Finite Group Databasebeta

← C7×#19212 #13196428

Subgroups — C7×#19212 #13196428

Order 1792.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3230C211yesmaximal, characteristic, complemented
2C7×#3208C211yesmaximal, characteristic
3C2×C7×#486C211yesmaximal, characteristic
4C7×#2960C211yesmaximal, characteristic, complemented
5C7×#1698C214yesmaximal, complemented
6C7×#1678C214yesmaximal, complemented
7C7×#1564C211yesmaximal, characteristic, complemented
8C7×#1548C211yesmaximal, characteristic, complemented
9C7×#1545C211yesmaximal, characteristic
10C7×#533C2211yescharacteristic
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×C7×#127C2211yescharacteristic
13C7×#500—21no—
14C7×#492—21no—
15C7×#487—21no—
16C7×#486—21no—
17C7×#486—21no—
18C7×#486—21no—
19C7×#486C2212yes—
20C7×#486C2212yes—
21C7×#484—22no—
22C7×#477C2214yescomplemented
23C7×#460C2214yes—
24C7×#450—22no—
25C7×#446—21no—
26C7×#446C2212yescomplemented
27C7×C8×Q8—21no—
28C7×C8×Q8C2212yescomplemented
29C7×#431C2211yescharacteristic
30C7×#431C2211yescharacteristic
31C7×#424—21no—
32C7×#423C2211yescharacteristic
33C2×C7×#102—21no—
34C2×C7×#102—21no—
35C7×#405—21no—
36C2×C4×C7×C8C2211yescharacteristic
37C7×#335C2212yescomplemented
38C7×#334C2212yescomplemented
39C7×#333C2214yescomplemented
40C7×#331C2214yes—
41#19212C711yesmaximal, characteristic, direct product factor, Sylow
42C22×C7×Q8—21no—
43C22×C7×D4—21no—
44C22×C7×D4—21no—
45C22×C7×D4—21no—
46C2×C7×SD16—24no—
47C2×C7×M16—21no—
48C22×C7×C8D411yescharacteristic
49C22×C7×C8D411yescharacteristic
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8D411yescharacteristic
54C7×C4⋊Q8D411yescharacteristic
55C7×C4⋊Q8C2312yes—
56C7×#127—21no—
57C7×#127—21no—
58C7×#127—21no—
59C7×#127C2312yes—
60C7×#125—21no—
61C7×C22⋊Q8—21no—
62C7×C22⋊Q8D412yes—
63C4×C7×Q8—21no—
64C4×C7×Q8C2312yes—
65C7×#117—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C42×C7C2311yescharacteristic
69C7×#107—24no—
70C7×C4⋊C8—22no—
71C7×C4⋊C8—22no—
72C7×C4⋊C8—21no—
73C7×C4⋊C8C2312yes—
74C7×C4⋊C8C2312yes—
75C7×C4⋊C8C2312yes—
76C7×#103—24no—
77C7×#102—22no—
78C7×#102—22no—
79C7×#102—22no—
80C7×#102—22no—
81C7×#102—24no—
82C7×C22⋊C8—21no—
83C7×C22⋊C8—21no—
84C7×#97—22no—
85C4×C7×C8—22no—
86C4×C7×C8—21no—
87C4×C7×C8—21no—
88C4×C7×C8D411yescharacteristic
89C4×C7×C8D411yescharacteristic
90C4×C7×C8C2311yescharacteristic
91C4×C7×C8C2311yescharacteristic
92#3230C2×C711yescharacteristic, complemented
93#3208C2×C711yescharacteristic
94C2×#486C2×C711yescharacteristic
95#2960C2×C711yescharacteristic, complemented
96#1698C2×C714yescomplemented
97#1678C2×C714yescomplemented
98#1564C2×C711yescharacteristic, complemented
99#1548C2×C711yescharacteristic, complemented
100#1545C2×C711yescharacteristic
101C24×C7—41no—
102C2×C7×Q8—21no—
103C2×C7×Q8—22no—
104C2×C7×Q8—41no—
105C2×C7×Q8—82no—
106C2×C7×Q8C2×D412yes—
107C2×C7×D4—22no—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4—41no—
111C2×C7×D4—41no—
112C2×C7×D4—41no—
113C2×C7×D4—42no—
114C2×C7×D4—42no—
115C2×C7×D4—82no—
116C2×C7×D4—82no—
117C22×C4×C7—41no—
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7#4111yescharacteristic
121C7×SD16—84no—
