Finite Group Databasebeta

← C7×#10033 #13187249

Subgroups — C7×#10033 #13187249

Order 1792.

550 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3135C211yesmaximal, characteristic, complemented
2C7×#3101C211yesmaximal, characteristic, complemented
3C7×#3100C211yesmaximal, characteristic
4C7×#1955C211yesmaximal, characteristic, complemented
5C7×#1481C211yesmaximal, characteristic, complemented
6C7×#1480C211yesmaximal, characteristic, complemented
7C7×#1275C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C7×#532C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#491—21no—
12C7×#490—21no—
13C7×#489—21no—
14C7×#489—21no—
15C7×#483—21no—
16C7×#480—21no—
17C7×#476—21no—
18C7×#474—21no—
19C7×#461—21no—
20C7×#459—21no—
21C7×#450—21no—
22C7×#447—21no—
23C4×C7×SD16—22no—
24C4×C7×D8—22no—
25C7×#431C2211yescharacteristic
26C7×#431C2211yescharacteristic
27C2×C7×C4⋊C8C2211yescharacteristic
28C7×#416—22no—
29C2×C7×C22⋊C8—22no—
30C2×C7×#97C2211yescharacteristic
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#10033C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41—21no—
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C2×C7×SD16—22no—
38C2×C7×D8—22no—
39C22×C7×C8—21no—
40C22×C7×C8D412yes—
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×C4⋊Q8D411yescharacteristic
44C7×#127D411yescharacteristic
45C7×#125—21no—
46C7×#124—21no—
47C7×C22⋊Q8—21no—
48C7×C22⋊Q8—21no—
49C7×#121—21no—
50C7×#121—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C7×#117—21no—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C7×C22⋊C4—21no—
62C2×C42×C7C2311yescharacteristic
63C7×#107—22no—
64C7×#106—22no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×#103—22no—
70C7×#103—22no—
71C7×#103—22no—
72C7×#102—22no—
73C7×#102—22no—
74C7×#102—22no—
75C7×#102—22no—
76C7×#102—22no—
77C7×C22⋊C8—21no—
78C7×C22⋊C8—21no—
79C7×C22⋊C8—44no—
80C7×#97—21no—
81C7×#97—21no—
82C4×C7×C8D412yes—
83C4×C7×C8D412yes—
84#3135C2×C711yescharacteristic, complemented
85#3101C2×C711yescharacteristic, complemented
86#3100C2×C711yescharacteristic
87#1955C2×C711yescharacteristic, complemented
88#1481C2×C711yescharacteristic, complemented
89#1480C2×C711yescharacteristic, complemented
90#1275C2×C711yescharacteristic, complemented
91C24×C7—21no—
92C7×#41—42no—
93C7×#41—42no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—42no—
103C2×C7×D4—41no—
104C22×C4×C7—21no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—42no—
109C22×C4×C7—42no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7#4111yescharacteristic
113C7×SD16—44no—
114C7×D8—44no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8—41no—
121C2×C7×C8—42no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C42×C7—41no—
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7C2×D411yescharacteristic
141C42×C7C2×D411yescharacteristic
142#533C22×C711yescharacteristic
143#532C22×C711yescharacteristic
144C2×C4×D4C22×C711yescharacteristic
145#491—21no—
146#490—21no—
147#489—21no—
148#489—21no—
149#483—21no—
150#480—21no—
151#476—21no—
152#474—21no—
153#461—21no—
154#459—21no—
155#450—21no—
156#447—21no—
157C4×SD16—22no—
158C4×D8—22no—
159#431C22×C711yescharacteristic
160#431C22×C711yescharacteristic
161C2×C4⋊C8C22×C711yescharacteristic
162#416—22no—
163C2×C22⋊C8—22no—
164C2×#97C22×C711yescharacteristic
165Q8⋊C8—22no—
166D4⋊C8—22no—
167C23×C7—22no—
168C23×C7—41no—
169C23×C7—42no—
170C23×C7—41no—
171C23×C7—42no—
172C23×C7—41no—
173C23×C7#12111yescharacteristic
174C7×Q8—42no—
175C7×Q8—42no—
176C7×Q8—42no—
177C7×D4—22no—
178C7×D4—22no—
179C7×D4—41no—
180C7×D4—41no—
181C7×D4—42no—
182C7×D4—42no—
183C7×D4—42no—
184C7×D4—82no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—82no—
193C2×C4×C7—81no—
194C2×C4×C7—81no—
195C2×C4×C7—81no—
196C2×C4×C7—81no—
197C2×C4×C7#12011yescharacteristic
