Finite Group Databasebeta

← #13251496

Subgroups — #13251496

Order 1792.

554 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2542C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3#11729530C212yesmaximal, complemented
4#11729357C211yesmaximal, characteristic
5#11725427C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×C22⋊Q8—21no—
8C2×C7×C22⋊Q8—21no—
9C2×C4×C7×D4C2211yescharacteristic
10C7×#399—21no—
11C7×#399—21no—
12C7×#390C412yes—
13C7×#384C412yes—
14C4×C7×C4⋊C4C2211yescharacteristic
15#89979—21no—
16C2×C22⋊(C7⋊C8)—21no—
17#89783—24no—
18#89698C2212yes—
19#89698C2212yes—
20#89342—24no—
21#89320—21no—
22#89320—21no—
23#9455—71nomaximal, Sylow
24C22×C7×Q8—21no—
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C7×C4⋊Q8—22no—
29C7×C4⋊Q8—22no—
30C7×C22⋊Q8—44no—
31C7×C22⋊Q8—44no—
32C4×C7×D4—22no—
33C4×C7×D4—22no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C412yes—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7C2×C412yes—
45C2×C42×C7C2311yescharacteristic
46C2×#1136—22no—
47C22×C7⋊C8—22no—
48C7×#95—21no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)—22no—
54C4⋊(C7⋊C8)D414yes—
55#6377—22no—
56#6377—22no—
57#2973—71no—
58#2542D711yescharacteristic
59#1556—72no—
60#1466—71no—
61#1285—72no—
62C24×C7C22⋊C411yescharacteristic
63C24×C7C22⋊C411yescharacteristic
64C2×C7×Q8—42no—
65C2×C7×Q8—42no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—41no—
71C22×C4×C7—22no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C7×C4⋊C4—24no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—24no—
87C7×C4⋊C4—44no—
88C7×C4⋊C4—44no—
89C7×C4⋊C4—42no—
90C7×C22⋊C4—22no—
91C7×C22⋊C4—22no—
92C7×C22⋊C4—44no—
93C7×C22⋊C4—44no—
94C42×C7—42no—
95C42×C7C2×D412yes—
96C42×C7#4112yes—
97#1136—82no—
98C2×C7⋊C8—24no—
99C2×C7⋊C8—42no—
100C2×C7⋊C8—82no—
101C2×C4⋊Q8C2×D711yescharacteristic
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×D4C2×D711yescharacteristic
105#435—144no—
106#432—72no—
107#432—72no—
108#407—141no—
109C2×C22⋊C8—141no—
110#399—21no—
111#399—21no—
112#390Dic712yes—
113#384Dic712yes—
114C4×C4⋊C4C2×D711yescharacteristic
115#332—144no—
116#323—141no—
117#323—141no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7#10112yes—
128C23×C7C2×C22⋊C411yescharacteristic
129C7×Q8—82no—
130C7×Q8—82no—
131C7×D4—44no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—24no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—81no—
149C2×C4×C7—81no—
150C2×C4×C7C4×D412yes—
151C2×C4×C7#12011yescharacteristic
152C2×C4×C7#12112yes—
153C2×C4×C7#12311yescharacteristic
154C7⋊C8—44no—
155C7⋊C8—82no—
156C22×Q8—21no—
157C22×D4C2×Dic711yescharacteristic
158C23×C4C22⋊D711yescharacteristic
159C23×C4C22⋊D711yescharacteristic
160C2×M16—142no—
161C22×C8—142no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C22⋊Q8—44no—
165C22⋊Q8—44no—
166C4×D4—22no—
167C4×D4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4C2×Dic712yes—
173C2×C4⋊C4C2×Dic711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C22⋊D711yescharacteristic
177C2×C22⋊C4C22⋊D711yescharacteristic
178C2×C42C2×Dic712yes—
179C2×C42C22×D711yescharacteristic
180C4⋊C8—74no—
181C4⋊C8—142no—
182C22⋊C8—142no—
183C22⋊C8—142no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186#97—142no—
187#97—142no—
188#95—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—22no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—41no—
195C22×C7—41no—
196C22×C7—41no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—81no—
200C22×C7—81no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10111yescharacteristic
203C22×C7#41311yescharacteristic
204C4×C7—21no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—44no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—82no—
211C4×C7#45312yes—
212C24C22⋊Dic711yescharacteristic
213C24C22⋊Dic711yescharacteristic
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C22⋊Dic711yescharacteristic
230C22×C4C22⋊Dic711yescharacteristic
231C22×C4C22×Dic711yescharacteristic
232C22×C4C2×C22⋊D711yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234M16—562no—
235C2×C8—144no—
236C2×C8—282no—
237C2×C8—562no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C42—42no—
249C42D4×D712yes—
250C42#115912yes—
251C2×C7—21no—
252C2×C7—21no—
253C2×C7—42no—
254C2×C7—81no—
255C2×C7#188711yescharacteristic
256C2×C7#190111yescharacteristic
257C2×C7#190911yescharacteristic
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#641012yes—
268C23C2×C22⋊Dic711yescharacteristic
269Q8—82no—
270Q8—82no—
271D4—44no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4D4×Dic712yes—
291C2×C4#649811yescharacteristic
292C2×C4#650011yescharacteristic
293C2×C4#650212yes—
294C8—284no—
295C8—562no—
296C7#945511yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22—81no—
309C22#8999211yescharacteristic
310C22C2×#641011yescharacteristic
311C22#9000911yescharacteristic
312C4—22no—
313C4—21no—
314C4—22no—
315C4—44no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4#8984912yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#1172945211yescharacteristic
325C2#1172951711yescharacteristic
326C2#1172961911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2542C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#2542D7
hypocenterC7#9455
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #11729530C2semidirect
5. #11725427C2semidirect
296. C7#9455semidirect

Maximal subgroups

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

subgroupindex
C7×#25422
#117397472
#117295302
#117293572
#117254272
#94557