Finite Group Databasebeta

← #13324635

Subgroups — #13324635

Order 1792.

624 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740736C211yesmaximal, characteristic
2#11740725C211yesmaximal, characteristic, complemented
3#11736783C211yesmaximal, characteristic, complemented
4C7×#1496C211yesmaximal, characteristic, complemented
5#11728548C211yesmaximal, characteristic, complemented
6#11728519C211yesmaximal, characteristic, complemented
7#11728496C211yesmaximal, characteristic, complemented
8#11725912C212yesmaximal, complemented
9#11725911C212yesmaximal, complemented
10#11725907C212yesmaximal, complemented
11#11725906C212yesmaximal, complemented
12C7×#532C2211yescharacteristic
13#90168C2211yescharacteristic
14C7×#432C2211yescharacteristic
15C7×#431C2211yescharacteristic
16#90006—22no—
17#89860C411yescharacteristic
18#89850C411yescharacteristic
19#89845C411yescharacteristic
20#89842C411yescharacteristic
21#89810C412yes—
22#89810C412yes—
23C4×#1141C2212yes—
24C4×#1141C2212yes—
25#89784C2212yes—
26#89783—22no—
27D4×C7⋊C8C2212yes—
28#89698C2211yescharacteristic
29#89697C2211yescharacteristic
30C2×C4×C7⋊C8C2211yescharacteristic
31#89610C2212yes—
32#89606C2212yes—
33#89499—22no—
34#89478—22no—
35C7×D4⋊C8C2212yes—
36C7×D4⋊C8C2212yes—
37#89275C2212yes—
38#89274C2212yes—
39#89254C2212yes—
40#89254C2212yes—
41#89250C2211yescharacteristic
42#89250C2211yescharacteristic
43#89249C2211yescharacteristic
44#89249C2211yescharacteristic
45#17989—71nomaximal, Sylow
46C2×C7×#41—21no—
47C2×#1157—21no—
48C7×C4⋊Q8C2×C411yescharacteristic
49C7×#127C2×C411yescharacteristic
50C7×#124—21no—
51C7×#121—21no—
52C7×#121C2×C412yes—
53C4×C7×D4—21no—
54C4×C7×D4C2312yes—
55C2×C42×C7C2311yescharacteristic
56C2×#1141C2×C412yes—
57C2×#1141C2×C412yes—
58#6493—21no—
59#6493C2×C412yes—
60C2×#1136—22no—
61C22×C7⋊C8—22no—
62C22×C7⋊C8C2×C411yescharacteristic
63C22×C7⋊C8C2×C411yescharacteristic
64C2×#1131—22no—
65C2×C8×D7—22no—
66#6438—21no—
67#6437C2×C411yescharacteristic
68C4×D28—21no—
69C4×D28C2312yes—
70#6432C2×C411yescharacteristic
71C7×C4⋊C8—21no—
72C7×C4⋊C8C2312yes—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×#97—21no—
76C4×C7×C8C2311yescharacteristic
77C4×C7×C8C2311yescharacteristic
78#6406C2×C412yes—
79#6406C2×C412yes—
80C22⋊(C7⋊C8)—21no—
81C22⋊(C7⋊C8)—22no—
82C22⋊(C7⋊C8)—22no—
83C22⋊(C7⋊C8)D412yes—
84(C2×D7)⋊C8—22no—
85(C2×D7)⋊C8—22no—
86#6383C2×C412yes—
87#6383C2×C412yes—
88#6381C2×C412yes—
89#6381C2×C412yes—
90C4⋊(C7⋊C8)D412yes—
91C4⋊(C7⋊C8)C2312yes—
92#6377—21no—
93C4×C7⋊C8C2×C411yescharacteristic
94C4×C7⋊C8C2×C411yescharacteristic
95C4×C7⋊C8C2311yescharacteristic
96C4×C7⋊C8C2311yescharacteristic
97C4×C7⋊C8C2311yescharacteristic
98C4×C7⋊C8C2311yescharacteristic
99#3169—71no—
100#2981—71no—
101#2981—71no—
102#1586—72no—
103#1586—72no—
104#1581—72no—
105#1581—72no—
106#1568—71no—
107#1496—71no—
108#1496D711yescharacteristic, complemented
109#1452—71no—
110C7×#41—82no—
111C2×C7×Q8—21no—
112C2×C7×D4—22no—
113C2×C7×D4—21no—
114C2×C7×D4—21no—
115C2×C7×D4C22×C412yes—
116C22×C4×C7—22no—
117C22×C4×C7C22×C411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7#4111yescharacteristic
120C2×C22⋊D7—22no—
121#1157—82no—
122C2×D28—21no—
