Finite Group Databasebeta

← #13263980

Subgroups — #13263980

Order 1792.

568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#11739731C211yesmaximal, characteristic
2C7×#1740C211yesmaximal, characteristic, complemented
3#11729325C211yesmaximal, characteristic, complemented
4#11725295C214yesmaximal, complemented
5C2×C7×#117—21no—
6C2×C7×#117C2211yescharacteristic
7C2×#6493—21no—
8C2×#6437C411yescharacteristic, complemented
9#89763—21no—
10#89763C412yescomplemented
11#89732—21no—
12#89732C2212yes—
13#89711—21no—
14#89708C411yescharacteristic, complemented
15#89697—21no—
16C7×#342—21no—
17C7×#342C2212yes—
18#89323—21no—
19#89320C2212yes—
20#11693—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C22×C22⋊D7—22no—
23C22×D28—21no—
24C22×D28—21no—
25C22×D28C2×C412yes—
26C7×#117—24no—
27C7×#117—41no—
28C7×#117—41no—
29C7×#117—42no—
30C2×C7×C4⋊C4—41no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4D411yescharacteristic
35C2×C7×C22⋊C4C2311yescharacteristic
36C2×C42×C7—41no—
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7D411yescharacteristic
39#6493—42no—
40#6493—42no—
41C2×(C2×D7)⋊C4—22no—
42C2×(C2×D7)⋊C4—22no—
43C2×(C2×D7)⋊C4—22no—
44C2×C4⋊Dic7—21no—
45#6437—41no—
46#6437—41no—
47C7⋊#95—21no—
48C7⋊#95C2×C412yes—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—41no—
51C4×C7⋊C8—41no—
52#2619—71no—
53#1740D711yescharacteristic, complemented
54#1458—71no—
55#1343—74no—
56C24×C7C2×D411yescharacteristic
57C23×D7—22no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—22no—
61C22×C4×C7—42no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7D812yes—
67C22×C4×C7SD1612yes—
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C2×C22⋊D7—42no—
71C2×C22⋊D7—42no—
72C2×C22⋊D7—82no—
73C2×C22⋊D7—82no—
74C22×Dic7—22no—
75C2×D28—42no—
76C2×D28—42no—
77C2×D28—41no—
78C2×D28—41no—
79C2×D28—42no—
80C2×D28—42no—
81C2×D28—82no—
82C2×D28—82no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—44no—
86C7×C22⋊C4—24no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—42no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—44no—
95(C2×D7)⋊C4—44no—
96(C2×D7)⋊C4—82no—
97(C2×D7)⋊C4—82no—
98C4⋊Dic7—42no—
99C2×C7⋊C8—42no—
100C2×#127—71no—
101C2×#121—141no—
102C2×#117—21no—
103C2×#117C2×D711yescharacteristic
104#431—141no—
105#394—72no—
106#394—141no—
107#392—72no—
108#392—141no—
109#390—141no—
110#382—71no—
111#342—21no—
112#342C2×D712yes—
113#324—141no—
114#323—72no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7C2×C22⋊C411yescharacteristic
120C23×C7C2×D811yescharacteristic
121C23×C7C2×SD1611yescharacteristic
122C22×D7—42no—
123C22×D7—42no—
124C22×D7—42no—
125C22×D7—42no—
126C22×D7—42no—
127C22×D7—42no—
128C22×D7—42no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—44no—
144C2×C4×C7—82no—
145C2×C4×C7—82no—
146C2×C4×C7#9911yescharacteristic
147C2×C4×C7#9911yescharacteristic
148C2×C4×C7#10212yes—
149C2×C4×C7#10212yes—
150C2×C4×C7#10412yes—
151C22⋊D7—84no—
152C22⋊D7—84no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156D28—84no—
157D28—84no—
158D28—82no—
159D28—82no—
160C7⋊C8—82no—
161C22×D4—72no—
162C22×D4—142no—
163C22×D4—141no—
164C22×D4—141no—
165C23×C4C22⋊D711yescharacteristic
166#127—281no—
167#127—281no—
168#121—282no—
169#121—282no—
170#117—24no—
171#117—41no—
172#117—41no—
173#117—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—141no—
176C2×C4⋊C4C4×D711yescharacteristic