122C7×M16—42no—
123C2×C7×C8—22no—
124C2×C7×C8—22no—
125C2×C7×C8—22no—
126C2×C7×C8—22no—
127C2×C7×C8—22no—
128C2×C7×C8C2×D411yescharacteristic
129C2×C7×C8C2×D411yescharacteristic
130C2×C7×C8C2×D411yescharacteristic
131C2×C7×C8C2×D411yescharacteristic
132C7×C4⋊C4—21no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—41no—
140C7×C4⋊C4—41no—
141C7×C4⋊C4C2×D412yes—
142C7×C22⋊C4—22no—
143C42×C7—22no—
144C42×C7C2×D411yescharacteristic
145C42×C7C2×D411yescharacteristic
146C42×C7#4111yescharacteristic
147C42×C7C2411yescharacteristic
148#533C22×C711yescharacteristic
149C2×C4⋊Q8C22×C711yescharacteristic
150C2×#127C22×C711yescharacteristic
151#500—21no—
152#492—21no—
153#487—21no—
154#486—21no—
155#486—21no—
156#486—21no—
157#486C22×C712yes—
158#486C22×C712yes—
159#484—22no—
160#477C22×C714yescomplemented
161#460C22×C714yes—
162#450—22no—
163#446—21no—
164#446C22×C712yescomplemented
165C8×Q8—21no—
166C8×Q8C22×C712yescomplemented
167#431C22×C711yescharacteristic
168#431C22×C711yescharacteristic
169#424—21no—
170#423C22×C711yescharacteristic
171C2×#102—21no—
172C2×#102—21no—
173#405—21no—
174C2×C4×C8C22×C711yescharacteristic
175#335C22×C712yescomplemented
176#334C22×C712yescomplemented
177#333C22×C714yescomplemented
178#331C22×C714yes—
179C23×C7—42no—
180C23×C7—82no—
181C23×C7—82no—
182C23×C7—82no—
183C23×C7#12111yescharacteristic
184C7×Q8—24no—
185C7×Q8—42no—
186C7×Q8—82no—
187C7×Q8—82no—
188C7×D4—42no—
189C7×D4—42no—
190C7×D4—44no—
191C7×D4—82no—
192C7×D4—82no—
193C7×D4—82no—
194C7×D4—82no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—22no—
199C2×C4×C7—42no—
200C2×C4×C7—82no—
201C2×C4×C7#12111yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×C4×C7#12111yescharacteristic
204C2×C4×C7#13512yes—
205C2×C4×C7#13712yes—
206C2×C4×C7C22×D411yescharacteristic
207C2×C4×C7C22×D411yescharacteristic
208C2×C4×C7C2×#4111yescharacteristic
209C7×C8—21no—
210C7×C8—21no—
211C7×C8—21no—
212C7×C8—21no—
213C7×C8—42no—
214C7×C8—42no—
215C7×C8—42no—
216C22×Q8—21no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—21no—
220C2×SD16—24no—
221C2×M16—21no—
222C22×C8C7×D411yescharacteristic
223C22×C8C7×D411yescharacteristic
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8C7×D411yescharacteristic
228C4⋊Q8C7×D411yescharacteristic
229C4⋊Q8C23×C712yes—
230#127—21no—
231#127—21no—
232#127—21no—
233#127C23×C712yes—
234#125—21no—
235C22⋊Q8—21no—
236C22⋊Q8C7×D412yes—
237C4×Q8—21no—
238C4×Q8C23×C712yes—
239#117—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C42C23×C711yescharacteristic
243#107—24no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8C23×C712yes—
248C4⋊C8C23×C712yes—
249C4⋊C8C23×C712yes—
250#103—24no—
251#102—22no—
252#102—22no—
253#102—22no—
254#102—22no—
255#102—24no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258#97—22no—
259C4×C8—22no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8C7×D411yescharacteristic
263C4×C8C7×D411yescharacteristic
264C4×C8C23×C711yescharacteristic
265C4×C8C23×C711yescharacteristic
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—82no—
269C22×C7—82no—
270C22×C7—82no—
271C22×C7—164no—
272C22×C7#46412yes—
273C22×C7C2×#12111yescharacteristic
