Finite Group Databasebeta

← #13058590

Subgroups — #13058590

Order 1344.

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

normal subgroups only

This group has 580 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
1C2×#66669C211yesmaximal, characteristic, complemented
2C2×D7×Dic6C211yesmaximal, characteristic, complemented
3#10621370C211yesmaximal, characteristic, complemented
4C7×#897C211yesmaximal, characteristic, complemented
5C3×#6458C211yesmaximal, characteristic, complemented
6#10621026C212yesmaximal, complemented
7C4×#4245C211yesmaximal, characteristic, complemented
8#10620996C212yesmaximal, complemented
9#10620992C212yesmaximal, complemented
10#10620970C212yesmaximal, complemented
11#10620930C211yesmaximal, characteristic, complemented
12#90324—31nomaximal, Hall
13C2×Dic42C2211yescharacteristic
14C2×C4×D3×C7C2211yescharacteristic
15C2×C7×Dic6C2211yescharacteristic
16C2×C3×D28C2211yescharacteristic
17C2×C3×D28C2212yes—
18C2×C3×C4×D7C2211yescharacteristic
19C2×#4248C2212yes—
20C2×#4245C2211yescharacteristic
21C2×#4245C2212yes—
22C2×Dic3×D7C2212yes—
23C2×Dic3×D7C2212yes—
24#66669—24no—
25D7×Dic6—24no—
26C3×C7×C4⋊C4C2211yescharacteristic
27#66629C2212yes—
28C4×Dic21C2211yescharacteristic
29C4×Dic21C2212yes—
30C7×(C2×D3)⋊C4C2212yes—
31C7×C4⋊Dic3C2212yes—
32C3×(C2×D7)⋊C4C2212yes—
33#66577C2211yescharacteristic
34(C2×D3)⋊Dic7C2211yescharacteristic
35(C2×D7)⋊Dic3C2212yes—
36(C2×D7)⋊Dic3C2212yes—
37(C2×D7)⋊Dic3C2211yescharacteristic
38C2×Q8×D7—31no—
39C2×#1157—31no—
40C7×C22⋊Q8—31no—
41#6510—31no—
42C4×C22⋊D7—31no—
43#6459—32no—
44#6458D311yescharacteristic, complemented
45#6456—32no—
46#6450—31no—
47#6448—32no—
48#6447—32no—
49#5598—71nomaximal, Hall
50C22×D3×C7C2311yescharacteristic
51C22×C3×D7C2312yes—
52C22×C3×D7C2311yescharacteristic
53C2×C3×C4×C7C2312yes—
54C2×C3×C4×C7C2311yescharacteristic
55C2×Dic21C2311yescharacteristic
56C2×Dic21C2311yescharacteristic
57C2×Dic21C2312yes—
58Dic42D414yescomplemented
59C2×C7×Dic3C2312yes—
60C2×C7×Dic3C2311yescharacteristic
61C4×D3×C7—22no—
62C7×Dic6—22no—
63C2×C3×Dic7C2311yescharacteristic
64C3×D28—24no—
65C3×D28—22no—
66C3×C4×D7—22no—
67#4248—24no—
68#4245—22no—
69#4245—24no—
70Dic3×D7—24no—
71Dic3×D7—24no—
72C2×C7×Q8—31no—
73C22×C4×C7—31no—
74C2×C22⋊D7—31no—
75C2×C22⋊D7—32no—
76Q8×D7—64no—
77#1157—64no—
78C2×D28C2×D312yes—
79C2×D28C2×D311yescharacteristic
80C2×C4×D7—32no—
81C2×C4×D7—32no—
82C2×C4×D7C2×D311yescharacteristic
83C2×Dic14—31no—
84C2×Dic14—32no—
85C7×C4⋊C4—32no—
86C7×C4⋊C4C2×D311yescharacteristic
87C7×C22⋊C4—32no—
88C22⋊Dic7—31no—
89(C2×D7)⋊C4—32no—
90(C2×D7)⋊C4—32no—
91(C2×D7)⋊C4—31no—
92(C2×D7)⋊C4C2×D312yes—
93Dic7⋊C4—32no—
94Dic7⋊C4—31no—
