Finite Group Databasebeta

← #13331770

Subgroups — #13331770

Order 1792.

924 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 179 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
1C7×#3548C211yesmaximal, characteristic
2#11740784C211yesmaximal, characteristic, complemented
3#11740766C212yesmaximal, complemented
4#11729667C214yesmaximal, complemented
5#11729666C214yesmaximal, complemented
6#11729588C211yesmaximal, characteristic, complemented
7#11729556C212yesmaximal, complemented
8C7×#562—22no—
9C7×#554—22no—
10C7×#550—22no—
11C7×#547C2214yes—
12C7×#540—22no—
13C7×#537—24no—
14C7×#533C2211yescharacteristic
15C7×#532C2212yes—
16#90161C2211yescharacteristic
17#89863—22no—
18#89851C2212yes—
19#89850C2212yes—
20#89847—22no—
21#89836—22no—
22#89814—24no—
23#89791C2214yes—
24#89779C2214yes—
25#89770—22no—
26#89696C2212yes—
27C2×C4×C7⋊C8C2211yescharacteristic
28#89342C2214yes—
29D4⋊(C7⋊C8)C2214yes—
30D4⋊(C7⋊C8)C2214yes—
31#18701—71nomaximal, Sylow
32C2×C7×#41—24no—
33C2×C7×#41—22no—
34C7×C4⋊Q8D412yes—
35C7×C4⋊Q8C2312yes—
36C7×#127—22no—
37C7×#127D412yes—
38C7×#125—22no—
39C7×#125D412yes—
40C7×#124—22no—
41C7×#123—28no—
42C7×C22⋊Q8—28no—
43C7×C22⋊Q8—22no—
44C7×C22⋊Q8—24no—
45C7×#121—28no—
46C7×#121—28no—
47C7×#121—24no—
48C7×#121—22no—
49C7×#121D414yes—
50C4×C7×Q8—22no—
51C4×C7×D4—24no—
52C4×C7×D4—22no—
53C4×C7×D4C2314yes—
54C7×#117D412yes—
55C2×C7×C4⋊C4—24no—
56C2×C42×C7C2311yescharacteristic
57C2×#1142—24no—
58#6489—21no—
59#6487—22no—
60C2×#1136—22no—
61C22×C7⋊C8—21no—
62#6433—21no—
63#6432C2312yes—
64C4×Dic14—22no—
65#6406—24no—
66#6384—24no—
67#6382—24no—
68#6381—24no—
69#6381—24no—
70C4⋊(C7⋊C8)—22no—
71C4⋊(C7⋊C8)C2314yes—
72C4×C7⋊C8—21no—
73C4×C7⋊C8C2311yescharacteristic
74C4×C7⋊C8C2311yescharacteristic
75#3548D711yescharacteristic
76#3254—72no—
77#3160—71no—
78#1654—74no—
79#1647—74no—
80#1561—71no—
81#1492—72no—
82C7×#41—82no—
83C7×#41—82no—
84C7×#41—84no—
85C7×#41—84no—
86C2×C7×Q8—22no—
87C2×C7×Q8—22no—
88C2×C7×D4—28no—
89C2×C7×D4—24no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4C2×D414yes—
93C22×C4×C7—28no—
94C22×C4×C7—24no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D412yes—
97C2×Dic14—22no—
98C7×C4⋊C4—28no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—24no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4C2×D414yes—
103C7×C4⋊C4C2×D414yes—
104C7×C22⋊C4—28no—
105C7×C22⋊C4—22no—
106C7×C22⋊C4—24no—
107C7×C22⋊C4—24no—
108C42×C7—22no—
109C42×C7C2×D412yes—
110C42×C7C2×D411yescharacteristic
111C42×C7C2411yescharacteristic
112C22⋊Dic7—41no—
113#1142—84no—
114C4⋊Dic7—22no—
115C4⋊Dic7—22no—
116Dic7⋊C4—42no—
117C4×Dic7—41no—
118#1136—82no—
119C2×C7⋊C8—24no—
120C2×C7⋊C8—21no—
121C2×C7⋊C8—21no—
122C2×C7⋊C8—24no—
123#562—22no—
124#554—22no—
125#550—22no—
126#547C2×D714yes—
127#540—22no—
128#537—24no—
129#533—71no—
130#533C2×D711yescharacteristic
131#532C2×D712yes—
132#500—72no—
133#499—142no—
134#498—72no—
135#477—142no—
136#474—74no—
137#468—144no—
138#461—74no—
139#460—142no—
140#427—142no—
141#423—72no—
142C2×C4×C8—71no—
143#332—74no—
144D4⋊C8—74no—
145D4⋊C8—74no—
146C23×C7—24no—
147C23×C7#12011yescharacteristic
148C7×Q8—42no—
149C7×Q8—42no—
150C7×Q8—82no—
151C7×D4—24no—
152C7×D4—24no—
153C7×D4—44no—
154C7×D4—82no—
155C7×D4—84no—
156C7×D4—88no—
157C2×C4×C7—22no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—24no—
