Finite Group Databasebeta

← #13321666

Subgroups — #13321666

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3502C211yesmaximal, characteristic
2#11739908C211yesmaximal, characteristic
3#11739881C211yesmaximal, characteristic
4#11729592C212yesmaximal
5#11729591C214yesmaximal
6#11729590C214yesmaximal
7#11729589C212yesmaximal
8C7×Q82C414yescomplemented
9C7×#557—24no—
10C7×#557C414yescomplemented
11C7×#544—21no—
12C7×#544—22no—
13C7×#533C2212yes—
14C7×#533C2214yes—
15C7×#519—21no—
16C2×C4×C7×Q8C2211yescharacteristic
17#90002—21no—
18#89796—24no—
19#89753—22no—
20C4⋊C4×Dic7—21no—
21#89709C2212yes—
22#89708C2211yescharacteristic
23C4×C4⋊Dic7C2211yescharacteristic
24#89704C2212yes—
25#89698C2214yes—
26#89697C2212yes—
27#89333C2214yes—
28#89333C2214yes—
29#89333C2218yes—
30#17632—71nomaximal, Sylow
31C22×C7×Q8D411yescharacteristic
32C7×C4⋊Q8—28no—
33C7×C4⋊Q8—24no—
34C7×C4⋊Q8C2×C412yes—
35C7×C4⋊Q8C2×C414yes—
36C7×C4⋊Q8C2×C412yes—
37C7×C4⋊Q8C2×C414yes—
38C7×#125—28no—
39C7×#125—24no—
40C7×#125—22no—
41C7×#125—24no—
42C7×C22⋊Q8—24no—
43C7×C22⋊Q8—22no—
44C7×C22⋊Q8C2314yes—
45C4×C7×Q8—24no—
46C4×C7×Q8—24no—
47C4×C7×Q8—22no—
48C4×C7×Q8—24no—
49C4×C7×Q8C2×C414yes—
50C4×C7×Q8C2×C418yes—
51C4×C7×Q8C2×C414yes—
52C4×C7×Q8D414yes—
53C7×#117—24no—
54C7×#117—22no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D412yes—
57C2×C42×C7C2311yescharacteristic
58C2×C42×C7C2312yes—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4×Dic7—22no—
62C2×C4×Dic7—21no—
63C7⋊#95—22no—
64C7⋊#95—21no—
65C7⋊#95—22no—
66C7⋊#95—22no—
67C7⋊#95C2311yescharacteristic
68C7⋊#95C2312yes—
69C7⋊#95C2311yescharacteristic
70C22⋊(C7⋊C8)C2314yes—
71C4⋊(C7⋊C8)—22no—
72C4⋊(C7⋊C8)—24no—
73#6377—24no—
74C4×C7⋊C8—22no—
75#3502D711yescharacteristic
76#2983—71no—
77#2356—71no—
78#1518—72no—
79#1515—74no—
80#1506—74no—
81#1503—72no—
82C2×C7×Q8—22no—
83C2×C7×Q8—24no—
84C2×C7×Q8—22no—
85C2×C7×Q8—22no—
86C2×C7×Q8C22⋊C414yes—
87C2×C7×Q8C22×C414yes—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2411yescharacteristic
93C22×Dic7—21no—
94C22×Dic7—22no—
95C22×Dic7—21no—
96C7×C4⋊C4—24no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—24no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—24no—
102C7×C4⋊C4—24no—
103C7×C4⋊C4—28no—
104C7×C4⋊C4—24no—
105C7×C4⋊C4C22⋊C418yes—
106C7×C4⋊C4C22⋊C414yes—
107C7×C4⋊C4C22×C414yes—
108C7×C22⋊C4—24no—
109C42×C7—28no—
110C42×C7—24no—
111C42×C7C22×C414yes—
112C42×C7C22×C412yes—
113C42×C7C2×D412yes—
114C42×C7C2×D414yes—
115C4⋊Dic7—42no—
116C4⋊Dic7—42no—
117C4×Dic7—44no—
118C4×Dic7—42no—
119C2×C7⋊C8—24no—
120Q82Dic714yescomplemented
121#557—24no—
122#557Dic714yescomplemented
123#544—21no—
124#544—22no—
125#533C2×D712yes—
126#533C2×D714yes—
127#519—21no—
128C2×C4×Q8C2×D711yescharacteristic
129#446—144no—
130#432—74no—
131#431—72no—
132#391—141no—
133#387—142no—
134#383—72no—
135#382—71no—
136C4×C4⋊C4—71no—
137C4×C4⋊C4—141no—
138#376—72no—
139#328—74no—
140#328—74no—
141#328—78no—
142C23×C7C22×D411yescharacteristic
143C7×Q8—28no—
144C7×Q8—44no—
145C7×Q8—84no—
146C7×Q8—82no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—24no—
150C2×C4×C7—24no—
151C2×C4×C7—42no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7C2×C22⋊C414yes—
155C2×C4×C7C2×C22⋊C412yes—
156C2×C4×C7C2×C22⋊C414yes—
157C2×C4×C7C2×C22⋊C412yes—
158C2×C4×C7#13712yes—
159C2×C4×C7#13712yes—
160C2×C4×C7C23×C411yescharacteristic
161C2×C4×C7C22×D411yescharacteristic
162C2×Dic7—22no—
163C2×Dic7—41no—
164C2×Dic7—41no—
165C2×Dic7—42no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—42no—
