Finite Group Databasebeta

← #13263183

Subgroups — #13263183

Order 1792.

574 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 77 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#11733927C211yesmaximal, characteristic
2#11725898C212yesmaximal, complemented
3C7×#1319C211yesmaximal, characteristic, complemented
4#11725449C211yesmaximal, characteristic, complemented
5#11725260C212yesmaximal, complemented
6C22×#1161C411yescharacteristic
7C2×D7×C4⋊C4C2211yescharacteristic
8C2×#6435C411yescharacteristic
9C7×#391C2211yescharacteristic
10#90002C411yescharacteristic
11#89760C2211yescharacteristic
12#89711C411yescharacteristic
13#89498—22no—
14#89439—22no—
15#89437—22no—
16C7×#324C2212yes—
17#89323C2212yes—
18#89264—22no—
19#11508—71nomaximal, Sylow
20C22×C7×Q8C2×C411yescharacteristic
21C2×#1161—22no—
22C2×#1161—22no—
23C2×#1161—42no—
24C2×#1161—42no—
25C22×D28—21no—
26C22×D28C2×C411yescharacteristic
27C22×C4×D7—21no—
28C22×C4×D7C2×C411yescharacteristic
29C22×C4×D7D412yes—
30C2×C7×C4⋊C4C2311yescharacteristic
31C2×C42×C7C2×C411yescharacteristic
32C2×(C2×D7)⋊C4—21no—
33C2×(C2×D7)⋊C4—22no—
34C2×(C2×D7)⋊C4—22no—
35C2×C4⋊Dic7C2×C411yescharacteristic
36C2×Dic7⋊C4—21no—
37C2×Dic7⋊C4D412yes—
38C2×C4×Dic7C2×C411yescharacteristic
39C2×#1131—42no—
40D7×C4⋊C4—42no—
41#6435—42no—
42C7×C22⋊C8—22no—
43C7×#95—22no—
44C7⋊#95—22no—
45C22⋊(C7⋊C8)—22no—
46(C2×D7)⋊C8—42no—
47#2311—71no—
48#1463—72no—
49#1319—71no—
50#1319D711yescharacteristic, complemented
51#1318—72no—
52C23×D7—21no—
53C23×D7C22⋊C411yescharacteristic
54C2×C7×Q8—21no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8M1612yescomplemented
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7C22×C411yescharacteristic
61C22×C4×C7C2×D412yes—
62C22×Dic7—21no—
63C22×Dic7C22⋊C411yescharacteristic
64C22×Dic7C22⋊C412yes—
65#1161—48no—
66#1161—48no—
67C2×D28—24no—
68C2×D28—21no—
69C2×D28—41no—
70C2×D28—41no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28M1612yescomplemented
74C2×C4×D7—22no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7—22no—
78C2×C4×D7—24no—
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7—42no—
82C2×C4×D7—42no—
83C2×C4×D7—82no—
84C2×C7×C8—42no—
85C7×C4⋊C4—41no—
86C42×C7—41no—
87(C2×D7)⋊C4—42no—
88(C2×D7)⋊C4—82no—
89(C2×D7)⋊C4—82no—
90C4⋊Dic7—41no—
91Dic7⋊C4—42no—
92Dic7⋊C4—42no—
93C4×Dic7—41no—
94C2×C7⋊C8—42no—
95#1131—84no—
96C22×#41—71no—
97C2×#117—71no—
98C22×C4⋊C4—71no—
99#407—142no—
100#391—71no—
101#391C2×D711yescharacteristic
102#390—71no—
103#390—71no—
104#386—142no—
105#380—142no—
106#324—72no—
107#324—142no—
108#324C2×D712yes—
109C23×C7C2×C22⋊C411yescharacteristic
110C22×D7—21no—
111C22×D7—21no—
112C22×D7—22no—
113C22×D7—41no—
114C22×D7—41no—
115C22×D7—42no—
116C22×D7—42no—
117C22×D7—41no—
118C22×D7—41no—
119C22×D7#9912yes—
120C7×Q8—42no—
121C7×Q8—42no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7C2×M1611yescharacteristic
133C2×C4×C7C2×M1611yescharacteristic
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—22no—
137C2×Dic7—42no—
138C2×Dic7—41no—
139C2×Dic7—41no—
140C2×Dic7—41no—
141C2×Dic7#10012yes—
142D28—42no—
143D28—42no—
144D28—48no—
145C4×D7—24no—
146C4×D7—24no—
147C4×D7—48no—
148C4×D7—44no—
149C4×D7—82no—
150C7×C8—82no—
151C7⋊C8—82no—
152C2×#41—142no—
153C2×#41—142no—
154C2×#41—282no—
155C2×#41—282no—
156C22×Q8C4×D711yescharacteristic
157C22×D4—71no—
158C22×D4—141no—
159C23×C4—71no—
160C23×C4—72no—
161C23×C4—141no—
