Finite Group Databasebeta

← #13251606

Subgroups — #13251606

Order 1792.

734 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2482C211yesmaximal, characteristic
2#11739528C211yesmaximal, characteristic
3#11729344C212yesmaximal, complemented
4#11729293C211yesmaximal, characteristic
5#11725434C212yesmaximal, complemented
6C2×C7×#117C2211yescharacteristic
7C22×C7×C22⋊C4C2211yescharacteristic
8C22×C22⋊Dic7—21no—
9C7×#399C412yescomplemented
10C7×#397C412yescomplemented
11C7×#388—22no—
12C7×#385—21no—
13C7×#381C2211yescharacteristic
14C7×#379—21no—
15C4×C22⋊Dic7—21no—
16C4×C22⋊Dic7C2212yes—
17#89979C2212yes—
18C22⋊C4×Dic7—24no—
19#89327—24no—
20#89320—22no—
21#9477—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C23×Dic7—22no—
26C7×#117—22no—
27C7×#117—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
30C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
31C2×C7×C22⋊C4—22no—
32C2×C7×C22⋊C4—22no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4—21no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C412yes—
38C2×C7×C22⋊C4D412yes—
39C2×C42×C7—21no—
40C2×C22⋊Dic7—22no—
41C2×C22⋊Dic7—28no—
42C2×C22⋊Dic7—22no—
43C2×C4×Dic7—22no—
44C2×C4×Dic7D414yes—
45C2×#1136—22no—
46C7×#95—22no—
47C7×#95—21no—
48C7×#95C2×C412yescomplemented
49C7⋊#95—22no—
50C7⋊#95—22no—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53#2482D711yescharacteristic
54#2273—71no—
55#1826—72no—
56#1462—71no—
57#1284—72no—
58C24×C7—21no—
59C24×C7—24no—
60C24×C7—21no—
61C24×C7—42no—
62C24×C7—81no—
63C24×C7C22⋊C412yes—
64C24×C7C2×D411yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—42no—
71C22×C4×C7—81no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7#4112yes—
77C22×Dic7—24no—
78C22×Dic7—28no—
79C22×Dic7—82no—
80C22×Dic7—82no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—82no—
84C7×C22⋊C4—24no—
85C7×C22⋊C4—24no—
86C7×C22⋊C4—42no—
87C7×C22⋊C4—44no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—82no—
91C42×C7—42no—
92C22⋊Dic7—28no—
93C22⋊Dic7—28no—
94C4×Dic7—44no—
95C4×Dic7—82no—
96#1136—44no—
97C2×C7⋊C8—42no—
98C2×#117C2×D711yescharacteristic
99C22×C22⋊C4—141no—
100C22×C22⋊C4C2×D711yescharacteristic
101#407—72no—
102#399Dic712yescomplemented
103#397Dic712yescomplemented
104#388—22no—
105#385—21no—
106#381C2×D711yescharacteristic
107#379—21no—
108C4×C22⋊C4—72no—
109C4×C22⋊C4—144no—
110C4×C22⋊C4—141no—
111#339—144no—
112#323—142no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7—24no—
116C23×C7—22no—
117C23×C7—22no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7—41no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—81no—
126C23×C7—81no—
127C23×C7—88no—
128C23×C7#10414yes—
129C23×C7C2×C22⋊C411yescharacteristic
130C23×C7#12011yescharacteristic
131C23×C7#12311yescharacteristic
132C2×C4×C7—24no—
133C2×C4×C7—22no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—81no—
147C2×C4×C7C4×D412yes—
148C2×C4×C7#12112yes—
149C2×Dic7—28no—
150C2×Dic7—44no—
151C2×Dic7—44no—
152C2×Dic7—88no—
153C7⋊C8—82no—
154C25C22⋊D711yescharacteristic
155C23×C4—142no—
156C23×C4C22⋊D711yescharacteristic
157C23×C4C22×D711yescharacteristic
158C2×M16—142no—
159#117—22no—
160#117—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×Dic711yescharacteristic, complemented
163C2×C4⋊C4C2×Dic711yescharacteristic, complemented
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—148no—
171C2×C22⋊C4—142no—
172C2×C22⋊C4—142no—
173C2×C22⋊C4C2×Dic712yes—
174C2×C22⋊C4C22⋊D712yes—
175C2×C42—21no—
176C2×C42—74no—
177C2×C42—142no—
178C22⋊C8—142no—
179C22⋊C8—142no—
180#95—22no—
181#95—21no—
182#95—142no—
183#95—142no—
184#95C2×Dic712yescomplemented
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—24no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—44no—
194C22×C7—44no—
195C22×C7—44no—
196C22×C7—44no—
197C22×C7—42no—
198C22×C7—42no—
199C22×C7—82no—
200C22×C7—84no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10412yes—
203C22×C7#45812yes—
204C4×C7—22no—
205C4×C7—22no—
206C4×C7—42no—
207C4×C7—82no—
208C4×C7—81no—
209C4×C7—81no—
210Dic7—48no—
211C24—21no—
212C24—24no—
213C24—21no—
214C24—42no—
215C24—81no—
216C24C22⋊Dic712yes—
217C24C2×C22⋊D711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—81no—
225C22×C4—144no—
226C22×C4—148no—
227C22×C4—562no—
228C22×C4—562no—
229C22×C4C22⋊Dic712yes—
230C22×C4D4×D712yes—
231C22×C4#115912yes—
232C22×C4C22×Dic711yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234M16—284no—
235C2×C8—282no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—82no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—44no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—82no—
246C22⋊C4—148no—
247C22⋊C4—148no—
248C42—42no—
249C42—284no—
250C42—562no—
251C2×C7—22no—
252C2×C7—22no—
253C2×C7—41no—
254C2×C7—44no—
255C2×C7#189011yescharacteristic
256C2×C7#189611yescharacteristic
257C2×C7#191111yescharacteristic
258C23—22no—
259C23—22no—
260C23—24no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—41no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—88no—
273C23#641114yes—
274C23#649811yescharacteristic
275C23#650011yescharacteristic
276C23C2×C22⋊Dic711yescharacteristic
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—148no—
293C2×C4—284no—
294C2×C4—284no—
295C2×C4—568no—
296C2×C4D4×Dic712yes—
297C2×C4#650212yes—
298C8—562no—
299C7#947711yescharacteristic, complemented, Sylow
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#8982512yes—
317C22#8999211yescharacteristic
318C22C2×#641112yes—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—288no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#1172945811yescharacteristic
331C2#1172946511yescharacteristic
332C2#1172960711yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2482C2
socleC22×C7#386
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2482D7
hypocenterC7#9477
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
3. #11729344C2semidirect
5. #11725434C2semidirect
9. C7×#399C4semidirect
10. C7×#397C4semidirect
29. C2×C7×C4⋊C4C2×C4semidirect
30. C2×C7×C4⋊C4C2×C4semidirect
48. C7×#95C2×C4semidirect
102. #399Dic7semidirect
103. #397Dic7semidirect
162. C2×C4⋊C4C2×Dic7semidirect
163. C2×C4⋊C4C2×Dic7semidirect
184. #95C2×Dic7semidirect
299. C7#9477semidirect

Maximal subgroups

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

subgroupindex
C7×#24822
#117395282
#117293442
#117292932
#117254342
#94777