C3×Dic7⋊C4C211yesmaximal, characteristic
5#10620529C211yesmaximal, characteristic
6#10620529C211yesmaximal, characteristic
7#10620529C211yesmaximal, characteristic
8#89751—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C2×#4248—22no—
17C2×#4248—22no—
18C2×#4248—22no—
19#66629—22no—
20C7×Dic3⋊C4—22no—
21C3×Dic7⋊C4—22no—
22C22×Dic14—31no—
23C2×C7×C4⋊C4—31no—
24C2×Dic7⋊C4—31no—
25C2×Dic7⋊C4D311yescharacteristic
26C7⋊#95—31no—
27C7⋊#95—31no—
28C7⋊#95—31no—
29#4979—71nomaximal, Hall
30C23×C3×C7C2311yescharacteristic
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×Dic21—21no—
35C2×Dic21—21no—
36C2×Dic21—21no—
37C2×Dic21—21no—
38C2×Dic21D412yes—
39C2×Dic21Q812yescomplemented
40C2×C7×Dic3—21no—
41C2×C7×Dic3—21no—
42C2×C7×Dic3—21no—
43C2×C7×Dic3—21no—
44C2×C7×Dic3D412yes—
45C2×C7×Dic3Q812yescomplemented
46C2×C3×Dic7—21no—
47C2×C3×Dic7—21no—
48C2×C3×Dic7—21no—
49C2×C3×Dic7—21no—
50C2×C3×Dic7D412yes—
51C2×C3×Dic7Q812yescomplemented
52#4248—28no—
53C22×C4×C7—31no—
54C22×C4×C7—31no—
55C22×C4×C7C2×D311yescharacteristic
56C22×Dic7—31no—
57C22×Dic7—31no—
58C22×Dic7C2×D311yescharacteristic
59C22×Dic7C2×D311yescharacteristic
60C2×Dic14—62no—
61C2×Dic14—62no—
62C2×Dic14—62no—
63C7×C4⋊C4—62no—
64Dic7⋊C4—22no—
65Dic7⋊C4—62no—
66C22×Dic6—71no—
67C2×C3×C4⋊C4—71no—
68C2×Dic3⋊C4—71no—
69C2×Dic3⋊C4D711yescharacteristic
70C3⋊#95—71no—
71C3⋊#95—71no—
72C3⋊#95—71no—
73C22×C3×C7C2×D411yescharacteristic
74C22×C3×C7C2×D411yescharacteristic
75C22×C3×C7C2×D411yescharacteristic
76C22×C3×C7C2×Q811yescharacteristic
77C22×C3×C7C2×Q811yescharacteristic
78C22×C3×C7C2×Q811yescharacteristic
79C22×C3×C7#4111yescharacteristic
80C3×C4×C7—41no—
81Dic21—22no—
82Dic21—22no—
83C3×Dic7—22no—
84C3×Dic7—22no—
85C7×Dic3—22no—
86C7×Dic3—22no—
87#395—211noSylow
88C23×C7C22×D311yescharacteristic
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—32no—
93C2×C4×C7—32no—
94C2×C4×C7—61no—
95C2×C4×C7—61no—
96C2×C4×C7—61no—
97C2×C4×C7—61no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—21no—
102C2×Dic7—32no—
103C2×Dic7—32no—
104C2×Dic7—61no—
105C2×Dic7—61no—
106C2×Dic7—61no—
107C2×Dic7—61no—
108C2×Dic7Dic612yescomplemented
109C2×Dic7#6712yes—
110Dic14—68no—
111C22×C3×C4—71no—
112C22×C3×C4—71no—
113C22×C3×C4C2×D711yescharacteristic
114C22×Dic3—71no—
115C22×Dic3—71no—
116C22×Dic3C2×D711yescharacteristic
117C22×Dic3C2×D711yescharacteristic
118C2×Dic6—142no—
119C2×Dic6—142no—
120C2×Dic6—142no—
121C3×C4⋊C4—142no—
122Dic3⋊C4—22no—
123Dic3⋊C4—142no—
124C2×C3×C7#12011yescharacteristic
125C2×C3×C7C22⋊Q811yescharacteristic
126C2×C3×C7C22⋊Q811yescharacteristic
127C2×C3×C7C22⋊Q811yescharacteristic
128C2×C3×C7C4⋊Q811yescharacteristic
129C2×C3×C7C4⋊Q811yescharacteristic
130C2×C3×C7C4⋊Q811yescharacteristic
131C22×Q8—211no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—71no—
134C2×C4⋊C4—211no—
135#95—211no—
136#95—211no—
137#95—211no—
138C22×C7C2×Dic611yescharacteristic
139C22×C7#23511yescharacteristic
140C22×C7D3×D411yescharacteristic
141C22×C7D3×D411yescharacteristic
142C22×C7D3×Q811yescharacteristic
143C22×C7D3×Q811yescharacteristic
144C22×C7C2×#6711yescharacteristic
145C4×C7—41no—
