Finite Group Databasebeta

← #13054260

Subgroups — #13054260

Order 1344.

436 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 110 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×D21⋊C4C211yesmaximal, characteristic
2C2×#66631C211yesmaximal, characteristic
3C2×#66575C211yesmaximal, characteristic
4C7×C3⋊#95C211yesmaximal, characteristic, complemented
5C3×C7⋊#95C211yesmaximal, characteristic, complemented
6#10620529C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89436—31nomaximal, Hall
9C23×D21C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C2×D21⋊C4—22no—
17C2×D21⋊C4—22no—
18C2×D21⋊C4C414yescomplemented
19#66631—22no—
20#66575—22no—
21C22×C4×D7—31no—
22C2×(C2×D7)⋊C4—31no—
23C2×(C2×D7)⋊C4—31no—
24C7×#95—31no—
25C7⋊#95—31no—
26C7⋊#95—31no—
27C7⋊#95D311yescharacteristic, complemented
28#4664—71nomaximal, Hall
29C23×C3×C7C2311yescharacteristic
30C22×D21—21no—
31C22×D21—21no—
32C22×D21—21no—
33C22×D21—21no—
34C22×D21C2×C412yes—
35C22×D21D412yes—
36C22×D21D412yes—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×Dic21—21no—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×C7×Dic3—21no—
44C2×C7×Dic3—21no—
45C2×C7×Dic3—21no—
46C2×C7×Dic3—21no—
47C2×C7×Dic3—21no—
48C2×C7×Dic3C2×C412yes—
49C2×C3×Dic7—21no—
50C2×C3×Dic7—21no—
51C2×C3×Dic7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7C2×C412yes—
55D21⋊C4—28no—
56C23×D7—31no—
57C22×C4×C7—31no—
58C22×C4×C7—31no—
59C22×C4×C7C2×D311yescharacteristic
60C22×Dic7—31no—
61C22×Dic7C2×D311yescharacteristic
62C22×Dic7C2×D311yescharacteristic
63C2×C4×D7—34no—
64C2×C4×D7—62no—
65C2×C4×D7—62no—
66(C2×D7)⋊C4—62no—
67(C2×D7)⋊C4—62no—
68C22×C4×D3—71no—
69C2×(C2×D3)⋊C4—71no—
70C2×(C2×D3)⋊C4—71no—
71C3×#95—71no—
72C3⋊#95—71no—
73C3⋊#95—71no—
74C3⋊#95D711yescharacteristic, complemented
75C22×C3×C7C22×C411yescharacteristic
76C22×C3×C7C2×D411yescharacteristic
77C22×C3×C7C2×D411yescharacteristic
78C22×C3×C7#4111yescharacteristic
79C22×C3×C7#4111yescharacteristic
80C22×C3×C7#4111yescharacteristic
81C22×C3×C7#4111yescharacteristic
82C2×D21—22no—
83C2×D21—22no—
84C2×D21—22no—
85C2×D21—22no—
86C2×D21—22no—
87C2×D21—22no—
88C2×D21C22⋊C414yes—
89C3×C4×C7—41no—
90Dic21—41no—
91C3×Dic7—22no—
92C3×Dic7—41no—
93C7×Dic3—22no—
94C7×Dic3—41no—
95#381—211noSylow
96C23×C7C22×D311yescharacteristic
97C22×D7—32no—
98C22×D7—32no—
99C22×D7—32no—
100C22×D7—61no—
101C22×D7—61no—
102C22×D7—61no—
103C22×D7—61no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—32no—
108C2×C4×C7—61no—
109C2×C4×C7—61no—
110C2×C4×C7—61no—
111C2×C4×C7—61no—
112C2×C4×C7—61no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C2×Dic7C4×D312yes—
122C4×D7—68no—
123C23×D3—71no—
124C22×C3×C4—71no—
125C22×C3×C4—71no—
126C22×C3×C4C2×D711yescharacteristic
127C22×Dic3—71no—
128C22×Dic3C2×D711yescharacteristic
129C22×Dic3C2×D711yescharacteristic
130C2×C4×D3—74no—
131C2×C4×D3—142no—
132C2×C4×D3—142no—
133(C2×D3)⋊C4—142no—
134(C2×D3)⋊C4—142no—
135C2×C3×C7C2×C22⋊C411yescharacteristic
136C2×C3×C7#11711yescharacteristic
137C2×C3×C7#11711yescharacteristic
138C2×C3×C7#12311yescharacteristic
139C2×C3×C7#12311yescharacteristic
140C2×C3×C7#12311yescharacteristic
141C2×C3×C7#12311yescharacteristic
142D21—24no—
143C23×C4—211no—
144C2×C22⋊C4—211no—
145C2×C22⋊C4—211no—
146#95—31no—
147#95—71no—
148#95—211no—
149#95—211no—
150C22×C7C2×C4×D311yescharacteristic
151C22×C7#23511yescharacteristic
152C22×C7#23511yescharacteristic
153C22×C7D3×D411yescharacteristic
154C22×C7D3×D411yescharacteristic
155C22×C7#23711yescharacteristic
156C22×C7#23911yescharacteristic
157C2×D7—34no—
158C2×D7—62no—
159C2×D7—62no—
