Finite Group Databasebeta

← #13509062

Subgroups — #13509062

Order 1792.

536 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3176C211yesmaximal, characteristic, complemented
2#11736024C211yesmaximal, characteristic, complemented
3#11729112C212yesmaximal, complemented
4#11729049C212yesmaximal, complemented
5#11728843C212yesmaximal, complemented
6#11728842C212yesmaximal, complemented
7#11728672C214yesmaximal, complemented
8#11727640C211yesmaximal, characteristic, complemented
9C7×#540—21no—
10#90439C2211yescharacteristic
11C2×#6464—21no—
12C7×#499C2211yescharacteristic, complemented
13C7×#493C2211yescharacteristic, complemented
14C7×#490—21no—
15C7×#469C2212yes—
16C7×#466—21no—
17C7×#442C2212yescomplemented
18C4×C7×D8—21no—
19C7×#405C2211yescharacteristic
20#89919C2214yes—
21#89897C2211yescharacteristic
22#89777—21no—
23#89489—21no—
24#89489C2212yes—
25#89472—21no—
26#89472C2212yes—
27#89358C2214yescomplemented
28#89348C2212yes—
29#89283C2214yes—
30#89282C2214yescomplemented
31#89261C2212yescomplemented
32#89258C2212yescomplemented
33#33692—71nomaximal, Sylow
34C2×#1157—21no—
35C2×C7×D8—21no—
36C2×C7×D8C2312yes—
37C2×C7×M16C2311yescharacteristic
38C22×C7×C8D411yescharacteristic
39C7×#127—21no—
40C7×#125—21no—
41C7×#123—42no—
42C7×#121—22no—
43C7×#121—42no—
44C4×C7×D4—22no—
45C7×#117D411yescharacteristic
46C2×#1136—21no—
47#6472—22no—
48#6471—22no—
49#6467—21no—
50#6467C2312yes—
51#6464—22no—
52#6464—22no—
53#6464—41no—
54#6464—41no—
55C2×#1131—41no—
56C2×C8×D7—41no—
57C7×#107—21no—
58C7×#106C2312yes—
59C7×#102—22no—
60C7×#102—22no—
61C7×#97D411yescharacteristic
62C7×#97C2311yescharacteristic
63C4×C7×C8D411yescharacteristic
64C4×C7×C8C2311yescharacteristic
65#6401—24no—
66#6400—24no—
67#6393—21no—
68#6393C2312yes—
69#6386C2314yes—
70#6379—22no—
71#6379—22no—
72#3176D711yescharacteristic, complemented
73#2955—71no—
74#2238—72no—
75#2231—72no—
76#2213—72no—
77#2190—72no—
78AGL(1,ℤ/16)—74no—
79#2166—71no—
80C2×C7×D4—22no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C22×C4×C7—42no—
84C22×C4×C7C2×D411yescharacteristic
85C2×C22⋊D7—41no—
86#1157—22no—
87#1157—22no—
88#1157—41no—
89#1157—41no—
90C2×D28—21no—
91C2×C4×D7—41no—
92C2×Dic14—21no—
93C7×D8—24no—
94C7×D8—41no—
95C7×D8—41no—
96C7×M16—22no—
97C2×C7×C8—21no—
98C2×C7×C8—21no—
99C2×C7×C8C2×D411yescharacteristic
100C2×C7×C8C2×D411yescharacteristic
101C2×C7×C8C2×D411yescharacteristic
102C2×C7×C8C2411yescharacteristic
103C7×C4⋊C4—21no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—42no—
106C7×C22⋊C4—21no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C42×C7C2×D411yescharacteristic
110C42×C7C2×D411yescharacteristic
111#1136—22no—
112#1136—41no—
113C2×C7⋊C8—41no—
114Dic28—21no—
115Dic28#4112yes—
116D56—21no—
117D56#4112yes—
118#1132—22no—
119#1132—22no—
120#1131—42no—
121#1131—42no—
122C8×D7—42no—
123C8×D7—42no—
124C7⋊C16—24no—
125#575—71no—
126C2×#131—141no—
127#540—21no—
128#509—74no—
129#499C2×D711yescharacteristic, complemented
130#493C2×D711yescharacteristic, complemented
131#490—21no—
132#469C2×D712yes—
133#466—21no—
134#444—72no—
135#444—141no—
136#443—72no—
137#443—141no—
138#442C2×D712yescomplemented
139C4×D8—21no—
140#430—71no—
141#421—141no—
142#405C2×D711yescharacteristic
143#365—74no—
144#364—72no—
145#361—74no—
146#360—74no—
147#349—72no—
148#347—72no—
149C23×C7—42no—
150C23×C7AGL(1,ℤ/8)11yescharacteristic
151C22×D7—41no—
152C7×D4—42no—
153C7×D4—42no—
154C7×D4—162no—
155C7×D4—162no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—42no—
161C2×C4×C7—162no—
162C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C7C22×D411yescharacteristic
164C22⋊D7—42no—
165C22⋊D7—42no—
166C2×Dic7—41no—
167D28—22no—
168D28—41no—
169C4×D7—42no—
170C4×D7—42no—
171Dic14—22no—
172Dic14—41no—
173C7×C8—21no—
174C7×C8—21no—
175C7×C8AGL(1,ℤ/8)11yescharacteristic
