Finite Group Databasebeta

← C7×#19217 #13196433

Subgroups — C7×#19217 #13196433

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3249C211yesmaximal, characteristic, complemented
2C7×#3246C211yesmaximal, characteristic
3C7×#3131C211yesmaximal, characteristic
4C7×#2960C211yesmaximal, characteristic, complemented
5C7×#1700C212yesmaximal, complemented
6C7×#1698C212yesmaximal, complemented
7C7×#1684C212yesmaximal, complemented
8C7×#1682C212yesmaximal, complemented
9C2×C7×#335C211yesmaximal, characteristic, complemented
10C7×#1550C211yesmaximal, characteristic, complemented
11C7×#1547C211yesmaximal, characteristic
12C7×#533C2211yescharacteristic
13C7×#533C2211yescharacteristic
14C7×#532C2211yescharacteristic
15C7×#488—21no—
16C7×#488—21no—
17C7×#487C2211yescharacteristic
18C7×#487C2211yescharacteristic
19C7×#486C2211yescharacteristic
20C7×#486C2211yescharacteristic
21C7×#477—21no—
22C7×#477C2212yes—
23C7×#476—21no—
24C7×#475C2212yes—
25C7×#467—22no—
26C7×#465—22no—
27C7×#462—21no—
28C7×#462C2212yes—
29C7×#461—21no—
30C7×#460C2212yes—
31C7×#446—21no—
32C7×#446C2212yes—
33C7×C8×Q8—21no—
34C7×C8×Q8C2212yes—
35C7×#424—21no—
36C7×#423C2211yescharacteristic
37C2×C7×C4⋊C8C2211yescharacteristic
38C2×C7×C4⋊C8C2211yescharacteristic
39C7×#416—21no—
40C7×#416—21no—
41C7×#405—21no—
42C2×C4×C7×C8C2211yescharacteristic
43C7×#335C2212yes—
44C7×#335C2212yes—
45C7×#333C2212yes—
46C7×#332C2212yes—
47C7×#332C2212yes—
48C7×#331C2212yes—
49#19217C711yesmaximal, characteristic, direct product factor, Sylow
50C2×C7×#41—21no—
51C2×C7×Q16—22no—
52C2×C7×SD16—22no—
53C2×C7×M16—21no—
54C22×C7×C8—21no—
55C22×C7×C8—21no—
56C22×C7×C8D411yescharacteristic
57C22×C7×C8D411yescharacteristic
58C7×C4⋊Q8D411yescharacteristic
59C7×C4⋊Q8D411yescharacteristic
60C7×C4⋊Q8C2311yescharacteristic
61C7×C4⋊Q8C2311yescharacteristic
62C7×C4⋊Q8C2311yescharacteristic
63C7×#127C2311yescharacteristic
64C7×#125—21no—
65C7×#125—21no—
66C7×#124—21no—
67C7×C22⋊Q8—21no—
68C7×C22⋊Q8—21no—
69C7×C22⋊Q8—21no—
70C7×C22⋊Q8D412yes—
71C7×#121—21no—
72C7×#121—21no—
73C4×C7×Q8—21no—
74C4×C7×Q8—21no—
75C4×C7×Q8—21no—
76C4×C7×Q8C2312yes—
77C4×C7×D4—21no—
78C4×C7×D4—21no—
79C7×#117—21no—
80C7×#117—21no—
81C2×C42×C7C2311yescharacteristic
82C7×#107—22no—
83C7×#106—22no—
84C7×C4⋊C8—22no—
85C7×C4⋊C8—22no—
86C7×C4⋊C8—21no—
87C7×C4⋊C8—21no—
88C7×C4⋊C8—21no—
89C7×C4⋊C8C2312yes—
90C7×C4⋊C8C2312yes—
91C7×C4⋊C8C2312yes—
92C7×#103—22no—
93C7×#103—22no—
94C7×#103—21no—
95C7×#103—21no—
96C7×#103—21no—
97C7×#103—21no—
98C7×#103—22no—
99C7×#102—21no—
100C7×#102—21no—
101C7×#102—21no—
102C7×#102—21no—
103C7×#102—22no—
104C7×#97—22no—
105C4×C7×C8—22no—
106C4×C7×C8D411yescharacteristic
107C4×C7×C8D411yescharacteristic
108C4×C7×C8C2311yescharacteristic
109C4×C7×C8C2311yescharacteristic
110#3249C2×C711yescharacteristic, complemented
111#3246C2×C711yescharacteristic
112#3131C2×C711yescharacteristic
113#2960C2×C711yescharacteristic, complemented
114#1700C2×C712yescomplemented
115#1698C2×C712yescomplemented
116#1684C2×C712yescomplemented
117#1682C2×C712yescomplemented
118C2×#335C2×C711yescharacteristic, complemented
119#1550C2×C711yescharacteristic, complemented
120#1547C2×C711yescharacteristic
121C7×#41—42no—
122C7×#41—42no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8C2×D412yes—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—41no—
