Finite Group Databasebeta

← #13059563

Subgroups — #13059563

Order 1344.

540 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 136 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×#66728C211yesmaximal, characteristic
2C2×D21⋊Q8C211yesmaximal, characteristic, complemented
3C7×#924C211yesmaximal, characteristic, complemented
4C3×#6487C211yesmaximal, characteristic, complemented
5#10621124C212yesmaximal, complemented
6#10621088C212yesmaximal, complemented
7#10620995C212yesmaximal, complemented
8#10620952C211yesmaximal, characteristic, complemented
9#10620945C211yesmaximal, characteristic, complemented
10#10620909C211yesmaximal, characteristic, complemented
11#10620890C212yesmaximal, complemented
12#90319—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×#4268C2211yescharacteristic
15C2×#4268C2211yescharacteristic
16#66728—24no—
17C2×D84C2211yescharacteristic
18C2×C4×D21C2211yescharacteristic
19C2×C4×D21C2211yescharacteristic
20C2×Dic42C2211yescharacteristic
21C2×C7×Dic6C2211yescharacteristic
22C2×C3×Dic14C2211yescharacteristic
23C2×#4248C2212yes—
24C2×D21⋊C4C2212yes—
25D21⋊Q8—24no—
26C7×#217C2212yes—
27C7×C4⋊Dic3C2211yescharacteristic
28C7×Dic3⋊C4C2212yes—
29C3×C22⋊Dic7C2212yes—
30C3×C4⋊Dic7C2211yescharacteristic
31C3×Dic7⋊C4C2212yes—
32Dic3⋊Dic7C2212yes—
33Dic7⋊Dic3C2212yes—
34#66575C2212yes—
35#66575C2212yes—
36#66575C2212yes—
37Dic3×Dic7C2212yes—
38C2×Q8×D7—31no—
39C2×#1157—31no—
40C7×C22⋊Q8—31no—
41#6509—31no—
42#6487D311yescharacteristic, complemented
43#6460—31no—
44#6457—32no—
45#6455—31no—
46#6451—32no—
47#6448—32no—
48#6446—32no—
49#5628—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×D21C2311yescharacteristic
53C2×C3×C4×C7—22no—
54C2×C3×C4×C7C2311yescharacteristic
55C2×C3×C4×C7C2311yescharacteristic
56#4268—22no—
57#4268—22no—
58C2×Dic21C2311yescharacteristic
59C2×Dic21C2311yescharacteristic
60D84—22no—
61C4×D21—22no—
62C4×D21D414yes—
63Dic42—22no—
64C2×C7×Dic3C2312yes—
65C2×C7×Dic3C2312yes—
66C7×Dic6—22no—
67C2×C3×Dic7C2312yes—
68C2×C3×Dic7C2312yes—
69C3×Dic14—22no—
70#4248—24no—
71D21⋊C4—24no—
72C2×C7×Q8—31no—
73C22×C4×C7C2×D311yescharacteristic
74C2×C22⋊D7—31no—
75C2×C22⋊D7—31no—
76Q8×D7—64no—
77#1157—64no—
78C2×D28—31no—
79C2×C4×D7—31no—
80C2×C4×D7—31no—
81C2×C4×D7—32no—
82C2×Dic14—31no—
83C2×Dic14—32no—
84C2×Dic14C2×D311yescharacteristic
85C7×C4⋊C4—32no—
86C7×C4⋊C4—31no—
87C7×C22⋊C4—32no—
88C22⋊Dic7C2×D312yes—
89(C2×D7)⋊C4—32no—
90(C2×D7)⋊C4—32no—
91(C2×D7)⋊C4—32no—
92C4⋊Dic7—32no—
93C4⋊Dic7C2×D311yescharacteristic
94Dic7⋊C4—32no—
95Dic7⋊C4C2×D312yes—
96C4×Dic7—32no—
97C2×D3×Q8—71no—
98C2×#235—71no—