176C7×C8AGL(1,ℤ/8)11yescharacteristic
177C7×C8C2×#4112yes—
178C7⋊C8—42no—
179C2×#41—141no—
180#137—142no—
181AGL(1,ℤ/8)—142no—
182#135—72no—
183#135—141no—
184C2×D8—21no—
185C2×D8C22×D712yes—
186#131—142no—
187#131—142no—
188#131—281no—
189#131—281no—
190C2×M16—141no—
191C2×M16—281no—
192C2×M16C22×D711yescharacteristic
193C22×C8—281no—
194C22×C8C22⋊D711yescharacteristic
195#127—21no—
196#125—21no—
197#123—42no—
198#121—22no—
199#121—42no—
200C4×D4—22no—
201#117C22⋊D711yescharacteristic
202SD32—144no—
203D16—144no—
204M32—74no—
205#108—72no—
206#108—141no—
207#107—21no—
208#106C22×D712yes—
209#104—142no—
210#104—142no—
211#102—22no—
212#102—22no—
213#97C22⋊D711yescharacteristic
214#97C22×D711yescharacteristic
215C4×C8C22⋊D711yescharacteristic
216C4×C8C22×D711yescharacteristic
217C22×C7—21no—
218C22×C7—41no—
219C22×C7—82no—
220C22×C7—82no—
221C22×C7—82no—
222C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
223C2×D7—42no—
224C2×D7—82no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—82no—
229C4×C7#52511yescharacteristic
230C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
231Dic7—42no—
232#41—142no—
233#41—142no—
234#41—281no—
235#41—281no—
236C2×Q8—141no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—141no—
241C2×D4—281no—
242C22×C4—42no—
243C22×C4—281no—
244C22×C4C2×C22⋊D711yescharacteristic
245Q16—72no—
246Q16—141no—
247SD16—142no—
248SD16—142no—
249D8—24no—
250D8—41no—
251D8—41no—
252D8—72no—
253D8—141no—
254M16—22no—
255M16—142no—
256M16—282no—
257M16—281no—
258M16—282no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—282no—
262C2×C8—281no—
263C2×C8—282no—
264C2×C8C2×C22⋊D711yescharacteristic
265C2×C8C2×C22⋊D711yescharacteristic
266C2×C8C2×C22⋊D711yescharacteristic
267C2×C8C23×D711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—42no—
271C22⋊C4—21no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C42C2×C22⋊D711yescharacteristic
275C42C2×C22⋊D711yescharacteristic
276C16—144no—
277C2×C7—21no—
278C2×C7—41no—
279C2×C7—162no—
280C2×C7#310911yescharacteristic
281D7—82no—
282C23—42no—
283C23—281no—
284C23#649511yescharacteristic
285Q8—142no—
286Q8—281no—
287D4—42no—
288D4—42no—
289D4—142no—
290D4—162no—
291D4—162no—
292D4—281no—
293D4—282no—
294D4—282no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—162no—
301C2×C4—282no—
302C2×C4—281no—
303C2×C4—282no—
304C2×C4#649511yescharacteristic
305C2×C4C22×C22⋊D711yescharacteristic
306C8—21no—
307C8—21no—
308C8—282no—
309C8#651211yescharacteristic
310C8#651211yescharacteristic
311C8D7×#4112yes—
312C7#3369211yescharacteristic, complemented, Sylow
313C22—21no—
314C22—41no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—282no—
319C22—562no—
320C22C2×#649511yescharacteristic
321C4—21no—
322C4—41no—
323C4—41no—
324C4—82no—
325C4—282no—
326C4C2×#651211yescharacteristic
327C4#9051911yescharacteristic
328C2—21no—
329C2—41no—
330C2—162no—
331C2—562no—
332C2#1174074511yescharacteristic
subgroupG/H
centerC2#11740745
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#3176C2
socleC2×C7#3109
hypercenter#3176D7
hypocenterC7#33692
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. C7×#3176C2semidirect
2. #11736024C2semidirect
3. #11729112C2semidirect
4. #11729049C2semidirect
5. #11728843C2semidirect
6. #11728842C2semidirect
7. #11728672C2semidirect
8. #11727640C2semidirect
12. C7×#499C22semidirect
13. C7×#493C22semidirect
17. C7×#442C22semidirect
27. #89358C22semidirect
30. #89282C22semidirect
31. #89261C22semidirect
32. #89258C22semidirect
72. #3176D7semidirect
129. #499C2×D7semidirect
130. #493C2×D7semidirect
138. #442C2×D7semidirect
312. C7#33692semidirect

Maximal subgroups

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

subgroupindex
C7×#31762
#117360242
#117291122
#117290492
#117288432
#117288422
#117286722
#117276402
#336927