Finite Group Databasebeta

← #13056629

Subgroups — #13056629

Order 1344.

380 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#66631C211yesmaximal, characteristic
3C2×#66631C211yesmaximal, characteristic
4C2×#66631C211yesmaximal, characteristic
5C2×C4×Dic21C211yesmaximal, characteristic, complemented
6#10620670C211yesmaximal, characteristic, complemented
7#10620670C211yesmaximal, characteristic, complemented
8#89762—31nomaximal, Hall
9C23×D21C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C3×C7×C4⋊C4—22no—
17#66631—22no—
18#66631—22no—
19#66631C414yescomplemented
20C4×Dic21—22no—
21C2×C7×C4⋊C4D311yescharacteristic, complemented
22C2×(C2×D7)⋊C4—31no—
23C2×(C2×D7)⋊C4—31no—
24C2×(C2×D7)⋊C4—31no—
25C2×C4×Dic7—31no—
26C7⋊#95—31no—
27C7⋊#95—31no—
28#4990—71nomaximal, Hall
29C23×C3×C7C2311yescharacteristic
30C22×D21—21no—
31C22×D21—21no—
32C22×D21—21no—
33C22×D21—21no—
34C22×D21—21no—
35C22×D21—21no—
36C22×D21C2×C412yes—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7—21no—
42C2×C3×C4×C7—21no—
43C2×C3×C4×C7—21no—
44C2×C3×C4×C7C2×C412yes—
45C2×C3×C4×C7D412yes—
46C2×Dic21—21no—
47C2×Dic21—21no—
48C2×Dic21—21no—
49C2×Dic21—21no—
50C2×Dic21—21no—
51C2×Dic21—21no—
52C2×Dic21—21no—
53C2×Dic21C2×C412yes—
54C2×Dic21D412yes—
55C23×D7—31no—
56C22×C4×C7C2×D311yescharacteristic
57C22×C4×C7C2×D311yescharacteristic
58C22×C4×C7C2×D311yescharacteristic
59C22×Dic7—31no—
60C22×Dic7—31no—
61C22×Dic7—31no—
62C7×C4⋊C4—22no—
63(C2×D7)⋊C4—34no—
64(C2×D7)⋊C4—62no—
65(C2×D7)⋊C4—62no—
66C4×Dic7—62no—
67C2×C3×C4⋊C4D711yescharacteristic, complemented
68C2×(C2×D3)⋊C4—71no—
69C2×(C2×D3)⋊C4—71no—
70C2×(C2×D3)⋊C4—71no—
71C2×C4×Dic3—71no—
72C3⋊#95—71no—
73C3⋊#95—71no—
74C22×C3×C7C22×C411yescharacteristic
75C22×C3×C7C2×D411yescharacteristic
76C22×C3×C7C2×D411yescharacteristic
77C22×C3×C7#4111yescharacteristic
78C22×C3×C7#4111yescharacteristic
79C22×C3×C7#4111yescharacteristic
80C22×C3×C7#4111yescharacteristic
81C2×D21—22no—
82C2×D21—22no—
83C2×D21—22no—
84C2×D21—41no—
85C2×D21—41no—
86C2×D21—41no—
87C2×D21—41no—
88C3×C4×C7—22no—
89C3×C4×C7—22no—
90C3×C4×C7—41no—
91Dic21—22no—
92Dic21—22no—
93Dic21—41no—
94#388—211noSylow
95C23×C7C22×D311yescharacteristic
96C22×D7—32no—
97C22×D7—61no—
98C22×D7—61no—
99C22×D7—61no—
100C22×D7—61no—
101C22×D7—61no—
102C22×D7—61no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7C4×D312yes—
111C2×C4×C7#6712yes—
112C2×Dic7—32no—
113C2×Dic7—32no—
114C2×Dic7—61no—
115C2×Dic7—61no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C23×D3—71no—
122C22×C3×C4C2×D711yescharacteristic
123C22×C3×C4C2×D711yescharacteristic
124C22×C3×C4C2×D711yescharacteristic
125C22×Dic3—71no—
126C22×Dic3—71no—
127C22×Dic3—71no—
128C3×C4⋊C4—22no—
129(C2×D3)⋊C4—74no—
130(C2×D3)⋊C4—142no—
131(C2×D3)⋊C4—142no—
132C4×Dic3—142no—
133C2×C3×C7#11711yescharacteristic
134C2×C3×C7C4×D411yescharacteristic
135C2×C3×C7C4×D411yescharacteristic
136C2×C3×C7#12111yescharacteristic
137C2×C3×C7#12311yescharacteristic
138C2×C3×C7#12411yescharacteristic
139C2×C3×C7#12611yescharacteristic
140D21—42no—
141C2×C4⋊C4D2111yescharacteristic, complemented
142C2×C22⋊C4—211no—
143C2×C22⋊C4—211no—
144C2×C22⋊C4—211no—
145C2×C42—211no—
146#95—211no—
147#95—211no—
148C22×C7C2×C4×D311yescharacteristic
149C22×C7#23511yescharacteristic
150C22×C7D3×D411yescharacteristic
151C22×C7#23711yescharacteristic
152C22×C7#23911yescharacteristic
153C22×C7#23911yescharacteristic
154C22×C7C2×#6711yescharacteristic
155C2×D7—62no—
156C2×D7—62no—
157C2×D7—62no—
158C2×D7—121no—
159C2×D7—121no—
