Finite Group Databasebeta

← C7×#10058 #13187274

Subgroups — C7×#10058 #13187274

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#487C211yesmaximal, characteristic, complemented
2C2×C7×#476C211yesmaximal, characteristic
3C7×#2547C211yesmaximal, characteristic
4C7×#1987C211yesmaximal, characteristic, complemented
5C2×C7×#332C211yesmaximal, characteristic, complemented
6C2×C7×D4⋊C8C211yesmaximal, characteristic
7C7×#1279C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#124—21no—
11C2×C4×C7×D4C2211yescharacteristic
12C7×#487—22no—
13C7×#487—22no—
14C7×#476—22no—
15C7×#476—22no—
16C2×C7×#106—21no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#103—22no—
20C2×C7×#102—21no—
21C2×C4×C7×C8C2211yescharacteristic
22C7×#397—22no—
23C7×#391—21no—
24C7×#380—22no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C7×#340—22no—
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10058C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8—21no—
32C22×C7×D4D411yescharacteristic
33C23×C4×C7—21no—
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C22×C7×C8—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8D412yes—
45C7×#124—42no—
46C7×#124—42no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4D412yes—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D412yes—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—21no—
59C2×C42×C7—21no—
60C2×C42×C7C2311yescharacteristic
61C7×#106—24no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8D412yes—
66C7×#103—28no—
67C7×#102—24no—
68C4×C7×C8—21no—
69C4×C7×C8D412yes—
70C7×#95—24no—
71C7×#95—21no—
72C2×#487C2×C711yescharacteristic, complemented
73C2×#476C2×C711yescharacteristic
74#2547C2×C711yescharacteristic
75#1987C2×C711yescharacteristic, complemented
76C2×#332C2×C711yescharacteristic, complemented
77C2×D4⋊C8C2×C711yescharacteristic
78#1279C2×C711yescharacteristic, complemented
79C24×C7—21no—
80C2×C7×Q8—22no—
81C2×C7×Q8—44no—
82C2×C7×Q8—41no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—41no—
87C2×C7×Q8—41no—
88C2×C7×Q8—41no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4SD1612yes—
95C22×C4×C7—21no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7—41no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C2×C7×C8—22no—
107C2×C7×C8—22no—
108C2×C7×C8—42no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C2×C7×C8—42no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C22⋊C4—48no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C42×C7—41no—
129C42×C7—41no—
130C42×C7C2×D411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7#4111yescharacteristic
134C2×C4⋊Q8C22×C711yescharacteristic
135C2×C4⋊Q8C22×C711yescharacteristic
136C2×#124—21no—
137C2×C4×D4C22×C711yescharacteristic
138#487—22no—
139#487—22no—
140#476—22no—
141#476—22no—
142C2×#106—21no—
143C2×C4⋊C8C22×C711yescharacteristic
144C2×C4⋊C8C22×C711yescharacteristic
145C2×#103—22no—
146C2×#102—21no—
147C2×C4×C8C22×C711yescharacteristic
148#397—22no—
149#391—21no—
150#380—22no—
151C4×C4⋊C4C22×C711yescharacteristic
152#340—22no—
153#332—22no—
154D4⋊C8—22no—
155C23×C7—22no—
156C23×C7—41no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7#12011yescharacteristic
161C7×Q8—44no—
162C7×Q8—82no—
163C7×Q8—81no—
164C7×Q8—81no—
165C7×Q8—81no—
166C7×Q8—81no—
167C7×D4—22no—
168C7×D4—22no—
169C7×D4—41no—
170C7×D4—41no—
171C2×C4×C7—22no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—82no—
182C2×C4×C7—82no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12011yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7#12111yescharacteristic
190C2×C4×C7#12711yescharacteristic
191C2×C4×C7C2×SD1611yescharacteristic
192C2×C4×C7#13511yescharacteristic
193C2×C4×C7#13512yes—
