Finite Group Databasebeta

← #13251493

Subgroups — #13251493

Order 1792.

554 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 103 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#11739824C211yesmaximal, characteristic
2#11739747C211yesmaximal, characteristic
3C7×#1466C211yesmaximal, characteristic
4#11728576C212yesmaximal, complemented
5#11725555C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487—21no—
8C2×#6487—21no—
9C2×#6432C2211yescharacteristic
10C7×#432C2212yes—
11#89993C412yes—
12#89979—21no—
13C2×C22⋊(C7⋊C8)—21no—
14#89783—24no—
15#89748C412yes—
16#89721—21no—
17#89721—21no—
18C4×C4⋊Dic7C2211yescharacteristic
19#89698C2212yes—
20C7×#323—21no—
21C7×#323—21no—
22#89281—24no—
23#9455—71nomaximal, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×Dic14—21no—
28C4×C7×D4—22no—
29C4×C7×D4—22no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C22⋊C4D411yescharacteristic
32C2×C7×C22⋊C4D411yescharacteristic
33C2×C42×C7C2311yescharacteristic
34C2×C22⋊Dic7—22no—
35C2×C22⋊Dic7—22no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C412yes—
39#6487—44no—
40#6487—44no—
41C2×Dic7⋊C4—22no—
42C2×Dic7⋊C4—22no—
43C2×C4×Dic7C2×C412yes—
44C2×#1136—22no—
45C22×C7⋊C8—22no—
46#6432—22no—
47#6432—22no—
48C7×C4⋊C8—22no—
49C7×C22⋊C8—22no—
50C7×#97—22no—
51C7⋊#95—21no—
52C22⋊(C7⋊C8)—22no—
53C22⋊(C7⋊C8)—22no—
54C22⋊(C7⋊C8)—22no—
55C4⋊(C7⋊C8)D414yes—
56#6377—22no—
57#2973—71no—
58#2542—71no—
59#1556—72no—
60#1466D711yescharacteristic
61#1285—72no—
62C24×C7C22⋊C411yescharacteristic
63C24×C7C22⋊C411yescharacteristic
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—42no—
72C22×C4×C7—42no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×Dic7—22no—
79C22×Dic7—22no—
80C2×Dic14—42no—
81C2×Dic14—42no—
82C2×Dic14—41no—
83C2×Dic14—41no—
84C2×C7×C8—42no—
85C7×C4⋊C4—22no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4—22no—
88C42×C7C2×D412yes—
89C42×C7#4112yes—
90C22⋊Dic7—44no—
91C22⋊Dic7—44no—
92C4⋊Dic7—24no—
93C4⋊Dic7—24no—
94C4⋊Dic7—42no—
95Dic7⋊C4—44no—
96Dic7⋊C4—44no—
97C4×Dic7—42no—
98#1136—82no—
99C2×C7⋊C8—24no—
100C2×C7⋊C8—82no—
101C2×C4⋊Q8—71no—
102C2×C22⋊Q8—141no—
103C2×C22⋊Q8—141no—
104C2×C4×D4C2×D711yescharacteristic
105#435—144no—
106#432—72no—
107#432C2×D712yes—
108#407—141no—
109C2×C22⋊C8—141no—
110#399—141no—
111#399—141no—
112#390—72no—
113#384—72no—
114C4×C4⋊C4—71no—
115#332—144no—
116#323—21no—
117#323—21no—
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×D4—44no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—22no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—41no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—81no—
141C2×C4×C7—81no—
142C2×C4×C7C4×D412yes—
143C2×C4×C7#12011yescharacteristic
144C2×C4×C7#12112yes—
145C2×C4×C7#12311yescharacteristic
146C2×Dic7—24no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152Dic14—82no—
153Dic14—82no—
154C7×C8—82no—
155C7⋊C8—44no—
156C22×Q8—141no—
157C22×D4C4×D711yescharacteristic
158C23×C4D2811yescharacteristic
159C23×C4D2811yescharacteristic
160C2×M16—142no—
161C22×C8—142no—
162C4⋊Q8—142no—
163C4⋊Q8—142no—
164C22⋊Q8—284no—
165C22⋊Q8—284no—
166C4×D4—22no—
167C4×D4—22no—
168C2×C4⋊C4—72no—
169C2×C4⋊C4—141no—
170C2×C4⋊C4—142no—
171C2×C4⋊C4—142no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4C4×D711yescharacteristic
174C2×C22⋊C4—142no—
175C2×C22⋊C4—142no—
176C2×C22⋊C4C22⋊D711yescharacteristic
177C2×C22⋊C4C22⋊D711yescharacteristic
178C2×C42—72no—
179C2×C42C22×D711yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—74no—
182C22⋊C8—22no—
183C22⋊C8—142no—
184C22⋊C8—142no—
185C22⋊C8—142no—
186#97—22no—
187#97—142no—
188#95—141no—
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—41no—
208C4×C7—41no—
209C4×C7#45312yes—
210Dic7—44no—
211Dic7—82no—
212C24(C2×D7)⋊C411yescharacteristic
213C24(C2×D7)⋊C411yescharacteristic
214C2×Q8—282no—
215C2×Q8—282no—
216C2×Q8—281no—
217C2×Q8—281no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—142no—
228C22×C4—142no—
229C22×C4(C2×D7)⋊C411yescharacteristic
230C22×C4(C2×D7)⋊C411yescharacteristic
231C22×C4C2×C4×D711yescharacteristic
232C22×C4C2×D2811yescharacteristic
233C22×C4C2×C22⋊D711yescharacteristic
234M16—562no—
235C2×C8—42no—
236C2×C8—144no—
237C2×C8—562no—
238C4⋊C4—22no—
239C4⋊C4—144no—
240C4⋊C4—144no—
241C4⋊C4—284no—
242C4⋊C4—284no—
243C4⋊C4—282no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—284no—
247C22⋊C4—284no—
248C42—282no—
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#639812yes—
268C23C2×(C2×D7)⋊C411yescharacteristic
269Q8—562no—
270Q8—562no—
271D4—44no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—144no—
285C2×C4—282no—
286C2×C4—282no—
287C2×C4—282no—
288C2×C4—282no—
289C2×C4—282no—
290C2×C4(C22⋊D7)⋊C412yes—
291C2×C4#644511yescharacteristic
292C2×C4#644911yescharacteristic
293C2×C4#650212yes—
294C8—82no—
295C8—284no—
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#8973011yescharacteristic
310C22#8990411yescharacteristic
311C22C2×#639811yescharacteristic
312C4—22no—
313C4—22no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—284no—
318C4—562no—
319C4#8951912yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—81no—
324C2#1172774011yescharacteristic
325C2#1172780011yescharacteristic
326C2#1172782411yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1466C2
socleC22×C7#386
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#1466D7
hypocenterC7#9455
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
4. #11728576C2semidirect
5. #11725555C2semidirect
296. C7#9455semidirect

Maximal subgroups

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

subgroupindex
#117398242
#117397472
C7×#14662
#117285762
#117255552
#94557