95C4×Dic7—32no—
96C4×Dic7—31no—
97C2×#237—71no—
98C22×Dic6—71no—
99C3×#121—71no—
100#939—72no—
101#936—71no—
102D4×Dic3—72no—
103C4×#67—71no—
104#897D711yescharacteristic, complemented
105#887—71no—
106#885—72no—
107#878—72no—
108C22×C3×C7C2411yescharacteristic
109C2×D3×C7—21no—
110C2×D3×C7—21no—
111C2×D3×C7—21no—
112C2×C3×D7—21no—
113C2×C3×D7—21no—
114C2×C3×D7—22no—
115C2×C3×D7—22no—
116C2×C3×D7—22no—
117C2×C3×D7#4112yescomplemented
118C3×C4×C7—22no—
119C3×C4×C7C2×D412yes—
120Dic21—21no—
121Dic21—21no—
122Dic21C2×D414yes—
123C3×Dic7—21no—
124C7×Dic3—21no—
125C7×Dic3—22no—
126#548—211noSylow
127C23×C7—31no—
128C22×D7C22×D312yes—
129C22×D7C22×D311yescharacteristic
130C7×Q8—62no—
131C2×C4×C7—31no—
132C2×C4×C7—32no—
133C2×C4×C7—62no—
134C2×C4×C7C22×D312yes—
135C2×C4×C7C22×D311yescharacteristic
136C22⋊D7—64no—
137C22⋊D7—62no—
138C2×Dic7—32no—
139C2×Dic7—31no—
140C2×Dic7—31no—
141C2×Dic7C22×D311yescharacteristic
142D28—24no—
143D28—22no—
144C4×D7—22no—
145C4×D7—64no—
146C4×D7—64no—
147Dic14—34no—
148Dic14—64no—
149C2×C3×D4—72no—
150C2×C3×D4—71no—
151C22×C3×C4—71no—
152C2×#67—72no—
153C2×#67—71no—
154C22×Dic3—72no—
155C22×Dic3—72no—
156#237—144no—
157C2×C4×D3C2×D711yescharacteristic
158C2×Dic6—71no—
159C2×Dic6—72no—
160C2×Dic6—144no—
161C2×Dic6C2×D711yescharacteristic
162C3×C4⋊C4C2×D711yescharacteristic
163C3×C22⋊C4—72no—
164#217—72no—
165#217—72no—
166#217—71no—
167(C2×D3)⋊C4—71no—
168(C2×D3)⋊C4C2×D712yes—
169C4⋊Dic3C2×D712yes—
170Dic3⋊C4—72no—
171Dic3⋊C4—71no—
172C4×Dic3—72no—
173C4×Dic3—71no—
174C2×C3×C7C22×D411yescharacteristic
175C2×C3×C7C2×#4111yescharacteristic
176C2×C3×C72−1+411yescharacteristic
177C3×D7—22no—
178C3×D7—42no—
179D3×C7—41no—
180C2×#41—211no—
181C22×Q8—211no—
182#124—212no—
183#124—211no—
184C22⋊Q8—31no—
185C22⋊Q8—212no—
186#121—71no—
187#121—212no—
188C4×Q8—211no—
189C4×D4—212no—
190C4×D4—211no—
191C22×C7—61no—
192C22×C7—61no—
193C22×C7—61no—
194C22×C7C23×D311yescharacteristic
195C2×D7—21no—
196C2×D7—21no—
197C2×D7—22no—
198C2×D7—22no—
199C2×D7—22no—
200C2×D7#23712yescomplemented
201C4×C7—22no—
202C4×C7—62no—
203C4×C7—61no—
204C4×C7D3×D412yes—
205Dic7—21no—
206Dic7—34no—
207Dic7—61no—
208Dic7—61no—
209C23×C3—71no—
210C23×C3—72no—
211C22×D3C22×D711yescharacteristic
212C3×D4—144no—
213C3×D4—142no—
214C2×C3×C4—71no—
215C2×C3×C4—142no—
216C2×C3×C4C22×D712yes—
217C2×C3×C4C22×D711yescharacteristic
218#67—144no—
219#67—142no—
220C2×Dic3—72no—
221C2×Dic3—71no—
222C2×Dic3—71no—
223C2×Dic3—144no—
224C2×Dic3—144no—
225C2×Dic3C22×D711yescharacteristic
226C2×Dic3C22×D712yes—
227C4×D3—22no—
228Dic6—22no—
229Dic6—74no—
230Dic6—144no—
231C3×C7#54811yescharacteristic, complemented, Hall
232#41—424no—
233C2×Q8—31no—
