Finite Group Databasebeta

← #13405911

Subgroups — #13405911

Order 1792.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3165C211yesmaximal, characteristic, complemented
2#11740795C211yesmaximal, characteristic
3#11740775C211yesmaximal, characteristic, complemented
4C2×#89859C211yesmaximal, characteristic
5#11729544C211yesmaximal, characteristic, complemented
6#11728430C212yesmaximal
7#11728427C212yesmaximal
8#11728424C212yesmaximal, complemented
9#11728423C212yesmaximal, complemented
10#11727629C211yesmaximal, characteristic, complemented
11C4×#6397C211yesmaximal, characteristic
12C7×#533C2211yescharacteristic
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×C4×D28C2211yescharacteristic
15C7×#501—22no—
16C7×#500—21no—
17C7×#498—21no—
18C7×#428C2212yes—
19C7×#428C2212yes—
20C4×C7×M16C2211yescharacteristic
21C2×#6397C2211yescharacteristic
22C2×#6397C2211yescharacteristic
23C2×#6396C2212yes—
24#89859—21no—
25#89859—21no—
26#89859—22no—
27#89859—21no—
28#89856—21no—
29#89838—22no—
30#89796—21no—
31#89796—22no—
32Q8×C7⋊C8—21no—
33#89773C2212yes—
34#89771C2212yes—
35#89769C2212yes—
36#89767C2212yes—
37C2×#6383C2212yes—
38C2×#6381C2212yes—
39#89696C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C4×#1136C2211yescharacteristic
42#89353C2212yes—
43#89349C2211yescharacteristic
44#89349C2211yescharacteristic
45#89325C2212yes—
46#89325C2212yes—
47#25489—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C2×C7×M16C2311yescharacteristic
52C2×C7×M16C2311yescharacteristic
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8—21no—
57C7×C4⋊Q8D411yescharacteristic
58C7×C4⋊Q8D411yescharacteristic
59C7×#125D412yes—
60C7×C22⋊Q8—21no—
61C7×C22⋊Q8—21no—
62C4×C7×Q8—21no—
63C4×C7×Q8—21no—
64C7×#117C2312yes—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2312yes—
67C2×C42×C7C2311yescharacteristic
68C2×(C2×D7)⋊C4—21no—
69C2×C4⋊Dic7C2311yescharacteristic
70C2×#1136C2311yescharacteristic
71C2×#1136C2312yes—
72C2×#1136C2311yescharacteristic
73C22×C7⋊C8C2312yes—
74C4×D28—21no—
75C4×D28—21no—
76C4×D28—21no—
77C4×D28—21no—
78C7×#107—22no—
79C7×#107—22no—
80C7×#106—22no—
81C7×#106—22no—
82C7×#97—21no—
83C4×C7×C8—21no—
84#6397—22no—
85#6397Q814yescomplemented
86#6396—24no—
87#6383—22no—
88#6383—22no—
89#6383—22no—
90#6383—22no—
91#6382—22no—
92#6381—22no—
93#6381—22no—
94#6381—22no—
95C4⋊(C7⋊C8)—21no—
96C4⋊(C7⋊C8)—21no—
97C4⋊(C7⋊C8)—22no—
98C4⋊(C7⋊C8)—22no—
99C4⋊(C7⋊C8)D411yescharacteristic
100C4⋊(C7⋊C8)D412yes—
101C4⋊(C7⋊C8)D411yescharacteristic
102#6377—21no—
103#6377—22no—
104C4×C7⋊C8—21no—
105C4×C7⋊C8—22no—
106#3165D711yescharacteristic, complemented
107#3077—71no—
108C2×#474—71no—
109#2962—71no—
110#2089—72no—
111#2063—72no—
112#1997—71no—
113#1952—72no—
114#1950—71no—
115#1931—72no—
116C4×#100—71no—
117C23×D7—21no—
118C2×C7×Q8—21no—
119C2×C7×Q8—21no—
120C2×C7×Q8—41no—
121C2×C7×Q8—41no—
122C2×C7×Q8—82no—
123C22×C4×C7—22no—
124C22×C4×C7#4111yescharacteristic
125C22×C4×C7#4111yescharacteristic
126C22×C4×C7C2411yescharacteristic
127C22×Dic7—21no—
128C2×D28—21no—
129C2×D28—21no—
130C2×D28—21no—
131C2×D28—21no—
132C2×D28—21no—
133C2×D28C2×Q812yes—
134C2×C4×D7—41no—
135C2×C4×D7—41no—
136C2×C4×D7—82no—
137C7×M16—21no—
138C7×M16—21no—
139C7×M16C2×Q812yes—
140C7×M16#4112yes—
141C2×C7×C8—21no—
