Finite Group Databasebeta

← #13053615

Subgroups — #13053615

Order 1344.

340 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 72 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
1#10621563C211yesmaximal, characteristic
2M16×D21C211yesmaximal, characteristic, complemented
3C4×D21⋊C4C211yesmaximal, characteristic, complemented
4C7×#825C211yesmaximal, characteristic, complemented
5C3×#6399C211yesmaximal, characteristic, complemented
6#10620514C211yesmaximal, characteristic, complemented
7#10620512C211yesmaximal, characteristic, complemented
8D7×#104—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12#66694—21no—
13C2×D21⋊C4—21no—
14#66684—21no—
15#66681C411yescharacteristic, complemented
16C4×D3×D7—21no—
17#66675—21no—
18D21⋊Q8C411yescharacteristic, complemented
19#66672C411yescharacteristic, complemented
20#66670C411yescharacteristic, complemented
21C3×C7×M16C2211yescharacteristic, complemented
22#66627C2211yescharacteristic, complemented
23#66622—21no—
24C8×D21—21no—
25C4×C7×Dic3C2211yescharacteristic, complemented
26C3×C4×Dic7C2211yescharacteristic, complemented
27Dic3×Dic7—21no—
28D7×#41—31no—
29D7×M16—31no—
30C42×D7—31no—
31C7×#104—31no—
32#6411—31no—
33#6399D311yescharacteristic, complemented
34#6379—31no—
35D3×#104—71nomaximal, Hall
36C22×D21D411yescharacteristic
37C2×D3×D7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39C2×Dic21D411yescharacteristic
40C4×D21C2×C411yescharacteristic
41C4×D21C2×C411yescharacteristic
42C4×D21D411yescharacteristic
43C4×D21D411yescharacteristic
44C7×#67—21no—
45C2×C7×Dic3—21no—
46C7×D12C2×C411yescharacteristic, complemented
47C4×D3×C7—21no—
48C7×Dic6C2×C411yescharacteristic, complemented
49C3×C22⋊D7—21no—
50C2×C3×Dic7—21no—
51C3×D28C2×C411yescharacteristic, complemented
52C3×C4×D7—21no—
53C3×Dic14C2×C411yescharacteristic, complemented
54#4248—21no—
55#4247—21no—
56#4246—21no—
57#4245—21no—
58D21⋊C4—24no—
59D21⋊C4—21no—
60D3×Dic7—21no—
61Dic3×D7—21no—
62C3×C7×C8—21no—
63C3⋊(C7⋊C8)—21no—
64C7×#41—31no—
65#1161—31no—
66Q8×D7—31no—
67#1159—31no—
68#1159—61no—
69D4×D7—31no—
70D4×D7—61no—
71#1157—61no—
72#1157C2×D311yescharacteristic
73C2×C4×D7—31no—
74C2×C4×D7—61no—
75C2×C4×D7—61no—
76C7×M16C2×D311yescharacteristic, complemented
77C42×C7—31no—
78C4×Dic7—61no—
79C4×Dic7C2×D311yescharacteristic, complemented
80#1136—31no—
81#1131—61no—
82C8×D7—61no—
83D3×#41—71no—
84D3×M16—71no—
85C42×D3—71no—
86C3×#104—71no—
87#837—71no—
88#825D711yescharacteristic, complemented
89#805—71no—
90C22×C3×C7C2×D411yescharacteristic
91C2×D21—22no—
92C2×D21C22⋊C411yescharacteristic
93C2×D21C22⋊C411yescharacteristic
94C2×D3×C7—21no—
95C2×C3×D7—21no—
96D3×D7—42no—
97C3×C4×C7C22×C411yescharacteristic
98C3×C4×C7C2×D411yescharacteristic
99Dic21C22⋊C411yescharacteristic
100Dic21C22⋊C411yescharacteristic
101C3×Dic7—22no—
102C3×Dic7—21no—
103C7×Dic3—22no—
104C7×Dic3—21no—
105C2×#104—211noSylow
106C22×D7—31no—
107C22×D7—61no—
108C7×Q8—31no—
109C7×D4—31no—
110C7×D4—61no—
111C2×C4×C7—61no—
112C2×C4×C7—61no—
113C2×C4×C7C22×D311yescharacteristic
114C22⋊D7—21no—
115C22⋊D7—61no—
116C22⋊D7—61no—
117C2×Dic7—21no—
118C2×Dic7—31no—
119C2×Dic7—61no—
120D28—61no—
121D28C4×D311yescharacteristic, complemented
122C4×D7—21no—
123C4×D7—31no—
124C4×D7—31no—
125C4×D7—31no—
126C4×D7—31no—
127C4×D7—61no—
128C4×D7—64no—
129C4×D7—61no—
130Dic14—61no—
131Dic14C4×D311yescharacteristic, complemented
132C7×C8—21no—
133C7⋊C8—61no—
134C3×#41—71no—
135#239—71no—
136D3×Q8—71no—
137#237—71no—
138#237—141no—
139D3×D4—71no—
140D3×D4—141no—
141#235—141no—
142#235C2×D711yescharacteristic
143C2×C4×D3—71no—
144C2×C4×D3—141no—
145C2×C4×D3—141no—
146C3×M16C2×D711yescharacteristic, complemented
147C3×C42—71no—
148C4×Dic3—141no—
149C4×Dic3C2×D711yescharacteristic, complemented
150#208—71no—
151ΓL(1,𝔽25)—141no—
152D3×C8—141no—
153C2×C3×C7—21no—
154C2×C3×C7C2×C22⋊C411yescharacteristic
155D21—21no—
156D21#10412yescomplemented
157C3×D7—41no—