146C4×C7—62no—
147C4×C7—62no—
148Dic7—22no—
149Dic7—22no—
150Dic7—62no—
151Dic7—62no—
152C23×C3C22×D711yescharacteristic
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—72no—
157C2×C3×C4—72no—
158C2×C3×C4—141no—
159C2×C3×C4—141no—
160C2×C3×C4—141no—
161C2×C3×C4—141no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—21no—
166C2×Dic3—72no—
167C2×Dic3—72no—
168C2×Dic3—141no—
169C2×Dic3—141no—
170C2×Dic3—141no—
171C2×Dic3—141no—
172C2×Dic3Dic1412yescomplemented
173C2×Dic3C22⋊D712yes—
174Dic6—148no—
175C3×C7#39511yescharacteristic, complemented, Hall
176C2×Q8—422no—
177C2×Q8—422no—
178C2×Q8—422no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—71no—
182C22×C4—71no—
183C22×C4—211no—
184C22×C4—211no—
185C22×C4D3×D711yescharacteristic
186C4⋊C4—62no—
187C4⋊C4—142no—
188C4⋊C4—422no—
189C2×C7#88811yescharacteristic
190C2×C7#88811yescharacteristic
191C2×C7#89611yescharacteristic
192C2×C7#89611yescharacteristic
193C2×C7#92411yescharacteristic
194C2×C7#93711yescharacteristic
195C2×C7#94411yescharacteristic
196C22×C3C2×Dic1411yescharacteristic
197C22×C3#115711yescharacteristic
198C22×C3D4×D711yescharacteristic
199C22×C3D4×D711yescharacteristic
200C22×C3Q8×D711yescharacteristic
201C22×C3Q8×D711yescharacteristic
202C22×C3C2×C22⋊D711yescharacteristic
203C3×C4—41no—
204C3×C4—142no—
205C3×C4—142no—
206Dic3—22no—
207Dic3—22no—
208Dic3—142no—
209Dic3—142no—
210C23C2×D3×D711yescharacteristic
211Q8—428no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—32no—
216C2×C4—32no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—72no—
222C2×C4—72no—
223C2×C4—141no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—141no—
227C2×C4—212no—
228C2×C4—212no—
229C2×C4—421no—
230C2×C4—421no—
231C2×C4—421no—
232C2×C4—421no—
233C7#497911yescharacteristic, complemented, Sylow
234C2×C3#645111yescharacteristic
235C2×C3#645111yescharacteristic
236C2×C3#645911yescharacteristic
237C2×C3#645911yescharacteristic
238C2×C3#648711yescharacteristic
239C2×C3#650011yescharacteristic
240C2×C3#650711yescharacteristic
241C22D7×Dic611yescharacteristic
242C22D3×Dic1411yescharacteristic
243C22D21⋊Q811yescharacteristic
244C22#6667511yescharacteristic
245C22D7×#6711yescharacteristic
246C22D3×C22⋊D711yescharacteristic
247C22#6669411yescharacteristic
248C4—41no—
249C4—62no—
250C4—62no—
251C4—142no—
252C4—142no—
253C4—422no—
254C4—422no—
255C3#8975111yescharacteristic, complemented, Sylow
256C2#1062088911yescharacteristic
257C2#1062089011yescharacteristic
258C2#1062089111yescharacteristic
259C2#1062091111yescharacteristic
260C2#1062091411yescharacteristic
261C2#1062093911yescharacteristic
262C2#1062113911yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#395
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
39. C2×Dic21Q8semidirect
45. C2×C7×Dic3Q8semidirect
51. C2×C3×Dic7Q8semidirect
108. C2×Dic7Dic6semidirect
172. C2×Dic3Dic14semidirect
175. C3×C7#395semidirect
233. C7#4979semidirect
255. C3#89751semidirect

Maximal subgroups

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

subgroupindex
C22×#42482
C2×#666292
C2×C7×Dic3⋊C42
C2×C3×Dic7⋊C42
#106205292
#897513
#49797