198C2×C4×C7#12011yescharacteristic
199C2×C4×C7#12011yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12111yescharacteristic
202C2×C4×C7#12711yescharacteristic
203C2×C4×C7#13512yes—
204C2×C4×C7AGL(1,ℤ/8)12yes—
205C7×C8—42no—
206C7×C8—42no—
207C7×C8—42no—
208C7×C8—81no—
209C2×#41—21no—
210C22×D4C7×D411yescharacteristic
211C23×C4—21no—
212C2×SD16—22no—
213C2×D8—22no—
214C22×C8—21no—
215C22×C8C7×D412yes—
216C4⋊Q8C7×D411yescharacteristic
217C4⋊Q8C7×D411yescharacteristic
218C4⋊Q8C7×D411yescharacteristic
219#127C7×D411yescharacteristic
220#125—21no—
221#124—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224#121—21no—
225#121—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234#117—21no—
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C42C23×C711yescharacteristic
238#107—22no—
239#106—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244#103—22no—
245#103—22no—
246#103—22no—
247#102—22no—
248#102—22no—
249#102—22no—
250#102—22no—
251#102—22no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—44no—
255#97—21no—
256#97—21no—
257C4×C8C7×D412yes—
258C4×C8C7×D412yes—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—44no—
262C22×C7—42no—
263C22×C7—82no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7#39211yescharacteristic
270C22×C7#46312yes—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—81no—
276C4×C7—81no—
277C4×C7—81no—
278C4×C7—81no—
279C4×C7#44912yes—
280C4×C7#49511yescharacteristic
281C4×C7#49611yescharacteristic
282C24—21no—
283#41—42no—
284#41—42no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—42no—
294C2×D4—41no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C2×C7×D411yescharacteristic
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C7×#4111yescharacteristic
304SD16—44no—
305D8—44no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—42no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—41no—
329C42C2×C7×D411yescharacteristic
330C42C2×C7×D411yescharacteristic
331C42C2×C7×D411yescharacteristic
332C42C2×C7×D411yescharacteristic
333C2×C7—22no—
334C2×C7—82no—
335C2×C7—161no—
336C2×C7#201011yescharacteristic
337C2×C7#201211yescharacteristic
338C2×C7#201711yescharacteristic
339C23—22no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23C7×#12111yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—42no—
349D4—22no—
350D4—22no—
351D4—41no—
352D4—41no—
353D4—42no—
354D4—42no—
355D4—42no—
356D4—82no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4C7×#12011yescharacteristic
370C2×C4C7×#12011yescharacteristic
371C2×C4C7×#12011yescharacteristic
372C2×C4C7×#12111yescharacteristic
373C2×C4C7×#12111yescharacteristic
374C2×C4C7×#12711yescharacteristic
375C2×C4C7×#13512yes—
376C2×C4C7×AGL(1,ℤ/8)12yes—
377C8—42no—
378C8—42no—
379C8—42no—
380C8—81no—
381C7#1003311yescharacteristic, direct product factor, Sylow
382C22—21no—
383C22—21no—
384C22—44no—
385C22—42no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C7×#39211yescharacteristic
393C22C7×#46312yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4C7×#44912yes—
403C4C7×#49511yescharacteristic
404C4C7×#49611yescharacteristic
405C2—22no—
406C2—82no—
407C2—161no—
408C2C7×#201011yescharacteristic
409C2C7×#201211yescharacteristic
410C2C7×#201711yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC7×#3100C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3135C2semidirect
2. C7×#3101C2semidirect
4. C7×#1955C2semidirect
5. C7×#1481C2semidirect
6. C7×#1480C2semidirect
7. C7×#1275C2semidirect
33. #10033C7direct (N × H)
84. #3135C2×C7semidirect
85. #3101C2×C7semidirect
87. #1955C2×C7semidirect
88. #1481C2×C7semidirect
89. #1480C2×C7semidirect
90. #1275C2×C7semidirect
381. C7#10033direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31352
C7×#31012
C7×#31002
C7×#19552
C7×#14812
C7×#14802
C7×#12752
#100337