Finite Group Databasebeta

← #13053616

Subgroups — #13053616

Order 1344.

308 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 70 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
3#10620948C211yesmaximal, characteristic, complemented
4C7×#825C211yesmaximal, characteristic, complemented
5C3×#6399C211yesmaximal, characteristic, complemented
6#10620514C211yesmaximal, characteristic, complemented
7#10620512C211yesmaximal, characteristic, complemented
8#89593—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12#66694—21no—
13#66684—21no—
14#66681C411yescharacteristic, complemented
15C4×D3×D7—21no—
16#66675—21no—
17D21⋊Q8C411yescharacteristic, complemented
18#66672C411yescharacteristic, complemented
19#66670C411yescharacteristic, complemented
20C3×C7×M16C2211yescharacteristic, complemented
21#66627C2211yescharacteristic, complemented
22#66622—21no—
23C8×D21—21no—
24C4×C7×Dic3C2211yescharacteristic, complemented
25C3×C4×Dic7C2211yescharacteristic, complemented
26#66577—21no—
27#66575—21no—
28D7×#41—31no—
29D7×M16—31no—
30#6435—31no—
31C7×#104—31no—
32#6411—31no—
33#6399D311yescharacteristic, complemented
34#6379—31no—
35#4821—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
47C4×D3×C7—21no—
48C7×Dic6C2×C411yescharacteristic
49C3×C22⋊D7—21no—
50C2×C3×Dic7—21no—
51C3×D28C2×C411yescharacteristic
52C3×C4×D7—21no—
53C3×Dic14C2×C411yescharacteristic
54#4248—21no—
55#4247—21no—
56#4246—21no—
57#4245—21no—
58D21⋊C4—21no—
59D3×Dic7—21no—
60Dic3×D7—21no—
61C3×C7×C8—21no—
62C3⋊(C7⋊C8)—21no—
63C7×#41—31no—
64#1161—31no—
65Q8×D7—31no—
66#1159—31no—
67#1159—61no—
68D4×D7—31no—
69D4×D7—61no—
70#1157—61no—
71#1157C2×D311yescharacteristic
72C2×C4×D7—31no—
73C2×C4×D7—61no—
74C7×M16C2×D311yescharacteristic, complemented
75C42×C7—31no—
76(C2×D7)⋊C4—61no—
77Dic7⋊C4—61no—
78C4×Dic7C2×D311yescharacteristic, complemented
79#1136—31no—
80#1131—61no—
81C8×D7—61no—
82D3×#41—71no—
83D3×M16—71no—
84#872—71no—
85C3×#104—71no—
86#837—71no—
87#825D711yescharacteristic, complemented
88#805—71no—
89C22×C3×C7C2×D411yescharacteristic
90C2×D21—41no—
91C2×D21C22⋊C411yescharacteristic
92C2×D21C22⋊C411yescharacteristic
93C2×D3×C7—21no—
94C2×C3×D7—21no—
95D3×D7—42no—
96C3×C4×C7C22×C411yescharacteristic
97C3×C4×C7C2×D411yescharacteristic
98Dic21C22⋊C411yescharacteristic
99Dic21C22⋊C411yescharacteristic
100C3×Dic7—21no—
101C3×Dic7—41no—
102C7×Dic3—21no—
103C7×Dic3—41no—
104#421—211noSylow
105C22×D7—31no—
106C22×D7—61no—
107C7×Q8—31no—
108C7×D4—31no—
109C7×D4—61no—
110C2×C4×C7—61no—
111C2×C4×C7—61no—
112C2×C4×C7C22×D311yescharacteristic
113C22⋊D7—21no—
114C22⋊D7—61no—
115C22⋊D7—61no—
116C2×Dic7—21no—
117C2×Dic7—31no—
118C2×Dic7—61no—
119D28—61no—
120D28C4×D311yescharacteristic
121C4×D7—21no—
122C4×D7—31no—
123C4×D7—31no—
124C4×D7—31no—
125C4×D7—31no—
126C4×D7—61no—
127C4×D7—61no—
128Dic14—61no—
129Dic14C4×D311yescharacteristic
130C7×C8—21no—
131C7⋊C8—61no—
132C3×#41—71no—
133#239—71no—
134D3×Q8—71no—
135#237—71no—
136#237—141no—
137D3×D4—71no—
138D3×D4—141no—
139#235—141no—
140#235C2×D711yescharacteristic
141C2×C4×D3—71no—
142C2×C4×D3—141no—
143C3×M16C2×D711yescharacteristic, complemented
144C3×C42—71no—
145(C2×D3)⋊C4—141no—
146Dic3⋊C4—141no—
147C4×Dic3C2×D711yescharacteristic, complemented
148#208—71no—
149ΓL(1,𝔽25)—141no—
150D3×C8—141no—
151C2×C3×C7—21no—
152C2×C3×C7C2×C22⋊C411yescharacteristic
153D21—21no—
154D21—21no—
155C3×D7—41no—
156D3×C7—41no—
