Finite Group Databasebeta

← D7×#825 #13053607

Subgroups — D7×#825 #13053607

Order 1344.

340 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 82 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
3C4×Dic3×D7C211yesmaximal, characteristic, complemented
4#10620664C211yesmaximal, characteristic, complemented
5C7×#825C211yesmaximal, characteristic, complemented
6#10620515C211yesmaximal, characteristic, complemented
7#10620513C211yesmaximal, characteristic, complemented
8D7×#104—31nomaximal, Hall
9#66728C2211yescharacteristic
10C7×#235C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12D7×#67—21no—
13#66686—21no—
14C2×Dic3×D7—21no—
15D7×D12C411yescharacteristic, complemented
16C4×D3×D7—21no—
17#66677C411yescharacteristic, complemented
18#66676C411yescharacteristic, complemented
19#66675—21no—
20D7×Dic6C411yescharacteristic, complemented
21C3×C7×M16C2211yescharacteristic, complemented
22C4×Dic21C2211yescharacteristic, complemented
23C4×C7×Dic3C2211yescharacteristic, complemented
24C3×#1136C2211yescharacteristic, complemented
25C3×#1131—21no—
26C3×C8×D7—21no—
27Dic3×Dic7—21no—
28D7×#41—31no—
29D7×M16D311yescharacteristic, complemented
30C42×D7—31no—
31C7×#104—31no—
32#6411—31no—
33#6399—31no—
34#6379—31no—
35C2×#825—71nomaximal, Hall
36C22×C3×D7D411yescharacteristic
37C2×D3×D7—21no—
38C2×C3×C4×C7C2311yescharacteristic
39#4268—21no—
40C2×Dic21—21no—
41D84C2×C411yescharacteristic, complemented
42C4×D21—21no—
43Dic42C2×C411yescharacteristic, complemented
44C7×#67—21no—
45C2×C7×Dic3—21no—
46C7×D12C2×C411yescharacteristic, complemented
47C4×D3×C7—21no—
48C7×Dic6C2×C411yescharacteristic, complemented
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—24no—
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—31no—
72#1157—61no—
73C2×C4×D7—61no—
74C2×C4×D7—61no—
75C2×C4×D7C2×D311yescharacteristic
76C7×M16C2×D311yescharacteristic, complemented
77C42×C7—31no—
78C4×Dic7—31no—
79C4×Dic7—61no—
80#1136C2×D311yescharacteristic, complemented
81#1131—21no—
82C8×D7—21no—
83C2×#235—71no—
84C2×C3×M16—71no—
85C2×C4×Dic3—71no—
86#825—71no—
87#825—71no—
88#825—71no—
89#825D711yescharacteristic, direct product factor
90C22×C3×C7C2×D411yescharacteristic
91C2×D21—21no—
92C2×D3×C7—21no—
93C2×C3×D7—22no—
94C2×C3×D7C22⋊C411yescharacteristic
95C2×C3×D7C22⋊C411yescharacteristic
96D3×D7—42no—
97C3×C4×C7C22×C411yescharacteristic
98C3×C4×C7C2×D411yescharacteristic
99Dic21—22no—
100Dic21—21no—
101C3×Dic7C22⋊C411yescharacteristic
102C3×Dic7C22⋊C411yescharacteristic
103C7×Dic3—22no—
104C7×Dic3—21no—
105C2×#104—211noSylow
106C22×D7—61no—
107C22×D7D1211yescharacteristic
108C7×Q8—31no—
109C7×D4—31no—
110C7×D4—61no—
111C2×C4×C7—61no—
112C2×C4×C7—61no—
113C2×C4×C7C22×D311yescharacteristic
114C22⋊D7—61no—
115C22⋊D7—61no—
116C22⋊D7—61no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C2×Dic7D1211yescharacteristic
120D28—31no—
121D28—61no—
122C4×D7—64no—
123C4×D7—61no—
124C4×D7—61no—
125C4×D7—61no—
126C4×D7C4×D311yescharacteristic
127C4×D7C4×D311yescharacteristic
128C4×D7#6711yescharacteristic
129C4×D7#6711yescharacteristic
130Dic14—31no—
131Dic14—61no—
132C7×C8—21no—
133C7⋊C8—21no—
134C22×C3×C4—71no—
135C2×#67—141no—
136C22×Dic3—141no—
137#235—71no—
138#235—71no—
139#235—71no—
140#235—141no—
141#235—141no—
142#235C2×D711yescharacteristic
143C2×D12—71no—
144C2×C4×D3—141no—
145C2×Dic6—71no—
146C3×M16—71no—
147C3×M16—141no—
148C3×M16C2×D711yescharacteristic, complemented
149C2×C3×C8—141no—
150C4×Dic3—71no—
151C4×Dic3—141no—
152C4×Dic3C2×D711yescharacteristic, complemented
153C2×C3×C7—21no—
154C2×C3×C7C2×C22⋊C411yescharacteristic
155D21—41no—
156C3×D7—21no—
157C3×D7#10412yescomplemented
158D3×C7—41no—
159C2×#41—211no—
160C2×M16—71no—
161C2×C42—211no—
162#104—31no—
163#104—211no—
