Finite Group Databasebeta

← #13250735

Subgroups — #13250735

Order 1792.

530 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3109C211yesmaximal, characteristic
2#11739770C211yesmaximal, characteristic, complemented
3#11729572C211yesmaximal, characteristic, complemented
4#11729566C211yesmaximal, characteristic, complemented
5#11729353C211yesmaximal, characteristic, complemented
6#11725556C211yesmaximal, characteristic, complemented
7#11725554C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#481—21no—
11C7×#476C412yescomplemented
12C7×#459C412yescomplemented
13C7×#447—21no—
14C7×#442—22no—
15C7×#432C2211yescharacteristic
16C2×#6411—22no—
17#89993—21no—
18#89984—21no—
19#89983—21no—
20#89757—21no—
21#89723—22no—
22C4×C4⋊Dic7C2211yescharacteristic
23#89704C2211yescharacteristic
24#89698C2211yescharacteristic
25C4×#1136C2211yescharacteristic
26#89332—21no—
27#89332—21no—
28D4⋊(C7⋊C8)—22no—
29#9329—71nomaximal, Sylow
30C2×C7×#41—21no—
31C22×C7×D4D411yescharacteristic
32C23×C4×C7—21no—
33C2×C7×D8—22no—
34C7×C4⋊Q8C2×C411yescharacteristic
35C7×#127C2×C411yescharacteristic
36C7×#124—21no—
37C7×#121—21no—
38C7×#121—21no—
39C4×C7×D4—21no—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C4×C7×D4—21no—
43C4×C7×D4C2×C412yes—
44C4×C7×D4D412yes—
45C2×C7×C4⋊C4D411yescharacteristic
46C2×C7×C22⋊C4—21no—
47C2×C42×C7C2311yescharacteristic
48C2×C22⋊Dic7—22no—
49C2×C22⋊Dic7—22no—
50C2×C4⋊Dic7—21no—
51C2×C4⋊Dic7—22no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7D412yes—
54C2×#1136—21no—
55C7×#106—22no—
56C7×C4⋊C8C2×C412yes—
57C7×#102—22no—
58C7×#102—22no—
59C7×C22⋊C8—21no—
60C7×#97—21no—
61#6411—44no—
62C7⋊#95—22no—
63C7⋊#95—21no—
64C7⋊#95—21no—
65C7⋊#95D411yescharacteristic
66C7⋊#95D411yescharacteristic
67C22⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)—21no—
69#6377—21no—
70#6377—21no—
71C4×C7⋊C8—21no—
72#3109D711yescharacteristic
73#2319—71no—
74#1980—71no—
75#1505—71no—
76#1490—71no—
77#1300—71no—
78#1285—71no—
79C24×C7—21no—
80C7×#41—42no—
81C7×#41—42no—
82C2×C7×Q8—21no—
83C2×C7×D4—21no—
84C2×C7×D4—21no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4—42no—
88C2×C7×D4—41no—
89C2×C7×D4C22⋊C412yes—
90C22×C4×C7—21no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—41no—
94C22×C4×C7—82no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×Dic7—21no—
99C22×Dic7—22no—
100C22×Dic7—21no—
101C7×D8—44no—
102C2×C7×C8—22no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4C22⋊C412yes—
107C7×C22⋊C4—22no—
108C7×C22⋊C4—22no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115C22⋊Dic7—82no—
116C22⋊Dic7—82no—
117C4⋊Dic7—42no—
118C4⋊Dic7—82no—
119C4×Dic7—42no—
120C4×Dic7—42no—
121#1136—22no—
122#1136—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125#532C2×D711yescharacteristic
126C2×C4×D4C2×D711yescharacteristic
127#481—21no—
128#476Dic712yescomplemented
129#459Dic712yescomplemented
130#447—21no—
131#442—22no—
132#432—71no—
133#432C2×D711yescharacteristic
134C2×#104—142no—
135C4×M16—71no—
136#390—141no—
137#389—141no—
138#388—142no—
139#381—141no—
140#380—141no—
141C4×C4⋊C4—71no—
142#376—71no—
143#327—141no—
144#327—141no—
145D4⋊C8—142no—
146C23×C7—22no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7#12011yescharacteristic
153C7×Q8—42no—
154C7×D4—22no—
155C7×D4—22no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—42no—
159C7×D4—81no—
160C7×D4—81no—
161C7×D4—81no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—22no—
165C2×C4×C7—22no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—41no—
169C2×C4×C7—82no—
170C2×C4×C7—82no—
171C2×C4×C7—82no—
172C2×C4×C7C2×C22⋊C411yescharacteristic
173C2×C4×C7C4×D411yescharacteristic
174C2×C4×C7C4×D411yescharacteristic
175C2×C4×C7#12111yescharacteristic
176C2×C4×C7#12111yescharacteristic
177C2×C4×C7#12311yescharacteristic
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—42no—
181C2×Dic7—41no—
182C2×Dic7—42no—