157C2×#41—211no—
158C2×M16—211no—
159#117—211no—
160#104—31no—
161#104—71no—
162#104—211no—
163#104—211no—
164C22×C7—61no—
165C22×C7D3×D411yescharacteristic
166C2×D7—21no—
167C2×D7—31no—
168C2×D7—31no—
169C2×D7—122no—
170C2×D7—121no—
171C4×C7—61no—
172C4×C7—121no—
173C4×C7C2×C4×D311yescharacteristic
174C4×C7D3×D411yescharacteristic
175Dic7—21no—
176Dic7—31no—
177Dic7—31no—
178Dic7—41no—
179C22×D3—71no—
180C22×D3—141no—
181C3×Q8—71no—
182C3×D4—71no—
183C3×D4—141no—
184C2×C3×C4—141no—
185C2×C3×C4—141no—
186C2×C3×C4C22×D711yescharacteristic
187#67—21no—
188#67—141no—
189#67—141no—
190C2×Dic3—21no—
191C2×Dic3—71no—
192C2×Dic3—141no—
193D12—141no—
194D12C4×D711yescharacteristic
195C4×D3—21no—
196C4×D3—71no—
197C4×D3—71no—
198C4×D3—71no—
199C4×D3—71no—
200C4×D3—141no—
201C4×D3—141no—
202Dic6—141no—
203Dic6C4×D711yescharacteristic
204C3×C8—21no—
205C3⋊C8—141no—
206C3×C7#42111yescharacteristic, complemented, Hall
207#41—31no—
208#41—71no—
209#41—211no—
210#41—211no—
211#41—421no—
212#41—421no—
213C2×Q8—211no—
214C2×D4—211no—
215C2×D4—421no—
216C22×C4—211no—
217C22×C4—421no—
218M16—211no—
219M16—421no—
220M16D3×D711yescharacteristic, complemented
221C2×C8—421no—
222C4⋊C4—421no—
223C22⋊C4—421no—
224C42—31no—
225C42—71no—
226C2×C7—21no—
227C2×C7—121no—
228C2×C7D3×C22⋊C411yescharacteristic
229D7—41no—
230D7—61no—
231D7—61no—
232C22×C3—141no—
233C22×C3D4×D711yescharacteristic
234C2×D3—21no—
235C2×D3—71no—
236C2×D3—71no—
237C2×D3—282no—
238C2×D3—281no—
239C3×C4—141no—
240C3×C4—281no—
241C3×C4C2×C4×D711yescharacteristic
242C3×C4D4×D711yescharacteristic
243Dic3—21no—
244Dic3—41no—
245Dic3—71no—
246Dic3—71no—
247C23—211no—
248C23—421no—
249Q8—31no—
250Q8—71no—
251Q8—421no—
252D4—31no—
253D4—61no—
254D4—71no—
255D4—141no—
256D4—421no—
257D4—421no—
258D4—421no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—211no—
264C2×C4—211no—
265C2×C4—211no—
266C2×C4—211no—
267C2×C4—211no—
268C2×C4—421no—
269C2×C4—421no—
270C2×C4—421no—
271C2×C4C2×D3×D711yescharacteristic
272C8—21no—
273C8—421no—
274C7#482111yescharacteristic, complemented, Sylow
275C2×C3—21no—
276C2×C3—281no—
277C2×C3D7×C22⋊C411yescharacteristic
278D3—41no—
279D3—141no—
280D3—141no—
281C22—61no—
282C22—141no—
283C22—211no—
284C22—211no—
285C22—841no—
286C22—842no—
287C22#6668111yescharacteristic
288C4—61no—
289C4—121no—
290C4—141no—
291C4—211no—
292C4—211no—
293C4—281no—
294C4C4×D3×D711yescharacteristic
295C4#6669411yescharacteristic
296C3#8959311yescharacteristic, complemented, Sylow
297C2—21no—
298C2—121no—
299C2—281no—
300C2—421no—
301C2—421no—
302C2#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×C7#421
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. #10620948C2semidirect
4. C7×#825C2semidirect
5. C3×#6399C2semidirect
6. #10620514C2semidirect
7. #10620512C2semidirect
14. #66681C4semidirect
17. D21⋊Q8C4semidirect
18. #66672C4semidirect
19. #66670C4semidirect
20. C3×C7×M16C22semidirect
21. #66627C22semidirect
24. C4×C7×Dic3C22semidirect
25. C3×C4×Dic7C22semidirect
33. #6399D3semidirect
74. C7×M16C2×D3semidirect
78. C4×Dic7C2×D3semidirect
87. #825D7semidirect
143. C3×M16C2×D7semidirect
147. C4×Dic3C2×D7semidirect
206. C3×C7#421semidirect
220. M16D3×D7semidirect
274. C7#4821semidirect
296. C3#89593semidirect

Maximal subgroups

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

subgroupindex
#106215632
M16×D212
#106209482
C7×#8252
C3×#63992
#106205142
#106205122
#895933
#48217