Finite Group Databasebeta

← #13264191

Subgroups — #13264191

Order 1792.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#11739837C211yesmaximal, characteristic
2C7×#1459C211yesmaximal, characteristic
3C4×#6407C211yesmaximal, characteristic
4#11725264C212yesmaximal, complemented
5#11725263C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487C411yescharacteristic
8C2×C7×C22⋊C8—21no—
9#89994—21no—
10C2×#6407—21no—
11#89983C411yescharacteristic
12#89982—21no—
13#89755C2211yescharacteristic
14#89751C2211yescharacteristic
15#89725C411yescharacteristic
16#89724—21no—
17#89719C411yescharacteristic
18#89705—21no—
19C4×#1136—21no—
20C7×#324C2212yes—
21C7×#323—21no—
22#89325C2212yes—
23#11739—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×Dic14C2×C411yescharacteristic
28C22×C7×C8—41no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4C2311yescharacteristic
32C2×C7×C22⋊C4C2×C411yescharacteristic
33C2×C7×C22⋊C4D411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7—21no—
38C2×C22⋊Dic7—21no—
39C2×C4⋊Dic7C2×C411yescharacteristic
40#6487—41no—
41#6487—41no—
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×#1136—22no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—41no—
48#6407—24no—
49C7⋊#95—21no—
50C7⋊#95—21no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95C2×C411yescharacteristic
55C7⋊#95C2×C411yescharacteristic
56#6377—41no—
57C4×C7⋊C8—41no—
58#2638—71no—
59C4×#100—71no—
60#1459D711yescharacteristic
61#1354—72no—
62#1346—72no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C22⋊C411yescharacteristic
65C2×C7×D4—22no—
66C2×C7×D4—22no—
67C2×C7×D4—41no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—81no—
73C22×C4×C7—81no—
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×Dic7—21no—
80C22×Dic7—21no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C2×Dic14—41no—
84C2×Dic14—81no—
85C2×C7×C8—42no—
86C2×C7×C8—82no—
87C7×C4⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C42×C7—21no—
91C42×C7—21no—
92C22⋊Dic7—42no—
93C22⋊Dic7—42no—
94C22⋊Dic7—81no—
95C22⋊Dic7—81no—
96C4⋊Dic7—41no—
97Dic7⋊C4—41no—
98Dic7⋊C4—41no—
99Dic7⋊C4—42no—
100Dic7⋊C4—81no—
101#1136—24no—
102#1136—42no—
103C2×C7⋊C8—42no—
104C2×C22⋊Q8—71no—
105C2×C4×D4C2×D711yescharacteristic
106C2×#100—141no—
107C2×C22⋊C8—21no—
108C4×M16—141no—
109#398—71no—
110#397—71no—
111#397—141no—
112#395—71no—
113#394—141no—
114#393—71no—
115#387—141no—
116#385—141no—
117#381—71no—
118#343—72no—
119#324C2×D712yes—
120#323—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7#10412yes—
131C23×C7C2×C22⋊C411yescharacteristic
132C7×D4—44no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—81no—
144C2×C4×C7—81no—
145C2×C4×C7—81no—
146C2×C4×C7#9911yescharacteristic
147C2×C4×C7#9911yescharacteristic
148C2×C4×C7#13512yes—
149C2×Dic7—41no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—41no—
153C2×Dic7—41no—
154C2×Dic7—41no—
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161Dic14—82no—
162C7×C8—82no—
163C7⋊C8—44no—
164C22×Q8—71no—
165C22×D4D2811yescharacteristic
166C23×C4C4×D711yescharacteristic
167C23×C4C22⋊D711yescharacteristic
168C2×M16—142no—
169C22×C8—41no—
170C22⋊Q8—281no—
171C22⋊Q8—281no—
172C4×D4—41no—
173C4×D4—41no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—141no—
176C2×C4⋊C4—141no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4C22×D711yescharacteristic
179C2×C22⋊C4—141no—
180C2×C22⋊C4—141no—
181C2×C22⋊C4—141no—
182C2×C22⋊C4—141no—
183C2×C22⋊C4C4×D711yescharacteristic
184C2×C22⋊C4C22⋊D711yescharacteristic
185C2×C42D2811yescharacteristic
186#100—144no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189#97—281no—
190C4×C8—281no—
191#95—71no—
192#95—71no—
193#95—141no—
194#95—141no—
195#95—141no—
196#95—141no—
197#95—141no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—22no—
201C22×C7—42no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—81no—
208C22×C7—81no—
209C22×C7—81no—
210C22×C7C2×#9911yescharacteristic
211C22×C7#41611yescharacteristic
212C22×C7C2×#10411yescharacteristic
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—22no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219Dic7—81no—
220Dic7—81no—
221Dic7—81no—
222Dic7—81no—
223C24(C2×D7)⋊C411yescharacteristic
224C24(C2×D7)⋊C411yescharacteristic
225C2×Q8—281no—
226C2×Q8—561no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—141no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4(C2×D7)⋊C411yescharacteristic
241C22×C4(C2×D7)⋊C411yescharacteristic
242C22×C4C2×C4×D711yescharacteristic
243C22×C4C2×D2811yescharacteristic
244C22×C4C2×C22⋊D711yescharacteristic
245M16—144no—
246M16—282no—
247C2×C8—42no—
248C2×C8—82no—
249C2×C8—282no—
250C4⋊C4—41no—
251C4⋊C4—281no—
252C4⋊C4—282no—
253C4⋊C4—281no—
254C4⋊C4—281no—
255C4⋊C4—561no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—282no—
259C22⋊C4—282no—
260C22⋊C4—561no—
261C22⋊C4—561no—
262C42—21no—
263C42—21no—
264C2×C7—21no—
265C2×C7—21no—
266C2×C7—42no—
267C2×C7—81no—
268C2×C7#150111yescharacteristic
269C2×C7#212111yescharacteristic
270C2×C7#213511yescharacteristic
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#639912yes—
281C23C2×(C2×D7)⋊C411yescharacteristic
282Q8—562no—
283D4—44no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—281no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—281no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4—281no—
307C2×C4—281no—
308C2×C4—281no—
309C2×C4#638011yescharacteristic
310C2×C4#638011yescharacteristic
311C2×C4#646712yes—
312C8—82no—
313C8—284no—
314C7#1173911yescharacteristic, complemented, Sylow
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×#638011yescharacteristic
328C22#8988511yescharacteristic
329C22C2×#639911yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—561no—
337C4—561no—
338C4—561no—
339C4—561no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#1172769611yescharacteristic
345C2#1172776911yescharacteristic
346C2#1172790411yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1459C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C4⋊C4C23
hypercenter#1459D7
hypocenterC7#11739
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
4. #11725264C2semidirect
5. #11725263C2semidirect
314. C7#11739semidirect

Maximal subgroups

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

subgroupindex
#117398372
C7×#14592
C4×#64072
#117252642
#117252632
#117397