177C2×C4⋊C4D2811yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—142no—
180C2×C22⋊C4—142no—
181C2×C22⋊C4—142no—
182C2×C22⋊C4C22⋊D711yescharacteristic
183C2×C22⋊C4C22×D711yescharacteristic
184C2×C42—41no—
185C2×C42C4×D711yescharacteristic
186C2×C42D2811yescharacteristic
187C4⋊C8—281no—
188C22⋊C8—142no—
189C4×C8—281no—
190#95—72no—
191#95—141no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—81no—
201C22×C7C2×#9911yescharacteristic
202C22×C7C2×#10211yescharacteristic
203C22×C7C2×#10411yescharacteristic
204C2×D7—82no—
205C2×D7—82no—
206C2×D7—82no—
207C2×D7—82no—
208C2×D7—82no—
209C2×D7—82no—
210C2×D7—82no—
211C4×C7—22no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—84no—
217C4×C7#32712yes—
218C4×C7#32712yes—
219Dic7—82no—
220C24—142no—
221C24C2×C22⋊D711yescharacteristic
222C2×D4—282no—
223C2×D4—282no—
224C2×D4—282no—
225C2×D4—282no—
226C2×D4—282no—
227C2×D4—281no—
228C2×D4—281no—
229C2×D4—282no—
230C2×D4—562no—
231C2×D4—562no—
232C2×D4—562no—
233C2×D4—562no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—142no—
239C22×C4(C2×D7)⋊C411yescharacteristic
240C22×C4(C2×D7)⋊C411yescharacteristic
241C22×C4(C2×D7)⋊C411yescharacteristic
242C22×C4(C2×D7)⋊C411yescharacteristic
243C22×C4#114112yes—
244C22×C4#114312yes—
245C22×C4C2×C4×D711yescharacteristic
246C22×C4C2×D2811yescharacteristic
247C2×C8—282no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—282no—
252C22⋊C4—24no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—42no—
256C22⋊C4—284no—
257C22⋊C4—562no—
258C22⋊C4—562no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—44no—
264C2×C7—21no—
265C2×C7—21no—
266C2×C7—41no—
267C2×C7—41no—
268C2×C7C2×#32711yescharacteristic
269C2×C7#212211yescharacteristic
270C2×C7#212811yescharacteristic
271D7—162no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23—282no—
277C23—282no—
278C23—282no—
279C23—282no—
280C23—282no—
281C23—282no—
282C23—282no—
283C23C2×(C2×D7)⋊C411yescharacteristic
284C23C2×#114111yescharacteristic
285C23C2×#114311yescharacteristic
286D4—564no—
287D4—564no—
288D4—564no—
289D4—564no—
290D4—562no—
291D4—562no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—282no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4#637912yes—
313C2×C4#638011yescharacteristic
314C2×C4#638011yescharacteristic
315C2×C4#638312yes—
316C2×C4#638312yes—
317C8—562no—
318C7#1169311yescharacteristic, complemented, Sylow
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—562no—
329C22—562no—
330C22—562no—
331C22—562no—
332C22—562no—
333C22—562no—
334C22—562no—
335C22C2×#637911yescharacteristic
336C22C2×#638011yescharacteristic
337C22C2×#638311yescharacteristic
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—84no—
344C4—562no—
345C4#8924612yes—
346C4#8924612yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—41no—
351C2—1122no—
352C2C2×#8924611yescharacteristic
353C2#1172781111yescharacteristic
354C2#1172781511yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#1740C2
socleC22×C7C2×#99
prime radical#11739731C2
cosocleC2×C7×C22⋊C4C23
hypercenter#1740D7
hypocenterC7#11693
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. C7×#1740C2semidirect
3. #11729325C2semidirect
4. #11725295C2semidirect
8. C2×#6437C4semidirect
10. #89763C4semidirect
14. #89708C4semidirect
53. #1740D7semidirect
318. C7#11693semidirect

Maximal subgroups

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

subgroupindex
#117397312
C7×#17402
#117293252
#117252952
#116937