Finite Group Databasebeta

← #13246346

Subgroups — #13246346

Order 1792.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3553C211yesmaximal, characteristic
2#11729510C212yesmaximal, complemented
3#11729506C212yesmaximal, complemented
4#11729494C211yesmaximal, characteristic, complemented
5#11729490C211yesmaximal, characteristic, complemented
6C7×#564C411yescharacteristic, complemented
7C7×#563C412yescomplemented
8C7×#562C411yescharacteristic, complemented
9C7×#554—22no—
10C7×#553—22no—
11C7×#552—22no—
12C7×#541—22no—
13C7×#539—22no—
14C7×#539—22no—
15C7×#529C2212yes—
16C2×C7×#125C2211yescharacteristic
17#89708C2211yescharacteristic
18#89698C2212yes—
19C2×C4×C7⋊C8C2211yescharacteristic
20#89338—22no—
21#89333—22no—
22#89332—22no—
23Q8⋊(C7⋊C8)—22no—
24D4⋊(C7⋊C8)—22no—
25#89324—22no—
26#8529—71nomaximal, Sylow
27C7×C4⋊Q8—22no—
28C7×#126—22no—
29C7×#126—22no—
30C7×#126—22no—
31C7×#126—22no—
32C7×#126—22no—
33C7×#125—22no—
34C7×#125—22no—
35C7×#125—22no—
36C7×#125—22no—
37C7×#125—22no—
38C7×#125—22no—
39C7×#125C2×C411yescharacteristic
40C7×#125C2×C411yescharacteristic
41C7×#125C2×C412yes—
42C7×#125D412yes—
43C7×#124C2×C412yes—
44C7×#123—22no—
45C7×#123—22no—
46C7×#123—22no—
47C7×#123—22no—
48C7×#123—22no—
49C7×C22⋊Q8—22no—
50C7×C22⋊Q8—22no—
51C7×C22⋊Q8—22no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8D412yes—
56C7×#121—22no—
57C7×#121—22no—
58C7×#121D412yes—
59C4×C7×Q8—22no—
60C4×C7×Q8D412yes—
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C4×C7×D4D412yes—
66C7×#117—22no—
67C7×#117—22no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4D412yes—
71C2×C42×C7C2311yescharacteristic
72C2×C4⋊Dic7—21no—
73C22×C7⋊C8—21no—
74C7⋊#95—22no—
75C22⋊(C7⋊C8)—22no—
76C4⋊(C7⋊C8)—22no—
77#6377—22no—
78C4×C7⋊C8—22no—
79#3553D711yescharacteristic
80#1991—71no—
81#1534—71no—
82#1512—72no—
83#1494—72no—
84C2×C7×Q8—22no—
85C2×C7×Q8C22⋊C412yes—
86C2×C7×D4—22no—
87C2×C7×D4—22no—
88C2×C7×D4C22⋊C412yes—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7C2×D411yescharacteristic
94C22×C4×C7C2×D412yes—
95C22×Dic7—41no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—22no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4C22⋊C412yes—
112C7×C4⋊C4C22⋊C412yes—
113C7×C4⋊C4C22⋊C412yes—
114C7×C4⋊C4C22⋊C412yes—
115C7×C22⋊C4—22no—
116C7×C22⋊C4—22no—
117C7×C22⋊C4—22no—
118C7×C22⋊C4—22no—
119C7×C22⋊C4—22no—
120C7×C22⋊C4—22no—
121C7×C22⋊C4—22no—
122C7×C22⋊C4—22no—
123C7×C22⋊C4—22no—
124C42×C7—22no—
125C42×C7—22no—
126C42×C7—22no—
127C42×C7C22×C411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D412yes—
130C4⋊Dic7—42no—
131C2×C7⋊C8—22no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×C7⋊C8—42no—
135#564Dic711yescharacteristic, complemented
136#563Dic712yescomplemented
137#562Dic711yescharacteristic, complemented
138#554—22no—
139#553—22no—
140#552—22no—
141#541—22no—
142#539—22no—
143#539—22no—
144#529C2×D712yes—
145C2×#125C2×D711yescharacteristic
146#432—72no—
147C2×C4×C8—71no—
148#382—71no—
149#340—142no—
150#330—142no—
151#328—142no—
152#327—142no—
153Q8⋊C8—142no—
154D4⋊C8—142no—
155C23×C7—22no—
156C23×C7#12011yescharacteristic
157C7×Q8—42no—
158C7×Q8—82no—
159C7×D4—42no—
160C7×D4—42no—
161C7×D4—42no—
162C7×D4—82no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—22no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—22no—
170C2×C4×C7—42no—
171C2×C4×C7—82no—
172C2×C4×C7—82no—
173C2×C4×C7—82no—