274C4×C7—21no—
275C4×C7—21no—
276C4×C7—42no—
277C4×C7—42no—
278C4×C7—82no—
279C4×C7#46412yes—
280C4×C7#47212yes—
281C4×C7C2×#13511yescharacteristic
282C4×C7C2×#13711yescharacteristic
283C4×C7#57712yes—
284C24—41no—
285C2×Q8—21no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—82no—
289C2×Q8C2×C7×D412yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—82no—
299C2×D4—82no—
300C22×C4—41no—
301C22×C4C2×C7×D411yescharacteristic
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C7×#4111yescharacteristic
304SD16—84no—
305M16—42no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8C2×C7×D411yescharacteristic
312C2×C8C2×C7×D411yescharacteristic
313C2×C8C2×C7×D411yescharacteristic
314C2×C8C2×C7×D411yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4C2×C7×D412yes—
325C22⋊C4—22no—
326C42—22no—
327C42C2×C7×D411yescharacteristic
328C42C2×C7×D411yescharacteristic
329C42C7×#4111yescharacteristic
330C42C24×C711yescharacteristic
331C2×C7—22no—
332C2×C7—162no—
333C2×C7C2×#46411yescharacteristic
334C2×C7#305711yescharacteristic
335C2×C7C2×#47211yescharacteristic
336C23—42no—
337C23—82no—
338C23—82no—
339C23—82no—
340C23C7×#12111yescharacteristic
341Q8—24no—
342Q8—42no—
343Q8—82no—
344Q8—82no—
345D4—42no—
346D4—42no—
347D4—44no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—82no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4C7×#12111yescharacteristic
359C2×C4C7×#12111yescharacteristic
360C2×C4C7×#12111yescharacteristic
361C2×C4C7×#13512yes—
362C2×C4C7×#13712yes—
363C2×C4C22×C7×D411yescharacteristic
364C2×C4C22×C7×D411yescharacteristic
365C2×C4C2×C7×#4111yescharacteristic
366C8—21no—
367C8—21no—
368C8—21no—
369C8—21no—
370C8—42no—
371C8—42no—
372C8—42no—
373C7#1921211yescharacteristic, direct product factor, Sylow
374C22—21no—
375C22—21no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—164no—
380C22C7×#46412yes—
381C22C2×C7×#12111yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4C7×#46412yes—
388C4C7×#47212yes—
389C4C2×C7×#13511yescharacteristic
390C4C2×C7×#13711yescharacteristic
391C4C7×#57712yes—
392C2—22no—
393C2—162no—
394C2C2×C7×#46411yescharacteristic
395C2C7×#305711yescharacteristic
396C2C2×C7×#47211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#121
commutator, Frattini, cosocleC42C24×C7
prime radicalC2×C7×#127C22
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×#3230C2semidirect
4. C7×#2960C2semidirect
5. C7×#1698C2semidirect
6. C7×#1678C2semidirect
7. C7×#1564C2semidirect
8. C7×#1548C2semidirect
22. C7×#477C22semidirect
26. C7×#446C22semidirect
28. C7×C8×Q8C22semidirect
37. C7×#335C22semidirect
38. C7×#334C22semidirect
39. C7×#333C22semidirect
41. #19212C7direct (N × H)
92. #3230C2×C7semidirect
95. #2960C2×C7semidirect
96. #1698C2×C7semidirect
97. #1678C2×C7semidirect
98. #1564C2×C7semidirect
99. #1548C2×C7semidirect
160. #477C22×C7semidirect
164. #446C22×C7semidirect
166. C8×Q8C22×C7semidirect
175. #335C22×C7semidirect
176. #334C22×C7semidirect
177. #333C22×C7semidirect
373. C7#19212direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32302
C7×#32082
C2×C7×#4862
C7×#29602
C7×#16982
C7×#16782
C7×#15642
C7×#15482
C7×#15452
#192127