Finite Group Databasebeta

← #13053612

Subgroups — #13053612

Order 1344.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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
1D3×#1157C211yesmaximal, characteristic
2D3×C7×M16C211yesmaximal, characteristic, complemented
3#10620933C211yesmaximal, characteristic, complemented
4#10620664C211yesmaximal, characteristic, complemented
5C3×#6399C211yesmaximal, characteristic, complemented
6#10620516C211yesmaximal, characteristic, complemented
7#10620511C211yesmaximal, characteristic, complemented
8#89911—31nomaximal, Hall
9#66728C2211yescharacteristic
10C2×C4×D3×C7C2211yescharacteristic
11C3×#1157C2211yescharacteristic
12D3×C22⋊D7—21no—
13#66683—21no—
14D3×D28C411yescharacteristic, complemented
15C4×D3×D7—21no—
16#66675—21no—
17#66673C411yescharacteristic, complemented
18D3×Dic14C411yescharacteristic, complemented
19#66669C411yescharacteristic, complemented
20C3×C7×M16C2211yescharacteristic, complemented
21C4×Dic21C2211yescharacteristic, complemented
22C7×#208C2211yescharacteristic, complemented
23C7×ΓL(1,𝔽25)—21no—
24D3×C7×C8—21no—
25C3×C4×Dic7C2211yescharacteristic, complemented
26Dic3⋊Dic7—21no—
27(C2×D3)⋊Dic7—21no—
28C2×#1157—31no—
29C2×C7×M16—31no—
30#6489—31no—
31#6399—31no—
32#6399—31no—
33#6399—31no—
34#6399D311yescharacteristic, complemented
35#4821—71nomaximal, Hall
36C22×D3×C7D411yescharacteristic
37C2×D3×D7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39#4268—21no—
40C2×Dic21—21no—
41D84C2×C411yescharacteristic
42C4×D21—21no—
43Dic42C2×C411yescharacteristic
44C2×C7×Dic3D411yescharacteristic
45C4×D3×C7C2×C411yescharacteristic
46C4×D3×C7C2×C411yescharacteristic
47C4×D3×C7D411yescharacteristic
48C4×D3×C7D411yescharacteristic
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—
62C7×C3⋊C8—21no—
63C22×C4×C7—31no—
64C2×C22⋊D7—61no—
65#1157—31no—
66#1157—31no—
67#1157—31no—
68#1157—61no—
69#1157—61no—
70#1157C2×D311yescharacteristic
71C2×D28—31no—
72C2×C4×D7—61no—
73C2×Dic14—31no—
74C7×M16—31no—
75C7×M16—61no—
76C7×M16C2×D311yescharacteristic, complemented
77C2×C7×C8—61no—
78C22⋊Dic7—61no—
79C4⋊Dic7—61no—
80C4×Dic7—31no—
81C4×Dic7C2×D311yescharacteristic, complemented
82D3×#41—71no—
83D3×M16D711yescharacteristic, complemented
84#872—71no—
85C3×#104—71no—
86#837—71no—
87#825—71no—
88#805—71no—
89C22×C3×C7C2×D411yescharacteristic
90C2×D21—21no—
91C2×D3×C7—41no—
92C2×D3×C7C22⋊C411yescharacteristic
93C2×D3×C7C22⋊C411yescharacteristic
94C2×C3×D7—21no—
95D3×D7—42no—
96C3×C4×C7C22×C411yescharacteristic
97C3×C4×C7C2×D411yescharacteristic
98Dic21—21no—
99Dic21—41no—
100C3×Dic7—21no—
101C3×Dic7—41no—
102C7×Dic3C22⋊C411yescharacteristic
103C7×Dic3C22⋊C411yescharacteristic
104#421—211noSylow
105C23×C7—31no—
106C22×D7—61no—
107C2×C4×C7—31no—
108C2×C4×C7—31no—
109C2×C4×C7—31no—
110C2×C4×C7—31no—
111C2×C4×C7—31no—
112C2×C4×C7C22×D311yescharacteristic
113C22⋊D7—21no—
114C22⋊D7—61no—
115C22⋊D7—61no—
116C22⋊D7—61no—
117C2×Dic7—21no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120D28—31no—
121D28—61no—
122D28C4×D311yescharacteristic
123C4×D7—21no—
124C4×D7—61no—
125C4×D7—61no—
126C4×D7—61no—
127Dic14—31no—
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—71no—
140#235—141no—
141C2×C4×D3—141no—
142C2×C4×D3C2×D711yescharacteristic
143C3×M16C2×D711yescharacteristic, complemented
144C3×C42—71no—
145(C2×D3)⋊C4—141no—
146Dic3⋊C4—141no—
147C4×Dic3—71no—
148#208C2×D711yescharacteristic, complemented
149ΓL(1,𝔽25)—21no—
150D3×C8—21no—
151C2×C3×C7—21no—
152C2×C3×C7C2×C22⋊C411yescharacteristic
153D21—41no—
154C3×D7—41no—
155D3×C7—21no—
156D3×C7—21no—
157C2×#41—211no—
158C2×M16—31no—
159#117—211no—
