Finite Group Databasebeta

← #13263259

Subgroups — #13263259

Order 1792.

442 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2322C211yesmaximal, characteristic
2#11729357C211yesmaximal, characteristic
3#11729356C211yesmaximal, characteristic
4#11725442C212yesmaximal, complemented
5#11725433C212yesmaximal, complemented
6C2×C7×#123C411yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C22×C7×C4⋊C4C411yescharacteristic
9C7×#389—21no—
10C7×#388—21no—
11C7×#387—21no—
12C7×#386C411yescharacteristic
13C7×#385—21no—
14C7×#385C2211yescharacteristic
15C7×#381C2211yescharacteristic
16C4×C7×C22⋊C4C411yescharacteristic
17#89979—21no—
18#89698—21no—
19#89323—21no—
20#89320—21no—
21#89320C2212yes—
22#89320C2212yes—
23#11528—71nomaximal, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C7×#123—22no—
30C7×#123—22no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—41no—
38C2×C7×C4⋊C4—41no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C7×C22⋊C4C2311yescharacteristic
46C2×C42×C7—21no—
47C2×C42×C7D411yescharacteristic
48C2×#1136—41no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—41no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2322D711yescharacteristic
60#1466—71no—
61#1465—71no—
62#1326—72no—
63#1317—72no—
64C24×C7C22⋊C411yescharacteristic
65C24×C7C22×C411yescharacteristic
66C2×C7×D4—21no—
67C2×C7×D4—21no—
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7—81no—
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7M1612yescomplemented
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C7×C4⋊C4—24no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—81no—
97C7×C22⋊C4—24no—
98C7×C22⋊C4—21no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—81no—
103C7×C22⋊C4M1612yescomplemented
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—81no—
107#1136—82no—
108C2×C7⋊C8—42no—
109C2×C7⋊C8—42no—
110C2×#123Dic711yescharacteristic
111C2×C4×D4C2×D711yescharacteristic
112C22×C4⋊C4Dic711yescharacteristic
113#432—141no—
114#407—141no—
115#389—21no—
116#388—21no—
117#387—21no—
118#386Dic711yescharacteristic
119#385—21no—
120#385C2×D711yescharacteristic
121#381C2×D711yescharacteristic
122C4×C22⋊C4Dic711yescharacteristic
123#324—141no—
124#323—72no—
125#323—72no—
126#323—141no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7C2×C22⋊C411yescharacteristic
137C23×C7C2×M1611yescharacteristic
138C23×C7C2×M1611yescharacteristic
139C7×D4—42no—
140C7×D4—42no—
141C2×C4×C7—22no—
142C2×C4×C7—22no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—42no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—81no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7—82no—
170C2×C4×C7#9911yescharacteristic
171C2×C4×C7#9911yescharacteristic
172C7⋊C8—82no—
173C7⋊C8—82no—
174C22×D4C2×Dic711yescharacteristic
175C23×C4C2×Dic711yescharacteristic
176C23×C4C2×Dic711yescharacteristic
177C23×C4C2×Dic711yescharacteristic
178C23×C4C22⋊D711yescharacteristic
179C2×M16—281no—
180#123—22no—
181#123—22no—
182C4×D4—41no—
183C4×D4—41no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C22⋊D711yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C2×Dic711yescharacteristic
194C2×C22⋊C4C2×Dic711yescharacteristic
195C2×C22⋊C4C22⋊D711yescharacteristic
196C2×C22⋊C4C22×D711yescharacteristic
197C2×C42—21no—
198C2×C42C22⋊D711yescharacteristic
199C4⋊C8—281no—
200C22⋊C8—142no—
201C22⋊C8—142no—
202C22⋊C8—281no—
203#97—281no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—42no—
216C22×C7—42no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—81no—
223C22×C7#40711yescharacteristic
224C22×C7C2×#9911yescharacteristic
225C22×C7#41311yescharacteristic
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—42no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7—42no—
234C4×C7—81no—
235C4×C7—81no—
236C24C22⋊Dic711yescharacteristic
237C24C22×Dic711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4#113612yescomplemented
258C22×C4C22⋊Dic711yescharacteristic
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4C22⋊Dic711yescharacteristic
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263M16—562no—
264C2×C8—282no—
265C2×C8—282no—
266C4⋊C4—24no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C22⋊C4—24no—
273C22⋊C4—21no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4#113612yescomplemented
279C42—21no—
280C42—21no—
281C42—81no—
282C2×C7—21no—
283C2×C7—21no—
284C2×C7—41no—
285C2×C7—41no—
286C2×C7—81no—
287C2×C7#146511yescharacteristic
288C2×C7#212111yescharacteristic
289C2×C7#212311yescharacteristic
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23C2×#113611yescharacteristic
300C23C2×#113611yescharacteristic
301C23C2×C22⋊Dic711yescharacteristic
302D4—42no—
303D4—42no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4#640811yescharacteristic
334C2×C4#640811yescharacteristic
335C8—562no—
336C8—562no—
337C7#1152811yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—81no—
352C22#8997911yescharacteristic
353C22C2×#640811yescharacteristic
354C22#9000911yescharacteristic
355C4—21no—
356C4—21no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—42no—
362C4—41no—
363C4—81no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2#1172935611yescharacteristic
371C2#1172960111yescharacteristic
372C2#1172962411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2322C2
socleC22×C7C2×#99
prime radicalC22×C7×D4C2×C4
cosocleC2×C7×C22⋊C4C23
hypercenter#2322D7
hypocenterC7#11528
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2322

Complements

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

NH ≅ G/Ntype
4. #11725442C2semidirect
5. #11725433C2semidirect
88. C22×C4×C7M16semidirect
103. C7×C22⋊C4M16semidirect
257. C22×C4#1136semidirect
278. C22⋊C4#1136semidirect
337. C7#11528semidirect

Maximal subgroups

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

subgroupindex
C7×#23222
#117293572
#117293562
#117254422
#117254332
#115287