Finite Group Databasebeta

← #13054806

Subgroups — #13054806

Order 1344.

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

normal subgroups only

This group has 308 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#41×D21C211yesmaximal, characteristic
2#10620948C211yesmaximal, characteristic, complemented
3#10620854C211yesmaximal, characteristic, complemented
4C7×#837C211yesmaximal, characteristic, complemented
5C3×#6411C211yesmaximal, characteristic, complemented
6#10620516C211yesmaximal, characteristic, complemented
7#10620515C211yesmaximal, characteristic, complemented
8#89593—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10#66732C411yescharacteristic, complemented
11Q8×D21C411yescharacteristic, complemented
12#66730—21no—
13#66730C411yescharacteristic, complemented
14D4×D21—21no—
15D4×D21C411yescharacteristic, complemented
16#66728—21no—
17#66728C2211yescharacteristic
18C2×C4×D21—21no—
19C2×C4×D21C2211yescharacteristic
20C4×C7×Dic3C2211yescharacteristic, complemented
21C7×#208C2211yescharacteristic, complemented
22C3×C4×Dic7C2211yescharacteristic, complemented
23C3×#1136C2211yescharacteristic, complemented
24#66577—21no—
25#66575—21no—
26#66559—21no—
27D21⋊C8—21no—
28D7×#41—31no—
29D7×M16—31no—
30#6435—31no—
31C7×#104—31no—
32#6411D311yescharacteristic, complemented
33#6399—31no—
34#6379—31no—
35#4821—71nomaximal, Hall
36C22×D21—21no—
37C22×D21D411yescharacteristic
38C3×C7×Q8C2×C411yescharacteristic
39C3×C7×D4—21no—
40C3×C7×D4C2×C411yescharacteristic
41C2×C3×C4×C7—21no—
42C2×C3×C4×C7C2311yescharacteristic
43#4268—21no—
44#4268—21no—
45#4268—21no—
46C2×Dic21—21no—
47C2×Dic21D411yescharacteristic
48D84—21no—
49D84C2×C411yescharacteristic
50C4×D21—21no—
51C4×D21—21no—
52C4×D21—21no—
53C4×D21C2×C411yescharacteristic
54C4×D21C2×C411yescharacteristic
55C4×D21D411yescharacteristic
56C4×D21D411yescharacteristic
57Dic42—21no—
58Dic42C2×C411yescharacteristic
59C2×C7×Dic3—21no—
60C2×C3×Dic7—21no—
61C3×C7⋊C8—21no—
62C7×C3⋊C8—21no—
63C7×#41C2×D311yescharacteristic
64#1161—31no—
65Q8×D7—31no—
66#1159—31no—
67#1159—61no—
68D4×D7—31no—
69D4×D7—61no—
70#1157—31no—
71#1157—61no—
72C2×C4×D7—31no—
73C2×C4×D7—61no—
74C7×M16—31no—
75C42×C7—31no—
76(C2×D7)⋊C4—61no—
77Dic7⋊C4—61no—
78C4×Dic7C2×D311yescharacteristic, complemented
79#1136C2×D311yescharacteristic, complemented
80#1131—61no—
81C8×D7—61no—
82D3×#41—71no—
83D3×M16—71no—
84#872—71no—
85C3×#104—71no—
86#837D711yescharacteristic, complemented
87#825—71no—
88#805—71no—
89C22×C3×C7—21no—
90C22×C3×C7C2×D411yescharacteristic
91C2×D21—21no—
92C2×D21—41no—
93C2×D21—42no—
94C2×D21C22⋊C411yescharacteristic
95C2×D21C22⋊C411yescharacteristic
96C3×C4×C7—21no—
97C3×C4×C7C22×C411yescharacteristic
98C3×C4×C7C2×D411yescharacteristic
99Dic21—21no—
100Dic21C22⋊C411yescharacteristic
101Dic21C22⋊C411yescharacteristic
102C3×Dic7—41no—
103C7×Dic3—41no—
104#421—211noSylow
105C22×D7—31no—
106C22×D7—61no—
107C7×Q8C4×D311yescharacteristic
108C7×D4—21no—
109C7×D4C4×D311yescharacteristic
110C2×C4×C7—21no—
111C2×C4×C7—61no—
112C2×C4×C7C22×D311yescharacteristic
113C22⋊D7—61no—
114C22⋊D7—61no—
115C22⋊D7—61no—
116C2×Dic7—21no—
117C2×Dic7—31no—
118C2×Dic7—61no—
119D28—31no—
120D28—61no—
121C4×D7—31no—
122C4×D7—31no—
123C4×D7—31no—
124C4×D7—31no—
125C4×D7—61no—
126C4×D7—61no—
127C4×D7—61no—
128Dic14—31no—
129Dic14—61no—
130C7×C8—61no—
131C7⋊C8—21no—
132C3×#41C2×D711yescharacteristic
133#239—71no—
134D3×Q8—71no—
135#237—71no—
136#237—141no—
137D3×D4—71no—
138D3×D4—141no—
139#235—71no—
140#235—141no—
141C2×C4×D3—71no—
142C2×C4×D3—141no—
143C3×M16—71no—
144C3×C42—71no—
145(C2×D3)⋊C4—141no—
146Dic3⋊C4—141no—
147C4×Dic3C2×D711yescharacteristic, complemented
148#208C2×D711yescharacteristic, complemented