164#104—211no—
165#104—211no—
166C22×C7—61no—
167C22×C7C2×D1211yescharacteristic
168C2×D7—22no—
169C2×D7—61no—
170C2×D7—122no—
171C2×D7(C2×D3)⋊C411yescharacteristic
172C2×D7(C2×D3)⋊C411yescharacteristic
173C4×C7—62no—
174C4×C7—61no—
175C4×C7C2×C4×D311yescharacteristic
176C4×C7C2×#6711yescharacteristic
177Dic7—62no—
178Dic7—61no—
179Dic7(C2×D3)⋊C411yescharacteristic
180Dic7(C2×D3)⋊C411yescharacteristic
181C23×C3—71no—
182C22×D3—141no—
183C2×C3×C4—71no—
184C2×C3×C4—71no—
185C2×C3×C4—71no—
186C2×C3×C4—71no—
187C2×C3×C4—71no—
188C2×C3×C4C22×D711yescharacteristic
189#67—21no—
190#67—141no—
191#67—141no—
192#67—141no—
193C2×Dic3—21no—
194C2×Dic3—141no—
195C2×Dic3—144no—
196C2×Dic3—141no—
197D12—71no—
198D12—141no—
199D12C4×D711yescharacteristic, complemented
200C4×D3—21no—
201C4×D3—141no—
202C4×D3—141no—
203C4×D3—141no—
204Dic6—71no—
205Dic6—141no—
206Dic6C4×D711yescharacteristic, complemented
207C3×C8—21no—
208C3×C8—141no—
209C3×C7C2×#10411yescharacteristic, complemented, Hall
210#41—31no—
211#41—211no—
212#41—211no—
213#41—211no—
214#41—421no—
215#41—421no—
216C2×Q8—211no—
217C2×D4—211no—
218C2×D4—421no—
219C22×C4—71no—
220C22×C4—421no—
221C22×C4—421no—
222M16—71no—
223M16—141no—
224M16D3×D711yescharacteristic, complemented
225C2×C8—141no—
226C42—31no—
227C42—211no—
228C42—421no—
229C2×C7—21no—
230C2×C7—121no—
231C2×C7C2×(C2×D3)⋊C411yescharacteristic
232D7—21no—
233D7—121no—
234D7#82512yesdirect product factor
235C22×C3—71no—
236C22×C3—71no—
237C22×C3—142no—
238C22×C3D4×D711yescharacteristic
239C2×D3—21no—
240C2×D3—141no—
241C2×D3—282no—
242C3×C4—71no—
243C3×C4—71no—
244C3×C4C2×C4×D711yescharacteristic
245C3×C4D4×D711yescharacteristic
246Dic3—22no—
247Dic3—21no—
248Dic3—142no—
249Dic3—141no—
250C23—71no—
251C23—421no—
252Q8—31no—
253Q8—211no—
254Q8—421no—
255D4—31no—
256D4—61no—
257D4—211no—
258D4—421no—
259D4—421no—
260D4—421no—
261D4—421no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—71no—
265C2×C4—71no—
266C2×C4—71no—
267C2×C4—71no—
268C2×C4—71no—
269C2×C4—421no—
270C2×C4—421no—
271C2×C4—424no—
272C2×C4—421no—
273C2×C4—421no—
274C2×C4—421no—
275C2×C4C2×D3×D711yescharacteristic
276C8—21no—
277C8—141no—
278C7C2×#82511yescharacteristic, complemented, Sylow
279C2×C3—21no—
280C2×C3—72no—
281C2×C3—141no—
282C2×C3D7×C22⋊C411yescharacteristic
283D3—41no—
284D3—281no—
285C22—61no—
286C22—71no—
287C22—71no—
288C22—142no—
289C22—421no—
290C22—842no—
291C22D7×D1211yescharacteristic
292C4—62no—
293C4—61no—
294C4—71no—
295C4—71no—
296C4—422no—
297C4—421no—
298C4C4×D3×D711yescharacteristic
299C4D7×#6711yescharacteristic
300C3D7×#10411yescharacteristic, complemented, Sylow
301C2—21no—
302C2—72no—
303C2—121no—
304C2—141no—
305C2—841no—
306C2D7×(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×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. C3×D7×M16C2semidirect
3. C4×Dic3×D7C2semidirect
4. #10620664C2semidirect
5. C7×#825C2semidirect
6. #10620515C2semidirect
7. #10620513C2semidirect
15. D7×D12C4semidirect
17. #66677C4semidirect
18. #66676C4semidirect
20. D7×Dic6C4semidirect
21. C3×C7×M16C22semidirect
22. C4×Dic21C22semidirect
23. C4×C7×Dic3C22semidirect
24. C3×#1136C22semidirect
29. D7×M16D3semidirect
41. D84C2×C4semidirect
43. Dic42C2×C4semidirect
46. C7×D12C2×C4semidirect
48. C7×Dic6C2×C4semidirect
76. C7×M16C2×D3semidirect
80. #1136C2×D3semidirect
89. #825D7direct (N × H)
148. C3×M16C2×D7semidirect
152. C4×Dic3C2×D7semidirect
157. C3×D7#104semidirect
199. D12C4×D7semidirect
206. Dic6C4×D7semidirect
209. C3×C7C2×#104semidirect
224. M16D3×D7semidirect
234. D7#825direct (N × H)
278. C7C2×#825semidirect
300. C3D7×#104semidirect

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
C4×Dic3×D72
#106206642
C7×#8252
#106205152
#106205132
D7×#1043
C2×#8257