Finite Group Databasebeta

← #13246710

Subgroups — #13246710

Order 1792.

460 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#89502C211yesmaximal, characteristic
2#11727736C212yesmaximal, complemented
3#11727722C212yesmaximal
4C7×#1293C211yesmaximal, characteristic, complemented
5#11725485C211yesmaximal, characteristic, complemented
6C2×C4×C22⋊D7C2211yescharacteristic
7C23×C7⋊C8C2211yescharacteristic
8C22×#1131C411yescharacteristic
9C7×#407C2212yes—
10C2×C7×C22⋊C8C2211yescharacteristic
11#89979C2212yes—
12C2×#6396—22no—
13C2×(C2×D7)⋊C8C411yescharacteristic
14C2×#6389C411yescharacteristic
15C2×Dic7⋊C8C411yescharacteristic
16#89502—22no—
17#89502—22no—
18#89353—22no—
19#89264—22no—
20#89263—22no—
21#8613—71nomaximal, Sylow
22C23×C4×C7C2311yescharacteristic
23C22×C22⋊D7Q811yescharacteristic
24C22×C4×D7C2×C411yescharacteristic
25C2×C7×M16—22no—
26C22×C7×C8C2×C411yescharacteristic
27C22×C7×C8C2×C411yescharacteristic
28C2×C22⋊Dic7Q811yescharacteristic
29C4×C22⋊D7—22no—
30C4×C22⋊D7—22no—
31C2×(C2×D7)⋊C4D411yescharacteristic
32C2×Dic7⋊C4D411yescharacteristic
33C2×C4×Dic7C2×C411yescharacteristic
34C2×#1136—22no—
35C22×C7⋊C8—42no—
36C22×C7⋊C8C2×C411yescharacteristic
37C22×C7⋊C8C2×C411yescharacteristic
38C22×C7⋊C8D412yes—
39C22×C7⋊C8D412yes—
40C2×#1131—21no—
41C2×#1131—21no—
42C2×#1131—42no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—22no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)—22no—
48#6396—44no—
49(C2×D7)⋊C8—22no—
50#6389—21no—
51#6389—21no—
52Dic7⋊C8—22no—
53C2×#435—71no—
54#1810—72no—
55#1465—72no—
56#1293—71no—
57#1293D711yescharacteristic, complemented
58C24×C7C2×Q811yescharacteristic
59C23×D7C4⋊C411yescharacteristic
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7C22×C411yescharacteristic
63C22×C4×C7C2×D412yes—
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7#4112yes—
66C2×C22⋊D7—22no—
67C2×C22⋊D7—42no—
68C22×Dic7C4⋊C411yescharacteristic
69C22×Dic7C4⋊C411yescharacteristic
70C22×Dic7C4⋊C411yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7—41no—
74C2×C4×D7—41no—
75C7×M16—42no—
76C7×M16—42no—
77C2×C7×C8—21no—
78C2×C7×C8—21no—
79C2×C7×C8—41no—
80C2×C7×C8—41no—
81C2×C7×C8—42no—
82C22⋊Dic7—22no—
83(C2×D7)⋊C4—22no—
84Dic7⋊C4—22no—
85C4×Dic7—21no—
86C4×Dic7—21no—
87#1136—42no—
88#1136—42no—
89C2×C7⋊C8—22no—
90C2×C7⋊C8—24no—
91C2×C7⋊C8—41no—
92C2×C7⋊C8—42no—
93C2×C7⋊C8—42no—
94C2×C7⋊C8C22⋊C411yescharacteristic
95C2×C7⋊C8C22⋊C412yes—
96C2×C7⋊C8C22⋊C411yescharacteristic
97#1131—42no—
98#1131—42no—
99C22×M16—71no—
100C23×C8—71no—
101C2×C4×D4—71no—
102#435—142no—
103#435—142no—
104C2×#108—142no—
105C2×C4⋊C8—71no—
106#407—72no—
107#407C2×D712yes—
108C2×C22⋊C8—71no—
109C2×C22⋊C8C2×D711yescharacteristic
110C2×#97—71no—
111#341—142no—
112#324—142no—
113#323—142no—
114C23×C7—22no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7C2×C4⋊C411yescharacteristic
118C23×C7C22⋊Q812yes—
119C22×D7—21no—
120C22×D7—22no—
121C22×D7—41no—
122C22×D7—41no—
123C2×C4×C7—22no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—21no—
127C2×C4×C7—41no—
128C2×C4×C7—42no—
129C2×C4×C7C2×C22⋊C411yescharacteristic
130C2×C4×C7#11711yescharacteristic
131C2×C4×C7#12112yes—
132C22⋊D7—44no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—22no—
137C2×Dic7—41no—
138C2×Dic7—41no—
139C4×D7—42no—
140C4×D7—41no—
141C4×D7—41no—
142C7×C8—41no—
