Finite Group Databasebeta

← #13264846

Subgroups — #13264846

Order 1792.

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

normal subgroups only

This group has 460 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
1C7×#2486C211yesmaximal, characteristic
2#11729371C211yesmaximal, characteristic
3#11729354C211yesmaximal, characteristic
4#11725456C214yesmaximal, complemented
5C2×C7×#123—21no—
6C2×C4×C7×D4C2211yescharacteristic
7C22×C7×C4⋊C4C411yescharacteristic
8C7×#398C2212yes—
9C7×#397C412yes—
10C7×#389—21no—
11C7×#381—21no—
12C7×#380—21no—
13C7×#380C411yescharacteristic
14C2×#6407—21no—
15C2×C22⋊(C7⋊C8)—21no—
16#89696—21no—
17#89325C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11881—71nomaximal, Sylow
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C23×C4×C7C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×#123—42no—
26C7×#123—42no—
27C4×C7×D4—41no—
28C4×C7×D4—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—41no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4C2×C412yes—
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×#1136—22no—
42C22×C7⋊C8—41no—
43C7×#95—22no—
44C7×#95—22no—
45C7×#95C2×C412yes—
46#6407—42no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—41no—
49C4⋊(C7⋊C8)—41no—
50C4⋊(C7⋊C8)—41no—
51#2486D711yescharacteristic
52#1800—71no—
53#1459—71no—
54#1346—74no—
55C24×C7C22⋊C411yescharacteristic
56C24×C7C22⋊C411yescharacteristic
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4D812yes—
60C2×C7×D4SD1612yes—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C7×C4⋊C4—41no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—42no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—44no—
88C42×C7—21no—
89C42×C7—21no—
90#1136—42no—
91#1136—42no—
92C2×C7⋊C8—42no—
93C2×C7⋊C8—42no—
94C2×C7⋊C8—82no—
95C2×#123—21no—
96C2×C4×D4C2×D711yescharacteristic
97C22×C4⋊C4Dic711yescharacteristic
98#423—141no—
99C2×#100—141no—
100C2×C22⋊C8—141no—
101#398C2×D712yes—
102#397Dic712yes—
103#389—21no—
104#381—21no—
105#380—21no—
106#380Dic711yescharacteristic
107#343—72no—
108#324—72no—
109#323—141no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7#10212yes—
119C23×C7#10212yes—
120C23×C7C2×C22⋊C411yescharacteristic
121C7×D4—42no—
122C7×D4—42no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—81no—
142C2×C4×C7—84no—
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7#9911yescharacteristic
145C2×C4×C7C2×D811yescharacteristic
146C2×C4×C7C2×SD1611yescharacteristic
147C7⋊C8—82no—
148C7⋊C8—82no—
149C22×D4C22⋊D711yescharacteristic
150C23×C4—21no—
151C23×C4C2×Dic711yescharacteristic
152C23×C4C22⋊D711yescharacteristic
153C2×M16—142no—
154C22×C8—281no—
155#123—42no—
156#123—42no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×Dic712yes—
165C2×C4⋊C4C22×D711yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4C2×Dic711yescharacteristic
169C2×C22⋊C4C22⋊D711yescharacteristic
170C2×C42C22⋊D711yescharacteristic
171C4⋊C8—281no—
172C4⋊C8—281no—
173#100—282no—
174C22⋊C8—142no—
175C22⋊C8—281no—
176#95—22no—
177#95—22no—
178#95C2×Dic712yes—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—22no—
182C22×C7—22no—
183C22×C7—42no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—81no—
191C22×C7C2×#9911yescharacteristic
192C22×C7C2×#10211yescharacteristic
193C22×C7#42111yescharacteristic
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—82no—
201C4×C7—82no—
202C4×C7#35212yes—
203C24C22⋊Dic711yescharacteristic
204C24C22⋊Dic711yescharacteristic
205C2×D4—41no—
206C2×D4—41no—
207C2×D4#114112yes—
208C2×D4#114212yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C22⋊Dic711yescharacteristic
221C22×C4C22⋊Dic711yescharacteristic
222C22×C4C22×Dic711yescharacteristic
223C22×C4C2×C22⋊D711yescharacteristic
224C22×C4C2×C22⋊D711yescharacteristic
225M16—282no—
226M16—282no—
227C2×C8—282no—
228C2×C8—282no—
229C2×C8—562no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C42—21no—
242C42—21no—
243C2×C7—21no—
244C2×C7—21no—
245C2×C7—42no—
246C2×C7—42no—
247C2×C7#151911yescharacteristic
248C2×C7C2×#35211yescharacteristic
249C2×C7#213511yescharacteristic
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#640612yes—
259C23#640612yes—
260C23C2×C22⋊Dic711yescharacteristic
261D4—42no—
262D4—42no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—84no—
283C2×C4#640811yescharacteristic
284C2×C4#640811yescharacteristic
285C2×C4C2×#114111yescharacteristic
286C2×C4C2×#114211yescharacteristic
287C8—562no—
288C8—562no—
289C7#1188111yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#640611yescharacteristic
303C22C2×#640811yescharacteristic
304C22#9000811yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#8933412yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#1172942911yescharacteristic
319C2C2×#8933411yescharacteristic
320C2#1172962711yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2486C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C4⋊C4C23
hypercenter#2486D7
hypocenterC7#11881
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #11725456C2semidirect
289. C7#11881semidirect

Maximal subgroups

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

subgroupindex
C7×#24862
#117293712
#117293542
#117254562
#118817