123C2×D28—21no—
124C2×D28C22×C412yes—
125C2×C4×D7—22no—
126C2×Dic14—21no—
127C2×C7×C8—22no—
128C2×C7×C8—21no—
129C2×C7×C8—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4C22×C412yes—
132C7×C22⋊C4—22no—
133C7×C22⋊C4—22no—
134C42×C7C22×C411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7#4111yescharacteristic
137C42×C7C2411yescharacteristic
138#1141—42no—
139#1141—42no—
140(C2×D7)⋊C4—22no—
141(C2×D7)⋊C4—22no—
142C4⋊Dic7—21no—
143C4⋊Dic7C22×C412yes—
144#1136—82no—
145C2×C7⋊C8—22no—
146C2×C7⋊C8—41no—
147C2×C7⋊C8—41no—
148C2×C7⋊C8—82no—
149C2×C7⋊C8C22×C411yescharacteristic
150C2×C7⋊C8C22×C411yescharacteristic
151C2×C7⋊C8C22×C411yescharacteristic
152C2×C7⋊C8C22×C411yescharacteristic
153C2×C7⋊C8C2×D412yes—
154C2×C7⋊C8C2×D412yes—
155#1131—82no—
156C8×D7—82no—
157#532—71no—
158#532C2×D711yescharacteristic
159#500—71no—
160#493—71no—
161#486—71no—
162#486—71no—
163#466—72no—
164#466—72no—
165C4×D8—72no—
166C4×D8—72no—
167#436—72no—
168#436—72no—
169#435—142no—
170#435—142no—
171C8×D4—72no—
172C8×D4—72no—
173#432—71no—
174#432C2×D711yescharacteristic
175#431—71no—
176#431C2×D711yescharacteristic
177#408—142no—
178#408—142no—
179C2×C4×C8—71no—
180#335—72no—
181#334—72no—
182D4⋊C8—72no—
183D4⋊C8—72no—
184D4⋊C8C2×D712yes—
185D4⋊C8C2×D712yes—
186#322—71no—
187#322—71no—
188#322—71no—
189#322—71no—
190C23×C7—22no—
191C23×C7C4×D411yescharacteristic
192C22×D7—22no—
193C7×Q8—81no—
194C7×D4—22no—
195C7×D4—22no—
196C7×D4—81no—
197C7×D4—81no—
198C7×D4—82no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—22no—
202C2×C4×C7—22no—
203C2×C4×C7—41no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7C4×D411yescharacteristic
207C2×C4×C7C4×D411yescharacteristic
208C2×C4×C7C4×D411yescharacteristic
209C2×C4×C7C23×C411yescharacteristic
210C2×C4×C7C22×D411yescharacteristic
211C2×C4×C7C2×#4111yescharacteristic
212C22⋊D7—82no—
213C2×Dic7—22no—
214D28—22no—
215D28—22no—
216D28—81no—
217D28—81no—
218C4×D7—42no—
219C4×D7—42no—
220Dic14—81no—
221C7×C8—41no—
222C7×C8—41no—
223C7×C8—42no—
224C7⋊C8—21no—
225C7⋊C8—21no—
226C7⋊C8—21no—
227C7⋊C8—21no—
228C7⋊C8—22no—
229C7⋊C8—22no—
230C7⋊C8—42no—
231C2×#41—21no—
232C2×#41—141no—
233C2×D8—72no—
234C2×D8—72no—
235C2×M16—142no—
236C2×M16—142no—
237C22×C8—71no—
238C22×C8—71no—
239C22×C8—142no—
240C22×C8—142no—
241C4⋊Q8—71no—
242C4⋊Q8C4×D711yescharacteristic
243#127—71no—
244#127C4×D711yescharacteristic
245#124—21no—
246#124—141no—
247#121—21no—
248#121—72no—
249#121—141no—
250#121C4×D712yes—
251C4×D4—21no—
252C4×D4—72no—
253C4×D4—141no—
254C4×D4C22×D712yes—
255C2×C42C22×D711yescharacteristic
256#107—72no—
257#107—72no—
258C4⋊C8—21no—
259C4⋊C8—72no—
260C4⋊C8—72no—
261C4⋊C8C22×D712yes—
262#102—72no—
263#102—72no—
264#102—72no—
265#102—72no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—72no—
269C22⋊C8—142no—
270C22⋊C8—142no—
271C22⋊C8—142no—
272C22⋊C8—142no—
273C22⋊C8—141no—
274#97—21no—
275#97—141no—
276C4×C8—71no—
277C4×C8—71no—
278C4×C8—71no—
279C4×C8—71no—
280C4×C8—71no—
281C4×C8—71no—
282C4×C8C22×D711yescharacteristic
283C4×C8C22×D711yescharacteristic
284C22×C7—21no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—42no—
288C22×C7—42no—
289C22×C7—42no—
290C22×C7C2×C4×D411yescharacteristic
