Finite Group Databasebeta

← C7×#9123 #13186339

Subgroups — C7×#9123 #13186339

Order 1792.

600 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#432C211yesmaximal, characteristic, complemented
2C7×#2850C211yesmaximal, characteristic
3C7×#1749C211yesmaximal, characteristic, complemented
4C2×C7×#329C212yesmaximal, complemented
5C7×#1295C212yesmaximal, complemented
6C2×C7×#124C2211yescharacteristic
7C2×C7×C22⋊Q8C411yescharacteristic
8C2×C7×#121C411yescharacteristic
9C22×C42×C7C2211yescharacteristic
10C7×#432—24no—
11C2×C7×C4⋊C8—21no—
12C2×C7×C22⋊C8—21no—
13C2×C7×C22⋊C8—21no—
14C2×C7×#97C2212yes—
15C2×C7×#97C2212yes—
16C7×#391C411yescharacteristic
17C7×#390C411yescharacteristic
18C7×#388—21no—
19C7×#388—21no—
20C7×#386—21no—
21C7×#386—21no—
22C7×#382C2211yescharacteristic
23C7×#336—21no—
24C7×#329—24no—
25#9123C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C22×C7×C8—22no—
33C22×C7×C8—22no—
34C7×#124—22no—
35C7×#124—22no—
36C7×C22⋊Q8—42no—
37C7×#121—42no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C42×C7—22no—
48C2×C42×C7—22no—
49C2×C42×C7C2×C411yescharacteristic
50C2×C42×C7D411yescharacteristic
51C2×C42×C7D411yescharacteristic
52C2×C42×C7C2311yescharacteristic
53C7×C4⋊C8—24no—
54C7×C22⋊C8—24no—
55C7×C22⋊C8—24no—
56C7×#97—22no—
57C7×#97—22no—
58C7×#97—22no—
59C7×#97D414yescomplemented
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C2×#432C2×C711yescharacteristic, complemented
65#2850C2×C711yescharacteristic
66#1749C2×C711yescharacteristic, complemented
67C2×#329C2×C712yescomplemented
68#1295C2×C712yescomplemented
69C24×C7—21no—
70C24×C7C22⋊C411yescharacteristic
71C2×C7×Q8—22no—
72C2×C7×Q8—41no—
73C2×C7×Q8—41no—
74C2×C7×D4—22no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C22×C4×C7—22no—
81C22×C4×C7—22no—
82C22×C4×C7—22no—
83C22×C4×C7—21no—
84C22×C4×C7—22no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—41no—
90C22×C4×C7—41no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C2×C7×C8—24no—
100C2×C7×C8—24no—
101C2×C7×C8—42no—
102C2×C7×C8—42no—
103C2×C7×C8—42no—
104C2×C7×C8—42no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—42no—
108C7×C22⋊C4—42no—
109C7×C22⋊C4—42no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—41no—
115C42×C7—22no—
116C42×C7—21no—
117C42×C7—22no—
118C42×C7—21no—
119C42×C7C2×D412yes—
120C42×C7#4112yes—
121C2×#124C22×C711yescharacteristic
122C2×C22⋊Q8C4×C711yescharacteristic
123C2×#121C4×C711yescharacteristic
124C22×C42C22×C711yescharacteristic
125#432—24no—
126C2×C4⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×#97C22×C712yes—
130C2×#97C22×C712yes—
131#391C4×C711yescharacteristic
132#390C4×C711yescharacteristic
133#388—21no—
134#388—21no—
135#386—21no—
136#386—21no—
137#382C22×C711yescharacteristic
138#336—21no—
139#329—24no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7#10412yes—
154C23×C7C2×C22⋊C411yescharacteristic
155C7×Q8—82no—
156C7×D4—82no—
157C7×D4—84no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—44no—
173C2×C4×C7—44no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7#10012yes—
186C2×C4×C7C4×D412yes—
187C2×C4×C7#12011yescharacteristic
188C2×C4×C7#12112yes—
189C2×C4×C7#12311yescharacteristic
190C7×C8—44no—
191C7×C8—44no—
192C22×Q8C2×C4×C711yescharacteristic
193C22×D4—21no—
194C22×D4C2×C4×C711yescharacteristic
195C23×C4C2×C4×C711yescharacteristic