169C2×Dic7—41no—
170C2×Dic7—42no—
171C7⋊C8—44no—
172C22×Q8C22⋊D711yescharacteristic
173C4⋊Q8—28no—
174C4⋊Q8—24no—
175C4⋊Q8C2×Dic712yes—
176C4⋊Q8C2×Dic714yes—
177C4⋊Q8C2×Dic712yes—
178C4⋊Q8C2×Dic714yes—
179#125—28no—
180#125—24no—
181#125—22no—
182#125—24no—
183C22⋊Q8—24no—
184C22⋊Q8—22no—
185C22⋊Q8C22×D714yes—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—22no—
189C4×Q8—24no—
190C4×Q8C2×Dic714yes—
191C4×Q8C2×Dic718yes—
192C4×Q8C2×Dic714yes—
193C4×Q8C22⋊D714yes—
194#117—24no—
195#117—22no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4C22⋊D711yescharacteristic
199C2×C4⋊C4C22⋊D712yes—
200C2×C42—142no—
201C2×C42—141no—
202C2×C42C22×D712yes—
203C2×C42C22×D711yescharacteristic
204C4⋊C8—142no—
205C4⋊C8—144no—
206C22⋊C8—74no—
207#97—144no—
208C4×C8—142no—
209#95—71no—
210#95—72no—
211#95—71no—
212#95—142no—
213#95—142no—
214#95—141no—
215#95—142no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7C22×C22⋊C411yescharacteristic
219C22×C7C2×#13711yescharacteristic
220C22×C7C2×#13711yescharacteristic
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—24no—
224C4×C7—24no—
225C4×C7—22no—
226C4×C7—44no—
227C4×C7—44no—
228C4×C7#41914yes—
229Dic7—42no—
230Dic7—81no—
231Dic7—82no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—24no—
236C2×Q8C22⋊Dic714yes—
237C2×Q8C22×Dic714yes—
238C22×C4—141no—
239C22×C4—142no—
240C22×C4—141no—
241C22×C4C2×C22⋊D712yes—
242C22×C4C2×C22⋊D712yes—
243C22×C4C2×C22⋊D711yescharacteristic
244C22×C4C2×C22⋊D711yescharacteristic
245C22×C4C23×D711yescharacteristic
246C2×C8—144no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—282no—
257C4⋊C4—282no—
258C4⋊C4C22⋊Dic718yes—
259C4⋊C4C22⋊Dic714yes—
260C4⋊C4C22×Dic714yes—
261C22⋊C4—24no—
262C42—28no—
263C42—24no—
264C42—284no—
265C42—282no—
266C42C22×Dic714yes—
267C42C22×Dic712yes—
268C42C2×C22⋊D712yes—
269C42C2×C22⋊D714yes—
270C2×C7—21no—
271C2×C7—21no—
272C2×C7#288411yescharacteristic
273C2×C7C2×#41911yescharacteristic
274C2×C7#290111yescharacteristic
275C23C22×C22⋊D711yescharacteristic
276Q8—28no—
277Q8—44no—
278Q8—84no—
279Q8—82no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—142no—
288C2×C4—281no—
289C2×C4—281no—
290C2×C4—282no—
291C2×C4—281no—
292C2×C4—281no—
293C2×C4—282no—
294C2×C4—281no—
295C2×C4—282no—
296C2×C4#650512yes—
297C2×C4#650512yes—
298C2×C4C2×C22⋊Dic714yes—
299C2×C4C2×C22⋊Dic712yes—
300C2×C4C2×C22⋊Dic714yes—
301C2×C4C2×C22⋊Dic712yes—
302C2×C4C23×Dic711yescharacteristic
303C2×C4C22×C22⋊D711yescharacteristic
304C8—284no—
305C7#1763211yescharacteristic, complemented, Sylow
306C22—21no—
307C22—21no—
308C22C2×#650511yescharacteristic
309C22C2×#650511yescharacteristic
310C22C22×C22⋊Dic711yescharacteristic
311C4—24no—
312C4—21no—
313C4—21no—
314C4—24no—
315C4—22no—
316C4—44no—
317C4—44no—
318C4—282no—
319C4—561no—
320C4—562no—
321C4#8999714yes—
322C2—21no—
323C2—21no—
324C2C2×#8999711yescharacteristic
325C2#1174156411yescharacteristic
326C2#1174156511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#3502C2
socleC22×C7C22×C22⋊C4
prime radicalC23×C7C22×D4
cosocleC22×C4×C7C24
hypercenter#3502D7
hypocenterC7#17632
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3502

Complements

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

NH ≅ G/Ntype
8. C7×Q82C4semidirect
10. C7×#557C4semidirect
120. Q82Dic7semidirect
122. #557Dic7semidirect
305. C7#17632semidirect

Maximal subgroups

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

subgroupindex
C7×#35022
#117399082
#117398812
#117295922
#117295912
#117295902
#117295892
#176327