162C2×M16—282no—
163#117—282no—
164C2×C4⋊C4—71no—
165C2×C4⋊C4—72no—
166C2×C4⋊C4—141no—
167C2×C4⋊C4—282no—
168C2×C4⋊C4C22×D711yescharacteristic
169C2×C22⋊C4—141no—
170C2×C22⋊C4—142no—
171C2×C22⋊C4—142no—
172C2×C42—71no—
173C2×C42C4×D711yescharacteristic
174C22⋊C8—22no—
175C22⋊C8—142no—
176C22⋊C8—282no—
177#95—22no—
178#95—142no—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7#40711yescharacteristic
182C22×C7C2×#9911yescharacteristic
183C22×C7C2×#10011yescharacteristic
184C2×D7—22no—
185C2×D7—22no—
186C2×D7—22no—
187C2×D7—42no—
188C2×D7—42no—
189C2×D7—44no—
190C2×D7—42no—
191C2×D7—82no—
192C2×D7—81no—
193C2×D7—81no—
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—42no—
197C4×C7—42no—
198C4×C7—81no—
199Dic7—22no—
200Dic7—42no—
201Dic7—81no—
202C24—71no—
203C24—141no—
204#41—288no—
205#41—288no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#113112yescomplemented
210C2×D4—72no—
211C2×D4—144no—
212C2×D4—141no—
213C2×D4—281no—
214C2×D4—281no—
215C2×D4—282no—
216C2×D4—282no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—71no—
220C22×C4—72no—
221C22×C4—142no—
222C22×C4—142no—
223C22×C4—142no—
224C22×C4—144no—
225C22×C4—142no—
226C22×C4—141no—
227C22×C4—141no—
228C22×C4—141no—
229C22×C4—282no—
230C22×C4—282no—
231C22×C4—562no—
232C22×C4C2×C4×D711yescharacteristic
233C22×C4D4×D712yes—
234M16—564no—
235C2×C8—42no—
236C2×C8—282no—
237C4⋊C4—41no—
238C4⋊C4—282no—
239C4⋊C4—281no—
240C4⋊C4—282no—
241C22⋊C4—282no—
242C22⋊C4—562no—
243C22⋊C4—562no—
244C42—41no—
245C42—281no—
246C2×C7—21no—
247C2×C7—21no—
248C2×C7#146211yescharacteristic
249C2×C7#212711yescharacteristic
250C2×C7#213311yescharacteristic
251D7—42no—
252D7—82no—
253C23—72no—
254C23—141no—
255C23—141no—
256C23—142no—
257C23—281no—
258C23—281no—
259C23—282no—
260C23—282no—
261C23—281no—
262C23—281no—
263C23D7×C22⋊C411yescharacteristic
264Q8—42no—
265Q8—42no—
266D4—282no—
267D4—282no—
268D4—288no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—72no—
280C2×C4—144no—
281C2×C4—144no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—142no—
285C2×C4—288no—
286C2×C4—284no—
287C2×C4—282no—
288C2×C4—281no—
289C2×C4—281no—
290C2×C4—281no—
291C2×C4—562no—
292C2×C4C2×#113111yescharacteristic
293C2×C4D7×M1611yescharacteristic
294C8—82no—
295C8—562no—
296C7#1150811yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—142no—
300C22—142no—
301C22—142no—
302C22—282no—
303C22—282no—
304C22—284no—
305C22—282no—
306C22—562no—
307C22—561no—
308C22—561no—
309C22#8949811yescharacteristic
310C22D7×#9911yescharacteristic
311C22D7×#10011yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—142no—
318C4—282no—
319C4—561no—
320C2—21no—
321C2—21no—
322C2—282no—
323C2—562no—
324C2#1172587411yescharacteristic
325C2#1172691111yescharacteristic
326C2#1172692511yescharacteristic
subgroupG/H
centerC22D7×#99
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1319C2
socleC22×C7C2×#99
prime radicalC22×#1161C4
cosocleC2×C7×C4⋊C4C23
hypercenter#1319D7
hypocenterC7#11508
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
2. #11725898C2semidirect
3. C7×#1319C2semidirect
4. #11725449C2semidirect
5. #11725260C2semidirect
50. #1319D7semidirect
57. C2×C7×Q8M16semidirect
73. C2×D28M16semidirect
209. C2×Q8#1131semidirect
296. C7#11508semidirect

Maximal subgroups

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

subgroupindex
#117339272
#117258982
C7×#13192
#117254492
#117252602
#115087