Finite Group Databasebeta

← #13053608

Subgroups — #13053608

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
1D7×#235C211yesmaximal, characteristic
2C3×D7×M16C211yesmaximal, characteristic, complemented
3#10620919C211yesmaximal, characteristic, complemented
4#10620664C211yesmaximal, characteristic, complemented
5C7×#825C211yesmaximal, characteristic, complemented
6#10620515C211yesmaximal, characteristic, complemented
7#10620513C211yesmaximal, characteristic, complemented
8#89593—31nomaximal, Hall
9#66728C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12D7×#67—21no—
13#66686—21no—
14D7×D12C411yescharacteristic, complemented
15C4×D3×D7—21no—
16#66677C411yescharacteristic, complemented
17#66676C411yescharacteristic, complemented
18#66675—21no—
19D7×Dic6C411yescharacteristic, complemented
20C3×C7×M16C2211yescharacteristic, complemented
21C4×Dic21C2211yescharacteristic, complemented
22C4×C7×Dic3C2211yescharacteristic, complemented
23C3×#1136C2211yescharacteristic, complemented
24C3×#1131—21no—
25C3×C8×D7—21no—
26Dic7⋊Dic3—21no—
27(C2×D7)⋊Dic3—21no—
28D7×#41—31no—
29D7×M16D311yescharacteristic, complemented
30#6435—31no—
31C7×#104—31no—
32#6411—31no—
33#6399—31no—
34#6379—31no—
35#5139—71nomaximal, Hall
36C22×C3×D7D411yescharacteristic
37C2×D3×D7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39#4268—21no—
40C2×Dic21—21no—
41D84C2×C411yescharacteristic
42C4×D21—21no—
43Dic42C2×C411yescharacteristic
44C7×#67—21no—
45C2×C7×Dic3—21no—
46C7×D12C2×C411yescharacteristic
47C4×D3×C7—21no—
48C7×Dic6C2×C411yescharacteristic
49C2×C3×Dic7D411yescharacteristic
50C3×C4×D7C2×C411yescharacteristic
51C3×C4×D7C2×C411yescharacteristic
52C3×C4×D7D411yescharacteristic
53C3×C4×D7D411yescharacteristic
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—31no—
71#1157—61no—
72C2×C4×D7—61no—
73C2×C4×D7C2×D311yescharacteristic
74C7×M16C2×D311yescharacteristic, complemented
75C42×C7—31no—
76(C2×D7)⋊C4—61no—
77Dic7⋊C4—61no—
78C4×Dic7—31no—
79#1136C2×D311yescharacteristic, complemented
80#1131—21no—
81C8×D7—21no—
82C2×#235—71no—
83C2×C3×M16—71no—
84#926—71no—
85#825—71no—
86#825—71no—
87#825—71no—
88#825D711yescharacteristic, complemented
89C22×C3×C7C2×D411yescharacteristic
90C2×D21—21no—
91C2×D3×C7—21no—
92C2×C3×D7—41no—
93C2×C3×D7C22⋊C411yescharacteristic
94C2×C3×D7C22⋊C411yescharacteristic
95D3×D7—42no—
96C3×C4×C7C22×C411yescharacteristic
97C3×C4×C7C2×D411yescharacteristic
98Dic21—21no—
99Dic21—41no—
100C3×Dic7C22⋊C411yescharacteristic
101C3×Dic7C22⋊C411yescharacteristic
102C7×Dic3—21no—
103C7×Dic3—41no—
104#421—211noSylow
105C22×D7—61no—
106C22×D7D1211yescharacteristic
107C7×Q8—31no—
108C7×D4—31no—
109C7×D4—61no—
110C2×C4×C7—61no—
111C2×C4×C7—61no—
112C2×C4×C7C22×D311yescharacteristic
113C22⋊D7—61no—
114C22⋊D7—61no—
115C22⋊D7—61no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7D1211yescharacteristic
119D28—31no—
120D28—61no—
121C4×D7—61no—
122C4×D7—61no—
123C4×D7—61no—
124C4×D7C4×D311yescharacteristic
125C4×D7C4×D311yescharacteristic
126C4×D7#6711yescharacteristic
127C4×D7#6711yescharacteristic
128Dic14—31no—
129Dic14—61no—
130C7×C8—21no—
131C7⋊C8—21no—
132C22×C3×C4—71no—
133C2×#67—141no—
134#235—71no—
135#235—71no—
136#235—71no—
137#235—141no—
138#235—141no—
139#235C2×D711yescharacteristic
140C2×D12—71no—
141C2×C4×D3—141no—
142C2×Dic6—71no—
143C3×M16—71no—
144C3×M16—141no—
145C3×M16C2×D711yescharacteristic, complemented
146C2×C3×C8—141no—
147#217—141no—
148C4⋊Dic3—141no—
149C4×Dic3—71no—
150C4×Dic3C2×D711yescharacteristic, complemented
151C2×C3×C7—21no—
152C2×C3×C7C2×C22⋊C411yescharacteristic
153D21—41no—
154C3×D7—21no—
155C3×D7—21no—