99C3×C22⋊Q8—71no—
100#946—71no—
101#924D711yescharacteristic, complemented
102#897—71no—
103#894—72no—
104#892—71no—
105#888—72no—
106#885—72no—
107#883—72no—
108C22×C3×C7—21no—
109C22×C3×C7—21no—
110C22×C3×C7—21no—
111C22×C3×C7C2411yescharacteristic
112C2×D21—21no—
113C2×D21—21no—
114C2×D21—21no—
115C2×D21—21no—
116C2×D21—21no—
117C2×D21C2×D412yes—
118C3×C4×C7—21no—
119C3×C4×C7C2×D412yes—
120Dic21—21no—
121Dic21C2×D412yes—
122C3×Dic7—22no—
123C3×Dic7—22no—
124C7×Dic3—22no—
125C7×Dic3—22no—
126#536—211noSylow
127C23×C7C22×D311yescharacteristic
128C22×D7—31no—
129C22×D7—31no—
130C7×Q8—62no—
131C2×C4×C7—22no—
132C2×C4×C7—32no—
133C2×C4×C7—32no—
134C2×C4×C7C22×D311yescharacteristic
135C2×C4×C7C22×D311yescharacteristic
136C22⋊D7—62no—
137C22⋊D7—62no—
138C2×Dic7—31no—
139C2×Dic7—31no—
140C2×Dic7C22×D312yes—
141C2×Dic7C22×D312yes—
142D28—62no—
143C4×D7—34no—
144C4×D7—64no—
145C4×D7—62no—
146Dic14—22no—
147Dic14—64no—
148Dic14—62no—
149C2×C3×Q8—71no—
150C22×C3×C4C2×D711yescharacteristic
151C2×#67—71no—
152C2×#67—71no—
153D3×Q8—144no—
154#235—144no—
155C2×D12—71no—
156C2×C4×D3—71no—
157C2×C4×D3—71no—
158C2×C4×D3—72no—
159C2×Dic6—71no—
160C2×Dic6—72no—
161C2×Dic6C2×D711yescharacteristic
162C3×C4⋊C4—72no—
163C3×C4⋊C4—71no—
164C3×C22⋊C4—72no—
165#217C2×D712yes—
166(C2×D3)⋊C4—72no—
167(C2×D3)⋊C4—72no—
168(C2×D3)⋊C4—72no—
169C4⋊Dic3—72no—
170C4⋊Dic3C2×D711yescharacteristic
171Dic3⋊C4—72no—
172Dic3⋊C4C2×D712yes—
173C4×Dic3—72no—
174C2×C3×C7—41no—
175C2×C3×C7C22×D411yescharacteristic
176C2×C3×C72−1+411yescharacteristic
177C2×C3×C72−1+411yescharacteristic
178D21—22no—
179D21—41no—
180C2×#41—211no—
181C22×Q8—211no—
182C4⋊Q8—212no—
183#124—212no—
184#123—212no—
185#123—212no—
186C22⋊Q8—31no—
187C22⋊Q8—71no—
188C22⋊Q8—211no—
189C22⋊Q8—211no—
190#117—211no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7C23×D311yescharacteristic
195C2×D7—32no—
196C2×D7—61no—
197C2×D7—61no—
198C2×D7—61no—
199C2×D7—61no—
200C2×D7—61no—
201C4×C7—21no—
202C4×C7—62no—
203C4×C7—62no—
204C4×C7D3×D412yes—
205Dic7—22no—
206Dic7—22no—
207Dic7—32no—
208Dic7—61no—
209C23×C3C22×D711yescharacteristic
210C22×D3—71no—
211C22×D3—71no—
212C3×Q8—142no—
213C2×C3×C4—22no—
214C2×C3×C4—72no—
215C2×C3×C4—72no—
216C2×C3×C4C22×D711yescharacteristic
217C2×C3×C4C22×D711yescharacteristic
218#67—142no—
219#67—142no—
220C2×Dic3—71no—
221C2×Dic3—71no—
222C2×Dic3C22×D712yes—
223C2×Dic3C22×D712yes—
224D12—142no—
225C4×D3—74no—
226C4×D3—144no—
227C4×D3—142no—
228Dic6—22no—
229Dic6—144no—
230Dic6—142no—
231C3×C7#53611yescharacteristic, complemented, Hall
232#41—424no—
233C2×Q8—31no—
234C2×Q8—71no—