183C2×Dic7—41no—
184C2×Dic7—42no—
185C2×Dic7—41no—
186C2×Dic7—41no—
187C7×C8—42no—
188C7⋊C8—42no—
189C7⋊C8—81no—
190C2×#41—21no—
191C22×D4C22⋊D711yescharacteristic
192C23×C4—21no—
193C2×D8—22no—
194C2×M16—141no—
195C4⋊Q8C2×Dic711yescharacteristic
196#127C2×Dic711yescharacteristic
197#124—21no—
198#121—21no—
199#121—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4C2×Dic712yes—
205C4×D4C22⋊D712yes—
206C2×C4⋊C4—141no—
207C2×C4⋊C4—142no—
208C2×C4⋊C4C22⋊D711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—142no—
211C2×C22⋊C4—142no—
212C2×C42—72no—
213C2×C42—141no—
214C2×C42C22×D711yescharacteristic
215#106—22no—
216C4⋊C8—141no—
217C4⋊C8C2×Dic712yes—
218#104—284no—
219#102—22no—
220#102—22no—
221C22⋊C8—21no—
222C22⋊C8—141no—
223#97—21no—
224#97—141no—
225#97—141no—
226C4×C8—141no—
227#95—71no—
228#95—71no—
229#95—142no—
230#95—141no—
231#95—141no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—42no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—82no—
238C22×C7—82no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7#38611yescharacteristic
243C22×C7#45311yescharacteristic
244C22×C7#45411yescharacteristic
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—42no—
251C4×C7—81no—
252C4×C7#42112yes—
253Dic7—81no—
254Dic7—82no—
255Dic7—81no—
256C24—21no—
257#41—42no—
258#41—42no—
259C2×Q8—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4C22⋊Dic712yes—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—82no—
272C22×C4—141no—
273C22×C4—141no—
274C22×C4—142no—
275C22×C4D4×D711yescharacteristic
276C22×C4D4×D711yescharacteristic
277C22×C4C2×C22⋊D711yescharacteristic
278D8—44no—
279M16—142no—
280M16—281no—
281C2×C8—22no—
282C2×C8—281no—
283C2×C8—281no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—282no—
288C4⋊C4—562no—
289C4⋊C4C22⋊Dic712yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—562no—
295C22⋊C4—562no—
296C42—282no—
297C42—282no—
298C42#115911yescharacteristic
299C42#115911yescharacteristic
300C42C22×Dic711yescharacteristic
301C42C2×C22⋊D711yescharacteristic
302C2×C7—21no—
303C2×C7—21no—
304C2×C7—82no—
305C2×C7—161no—
306C2×C7#188311yescharacteristic
307C2×C7#189711yescharacteristic
308C2×C7#190411yescharacteristic
309C23—22no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23#650011yescharacteristic
316Q8—42no—
317D4—22no—
318D4—22no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—281no—
336C2×C4—281no—
337C2×C4—281no—
338C2×C4—282no—
339C2×C4—282no—
340C2×C4—281no—
341C2×C4—282no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4D4×Dic711yescharacteristic
345C2×C4D4×Dic711yescharacteristic
346C2×C4#649811yescharacteristic
347C2×C4#650211yescharacteristic
348C2×C4#650211yescharacteristic
349C2×C4C2×C22⋊Dic711yescharacteristic
350C8—42no—
351C8—282no—
352C8—561no—
353C7#932911yescharacteristic, complemented, Sylow
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#8997011yescharacteristic
365C22#8997311yescharacteristic
366C22#8999211yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—561no—
375C4—562no—
376C4—561no—
377C4#9000812yes—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—161no—
382C2#1172946611yescharacteristic
383C2#1172984211yescharacteristic
384C2#1172984311yescharacteristic
subgroupG/H
centerC22#89992
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#3109C2
socleC22×C7#386
cosocleC2×C42×C7C23
hypercenter#3109D7
hypocenterC7#9329
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
2. #11739770C2semidirect
3. #11729572C2semidirect
4. #11729566C2semidirect
5. #11729353C2semidirect
6. #11725556C2semidirect
7. #11725554C2semidirect
11. C7×#476C4semidirect
12. C7×#459C4semidirect
128. #476Dic7semidirect
129. #459Dic7semidirect
353. C7#9329semidirect

Maximal subgroups

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

subgroupindex
C7×#31092
#117397702
#117295722
#117295662
#117293532
#117255562
#117255542
#93297