156D3×C7—41no—
157C2×#41—211no—
158C2×M16—71no—
159#117—211no—
160#104—31no—
161#104—211no—
162#104—211no—
163#104—211no—
164C22×C7—61no—
165C22×C7C2×D1211yescharacteristic
166C2×D7—41no—
167C2×D7—61no—
168C2×D7—122no—
169C2×D7(C2×D3)⋊C411yescharacteristic
170C2×D7(C2×D3)⋊C411yescharacteristic
171C4×C7—61no—
172C4×C7—121no—
173C4×C7C2×C4×D311yescharacteristic
174C4×C7C2×#6711yescharacteristic
175Dic7—61no—
176Dic7—121no—
177Dic7(C2×D3)⋊C411yescharacteristic
178Dic7(C2×D3)⋊C411yescharacteristic
179C23×C3—71no—
180C22×D3—141no—
181C2×C3×C4—71no—
182C2×C3×C4—71no—
183C2×C3×C4—71no—
184C2×C3×C4—71no—
185C2×C3×C4—71no—
186C2×C3×C4C22×D711yescharacteristic
187#67—21no—
188#67—141no—
189#67—141no—
190#67—141no—
191C2×Dic3—21no—
192C2×Dic3—141no—
193C2×Dic3—141no—
194D12—71no—
195D12—141no—
196D12C4×D711yescharacteristic
197C4×D3—21no—
198C4×D3—141no—
199C4×D3—141no—
200C4×D3—141no—
201Dic6—71no—
202Dic6—141no—
203Dic6C4×D711yescharacteristic
204C3×C8—21no—
205C3×C8—141no—
206C3×C7#42111yescharacteristic, complemented, Hall
207#41—31no—
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—71no—
217C22×C4—421no—
218M16—71no—
219M16—141no—
220M16D3×D711yescharacteristic, complemented
221C2×C8—141no—
222C4⋊C4—421no—
223C22⋊C4—421no—
224C42—31no—
225C42—211no—
226C2×C7—21no—
227C2×C7—121no—
228C2×C7C2×(C2×D3)⋊C411yescharacteristic
229D7—21no—
230D7—21no—
231D7—121no—
232C22×C3—71no—
233C22×C3—71no—
234C22×C3—281no—
235C22×C3D4×D711yescharacteristic
236C2×D3—21no—
237C2×D3—141no—
238C2×D3—282no—
239C3×C4—71no—
240C3×C4—71no—
241C3×C4C2×C4×D711yescharacteristic
242C3×C4D4×D711yescharacteristic
243Dic3—21no—
244Dic3—41no—
245Dic3—141no—
246Dic3—281no—
247C23—71no—
248C23—421no—
249Q8—31no—
250Q8—211no—
251Q8—421no—
252D4—31no—
253D4—61no—
254D4—211no—
255D4—421no—
256D4—421no—
257D4—421no—
258D4—421no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—71no—
262C2×C4—71no—
263C2×C4—71no—
264C2×C4—71no—
265C2×C4—71no—
266C2×C4—421no—
267C2×C4—421no—
268C2×C4—421no—
269C2×C4—421no—
270C2×C4—421no—
271C2×C4C2×D3×D711yescharacteristic
272C8—21no—
273C8—141no—
274C7#513911yescharacteristic, complemented, Sylow
275C2×C3—21no—
276C2×C3—141no—
277C2×C3—141no—
278C2×C3D7×C22⋊C411yescharacteristic
279D3—41no—
280D3—281no—
281C22—61no—
282C22—71no—
283C22—71no—
284C22—281no—
285C22—421no—
286C22—842no—
287C22D7×D1211yescharacteristic
288C4—61no—
289C4—71no—
290C4—71no—
291C4—121no—
292C4—421no—
293C4—841no—
294C4C4×D3×D711yescharacteristic
295C4D7×#6711yescharacteristic
296C3#8959311yescharacteristic, complemented, Sylow
297C2—21no—
298C2—121no—
299C2—141no—
300C2—141no—
301C2—841no—
302C2D7×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC4D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×M16C22
socleC2×C3×C7C2×C22⋊C4
prime radicalD7×#235C2
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. C3×D7×M16C2semidirect
3. #10620919C2semidirect
4. #10620664C2semidirect
5. C7×#825C2semidirect
6. #10620515C2semidirect
7. #10620513C2semidirect
14. D7×D12C4semidirect
16. #66677C4semidirect
17. #66676C4semidirect
19. D7×Dic6C4semidirect
20. C3×C7×M16C22semidirect
21. C4×Dic21C22semidirect
22. C4×C7×Dic3C22semidirect
23. C3×#1136C22semidirect
29. D7×M16D3semidirect
74. C7×M16C2×D3semidirect
79. #1136C2×D3semidirect
88. #825D7semidirect
145. C3×M16C2×D7semidirect
150. C4×Dic3C2×D7semidirect
206. C3×C7#421semidirect
220. M16D3×D7semidirect
274. C7#5139semidirect
296. C3#89593semidirect

Maximal subgroups

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

subgroupindex
D7×#2352
C3×D7×M162
#106209192
#106206642
C7×#8252
#106205152
#106205132
#895933
#51397