235C2×Q8—212no—
236C2×Q8—211no—
237C2×Q8—424no—
238C2×D4—211no—
239C2×D4—211no—
240C2×D4—211no—
241C22×C4—212no—
242C22×C4—211no—
243C22×C4—211no—
244C22×C4D3×D711yescharacteristic
245C4⋊C4—31no—
246C4⋊C4—32no—
247C4⋊C4—72no—
248C4⋊C4—71no—
249C4⋊C4—212no—
250C4⋊C4—212no—
251C22⋊C4—32no—
252C22⋊C4—72no—
253C22⋊C4—212no—
254C22⋊C4—212no—
255C22⋊C4—212no—
256C42—212no—
257C2×C7—41no—
258C2×C7C2×D3×D411yescharacteristic
259C2×C7#100711yescharacteristic
260C2×C7#101011yescharacteristic
261D7—62no—
262D7—121no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3C23×D711yescharacteristic
267C2×D3—72no—
268C2×D3—141no—
269C2×D3—141no—
270C2×D3—141no—
271C2×D3—141no—
272C2×D3—141no—
273C3×C4—21no—
274C3×C4—142no—
275C3×C4—142no—
276C3×C4D4×D712yes—
277Dic3—22no—
278Dic3—22no—
279Dic3—72no—
280Dic3—141no—
281C23—211no—
282C23—211no—
283C23C2×D3×D711yescharacteristic
284Q8—62no—
285Q8—142no—
286Q8—422no—
287Q8—424no—
288D4—422no—
289D4—422no—
290D4—422no—
291C2×C4—22no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—72no—
295C2×C4—72no—
296C2×C4—211no—
297C2×C4—214no—
298C2×C4—211no—
299C2×C4—424no—
300C2×C4—422no—
301C2×C4C2×D3×D711yescharacteristic
302C2×C4C2×D3×D711yescharacteristic
303C7#562811yescharacteristic, complemented, Sylow
304C2×C3—41no—
305C2×C3C2×D4×D711yescharacteristic
306C2×C3#655111yescharacteristic
307C2×C3#655411yescharacteristic
308D3—142no—
309D3—281no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—212no—
314C22—421no—
315C22—421no—
316C22—421no—
317C22—421no—
318C22—421no—
319C22C22×D3×D711yescharacteristic
320C4—21no—
321C4—62no—
322C4—62no—
323C4—142no—
324C4—142no—
325C4—212no—
326C4—421no—
327C4#6669412yes—
328C3#9031911yescharacteristic, complemented, Sylow
329C2—41no—
330C2—422no—
331C2—841no—
332C2#1062154711yescharacteristic
333C2#1062155111yescharacteristic
334C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC2×#66728C2
hypercenterC22×C4D3×D7
hypocenterC3×C7#536
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×D21⋊Q8C2semidirect
3. C7×#924C2semidirect
4. C3×#6487C2semidirect
5. #10621124C2semidirect
6. #10621088C2semidirect
7. #10620995C2semidirect
8. #10620952C2semidirect
9. #10620945C2semidirect
10. #10620909C2semidirect
11. #10620890C2semidirect
42. #6487D3semidirect
101. #924D7semidirect
231. C3×C7#536semidirect
303. C7#5628semidirect
328. C3#90319semidirect

Maximal subgroups

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

subgroupindex
C2×#667282
C2×D21⋊Q82
C7×#9242
C3×#64872
#106211242
#106210882
#106209952
#106209522
#106209452
#106209092
#106208902
#903193
#56287