132C22×C4×C7—41no—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7#4111yescharacteristic
136C7×Q16—82no—
137C7×SD16—82no—
138C7×M16—42no—
139C2×C7×C8—22no—
140C2×C7×C8—22no—
141C2×C7×C8—22no—
142C2×C7×C8—22no—
143C2×C7×C8—22no—
144C2×C7×C8—81no—
145C2×C7×C8—81no—
146C2×C7×C8C2×D411yescharacteristic
147C2×C7×C8C2×D411yescharacteristic
148C2×C7×C8C2×D411yescharacteristic
149C2×C7×C8C2×D411yescharacteristic
150C7×C4⋊C4—21no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—41no—
162C7×C4⋊C4—41no—
163C7×C4⋊C4C2×D412yes—
164C7×C22⋊C4—22no—
165C7×C22⋊C4—41no—
166C7×C22⋊C4—41no—
167C7×C22⋊C4—41no—
168C42×C7—22no—
169C42×C7—41no—
170C42×C7C2×D411yescharacteristic
171C42×C7C2×D411yescharacteristic
172C42×C7#4111yescharacteristic
173C42×C7C2411yescharacteristic
174#533C22×C711yescharacteristic
175#533C22×C711yescharacteristic
176#532C22×C711yescharacteristic
177#488—21no—
178#488—21no—
179#487C22×C711yescharacteristic
180#487C22×C711yescharacteristic
181#486C22×C711yescharacteristic
182#486C22×C711yescharacteristic
183#477—21no—
184#477C22×C712yes—
185#476—21no—
186#475C22×C712yes—
187#467—22no—
188#465—22no—
189#462—21no—
190#462C22×C712yes—
191#461—21no—
192#460C22×C712yes—
193#446—21no—
194#446C22×C712yes—
195C8×Q8—21no—
196C8×Q8C22×C712yes—
197#424—21no—
198#423C22×C711yescharacteristic
199C2×C4⋊C8C22×C711yescharacteristic
200C2×C4⋊C8C22×C711yescharacteristic
201#416—21no—
202#416—21no—
203#405—21no—
204C2×C4×C8C22×C711yescharacteristic
205#335C22×C712yes—
206#335C22×C712yes—
207#333C22×C712yes—
208#332C22×C712yes—
209#332C22×C712yes—
210#331C22×C712yes—
211C23×C7—41no—
212C23×C7#12111yescharacteristic
213C7×Q8—22no—
214C7×Q8—22no—
215C7×Q8—42no—
216C7×Q8—41no—
217C7×Q8—41no—
218C7×Q8—41no—
219C7×Q8—41no—
220C7×Q8—81no—
221C7×D4—41no—
222C7×D4—41no—
223C7×D4—41no—
224C7×D4—41no—
225C7×D4—81no—
226C7×D4—82no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7—82no—
236C2×C4×C7#12111yescharacteristic
237C2×C4×C7#12111yescharacteristic
238C2×C4×C7#12111yescharacteristic
239C2×C4×C7#13512yes—
240C2×C4×C7#13712yes—
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C2×C4×C7C2×#4111yescharacteristic
244C7×C8—21no—
245C7×C8—21no—
246C7×C8—21no—
247C7×C8—21no—
248C7×C8—42no—
249C7×C8—42no—
250C7×C8—42no—
251C2×#41—21no—
252C2×Q16—22no—
253C2×SD16—22no—
254C2×M16—21no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8C7×D411yescharacteristic
258C22×C8C7×D411yescharacteristic
259C4⋊Q8C7×D411yescharacteristic
260C4⋊Q8C7×D411yescharacteristic
261C4⋊Q8C23×C711yescharacteristic
262C4⋊Q8C23×C711yescharacteristic
263C4⋊Q8C23×C711yescharacteristic
264#127C23×C711yescharacteristic
265#125—21no—
266#125—21no—
267#124—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8C7×D412yes—
272#121—21no—
273#121—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8C23×C712yes—
278C4×D4—21no—
279C4×D4—21no—
280#117—21no—
281#117—21no—
282C2×C42C23×C711yescharacteristic
283#107—22no—
284#106—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8C23×C712yes—
291C4⋊C8C23×C712yes—
292C4⋊C8C23×C712yes—
293#103—22no—
294#103—22no—
295#103—21no—
296#103—21no—
297#103—21no—
298#103—21no—
299#103—22no—
300#102—21no—
301#102—21no—
302#102—21no—
303#102—21no—
304#102—22no—
305#97—22no—
306C4×C8—22no—
307C4×C8C7×D411yescharacteristic