234C2×Q8—212no—
235C2×Q8—211no—
236C2×Q8—424no—
237C2×D4—72no—
238C2×D4—71no—
239C2×D4—211no—
240C2×D4—212no—
241C22×C4—31no—
242C22×C4—71no—
243C22×C4—212no—
244C22×C4—212no—
245C4⋊C4—32no—
246C4⋊C4—211no—
247C4⋊C4—212no—
248C4⋊C4D3×D711yescharacteristic
249C22⋊C4—32no—
250C22⋊C4—72no—
251C22⋊C4—211no—
252C22⋊C4—212no—
253C22⋊C4—212no—
254C22⋊C4—211no—
255C42—212no—
256C42—211no—
257C2×C7—121no—
258C2×C7C2×D3×D411yescharacteristic
259C2×C7C2×#23711yescharacteristic
260C2×C7#101011yescharacteristic
261D7—22no—
262D7—42no—
263C22×C3—72no—
264C22×C3—141no—
265C22×C3—141no—
266C22×C3—142no—
267C22×C3—142no—
268C22×C3—142no—
269C22×C3C23×D711yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C3×C4—22no—
274C3×C4—141no—
275C3×C4D4×D712yes—
276Dic3—22no—
277Dic3—21no—
278Dic3—74no—
279Dic3—141no—
280Dic3—141no—
281C23—31no—
282C23—71no—
283C23—72no—
284Q8—62no—
285Q8—214no—
286Q8—424no—
287D4—144no—
288D4—142no—
289D4—424no—
290D4—422no—
291C2×C4—32no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—71no—
295C2×C4—142no—
296C2×C4—212no—
297C2×C4—211no—
298C2×C4—211no—
299C2×C4—424no—
300C2×C4—424no—
301C2×C4C2×D3×D712yes—
302C2×C4C2×D3×D711yescharacteristic
303C7#559811yescharacteristic, complemented, Sylow
304C2×C3—142no—
305C2×C3—282no—
306C2×C3C2×D4×D711yescharacteristic
307C2×C3#655111yescharacteristic
308C2×C3D7×#4111yescharacteristic
309D3—41no—
310C22—61no—
311C22—61no—
312C22—61no—
313C22—72no—
314C22—141no—
315C22—141no—
316C22—142no—
317C22—142no—
318C22—142no—
319C22C22×D3×D711yescharacteristic
320C4—22no—
321C4—61no—
322C4—62no—
323C4—141no—
324C4—214no—
325C4—421no—
326C4—421no—
327C4#6668112yes—
328C3#9032411yescharacteristic, complemented, Sylow
329C2—121no—
330C2—142no—
331C2—282no—
332C2C2×#6668111yescharacteristic
333C2D7×#23711yescharacteristic
334C2#1062157511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C4⋊C4C22
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#548
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#66669C2semidirect
2. C2×D7×Dic6C2semidirect
3. #10621370C2semidirect
4. C7×#897C2semidirect
5. C3×#6458C2semidirect
6. #10621026C2semidirect
7. C4×#4245C2semidirect
8. #10620996C2semidirect
9. #10620992C2semidirect
10. #10620970C2semidirect
11. #10620930C2semidirect
44. #6458D3semidirect
58. Dic42D4semidirect
104. #897D7semidirect
117. C2×C3×D7#41semidirect
200. C2×D7#237semidirect
231. C3×C7#548semidirect
303. C7#5598semidirect
328. C3#90324semidirect

Maximal subgroups

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

subgroupindex
C2×#666692
C2×D7×Dic62
#106213702
C7×#8972
C3×#64582
#106210262
C4×#42452
#106209962
#106209922
#106209702
#106209302
#903243
#55987