158D3×C7—41no—
159C2×#41—211no—
160C2×M16—211no—
161C2×C42—211no—
162#104—31no—
163#104—71no—
164#104—211no—
165#104—211no—
166C22×C7—61no—
167C22×C7D3×D411yescharacteristic
168C2×D7—21no—
169C2×D7—31no—
170C2×D7—31no—
171C2×D7—62no—
172C2×D7—122no—
173C4×C7—62no—
174C4×C7—61no—
175C4×C7C2×C4×D311yescharacteristic
176C4×C7D3×D411yescharacteristic
177Dic7—22no—
178Dic7—21no—
179Dic7—31no—
180Dic7—31no—
181C22×D3—71no—
182C22×D3—141no—
183C3×Q8—71no—
184C3×D4—71no—
185C3×D4—141no—
186C2×C3×C4—141no—
187C2×C3×C4—141no—
188C2×C3×C4C22×D711yescharacteristic
189#67—21no—
190#67—141no—
191#67—141no—
192C2×Dic3—21no—
193C2×Dic3—71no—
194C2×Dic3—141no—
195D12—141no—
196D12C4×D711yescharacteristic, complemented
197C4×D3—21no—
198C4×D3—71no—
199C4×D3—71no—
200C4×D3—71no—
201C4×D3—71no—
202C4×D3—141no—
203C4×D3—144no—
204C4×D3—141no—
205Dic6—141no—
206Dic6C4×D711yescharacteristic, complemented
207C3×C8—21no—
208C3⋊C8—141no—
209C3×C7C2×#10411yescharacteristic, complemented, Hall
210#41—31no—
211#41—71no—
212#41—211no—
213#41—211no—
214#41—421no—
215#41—421no—
216C2×Q8—211no—
217C2×D4—211no—
218C2×D4—421no—
219C22×C4—211no—
220C22×C4—421no—
221C22×C4—421no—
222M16—211no—
223M16—421no—
224M16D3×D711yescharacteristic, complemented
225C2×C8—421no—
226C42—31no—
227C42—71no—
228C42—421no—
229C2×C7—21no—
230C2×C7—121no—
231C2×C7D3×C22⋊C411yescharacteristic
232D7—32no—
233D7—41no—
234D7—61no—
235C22×C3—141no—
236C22×C3D4×D711yescharacteristic
237C2×D3—21no—
238C2×D3—71no—
239C2×D3—71no—
240C2×D3—142no—
241C2×D3—282no—
242C3×C4—142no—
243C3×C4—141no—
244C3×C4C2×C4×D711yescharacteristic
245C3×C4D4×D711yescharacteristic
246Dic3—22no—
247Dic3—21no—
248Dic3—71no—
249Dic3—71no—
250C23—211no—
251C23—421no—
252Q8—31no—
253Q8—71no—
254Q8—421no—
255D4—31no—
256D4—61no—
257D4—71no—
258D4—141no—
259D4—421no—
260D4—421no—
261D4—421no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—211no—
267C2×C4—211no—
268C2×C4—211no—
269C2×C4—211no—
270C2×C4—211no—
271C2×C4—421no—
272C2×C4—424no—
273C2×C4—421no—
274C2×C4—421no—
275C2×C4C2×D3×D711yescharacteristic
276C8—21no—
277C8—421no—
278C7D3×#10411yescharacteristic, complemented, Sylow
279C2×C3—21no—
280C2×C3—281no—
281C2×C3D7×C22⋊C411yescharacteristic
282D3—41no—
283D3—72no—
284D3—141no—
285C22—61no—
286C22—141no—
287C22—211no—
288C22—211no—
289C22—422no—
290C22—842no—
291C22#6668111yescharacteristic
292C4—62no—
293C4—61no—
294C4—142no—
295C4—141no—
296C4—211no—
297C4—211no—
298C4C4×D3×D711yescharacteristic
299C4#6669411yescharacteristic
300C3D7×#10411yescharacteristic, complemented, Sylow
301C2—21no—
302C2—121no—
303C2—212no—
304C2—281no—
305C2—421no—
306C2#1062103511yescharacteristic
subgroupG/H
centerC4#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7C2×C22⋊C4
prime radical#10621563C2
cosocleC2×C3×C4×C7C23
hypercenterM16D3×D7
hypocenterC3×C7C2×#104
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. M16×D21C2semidirect
3. C4×D21⋊C4C2semidirect
4. C7×#825C2semidirect
5. C3×#6399C2semidirect
6. #10620514C2semidirect
7. #10620512C2semidirect
15. #66681C4semidirect
18. D21⋊Q8C4semidirect
19. #66672C4semidirect
20. #66670C4semidirect
21. C3×C7×M16C22semidirect
22. #66627C22semidirect
25. C4×C7×Dic3C22semidirect
26. C3×C4×Dic7C22semidirect
33. #6399D3semidirect
46. C7×D12C2×C4semidirect
48. C7×Dic6C2×C4semidirect
51. C3×D28C2×C4semidirect
53. C3×Dic14C2×C4semidirect
76. C7×M16C2×D3semidirect
79. C4×Dic7C2×D3semidirect
88. #825D7semidirect
121. D28C4×D3semidirect
131. Dic14C4×D3semidirect
146. C3×M16C2×D7semidirect
149. C4×Dic3C2×D7semidirect
156. D21#104semidirect
196. D12C4×D7semidirect
206. Dic6C4×D7semidirect
209. C3×C7C2×#104semidirect
224. M16D3×D7semidirect
278. C7D3×#104semidirect
300. C3D7×#104semidirect

Maximal subgroups

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

subgroupindex
#106215632
M16×D212
C4×D21⋊C42
C7×#8252
C3×#63992
#106205142
#106205122
D7×#1043
D3×#1047