308C4×C8C7×D411yescharacteristic
309C4×C8C23×C711yescharacteristic
310C4×C8C23×C711yescharacteristic
311C22×C7—21no—
312C22×C7—21no—
313C22×C7—81no—
314C22×C7—81no—
315C22×C7—81no—
316C22×C7#46412yes—
317C22×C7C2×#12111yescharacteristic
318C4×C7—21no—
319C4×C7—21no—
320C4×C7—21no—
321C4×C7—42no—
322C4×C7—42no—
323C4×C7—81no—
324C4×C7—81no—
325C4×C7—81no—
326C4×C7#46412yes—
327C4×C7C2×#13511yescharacteristic
328C4×C7C2×#13711yescharacteristic
329C4×C7#57611yescharacteristic
330C4×C7#57811yescharacteristic
331#41—42no—
332#41—42no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8C2×C7×D412yes—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—41no—
342C22×C4—41no—
343C22×C4C2×C7×D411yescharacteristic
344C22×C4C2×C7×D411yescharacteristic
345C22×C4C7×#4111yescharacteristic
346Q16—82no—
347SD16—82no—
348M16—42no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—81no—
355C2×C8—81no—
356C2×C8C2×C7×D411yescharacteristic
357C2×C8C2×C7×D411yescharacteristic
358C2×C8C2×C7×D411yescharacteristic
359C2×C8C2×C7×D411yescharacteristic
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4C2×C7×D412yes—
374C22⋊C4—22no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C42—22no—
379C42—41no—
380C42C2×C7×D411yescharacteristic
381C42C2×C7×D411yescharacteristic
382C42C7×#4111yescharacteristic
383C42C24×C711yescharacteristic
384C2×C7—22no—
385C2×C7—161no—
386C2×C7C2×#46411yescharacteristic
387C2×C7#306111yescharacteristic
388C2×C7#306711yescharacteristic
389C23—41no—
390C23C7×#12111yescharacteristic
391Q8—22no—
392Q8—22no—
393Q8—42no—
394Q8—41no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398Q8—81no—
399D4—41no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—81no—
404D4—82no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—82no—
414C2×C4C7×#12111yescharacteristic
415C2×C4C7×#12111yescharacteristic
416C2×C4C7×#12111yescharacteristic
417C2×C4C7×#13512yes—
418C2×C4C7×#13712yes—
419C2×C4C22×C7×D411yescharacteristic
420C2×C4C22×C7×D411yescharacteristic
421C2×C4C2×C7×#4111yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—21no—
426C8—42no—
427C8—42no—
428C8—42no—
429C7#1921711yescharacteristic, direct product factor, Sylow
430C22—21no—
431C22—21no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22C7×#46412yes—
436C22C2×C7×#12111yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4C7×#46412yes—
446C4C2×C7×#13511yescharacteristic
447C4C2×C7×#13711yescharacteristic
448C4C7×#57611yescharacteristic
449C4C7×#57811yescharacteristic
450C2—22no—
451C2—161no—
452C2C2×C7×#46411yescharacteristic
453C2C7×#306111yescharacteristic
454C2C7×#306711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#121
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3249C2semidirect
4. C7×#2960C2semidirect
5. C7×#1700C2semidirect
6. C7×#1698C2semidirect
7. C7×#1684C2semidirect
8. C7×#1682C2semidirect
9. C2×C7×#335C2semidirect
10. C7×#1550C2semidirect
49. #19217C7direct (N × H)
110. #3249C2×C7semidirect
113. #2960C2×C7semidirect
114. #1700C2×C7semidirect
115. #1698C2×C7semidirect
116. #1684C2×C7semidirect
117. #1682C2×C7semidirect
118. C2×#335C2×C7semidirect
119. #1550C2×C7semidirect
429. C7#19217direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32492
C7×#32462
C7×#31312
C7×#29602
C7×#17002
C7×#16982
C7×#16842
C7×#16822
C2×C7×#3352
C7×#15502
C7×#15472
#192177