143C7×C8—41no—
144C7×C8—82no—
145C7⋊C8—21no—
146C7⋊C8—22no—
147C7⋊C8—82no—
148C7⋊C8#9912yes—
149C22×D4—71no—
150C23×C4—71no—
151C23×C4C22×D711yescharacteristic
152C2×M16—22no—
153C2×M16—141no—
154C2×M16—141no—
155C2×M16—142no—
156C2×M16—282no—
157C22×C8—71no—
158C22×C8—72no—
159C22×C8—71no—
160C22×C8—72no—
161C22×C8—282no—
162C22×C8C4×D711yescharacteristic
163C22×C8C4×D711yescharacteristic
164C4×D4—142no—
165C4×D4—142no—
166C2×C4⋊C4—71no—
167C2×C22⋊C4—71no—
168C2×C22⋊C4—71no—
169C2×C42—71no—
170#108—284no—
171C4⋊C8—142no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—142no—
176C22⋊C8—142no—
177C22⋊C8—142no—
178#97—141no—
179#97—141no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—22no—
183C22×C7—22no—
184C22×C7—22no—
185C22×C7—44no—
186C22×C7#38011yescharacteristic
187C22×C7#43012yes—
188C2×D7—41no—
189C2×D7—41no—
190C2×D7—41no—
191C2×D7—82no—
192C4×C7—22no—
193C4×C7—21no—
194C4×C7C2×#9911yescharacteristic
195C4×C7#41011yescharacteristic
196Dic7—41no—
197Dic7—41no—
198Dic7—41no—
199C24—71no—
200C24Q8×D711yescharacteristic
201C2×D4—142no—
202C2×D4—282no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—71no—
206C22×C4—71no—
207C22×C4—71no—
208C22×C4—141no—
209C22×C4—141no—
210C22×C4—281no—
211C22×C4—281no—
212C22×C4C2×C4×D711yescharacteristic
213C22×C4#115712yes—
214C22×C4D4×D712yes—
215C22×C4D4×D711yescharacteristic
216M16—42no—
217M16—42no—
218M16—282no—
219M16—282no—
220M16—282no—
221M16—282no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—71no—
228C2×C8—71no—
229C2×C8—72no—
230C2×C8—144no—
231C2×C8—142no—
232C2×C8—281no—
233C2×C8—282no—
234C2×C8—282no—
235C4⋊C4—142no—
236C22⋊C4—142no—
237C22⋊C4—142no—
238C42—141no—
239C42—141no—
240C2×C7—22no—
241C2×C7—42no—
242C2×C7#179911yescharacteristic
243C2×C7#181111yescharacteristic
244C2×C7#181311yescharacteristic
245D7—81no—
246C23—22no—
247C23—41no—
248C23—41no—
249C23—141no—
250C23—142no—
251C23—281no—
252C23—281no—
253C23D7×C4⋊C411yescharacteristic
254C23#645912yes—
255D4—284no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—142no—
263C2×C4—141no—
264C2×C4—141no—
265C2×C4—141no—
266C2×C4—282no—
267C2×C4—281no—
268C2×C4—281no—
269C2×C4—281no—
270C2×C4—281no—
271C2×C4#643511yescharacteristic
272C2×C4D7×C22⋊C411yescharacteristic
273C2×C4#644712yes—
274C8—41no—
275C8—41no—
276C8—72no—
277C8—82no—
278C8—142no—
279C8—141no—
280C8—562no—
281C7#861311yescharacteristic, complemented, Sylow
282C22—21no—
283C22—21no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—44no—
288C22—281no—
289C22—281no—
290C22—281no—
291C22—562no—
292C22#8943911yescharacteristic
293C22#8966512yes—
294C4—22no—
295C4—21no—
296C4—281no—
297C4—281no—
298C4—281no—
299C4#8951211yescharacteristic
300C4D7×#9911yescharacteristic
301C2—22no—
302C2—42no—
303C2—561no—
304C2#1172647811yescharacteristic
305C2#1172669411yescharacteristic
306C2#1172671311yescharacteristic
subgroupG/H
centerC22#89439
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1293C2
socleC22×C7#380
prime radicalC22×C22⋊D7Q8
cosocleC23×C4×C7C23
hypercenter#1293D7
hypocenterC7#8613
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #11727736C2semidirect
4. C7×#1293C2semidirect
5. #11725485C2semidirect
57. #1293D7semidirect
281. C7#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#895022
#117277362
#117277222
C7×#12932
#117254852
#86137