161C2×C4×C7—24no—
162C2×C4×C7—44no—
163C2×C4×C7—44no—
164C2×C4×C7—88no—
165C2×C4×C7#12012yes—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7C22×D411yescharacteristic
168C2×C4×C7C22×D412yes—
169C2×Dic7—42no—
170Dic14—42no—
171Dic14—42no—
172C7⋊C8—22no—
173C7⋊C8—22no—
174C7⋊C8—44no—
175C2×#41—24no—
176C2×#41—22no—
177C2×SD16—144no—
178C2×M16—142no—
179C22×C8—141no—
180C4⋊Q8—72no—
181C4⋊Q8C22⋊D712yes—
182C4⋊Q8C22×D712yes—
183#127—22no—
184#127C22⋊D712yes—
185#125—22no—
186#125—141no—
187#125C22⋊D712yes—
188#124—22no—
189#123—28no—
190C22⋊Q8—28no—
191C22⋊Q8—22no—
192C22⋊Q8—24no—
193C22⋊Q8—142no—
194#121—28no—
195#121—28no—
196#121—24no—
197#121—22no—
198#121C22⋊D714yes—
199C4×Q8—22no—
200C4×Q8—142no—
201C4×D4—24no—
202C4×D4—22no—
203C4×D4C22×D714yes—
204#117—141no—
205#117C22⋊D712yes—
206C2×C4⋊C4—24no—
207C2×C42C22×D711yescharacteristic
208#107—144no—
209#107—144no—
210#106—144no—
211C4⋊C8—74no—
212C4⋊C8—142no—
213#103—144no—
214#102—144no—
215C4×C8—71no—
216C4×C8—71no—
217C4×C8—141no—
218C22×C7—22no—
219C22×C7—21no—
220C22×C7—44no—
221C22×C7—44no—
222C22×C7—44no—
223C22×C7C2×#12011yescharacteristic
224C4×C7—22no—
225C4×C7—21no—
226C4×C7—44no—
227C4×C7—44no—
228C4×C7#45612yes—
229C4×C7#57612yes—
230C4×C7#57712yes—
231Dic7—82no—
232#41—82no—
233#41—82no—
234#41—84no—
235#41—84no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—142no—
239C2×D4—28no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4C2×C22⋊D714yes—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4C2×C22⋊D711yescharacteristic
247C22×C4C2×C22⋊D712yes—
248SD16—564no—
249M16—562no—
250C2×C8—144no—
251C2×C8—141no—
252C2×C8—141no—
253C2×C8—144no—
254C4⋊C4—28no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—282no—
261C4⋊C4C2×C22⋊D714yes—
262C4⋊C4C2×C22⋊D714yes—
263C22⋊C4—28no—
264C22⋊C4—22no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—281no—
268C42—22no—
269C42—281no—
270C42C2×C22⋊D711yescharacteristic
271C42C2×C22⋊D712yes—
272C42C23×D711yescharacteristic
273C2×C7—41no—
274C2×C7—84no—
275C2×C7#301212yes—
276C2×C7C2×#45611yescharacteristic
277C23—24no—
278C23#651611yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—82no—
282Q8—282no—
283Q8—282no—
284D4—24no—
285D4—24no—
286D4—44no—
287D4—82no—
288D4—84no—
289D4—88no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—88no—
298C2×C4—282no—
299C2×C4#651612yes—
300C2×C4#651611yescharacteristic
301C2×C4C22×C22⋊D711yescharacteristic
302C2×C4C22×C22⋊D712yes—
303C8—142no—
304C8—142no—
305C8—284no—
306C7#1870111yescharacteristic, complemented, Sylow
307C22—22no—
308C22—21no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22C2×#651611yescharacteristic
313C4—22no—
314C4—21no—
315C4—44no—
316C4—44no—
317C4—562no—
318C4#9001812yes—
319C4#9052612yes—
320C4#9052712yes—
321C2—41no—
322C2—84no—
323C2#1174154712yes—
324C2C2×#9001811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3548C2
socleC22×C7C2×#120
hypercenter#3548D7
hypocenterC7#18701
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
2. #11740784C2semidirect
3. #11740766C2semidirect
4. #11729667C2semidirect
5. #11729666C2semidirect
6. #11729588C2semidirect
7. #11729556C2semidirect
306. C7#18701semidirect

Maximal subgroups

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

subgroupindex
C7×#35482
#117407842
#117407662
#117296672
#117296662
#117295882
#117295562
#187017