160C2×D7—62no—
161C2×D7—62no—
162C2×D7—62no—
163C2×D7—62no—
164C4×C7—41no—
165C4×C7—62no—
166C4×C7—121no—
167Dic7—22no—
168Dic7—41no—
169Dic7—121no—
170C23×C3C22×D711yescharacteristic
171C22×D3—72no—
172C22×D3—72no—
173C22×D3—72no—
174C22×D3—141no—
175C22×D3—141no—
176C22×D3—141no—
177C22×D3—141no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—72no—
182C2×C3×C4—141no—
183C2×C3×C4—141no—
184C2×C3×C4—141no—
185C2×C3×C4—141no—
186C2×C3×C4—141no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—141no—
193C2×Dic3—141no—
194C2×Dic3—141no—
195C2×Dic3C4×D712yes—
196C4×D3—148no—
197C3×C7#38111yescharacteristic, complemented, Hall
198C24—211no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—214no—
204C22×C4—211no—
205C22×C4—422no—
206C22×C4—422no—
207C22×C4D3×D711yescharacteristic
208C22⋊C4—422no—
209C22⋊C4—422no—
210C2×C7#87211yescharacteristic
211C2×C7D3×C22⋊C411yescharacteristic
212C2×C7#88311yescharacteristic
213C2×C7#88311yescharacteristic
214C2×C7#89211yescharacteristic
215C2×C7#89411yescharacteristic
216C2×C7#89411yescharacteristic
217D7—64no—
218C22×C3C2×C4×D711yescharacteristic
219C22×C3#115711yescharacteristic
220C22×C3#115711yescharacteristic
221C22×C3D4×D711yescharacteristic
222C22×C3D4×D711yescharacteristic
223C22×C3#115911yescharacteristic
224C22×C3#116111yescharacteristic
225C2×D3—74no—
226C2×D3—142no—
227C2×D3—142no—
228C2×D3—142no—
229C2×D3—142no—
230C2×D3—142no—
231C2×D3—142no—
232C3×C4—41no—
233C3×C4—142no—
234C3×C4—281no—
235Dic3—22no—
236Dic3—41no—
237Dic3—281no—
238C23—212no—
239C23—212no—
240C23—212no—
241C23—421no—
242C23—421no—
243C23—421no—
244C23—421no—
245C23C2×D3×D711yescharacteristic
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—32no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—72no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4—141no—
260C2×C4—141no—
261C2×C4—428no—
262C2×C4—421no—
263C2×C4—421no—
264C2×C4—421no—
265C7#466411yescharacteristic, complemented, Sylow
266C2×C3#643511yescharacteristic
267C2×C3D7×C22⋊C411yescharacteristic
268C2×C3#644611yescharacteristic
269C2×C3#644611yescharacteristic
270C2×C3#645511yescharacteristic
271C2×C3#645711yescharacteristic
272C2×C3#645711yescharacteristic
273D3—144no—
274C22—214no—
275C22—422no—
276C22—422no—
277C22—422no—
278C22—422no—
279C22—422no—
280C22—422no—
281C22#6667311yescharacteristic
282C22#6667711yescharacteristic
283C22C4×D3×D711yescharacteristic
284C22#6668111yescharacteristic
285C22#6668311yescharacteristic
286C22#6668611yescharacteristic
287C22#6669411yescharacteristic
288C4—41no—
289C4—62no—
290C4—121no—
291C4—142no—
292C4—281no—
293C4—841no—
294C3#8943611yescharacteristic, complemented, Sylow
295C2—424no—
296C2#1062091811yescharacteristic
297C2#1062096411yescharacteristic
298C2#1062096511yescharacteristic
299C2#1062099511yescharacteristic
300C2#1062100011yescharacteristic
301C2#1062103511yescharacteristic
302C2#1062112411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutatorC22×C3×C7C22×C4
FittingC22×C3×C4×C7C22
socle, cosocleC23×C3×C7C23
prime radicalC23×D21C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#381
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C7×C3⋊#95C2semidirect
5. C3×C7⋊#95C2semidirect
6. #10620529C2semidirect
7. #10620529C2semidirect
18. C2×D21⋊C4C4semidirect
27. C7⋊#95D3semidirect
74. C3⋊#95D7semidirect
197. C3×C7#381semidirect
265. C7#4664semidirect
294. C3#89436semidirect

Maximal subgroups

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

subgroupindex
C22×D21⋊C42
C2×#666312
C2×#665752
C7×C3⋊#952
C3×C7⋊#952
#106205292
#894363
#46647