194C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C7#13711yescharacteristic
196C2×C4×C7#13712yes—
197C7×C8—42no—
198C7×C8—42no—
199C7×C8—81no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×D4C7×D411yescharacteristic
203C23×C4—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C7×D412yes—
215#124—42no—
216#124—42no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C7×D411yescharacteristic
225C2×C4⋊C4C7×D412yes—
226C2×C4⋊C4C7×D412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—21no—
229C2×C42—21no—
230C2×C42C23×C711yescharacteristic
231#106—24no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8C7×D412yes—
236#103—28no—
237#102—24no—
238C4×C8—21no—
239C4×C8C7×D412yes—
240#95—24no—
241#95—21no—
242C22×C7—44no—
243C22×C7—42no—
244C22×C7—82no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7#39211yescharacteristic
248C22×C7#45011yescharacteristic
249C22×C7#45212yes—
250C22×C7#45211yescharacteristic
251C22×C7#45412yes—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—82no—
255C4×C7—82no—
256C4×C7—82no—
257C4×C7—81no—
258C4×C7#44811yescharacteristic
259C4×C7#44911yescharacteristic
260C4×C7#46412yes—
261C4×C7#46511yescharacteristic
262C4×C7#46811yescharacteristic
263C4×C7#49511yescharacteristic
264C4×C7#49711yescharacteristic
265C24—21no—
266C2×Q8—22no—
267C2×Q8—44no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4C7×SD1612yes—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291C22×C4C2×C7×D411yescharacteristic
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C22⋊C4—48no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C42—41no—
315C42—41no—
316C42C2×C7×D411yescharacteristic
317C42C2×C7×D411yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C7×#4111yescharacteristic
320C2×C7—82no—
321C2×C7#163512yes—
322C2×C7#163812yes—
323C2×C7#200011yescharacteristic
324C2×C7#200311yescharacteristic
325C2×C7#201511yescharacteristic
326C23—22no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23C7×#12011yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338D4—22no—
339D4—22no—
340D4—41no—
341D4—41no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4C7×#12011yescharacteristic
357C2×C4C7×#12011yescharacteristic
358C2×C4C7×#12111yescharacteristic
359C2×C4C7×#12111yescharacteristic
360C2×C4C7×#12111yescharacteristic
361C2×C4C7×#12711yescharacteristic
362C2×C4C2×C7×SD1611yescharacteristic
363C2×C4C7×#13512yes—
364C2×C4C7×#13511yescharacteristic
365C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
366C2×C4C7×#13711yescharacteristic
367C2×C4C7×#13712yes—
368C8—42no—
369C8—42no—
370C8—81no—
371C7#1005811yescharacteristic, direct product factor, Sylow
372C22—42no—
373C22—44no—
374C22—81no—
375C22—81no—
376C22—82no—
377C22C7×#39211yescharacteristic
378C22C7×#45011yescharacteristic
379C22C7×#45211yescharacteristic
380C22C7×#45212yes—
381C22C7×#45412yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—81no—
387C4—82no—
388C4C7×#44811yescharacteristic
389C4C7×#44911yescharacteristic
390C4C7×#46412yes—
391C4C7×#46511yescharacteristic
392C4C7×#46811yescharacteristic
393C4C7×#49511yescharacteristic
394C4C7×#49711yescharacteristic
395C2—82no—
396C2C7×#163512yes—
397C2C7×#163812yes—
398C2C7×#200011yescharacteristic
399C2C7×#200311yescharacteristic
400C2C7×#201511yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#487C2semidirect
4. C7×#1987C2semidirect
5. C2×C7×#332C2semidirect
7. C7×#1279C2semidirect
29. #10058C7direct (N × H)
72. C2×#487C2×C7semidirect
75. #1987C2×C7semidirect
76. C2×#332C2×C7semidirect
78. #1279C2×C7semidirect
371. C7#10058direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4872
C2×C7×#4762
C7×#25472
C7×#19872
C2×C7×#3322
C2×C7×D4⋊C82
C7×#12792
#100587