160C2×D7—121no—
161C2×D7—121no—
162C4×C7—22no—
163C4×C7—22no—
164C4×C7—41no—
165Dic7—62no—
166Dic7—62no—
167Dic7—121no—
168C23×C3C22×D711yescharacteristic
169C22×D3—72no—
170C22×D3—141no—
171C22×D3—141no—
172C22×D3—141no—
173C22×D3—141no—
174C22×D3—141no—
175C22×D3—141no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C4×D712yes—
184C2×C3×C4C22⋊D712yes—
185C2×Dic3—72no—
186C2×Dic3—72no—
187C2×Dic3—141no—
188C2×Dic3—141no—
189C2×Dic3—141no—
190C2×Dic3—141no—
191C2×Dic3—141no—
192C2×Dic3—141no—
193C2×Dic3—141no—
194C3×C7#38811yescharacteristic, complemented, Hall
195C24—211no—
196C22×C4—211no—
197C22×C4—211no—
198C22×C4—211no—
199C22×C4C2×D2111yescharacteristic
200C22×C4C2×D2111yescharacteristic
201C22×C4C2×D2111yescharacteristic
202C4⋊C4—22no—
203C22⋊C4—214no—
204C22⋊C4—422no—
205C22⋊C4—422no—
206C42—422no—
207C2×C7#89211yescharacteristic
208C2×C7#89311yescharacteristic
209C2×C7#89411yescharacteristic
210C2×C7#89811yescharacteristic
211C2×C7C4×#6711yescharacteristic
212C2×C7#93911yescharacteristic
213C2×C7#94711yescharacteristic
214D7—122no—
215C22×C3C2×C4×D711yescharacteristic
216C22×C3#115711yescharacteristic
217C22×C3D4×D711yescharacteristic
218C22×C3#115911yescharacteristic
219C22×C3#116111yescharacteristic
220C22×C3#116111yescharacteristic
221C22×C3C2×C22⋊D711yescharacteristic
222C2×D3—142no—
223C2×D3—142no—
224C2×D3—142no—
225C2×D3—281no—
226C2×D3—281no—
227C2×D3—281no—
228C2×D3—281no—
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—41no—
232Dic3—142no—
233Dic3—142no—
234Dic3—281no—
235C23—212no—
236C23—421no—
237C23—421no—
238C23—421no—
239C23—421no—
240C23—421no—
241C23—421no—
242C23C22×D2111yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—212no—
251C2×C4—212no—
252C2×C4—421no—
253C2×C4—421no—
254C2×C4—421no—
255C2×C4—421no—
256C2×C4—421no—
257C2×C4—421no—
258C2×C4—421no—
259C2×C4C4×D2112yes—
260C2×C4#426812yes—
261C7#499011yescharacteristic, complemented, Sylow
262C2×C3#645511yescharacteristic
263C2×C3#645611yescharacteristic
264C2×C3#645711yescharacteristic
265C2×C3#646111yescharacteristic
266C2×C3C4×C22⋊D711yescharacteristic
267C2×C3#650211yescharacteristic
268C2×C3#651011yescharacteristic
269D3—282no—
270C22—422no—
271C22—422no—
272C22—422no—
273C22—841no—
274C22—841no—
275C22—841no—
276C22—841no—
277C22C2×C4×D2111yescharacteristic
278C22#6672811yescharacteristic
279C22D4×D2111yescharacteristic
280C22#6673011yescharacteristic
281C22#6673211yescharacteristic
282C22#6673211yescharacteristic
283C22C2×#426811yescharacteristic
284C4—22no—
285C4—22no—
286C4—41no—
287C4—422no—
288C4—422no—
289C4—841no—
290C3#8976211yescharacteristic, complemented, Sylow
291C2—842no—
292C2#1062137511yescharacteristic
293C2#1062137611yescharacteristic
294C2#1062137711yescharacteristic
295C2#1062138111yescharacteristic
296C2C4×#426811yescharacteristic
297C2#1062142211yescharacteristic
298C2#1062143011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutatorC22×C3×C7C22×C4
FittingC2×C3×C7×C4⋊C4C2
socle, cosocleC23×C3×C7C23
prime radicalC23×D21C22
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#388
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C7×C4⋊C4C2semidirect
5. C2×C4×Dic21C2semidirect
6. #10620670C2semidirect
7. #10620670C2semidirect
19. #66631C4semidirect
21. C2×C7×C4⋊C4D3semidirect
67. C2×C3×C4⋊C4D7semidirect
141. C2×C4⋊C4D21semidirect
194. C3×C7#388semidirect
261. C7#4990semidirect
290. C3#89762semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
C2×#666312
C2×C4×Dic212
#106206702
#897623
#49907