291C2×D7—42no—
292C2×D7—42no—
293C2×D7—42no—
294C4×C7—21no—
295C4×C7—21no—
296C4×C7—21no—
297C4×C7—21no—
298C4×C7—42no—
299C4×C7#56811yescharacteristic
300C4×C7#56811yescharacteristic
301C4×C7#57612yes—
302Dic7—42no—
303#41—82no—
304#41—562no—
305C2×Q8—21no—
306C2×Q8—141no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—72no—
311C2×D4—142no—
312C2×D4—141no—
313C2×D4—141no—
314C2×D4C2×C4×D712yes—
315C22×C4—22no—
316C22×C4—142no—
317C22×C4C2×C4×D711yescharacteristic
318C22×C4D4×D711yescharacteristic
319C22×C4#115911yescharacteristic
320D8—282no—
321D8—282no—
322M16—562no—
323M16—562no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—71no—
328C2×C8—71no—
329C2×C8—71no—
330C2×C8—71no—
331C2×C8—72no—
332C2×C8—72no—
333C2×C8—142no—
334C2×C8—281no—
335C2×C8—281no—
336C2×C8—562no—
337C2×C8—562no—
338C4⋊C4—21no—
339C4⋊C4—72no—
340C4⋊C4—141no—
341C4⋊C4C2×C4×D712yes—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—142no—
345C22⋊C4—142no—
346C42C2×C4×D711yescharacteristic
347C42D4×D711yescharacteristic
348C42#115911yescharacteristic
349C42C23×D711yescharacteristic
350C2×C7—41no—
351C2×C7—82no—
352C2×C7#293211yescharacteristic
353C2×C7#294811yescharacteristic
354C2×C7#295711yescharacteristic
355D7—82no—
356C23—22no—
357C23—142no—
358C23(C22⋊D7)⋊C411yescharacteristic
359Q8—81no—
360Q8—561no—
361D4—22no—
362D4—22no—
363D4—81no—
364D4—81no—
365D4—82no—
366D4—142no—
367D4—142no—
368D4—561no—
369D4—561no—
370D4—562no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—142no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4(C22⋊D7)⋊C411yescharacteristic
382C2×C4(C22⋊D7)⋊C411yescharacteristic
383C2×C4(C22⋊D7)⋊C411yescharacteristic
384C2×C4C22×C4×D711yescharacteristic
385C2×C4C2×D4×D711yescharacteristic
386C2×C4C2×#115911yescharacteristic
387C8—41no—
388C8—41no—
389C8—42no—
390C8—141no—
391C8—141no—
392C8—141no—
393C8—141no—
394C8—142no—
395C8—142no—
396C8—282no—
397C7#1798911yescharacteristic, complemented, Sylow
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—282no—
405C22—282no—
406C22—282no—
407C22C2×(C22⋊D7)⋊C411yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—42no—
413C4—282no—
414C4#9044111yescharacteristic
415C4#9044111yescharacteristic
416C4#9046512yes—
417C2—41no—
418C2—82no—
419C2—562no—
420C2#1173610311yescharacteristic
421C2#1173636011yescharacteristic
422C2#1173643511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1496C2
socleC22×C7C2×C4×D4
prime radical#11740736C2
cosocleC42×C7C24
hypercenter#1496D7
hypocenterC7#17989
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1496

Complements

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

NH ≅ G/Ntype
2. #11740725C2semidirect
3. #11736783C2semidirect
4. C7×#1496C2semidirect
5. #11728548C2semidirect
6. #11728519C2semidirect
7. #11728496C2semidirect
8. #11725912C2semidirect
9. #11725911C2semidirect
10. #11725907C2semidirect
11. #11725906C2semidirect
108. #1496D7semidirect
397. C7#17989semidirect

Maximal subgroups

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

subgroupindex
#117407362
#117407252
#117367832
C7×#14962
#117285482
#117285192
#117284962
#117259122
#117259112
#117259072
#117259062
#179897