142C2×C7×C8—21no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—41no—
150C7×C4⋊C4—41no—
151C7×C4⋊C4C2×D412yes—
152C7×C4⋊C4C2×D412yes—
153C7×C22⋊C4—22no—
154C42×C7—22no—
155C42×C7C2×D411yescharacteristic
156C42×C7C2×D411yescharacteristic
157C42×C7C2×D411yescharacteristic
158C42×C7C2×D411yescharacteristic
159(C2×D7)⋊C4—41no—
160(C2×D7)⋊C4—41no—
161C4⋊Dic7—21no—
162C4⋊Dic7—21no—
163#1136—21no—
164#1136—22no—
165#1136—21no—
166#1136—21no—
167#1136—22no—
168#1136C2×Q812yes—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—21no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8C2×D412yes—
175C2×C7⋊C8C2×D412yes—
176#533C2×D711yescharacteristic
177C2×C4⋊Q8C2×D711yescharacteristic
178C2×C4×D4—71no—
179#501—22no—
180#500—21no—
181#498—21no—
182#478—142no—
183#476—141no—
184#474—141no—
185#474—142no—
186#474—141no—
187#474—141no—
188#446—142no—
189#446—141no—
190C8×Q8—141no—
191C2×#108—72no—
192#428—72no—
193#428C2×D712yes—
194#428C2×D712yes—
195C2×#107—72no—
196#424—72no—
197#423—71no—
198C2×C4⋊C8—71no—
199#418—72no—
200C2×#102—72no—
201C2×#100—71no—
202C2×#100—71no—
203#405—72no—
204C4×M16—71no—
205C4×M16C2×D711yescharacteristic
206#343—71no—
207#343—71no—
208#343—72no—
209#343—72no—
210#341—72no—
211C23×C7C2×#4111yescharacteristic
212C22×D7—22no—
213C22×D7—41no—
214C22×D7—41no—
215C22×D7—81no—
216C22×D7—81no—
217C7×Q8—42no—
218C7×Q8—81no—
219C7×Q8—82no—
220C7×Q8—82no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—82no—
227C2×C4×C7#12111yescharacteristic
228C2×C4×C7#12111yescharacteristic
229C2×C4×C7#12111yescharacteristic
230C2×C4×C7#12111yescharacteristic
231C2×C4×C7C22×D411yescharacteristic
232C2×C4×C7C22×D411yescharacteristic
233C2×C4×C7C22×Q811yescharacteristic
234C2×C4×C7C2×#4111yescharacteristic
235C2×C4×C72−1+411yescharacteristic
236C2×C4×C72−1+411yescharacteristic
237C2×Dic7—41no—
238C2×Dic7—81no—
239D28—22no—
240D28—22no—
241D28—41no—
242D28—41no—
243C4×D7—82no—
244C4×D7—82no—
245C7×C8—22no—
246C7×C8—22no—
247C7⋊C8—22no—
248C7⋊C8—22no—
249C7⋊C8—22no—
250C7⋊C8—41no—
251C7⋊C8—42no—
252C22×Q8—21no—
253C22×D4—71no—
254C23×C4—141no—
255C2×M16—71no—
256C2×M16—72no—
257C2×M16—71no—
258C2×M16C22×D711yescharacteristic
259C2×M16C22×D711yescharacteristic
260C22×C8—72no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8C22⋊D711yescharacteristic
266C4⋊Q8C22⋊D711yescharacteristic
267#125C22⋊D712yes—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—141no—
273C4×D4—141no—
274C4×D4—141no—
275C4×D4—141no—
276#117C22×D712yes—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—71no—
279C2×C4⋊C4C22×D712yes—
280C2×C22⋊C4—141no—
281C2×C42C22×D711yescharacteristic
282#108—144no—
283#107—22no—
284#107—22no—
285#107—142no—
286#107—142no—
287#107—142no—
288#106—22no—
289#106—22no—
290#106—142no—
291C4⋊C8—71no—
292C4⋊C8—71no—
293C4⋊C8—72no—
294C4⋊C8—142no—
295C4⋊C8—142no—
296C4⋊C8—141no—
297C4⋊C8—141no—
298#102—142no—
299#102—142no—
300#102—142no—
301#102—142no—
302#100—74no—
303#100—142no—
304#97—21no—
305#97—142no—
306#97—141no—
307C4×C8—21no—
308C4×C8—142no—
309C4×C8—141no—
310C22×C7—21no—
311C22×C7—21no—
312C22×C7C2×#12111yescharacteristic
313C22×C7#53311yescharacteristic
314C22×C7#54111yescharacteristic
315C2×D7—42no—
316C2×D7—82no—
317C2×D7—81no—
318C2×D7—81no—
319C2×D7—82no—
320C4×C7—21no—
321C4×C7—21no—
