Finite Group Databasebeta

← #13264497

Subgroups — #13264497

Order 1792.

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

normal subgroups only

This group has 488 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×#1464C211yesmaximal, characteristic, complemented
3#11729343C211yesmaximal, characteristic, complemented
4#11725262C214yesmaximal, complemented
5C2×C7×#117C2211yescharacteristic
6C2×#6493—21no—
7C2×#6437C411yescharacteristic
8C7×#432—21no—
9#89979—21no—
10#89763—21no—
11#89763C2212yes—
12#89732—21no—
13#89732C412yes—
14#89711—21no—
15#89708C411yescharacteristic
16C7×#324C2212yes—
17C7×#323—21no—
18#89325—21no—
19#89325C2212yes—
20#11806—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⋊C4D411yescharacteristic
29C2×C7×C4⋊C4C2311yescharacteristic
30C2×C7×C22⋊C4C2×C411yescharacteristic
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C42×C7C2×C411yescharacteristic
33C2×C42×C7D411yescharacteristic
34#6493—44no—
35C2×(C2×D7)⋊C4—22no—
36C2×(C2×D7)⋊C4—22no—
37C2×(C2×D7)⋊C4—22no—
38C2×C4⋊Dic7—21no—
39C2×#1136—22no—
40#6437—41no—
41#6437—41no—
42C7×C4⋊C8—41no—
43C7×C22⋊C8—22no—
44C7×#97—41no—
45C7⋊#95—21no—
46C7⋊#95C2×C412yes—
47C22⋊(C7⋊C8)—41no—
48C22⋊(C7⋊C8)—41no—
49#2619—71no—
50#1822—71no—
51#1464D711yescharacteristic, complemented
52#1345—74no—
53C24×C7C22⋊C411yescharacteristic
54C23×D7—22no—
55C22×C4×C7—21no—
56C22×C4×C7—21no—
57C22×C4×C7—22no—
58C22×C4×C7—41no—
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22×C411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C2×C22⋊D7—44no—
66C2×C22⋊D7—82no—
67C2×C22⋊D7—82no—
68C22×Dic7—22no—
69C2×D28—42no—
70C2×D28—42no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28—41no—
74C2×D28—41no—
75C2×D28—82no—
76C2×D28—82no—
77C2×C7×C8—42no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C42×C7—21no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86(C2×D7)⋊C4—44no—
87(C2×D7)⋊C4—82no—
88(C2×D7)⋊C4—82no—
89C4⋊Dic7—42no—
90#1136—42no—
91#1136—42no—
92C2×C7⋊C8—42no—
93C2×#127—71no—
94C2×#121—141no—
95C2×#117C2×D711yescharacteristic
96#432—21no—
97#407—141no—
98#394—72no—
99#394—141no—
100#392—72no—
101#392—141no—
102#390—141no—
103#382—71no—
104#343—72no—
105#343—141no—
106#324C2×D712yes—
107#323—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—41no—
112C23×C7—41no—
113C23×C7C2×C22⋊C411yescharacteristic
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—42no—
130C2×C4×C7—42no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7#9911yescharacteristic
136C2×C4×C7#9911yescharacteristic
137C2×C4×C7AGL(1,ℤ/8)12yes—
138C22⋊D7—88no—
139C2×Dic7—42no—
140C2×Dic7—42no—
141C2×Dic7—42no—
142D28—82no—
143D28—82no—
144D28—84no—
145D28—84no—
146C7×C8—82no—
147C7⋊C8—82no—
148C22×D4—72no—
149C22×D4—142no—
150C22×D4—141no—
151C22×D4—141no—
152C23×C4D2811yescharacteristic
153C2×M16—142no—
154#127—281no—
155#127—281no—
156#121—284no—
157#117—41no—
158#117—41no—
159C2×C4⋊C4—141no—
160C2×C4⋊C4D2811yescharacteristic
161C2×C4⋊C4C22×D711yescharacteristic
162C2×C22⋊C4—142no—
163C2×C22⋊C4—142no—
164C2×C22⋊C4—142no—
165C2×C22⋊C4C4×D711yescharacteristic
166C2×C22⋊C4C22⋊D711yescharacteristic
167C2×C42C4×D711yescharacteristic
168C2×C42C22⋊D711yescharacteristic
169C4⋊C8—41no—
170C22⋊C8—22no—
171C22⋊C8—281no—
172C22⋊C8—281no—
173#97—41no—
174#95—72no—
175#95—141no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7—42no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—81no—
183C22×C7C2×#9911yescharacteristic
184C22×C7#41811yescharacteristic
185C22×C7#42111yescharacteristic
186C2×D7—82no—
187C2×D7—82no—
188C2×D7—82no—
189C2×D7—82no—
190C2×D7—82no—
191C2×D7—82no—
192C2×D7—82no—
193C4×C7—22no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—41no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200Dic7—82no—
201C24—142no—
202C24(C2×D7)⋊C411yescharacteristic
203C2×D4—282no—
204C2×D4—282no—
205C2×D4—282no—
206C2×D4—282no—
207C2×D4—284no—
208C2×D4—281no—
209C2×D4—281no—
210C2×D4—562no—
211C2×D4—562no—
212C2×D4—562no—
213C2×D4—562no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—142no—
219C22×C4(C2×D7)⋊C411yescharacteristic
220C22×C4(C2×D7)⋊C411yescharacteristic
221C22×C4(C2×D7)⋊C411yescharacteristic
222C22×C4C2×C4×D711yescharacteristic
223C22×C4C2×D2811yescharacteristic
224C22×C4C2×C22⋊D711yescharacteristic
225M16—282no—
226M16—282no—
227C2×C8—42no—
228C2×C8—282no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—282no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—284no—
235C22⋊C4—562no—
236C22⋊C4—562no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C2×C7—21no—
242C2×C7—21no—
243C2×C7—81no—
244C2×C7#151011yescharacteristic
245C2×C7#212211yescharacteristic
246C2×C7#213411yescharacteristic
247D7—162no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—282no—
254C23—282no—
255C23—282no—
256C23—282no—
257C23—282no—
258C23—282no—
259C23—282no—
260C23C2×(C2×D7)⋊C411yescharacteristic
261D4—562no—
262D4—562no—
263D4—564no—
264D4—564no—
265D4—568no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—282no—
281C2×C4—282no—
282C2×C4—282no—
283C2×C4#638011yescharacteristic
284C2×C4#638011yescharacteristic
285C2×C4#647112yes—
286C8—82no—
287C8—562no—
288C7#1180611yescharacteristic, complemented, Sylow
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—562no—
297C22—562no—
298C22—562no—
299C22—562no—
300C22—562no—
301C22—562no—
302C22—562no—
303C22C2×#638011yescharacteristic
304C22#8990311yescharacteristic
305C22#8991111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—562no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—1122no—
318C2#1172773211yescharacteristic
319C2#1172781111yescharacteristic
320C2#1172790311yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1464C2
socleC22×C7C2×#99
prime radical#11739731C2
cosocleC2×C7×C4⋊C4C23
hypercenter#1464D7
hypocenterC7#11806
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. C7×#1464C2semidirect
3. #11729343C2semidirect
4. #11725262C2semidirect
51. #1464D7semidirect
288. C7#11806semidirect

Maximal subgroups

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

subgroupindex
#117397312
C7×#14642
#117293432
#117252622
#118067