149ΓL(1,𝔽25)—141no—
150D3×C8—141no—
151C2×C3×C7—21no—
152C2×C3×C7—41no—
153C2×C3×C7C2×C22⋊C411yescharacteristic
154D21—21no—
155D21—21no—
156D21—41no—
157C2×#41—211no—
158C2×M16—211no—
159#117—211no—
160#104—31no—
161#104—71no—
162#104—211no—
163#104—211no—
164C22×C7—21no—
165C22×C7D3×D411yescharacteristic
166C2×D7—31no—
167C2×D7—31no—
168C2×D7—61no—
169C2×D7—121no—
170C2×D7—122no—
171C4×C7—21no—
172C4×C7—121no—
173C4×C7C2×C4×D311yescharacteristic
174C4×C7D3×D411yescharacteristic
175Dic7—31no—
176Dic7—31no—
177Dic7—41no—
178Dic7—61no—
179C22×D3—71no—
180C22×D3—141no—
181C3×Q8C4×D711yescharacteristic
182C3×D4—21no—
183C3×D4C4×D711yescharacteristic
184C2×C3×C4—21no—
185C2×C3×C4—141no—
186C2×C3×C4C22×D711yescharacteristic
187#67—141no—
188#67—141no—
189#67—141no—
190C2×Dic3—21no—
191C2×Dic3—71no—
192C2×Dic3—141no—
193D12—71no—
194D12—141no—
195C4×D3—71no—
196C4×D3—71no—
197C4×D3—71no—
198C4×D3—71no—
199C4×D3—141no—
200C4×D3—141no—
201C4×D3—141no—
202Dic6—71no—
203Dic6—141no—
204C3×C8—141no—
205C3⋊C8—21no—
206C3×C7#42111yescharacteristic, complemented, Hall
207#41—211no—
208#41—211no—
209#41—211no—
210#41—421no—
211#41—421no—
212#41D3×D711yescharacteristic
213C2×Q8—211no—
214C2×D4—211no—
215C2×D4—421no—
216C22×C4—211no—
217C22×C4—421no—
218M16—31no—
219M16—71no—
220M16—421no—
221C2×C8—421no—
222C4⋊C4—421no—
223C22⋊C4—421no—
224C42—31no—
225C42—71no—
226C2×C7—21no—
227C2×C7—41no—
228C2×C7D3×C22⋊C411yescharacteristic
229D7—61no—
230D7—61no—
231D7—121no—
232C22×C3—21no—
233C22×C3D4×D711yescharacteristic
234C2×D3—71no—
235C2×D3—71no—
236C2×D3—141no—
237C2×D3—281no—
238C2×D3—282no—
239C3×C4—21no—
240C3×C4—281no—
241C3×C4C2×C4×D711yescharacteristic
242C3×C4D4×D711yescharacteristic
243Dic3—41no—
244Dic3—71no—
245Dic3—71no—
246Dic3—141no—
247C23—211no—
248C23—421no—
249Q8—211no—
250Q8—421no—
251Q8D21⋊C411yescharacteristic
252D4—21no—
253D4—211no—
254D4—421no—
255D4—421no—
256D4—421no—
257D4—421no—
258D4D21⋊C411yescharacteristic
259C2×C4—21no—
260C2×C4—61no—
261C2×C4—141no—
262C2×C4—211no—
263C2×C4—211no—
264C2×C4—211no—
265C2×C4—211no—
266C2×C4—211no—
267C2×C4—421no—
268C2×C4—421no—
269C2×C4—421no—
270C2×C4—421no—
271C2×C4C2×D3×D711yescharacteristic
272C8—61no—
273C8—141no—
274C7#482111yescharacteristic, complemented, Sylow
275C2×C3—21no—
276C2×C3—41no—
277C2×C3D7×C22⋊C411yescharacteristic
278D3—141no—
279D3—141no—
280D3—281no—
281C22—21no—
282C22—211no—
283C22—211no—
284C22—421no—
285C22—841no—
286C22—842no—
287C22#6669411yescharacteristic
288C4—21no—
289C4—121no—
290C4—211no—
291C4—211no—
292C4—281no—
293C4—421no—
294C4C2×D21⋊C411yescharacteristic
295C4#6669411yescharacteristic
296C3#8959311yescharacteristic, complemented, Sylow
297C2—21no—
298C2—41no—
299C2—421no—
300C2—421no—
301C2—841no—
302C2#1062112311yescharacteristic
subgroupG/H
centerC4#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7C2×C22⋊C4
prime radical#41×D21C2
cosocleC2×C3×C4×C7C23
hypercenter#41D3×D7
hypocenterC3×C7#421
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. #10620948C2semidirect
3. #10620854C2semidirect
4. C7×#837C2semidirect
5. C3×#6411C2semidirect
6. #10620516C2semidirect
7. #10620515C2semidirect
10. #66732C4semidirect
11. Q8×D21C4semidirect
13. #66730C4semidirect
15. D4×D21C4semidirect
20. C4×C7×Dic3C22semidirect
21. C7×#208C22semidirect
22. C3×C4×Dic7C22semidirect
23. C3×#1136C22semidirect
32. #6411D3semidirect
78. C4×Dic7C2×D3semidirect
79. #1136C2×D3semidirect
86. #837D7semidirect
147. C4×Dic3C2×D7semidirect
148. #208C2×D7semidirect
206. C3×C7#421semidirect
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
#41×D212
#106209482
#106208542
C7×#8372
C3×#64112
#106205162
#106205152
#895933
#48217