160#104—71no—
161#104—211no—
162#104—211no—
163#104—211no—
164C22×C7—31no—
165C22×C7—31no—
166C22×C7—121no—
167C22×C7D3×D411yescharacteristic
168C2×D7—21no—
169C2×D7—61no—
170C2×D7—122no—
171C4×C7—31no—
172C4×C7—31no—
173C4×C7C2×C4×D311yescharacteristic
174C4×C7D3×D411yescharacteristic
175Dic7—21no—
176Dic7—41no—
177Dic7—61no—
178Dic7—121no—
179C22×D3—141no—
180C22×D3D2811yescharacteristic
181C3×Q8—71no—
182C3×D4—71no—
183C3×D4—141no—
184C2×C3×C4—141no—
185C2×C3×C4—141no—
186C2×C3×C4C22×D711yescharacteristic
187#67—141no—
188#67—141no—
189#67—141no—
190C2×Dic3—141no—
191C2×Dic3—141no—
192C2×Dic3D2811yescharacteristic
193D12—71no—
194D12—141no—
195C4×D3—141no—
196C4×D3—141no—
197C4×D3—141no—
198C4×D3C4×D711yescharacteristic
199C4×D3C4×D711yescharacteristic
200C4×D3C22⋊D711yescharacteristic
201C4×D3C22⋊D711yescharacteristic
202Dic6—71no—
203Dic6—141no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3×C7#42111yescharacteristic, complemented, Hall
207#41—71no—
208#41—211no—
209#41—211no—
210#41—211no—
211#41—421no—
212#41—421no—
213C2×Q8—211no—
214C2×D4—211no—
215C2×D4—421no—
216C22×C4—31no—
217C22×C4—421no—
218M16—31no—
219M16—61no—
220M16D3×D711yescharacteristic, complemented
221C2×C8—61no—
222C4⋊C4—421no—
223C22⋊C4—421no—
224C42—71no—
225C42—211no—
226C2×C7—21no—
227C2×C7—61no—
228C2×C7—61no—
229C2×C7D3×C22⋊C411yescharacteristic
230D7—41no—
231D7—121no—
232C22×C3—141no—
233C22×C3C2×D2811yescharacteristic
234C2×D3—41no—
235C2×D3—141no—
236C2×D3—282no—
237C2×D3(C2×D7)⋊C411yescharacteristic
238C2×D3(C2×D7)⋊C411yescharacteristic
239C3×C4—141no—
240C3×C4—281no—
241C3×C4C2×C4×D711yescharacteristic
242C3×C4C2×C22⋊D711yescharacteristic
243Dic3—141no—
244Dic3—281no—
245Dic3(C2×D7)⋊C411yescharacteristic
246Dic3(C2×D7)⋊C411yescharacteristic
247C23—31no—
248C23—421no—
249Q8—71no—
250Q8—211no—
251Q8—421no—
252D4—71no—
253D4—141no—
254D4—211no—
255D4—421no—
256D4—421no—
257D4—421no—
258D4—421no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—421no—
267C2×C4—421no—
268C2×C4—421no—
269C2×C4—421no—
270C2×C4—421no—
271C2×C4C2×D3×D711yescharacteristic
272C8—21no—
273C8—61no—
274C7#482111yescharacteristic, complemented, Sylow
275C2×C3—21no—
276C2×C3—281no—
277C2×C3C2×(C2×D7)⋊C411yescharacteristic
278D3—21no—
279D3—21no—
280D3—281no—
281C22—31no—
282C22—31no—
283C22—121no—
284C22—141no—
285C22—421no—
286C22—842no—
287C22D3×D2811yescharacteristic
288C4—31no—
289C4—31no—
290C4—141no—
291C4—281no—
292C4—421no—
293C4—841no—
294C4C4×D3×D711yescharacteristic
295C4D3×C22⋊D711yescharacteristic
296C3#8991111yescharacteristic, complemented, Sylow
297C2—21no—
298C2—61no—
299C2—61no—
300C2—281no—
301C2—841no—
302C2D3×(C2×D7)⋊C411yescharacteristic
subgroupG/H
centerC4D3×C22⋊D7
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7C2×C22⋊C4
prime radicalD3×#1157C2
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. D3×C7×M16C2semidirect
3. #10620933C2semidirect
4. #10620664C2semidirect
5. C3×#6399C2semidirect
6. #10620516C2semidirect
7. #10620511C2semidirect
14. D3×D28C4semidirect
17. #66673C4semidirect
18. D3×Dic14C4semidirect
19. #66669C4semidirect
20. C3×C7×M16C22semidirect
21. C4×Dic21C22semidirect
22. C7×#208C22semidirect
25. C3×C4×Dic7C22semidirect
34. #6399D3semidirect
76. C7×M16C2×D3semidirect
81. C4×Dic7C2×D3semidirect
83. D3×M16D7semidirect
143. C3×M16C2×D7semidirect
148. #208C2×D7semidirect
206. C3×C7#421semidirect
220. M16D3×D7semidirect
274. C7#4821semidirect
296. C3#89911semidirect

Maximal subgroups

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

subgroupindex
D3×#11572
D3×C7×M162
#106209332
#106206642
C3×#63992
#106205162
#106205112
#899113
#48217