Finite Group Databasebeta

← #13264469

Subgroups — #13264469

Order 1792.

502 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 79 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×#1820C211yesmaximal, characteristic, complemented
3#11729326C211yesmaximal, characteristic, complemented
4#11725295C214yesmaximal, complemented
5C2×C7×#117C2211yescharacteristic
6C2×#6493—21no—
7C2×#6437C411yescharacteristic, complemented
8C7×#381—21no—
9#89763—21no—
10#89763C412yescomplemented
11#89732—21no—
12#89732C2212yes—
13#89711—21no—
14#89708C411yescharacteristic, complemented
15#89698—21no—
16C7×#342—21no—
17C7×#342C2212yes—
18#89323—21no—
19#89320C2212yes—
20#11800—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C22×C22⋊D7—22no—
23C22×D28—21no—
24C22×D28—21no—
25C22×D28C2×C412yes—
26C7×#117—41no—
27C7×#117—41no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C22⋊C4—22no—
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C7×C22⋊C4C2311yescharacteristic
33C2×C42×C7C2×C411yescharacteristic
34C2×C42×C7D411yescharacteristic
35#6493—44no—
36C2×(C2×D7)⋊C4—22no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×C4⋊Dic7—21no—
40#6437—41no—
41#6437—41no—
42C7×#95—41no—
43C7×#95—41no—
44C7⋊#95—21no—
45C7⋊#95C2×C412yes—
46C22⋊(C7⋊C8)—22no—
47C4⋊(C7⋊C8)—41no—
48#6377—41no—
49#2619—71no—
50#1820D711yescharacteristic, complemented
51#1464—71no—
52#1343—74no—
53C24×C7C2×D411yescharacteristic
54C23×D7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—21no—
57C22×C4×C7—21no—
58C22×C4×C7—41no—
59C22×C4×C7—42no—
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22×C411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C2×C22⋊D7—44no—
67C2×C22⋊D7—82no—
68C2×C22⋊D7—82no—
69C22×Dic7—22no—
70C2×D28—42no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28—42no—
74C2×D28—41no—
75C2×D28—41no—
76C2×D28—82no—
77C2×D28—82no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C7×C22⋊C4—42no—
83C7×C22⋊C4—42no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88(C2×D7)⋊C4—44no—
89(C2×D7)⋊C4—82no—
90(C2×D7)⋊C4—82no—
91C4⋊Dic7—42no—
92C2×C7⋊C8—42no—
93C2×#127—71no—
94C2×#121—141no—
95C2×#117C2×D711yescharacteristic
96#432—141no—
97#394—72no—
98#394—141no—
99#392—72no—
100#392—141no—
101#390—141no—
102#382—71no—
103#381—21no—
104#342—21no—
105#342C2×D712yes—
106#324—141no—
107#323—72no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—41no—
111C23×C7—41no—
112C23×C7C2×C22⋊C411yescharacteristic
113C23×C7AGL(1,ℤ/8)12yes—
114C22×D7—42no—
115C22×D7—42no—
116C22×D7—42no—
117C22×D7—42no—
118C22×D7—42no—
119C22×D7—42no—
120C22×D7—42no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—82no—
137C2×C4×C7—82no—
138C2×C4×C7—82no—
139C2×C4×C7#9911yescharacteristic
140C2×C4×C7#9911yescharacteristic
141C22⋊D7—88no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7—42no—
145D28—82no—
146D28—82no—
147D28—84no—
148D28—84no—
149C7⋊C8—82no—
150C22×D4—72no—
151C22×D4—142no—
152C22×D4—141no—
153C22×D4—141no—
154C23×C4C22⋊D711yescharacteristic
155#127—281no—
156#127—281no—
157#121—284no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—141no—
161C2×C4⋊C4C4×D711yescharacteristic
162C2×C4⋊C4D2811yescharacteristic
163C2×C22⋊C4—22no—
164C2×C22⋊C4—142no—
165C2×C22⋊C4—142no—
166C2×C22⋊C4—142no—
167C2×C22⋊C4C22⋊D711yescharacteristic
168C2×C22⋊C4C22×D711yescharacteristic
169C2×C42C4×D711yescharacteristic
170C2×C42D2811yescharacteristic
171C4⋊C8—281no—
172C22⋊C8—142no—
173#97—281no—
174#95—41no—
175#95—41no—
176#95—72no—
177#95—141no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—22no—
181C22×C7—42no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—81no—
185C22×C7C2×#9911yescharacteristic
186C22×C7#41811yescharacteristic
187C22×C7#42111yescharacteristic
188C2×D7—82no—
189C2×D7—82no—
190C2×D7—82no—
191C2×D7—82no—
192C2×D7—82no—
193C2×D7—82no—
194C2×D7—82no—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—162no—
203Dic7—82no—
204C24—142no—
205C24C2×C22⋊D711yescharacteristic
206C2×D4—282no—
207C2×D4—282no—
208C2×D4—282no—
209C2×D4—282no—
210C2×D4—284no—
211C2×D4—281no—
212C2×D4—281no—
213C2×D4—562no—
214C2×D4—562no—
215C2×D4—562no—
216C2×D4—562no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—142no—
223C22×C4(C2×D7)⋊C411yescharacteristic
224C22×C4(C2×D7)⋊C411yescharacteristic
225C22×C4(C2×D7)⋊C411yescharacteristic
226C22×C4(C2×D7)⋊C411yescharacteristic
227C22×C4C2×C4×D711yescharacteristic
228C22×C4C2×D2811yescharacteristic
229C2×C8—282no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—282no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—284no—
238C22⋊C4—562no—
239C22⋊C4—562no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C2×C7—21no—
245C2×C7—21no—
246C2×C7—42no—
247C2×C7#151011yescharacteristic
248C2×C7#212211yescharacteristic
249C2×C7#212811yescharacteristic
250D7—162no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—282no—
256C23—282no—
257C23—282no—
258C23—282no—
259C23—282no—
260C23—282no—
261C23—282no—
262C23C2×(C2×D7)⋊C411yescharacteristic
263C23#651212yes—
264D4—562no—
265D4—562no—
266D4—564no—
267D4—564no—
268D4—568no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—282no—
288C2×C4—282no—
289C2×C4—282no—
290C2×C4#638011yescharacteristic
291C2×C4#638011yescharacteristic
292C8—562no—
293C7#1180011yescharacteristic, complemented, Sylow
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—562no—
302C22—562no—
303C22—562no—
304C22—562no—
305C22—562no—
306C22—562no—
307C22—562no—
308C22C2×#638011yescharacteristic
309C22#8977311yescharacteristic
310C22#8977711yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—562no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—1122no—
324C2#1172728511yescharacteristic
325C2#1172781111yescharacteristic
326C2#1172781511yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#1820C2
socleC22×C7C2×#99
prime radical#11739731C2
cosocleC2×C7×C22⋊C4C23
hypercenter#1820D7
hypocenterC7#11800
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. C7×#1820C2semidirect
3. #11729326C2semidirect
4. #11725295C2semidirect
7. C2×#6437C4semidirect
10. #89763C4semidirect
14. #89708C4semidirect
50. #1820D7semidirect
293. C7#11800semidirect

Maximal subgroups

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

subgroupindex
#117397312
C7×#18202
#117293262
#117252952
#118007