174C2×C4×C7—82no—
175C2×C4×C7C2×C22⋊C411yescharacteristic
176C2×C4×C7C2×C22⋊C412yes—
177C2×C4×C7#12012yes—
178C2×C4×C7#12011yescharacteristic
179C2×Dic7—82no—
180C2×Dic7—81no—
181C7⋊C8—42no—
182C7⋊C8—82no—
183C22×C8—141no—
184C4⋊Q8—22no—
185#126—22no—
186#126—22no—
187#126—22no—
188#126—22no—
189#126—22no—
190#125—22no—
191#125—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125C2×Dic711yescharacteristic
197#125C2×Dic711yescharacteristic
198#125C2×Dic712yes—
199#125C22⋊D712yes—
200#124C2×Dic712yes—
201#123—22no—
202#123—22no—
203#123—22no—
204#123—22no—
205#123—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8C22⋊D712yes—
213#121—22no—
214#121—22no—
215#121C22⋊D712yes—
216C4×Q8—22no—
217C4×Q8C22⋊D712yes—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4C22⋊D712yes—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—141no—
228C2×C4⋊C4C22⋊D712yes—
229C2×C42C22×D711yescharacteristic
230C4⋊C8—142no—
231C22⋊C8—142no—
232#97—142no—
233C4×C8—142no—
234#95—142no—
235C22×C7—22no—
236C22×C7—42no—
237C22×C7—42no—
238C22×C7—42no—
239C22×C7#37911yescharacteristic
240C22×C7#45411yescharacteristic
241C22×C7#45611yescharacteristic
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—22no—
245C4×C7—22no—
246C4×C7—42no—
247C4×C7—42no—
248C4×C7—42no—
249C4×C7—42no—
250C4×C7—42no—
251Dic7—161no—
252C2×Q8—22no—
253C2×Q8C22⋊Dic712yes—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4C22⋊Dic712yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—281no—
262C22×C4C2×C22⋊D711yescharacteristic
263C22×C4C2×C22⋊D712yes—
264C2×C8—142no—
265C2×C8—281no—
266C2×C8—281no—
267C2×C8—282no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—282no—
284C4⋊C4C22⋊Dic712yes—
285C4⋊C4C22⋊Dic712yes—
286C4⋊C4C22⋊Dic712yes—
287C4⋊C4C22⋊Dic712yes—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42C22×Dic711yescharacteristic
301C42C2×C22⋊D712yes—
302C42C2×C22⋊D711yescharacteristic
303C2×C7—21no—
304C2×C7—21no—
305C2×C7—82no—
306C2×C7#179311yescharacteristic
307C2×C7#179412yes—
308C23—22no—
309C23#651611yescharacteristic
310Q8—42no—
311Q8—82no—
312D4—42no—
313D4—42no—
314D4—42no—
315D4—82no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—562no—
329C2×C4—561no—
330C2×C4C2×C22⋊Dic711yescharacteristic
331C2×C4C2×C22⋊Dic712yes—
332C2×C4#651612yes—
333C2×C4#651611yescharacteristic
334C8—282no—
335C8—562no—
336C7#852911yescharacteristic, complemented, Sylow
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#9001711yescharacteristic
342C22#9001811yescharacteristic
343C22#9001911yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—1121no—
354C2—21no—
355C2—21no—
356C2—82no—
357C2#1172989711yescharacteristic
358C2#1172990012yes—
subgroupG/H
centerC22#90019
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#3553C2
socleC22×C7#379
cosocleC2×C42×C7C23
hypercenter#3553D7
hypocenterC7#8529
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3553

Complements

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

NH ≅ G/Ntype
2. #11729510C2semidirect
3. #11729506C2semidirect
4. #11729494C2semidirect
5. #11729490C2semidirect
6. C7×#564C4semidirect
7. C7×#563C4semidirect
8. C7×#562C4semidirect
135. #564Dic7semidirect
136. #563Dic7semidirect
137. #562Dic7semidirect
336. C7#8529semidirect

Maximal subgroups

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

subgroupindex
C7×#35532
#117295102
#117295062
#117294942
#117294902
#85297