322C4×C7—21no—
323C4×C7—42no—
324C4×C7—42no—
325C4×C7—82no—
326C4×C7#47212yes—
327C4×C7#54711yescharacteristic
328C4×C7#54811yescharacteristic
329C4×C7D4×Q811yescharacteristic
330C4×C7D4×Q811yescharacteristic
331Dic7—81no—
332C24—141no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—82no—
338C2×D4—72no—
339C2×D4—141no—
340C2×D4—141no—
341C2×D4—141no—
342C2×D4—141no—
343C2×D4—141no—
344C22×C4—22no—
345C22×C4—141no—
346C22×C4—281no—
347C22×C4—281no—
348C22×C4—562no—
349C22×C4#115911yescharacteristic
350C22×C4#115911yescharacteristic
351C22×C4C23×D711yescharacteristic
352M16—21no—
353M16—21no—
354M16—72no—
355M16—141no—
356M16—141no—
357M16—142no—
358M16—141no—
359M16—142no—
360M16Q8×D712yes—
361M16#116112yes—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—72no—
365C2×C8—72no—
366C2×C8—141no—
367C2×C8—142no—
368C2×C8—142no—
369C2×C8—141no—
370C2×C8—142no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—141no—
380C4⋊C4—141no—
381C4⋊C4C2×C22⋊D712yes—
382C4⋊C4C2×C22⋊D712yes—
383C22⋊C4—22no—
384C22⋊C4—281no—
385C22⋊C4—281no—
386C42—22no—
387C42D4×D711yescharacteristic
388C42D4×D711yescharacteristic
389C42C2×C22⋊D711yescharacteristic
390C42C2×C22⋊D711yescharacteristic
391C2×C7—21no—
392C2×C7—21no—
393C2×C7#242911yescharacteristic
394C2×C7C2×#47211yescharacteristic
395C2×C7#306911yescharacteristic
396D7—82no—
397C23—142no—
398C23—281no—
399C23—281no—
400C23—561no—
401C23—561no—
402C23C2×#115911yescharacteristic
403Q8—42no—
404Q8—81no—
405Q8—82no—
406Q8—82no—
407D4—142no—
408D4—142no—
409D4—281no—
410D4—281no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—281no—
418C2×C4—562no—
419C2×C4—562no—
420C2×C4—561no—
421C2×C4#650211yescharacteristic
422C2×C4#650211yescharacteristic
423C2×C4#650211yescharacteristic
424C2×C4#650211yescharacteristic
425C2×C4C2×D4×D711yescharacteristic
426C2×C4C2×Q8×D711yescharacteristic
427C2×C4C2×#116111yescharacteristic
428C2×C4#655111yescharacteristic
429C2×C4#655111yescharacteristic
430C2×C4C22×C22⋊D711yescharacteristic
431C8—22no—
432C8—22no—
433C8—142no—
434C8—142no—
435C8—142no—
436C8—281no—
437C8—282no—
438C7#2548911yescharacteristic, complemented, Sylow
439C22—21no—
440C22—21no—
441C22—282no—
442C22—562no—
443C22—562no—
444C22—561no—
445C22—561no—
446C22#9041911yescharacteristic
447C22#9042211yescharacteristic
448C22C2×#650211yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—561no—
456C4#8997112yes—
457C4#9041711yescharacteristic
458C4#9041811yescharacteristic
459C4Q8×C22⋊D711yescharacteristic
460C4#9050811yescharacteristic
461C2—21no—
462C2—21no—
463C2—562no—
464C2#1174069911yescharacteristic
465C2C2×#8997111yescharacteristic
466C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3165C2
socleC22×C7C2×#121
prime radicalC22×D28C23
hypercenter#3165D7
hypocenterC7#25489
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. C7×#3165C2semidirect
3. #11740775C2semidirect
5. #11729544C2semidirect
8. #11728424C2semidirect
9. #11728423C2semidirect
10. #11727629C2semidirect
85. #6397Q8semidirect
106. #3165D7semidirect
438. C7#25489semidirect

Maximal subgroups

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

subgroupindex
C7×#31652
#117407952
#117407752
C2×#898592
#117295442
#117284302
#117284272
#117284242
#117284232
#117276292
C4×#63972
#254897