196C23×C4C7×D411yescharacteristic
197C23×C4C7×D411yescharacteristic
198C22×C8—22no—
199C22×C8—22no—
200#124—22no—
201#124—22no—
202C22⋊Q8—42no—
203#121—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×C4×C711yescharacteristic
206C2×C4⋊C4C2×C4×C711yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C2×C4×C711yescharacteristic
216C2×C42C7×D411yescharacteristic
217C2×C42C7×D411yescharacteristic
218C2×C42C23×C711yescharacteristic
219C4⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222#97—22no—
223#97—22no—
224#97—22no—
225#97C7×D414yescomplemented
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230C22×C7—22no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7#38611yescharacteristic
245C22×C7C2×#10011yescharacteristic
246C22×C7#41311yescharacteristic
247C22×C7C2×#10411yescharacteristic
248C22×C7#42111yescharacteristic
249C22×C7#42111yescharacteristic
250C22×C7#42111yescharacteristic
251C4×C7—22no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—22no—
255C4×C7—22no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—81no—
259C4×C7—81no—
260C4×C7—81no—
261C24—21no—
262C24C7×C22⋊C411yescharacteristic
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C7×C22⋊C411yescharacteristic
286C22×C4C7×C22⋊C411yescharacteristic
287C22×C4C7×C22⋊C411yescharacteristic
288C22×C4C22×C4×C711yescharacteristic
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C42—22no—
308C42—21no—
309C42—22no—
310C42—21no—
311C42C2×C7×D412yes—
312C42C7×#4112yes—
313C2×C7—42no—
314C2×C7—161no—
315C2×C7#152711yescharacteristic
316C2×C7#153211yescharacteristic
317C2×C7#153511yescharacteristic
318C2×C7#153611yescharacteristic
319C2×C7#187111yescharacteristic
320C2×C7#189711yescharacteristic
321C2×C7#190011yescharacteristic
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23C7×#10412yes—
336C23C2×C7×C22⋊C411yescharacteristic
337Q8—82no—
338D4—82no—
339D4—84no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4C7×#10012yes—
368C2×C4C4×C7×D412yes—
369C2×C4C7×#12011yescharacteristic
370C2×C4C7×#12112yes—
371C2×C4C7×#12311yescharacteristic
372C8—44no—
373C8—44no—
374C7#912311yescharacteristic, direct product factor, Sylow
375C22—22no—
376C22—22no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C7×#38611yescharacteristic
390C22C2×C7×#10011yescharacteristic
391C22C7×#41311yescharacteristic
392C22C2×C7×#10411yescharacteristic
393C22C7×#42111yescharacteristic
394C22C7×#42111yescharacteristic
395C22C7×#42111yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—81no—
404C4—81no—
405C4—81no—
406C2—42no—
407C2—161no—
408C2C7×#152711yescharacteristic
409C2C7×#153211yescharacteristic
410C2C7×#153511yescharacteristic
411C2C7×#153611yescharacteristic
412C2C7×#187111yescharacteristic
413C2C7×#189711yescharacteristic
414C2C7×#190011yescharacteristic
subgroupG/H
center, socleC23×C7C2×C22⋊C4
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C22×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#432C2semidirect
3. C7×#1749C2semidirect
4. C2×C7×#329C2semidirect
5. C7×#1295C2semidirect
25. #9123C7direct (N × H)
59. C7×#97D4semidirect
64. C2×#432C2×C7semidirect
66. #1749C2×C7semidirect
67. C2×#329C2×C7semidirect
68. #1295C2×C7semidirect
225. #97C7×D4semidirect
374. C7#9123direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4322
C7×#28502
C7×#17492
C2×C7×#3292
C7×#12952
#91237