Finite Group Databasebeta

← #13264592

Subgroups — #13264592

Order 1792.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2633C211yesmaximal, characteristic
2#11729373C211yesmaximal, characteristic
3#11729356C211yesmaximal, characteristic
4#11725457C212yesmaximal, complemented
5#11725455C212yesmaximal, complemented
6C2×C7×#125C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C7×#399C411yescharacteristic, complemented
9C7×#399C2211yescharacteristic
10C7×#398—21no—
11C7×#397C411yescharacteristic, complemented
12C7×#397C2211yescharacteristic
13C7×#394—21no—
14C7×#385—21no—
15C7×#385—21no—
16C7×#383C411yescharacteristic, complemented
17C2×#6408—21no—
18#89984—21no—
19#89979—21no—
20#89323—21no—
21#89321C2212yes—
22#89320C2212yes—
23#11827—71nomaximal, Sylow
24C22×C7×D4D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C7×#125—41no—
28C7×#125—41no—
29C4×C7×D4—41no—
30C4×C7×D4—41no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4C2311yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C22⋊Dic7—22no—
46C2×C4⋊Dic7—41no—
47C2×#1136—41no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7×#95—21no—
52C7×#95C2×C411yescharacteristic
53C7×#95C2×C411yescharacteristic
54#6408—42no—
55C7⋊#95—41no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)—41no—
58#2633D711yescharacteristic
59#1799—71no—
60#1465—71no—
61#1352—72no—
62#1344—72no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C2×D411yescharacteristic
65C2×C7×D4—21no—
66C2×C7×D4—21no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C22×C4×C7—21no—
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—81no—
78C22×C4×C7—81no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×Dic7—42no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—81no—
92C7×C4⋊C4—81no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—41no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—81no—
99C7×C22⋊C4—81no—
100C42×C7—21no—
101C42×C7—21no—
102C22⋊Dic7—42no—
103C22⋊Dic7—42no—
104C4⋊Dic7—82no—
105#1136—82no—
106C2×C7⋊C8—42no—
107C2×#125Dic711yescharacteristic, complemented
108C2×C4×D4C2×D711yescharacteristic
109C2×#99—141no—
110#407—141no—
111#399Dic711yescharacteristic, complemented
112#399C2×D711yescharacteristic
113#398—21no—
114#397Dic711yescharacteristic, complemented
115#397C2×D711yescharacteristic
116#394—21no—
117#385—21no—
118#385—21no—
119#383Dic711yescharacteristic, complemented
120#380—141no—
121#342—72no—
122#324—141no—
123#323—72no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7C2×C22⋊C411yescharacteristic
134C23×C7AGL(1,ℤ/8)11yescharacteristic
135C23×C7#13711yescharacteristic
136C7×D4—42no—
137C7×D4—42no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7#9911yescharacteristic
165C2×C4×C7#9911yescharacteristic
166C2×Dic7—82no—
167C2×Dic7—82no—
168C2×Dic7—82no—
169C7⋊C8—82no—
170C22×D4C22⋊D711yescharacteristic
171C23×C4C22⋊D711yescharacteristic
172C23×C4C22⋊D711yescharacteristic
173C2×M16—281no—
174#125—41no—
175#125—41no—
176C4×D4—41no—
177C4×D4—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—281no—
183C2×C4⋊C4C2×Dic711yescharacteristic
184C2×C4⋊C4C2×Dic711yescharacteristic
185C2×C4⋊C4C22⋊D711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—142no—
191C2×C22⋊C4C2×Dic711yescharacteristic
192C2×C22⋊C4C22×D711yescharacteristic
193C2×C42C2×Dic711yescharacteristic
194#99—282no—
195C22⋊C8—142no—
196C22⋊C8—281no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—281no—
202#95C2×Dic711yescharacteristic
203#95C2×Dic711yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—42no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7C2×#9911yescharacteristic
219C22×C7#41711yescharacteristic
220C22×C7#42111yescharacteristic
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7—81no—
231Dic7—162no—
232C24C22⋊Dic711yescharacteristic
233C24C2×C22⋊D711yescharacteristic
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—282no—
249C22×C4C22⋊Dic711yescharacteristic
250C22×C4C22⋊Dic711yescharacteristic
251C22×C4C22⋊Dic711yescharacteristic
252C22×C4C22×Dic711yescharacteristic
253C22×C4C2×C22⋊D711yescharacteristic
254M16—562no—
255C2×C8—282no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—562no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C22⋊C4—282no—
273C22⋊C4—282no—
274C42—21no—
275C42—21no—
276C2×C7—21no—
277C2×C7—21no—
278C2×C7—41no—
279C2×C7—41no—
280C2×C7—81no—
281C2×C7#151311yescharacteristic
282C2×C7#212311yescharacteristic
283C2×C7#212711yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#651211yescharacteristic
294C23#651411yescharacteristic
295C23C2×C22⋊Dic711yescharacteristic
296D4—42no—
297D4—42no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—562no—
325C2×C4—562no—
326C2×C4—562no—
327C2×C4#640811yescharacteristic
328C2×C4#640811yescharacteristic
329C8—562no—
330C7#1182711yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22C2×#640811yescharacteristic
346C22#9000411yescharacteristic
347C22#9000811yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—1122no—
359C2—21no—
360C2—21no—
361C2—41no—
362C2—41no—
363C2—81no—
364C2#1172946311yescharacteristic
365C2#1172961411yescharacteristic
366C2#1172962411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2633C2
socleC22×C7C2×#99
prime radicalC22×C7×D4D4
cosocleC2×C7×C22⋊C4C23
hypercenter#2633D7
hypocenterC7#11827
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2633

Complements

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

NH ≅ G/Ntype
4. #11725457C2semidirect
5. #11725455C2semidirect
6. C2×C7×#125C4semidirect
8. C7×#399C4semidirect
11. C7×#397C4semidirect
16. C7×#383C4semidirect
107. C2×#125Dic7semidirect
111. #399Dic7semidirect
114. #397Dic7semidirect
119. #383Dic7semidirect
330. C7#11827semidirect

Maximal subgroups

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

subgroupindex
C7×#26332
#117293732
#117293562
#117254572
#117254552
#118277