Finite Group Databasebeta

← C7×#10072 #13187288

Subgroups — C7×#10072 #13187288

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#487C211yesmaximal, characteristic, complemented
2C2×C7×#474C211yesmaximal, characteristic
3C7×#2563C211yesmaximal, characteristic
4C7×#1981C211yesmaximal, characteristic, complemented
5C2×C7×#333C211yesmaximal, characteristic, complemented
6C2×C7×D4⋊C8C211yesmaximal, characteristic
7C7×#1279C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#487—22no—
12C7×#487—22no—
13C7×#474—22no—
14C7×#474—22no—
15C2×C7×#107—21no—
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#103—22no—
19C2×C7×#103—22no—
20C2×C7×#102—21no—
21C2×C4×C7×C8C2211yescharacteristic
22C7×#393—22no—
23C7×#390—21no—
24C7×#382—21no—
25C7×#380—22no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#333—22no—
28C7×D4⋊C8—22no—
29#10072C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8D412yes—
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—
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4D412yes—
54C2×C7×C4⋊C4D411yescharacteristic
55C2×C7×C22⋊C4—21no—
56C2×C7×C22⋊C4—24no—
57C2×C42×C7—21no—
58C2×C42×C7C2311yescharacteristic
59C7×#107—24no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8D412yes—
64C7×#103—28no—
65C7×#103—44no—
66C7×#102—24no—
67C4×C7×C8—21no—
68C4×C7×C8D412yes—
69C7×#95—24no—
70C7×#95—21no—
71C2×#487C2×C711yescharacteristic, complemented
72C2×#474C2×C711yescharacteristic
73#2563C2×C711yescharacteristic
74#1981C2×C711yescharacteristic, complemented
75C2×#333C2×C711yescharacteristic, complemented
76C2×D4⋊C8C2×C711yescharacteristic
77#1279C2×C711yescharacteristic, complemented
78C24×C7—21no—
79C2×C7×Q8—22no—
80C2×C7×Q8—22no—
81C2×C7×Q8—22no—
82C2×C7×Q8—44no—
83C2×C7×Q8—41no—
84C2×C7×Q8—42no—
85C2×C7×Q8—42no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4Q1612yes—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—41no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7—42no—
99C22×C4×C7—42no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C2×C7×C8—22no—
104C2×C7×C8—22no—
105C2×C7×C8—42no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8—42no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—84no—
125C42×C7—41no—
126C42×C7—41no—
127C42×C7C2×D411yescharacteristic
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C2×C4⋊Q8C22×C711yescharacteristic
132C2×C4⋊Q8C22×C711yescharacteristic
133C2×C4×D4C22×C711yescharacteristic
134#487—22no—
135#487—22no—
136#474—22no—
137#474—22no—
138C2×#107—21no—
139C2×C4⋊C8C22×C711yescharacteristic
140C2×C4⋊C8C22×C711yescharacteristic
141C2×#103—22no—
142C2×#103—22no—
143C2×#102—21no—
144C2×C4×C8C22×C711yescharacteristic
145#393—22no—
146#390—21no—
147#382—21no—
148#380—22no—
149C4×C4⋊C4C22×C711yescharacteristic
150#333—22no—
151D4⋊C8—22no—
152C23×C7—22no—
153C23×C7—41no—
154C23×C7—42no—
155C23×C7—41no—
156C23×C7—42no—
157C23×C7#12011yescharacteristic
158C7×Q8—44no—
159C7×Q8—44no—
160C7×Q8—82no—
161C7×Q8—81no—
162C7×Q8—81no—
163C7×D4—22no—
164C7×D4—22no—
165C7×D4—41no—
166C7×D4—41no—
167C2×C4×C7—22no—
168C2×C4×C7—41no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—44no—
176C2×C4×C7—82no—
177C2×C4×C7—82no—
178C2×C4×C7—81no—
179C2×C4×C7—81no—
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12111yescharacteristic
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12711yescharacteristic
186C2×C4×C7C2×Q1611yescharacteristic
187C2×C4×C7#13511yescharacteristic
188C2×C4×C7#13512yes—
189C2×C4×C7AGL(1,ℤ/8)12yes—
190C2×C4×C7#13711yescharacteristic
191C2×C4×C7#13711yescharacteristic
192C7×C8—42no—
193C7×C8—42no—
194C7×C8—81no—
195C22×Q8—21no—
196C22×Q8C7×D412yes—
197C22×D4C7×D411yescharacteristic
198C23×C4—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C7×D412yes—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C7×D411yescharacteristic
219C2×C4⋊C4C7×D412yes—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—24no—
222C2×C42—21no—
223C2×C42C23×C711yescharacteristic
224#107—24no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8C7×D412yes—
229#103—28no—
230#103—44no—
231#102—24no—
232C4×C8—21no—
233C4×C8C7×D412yes—
234#95—24no—
235#95—21no—
236C22×C7—44no—
237C22×C7—42no—
238C22×C7—82no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7#39211yescharacteristic
242C22×C7#44912yes—
243C22×C7#45111yescharacteristic
244C22×C7#45211yescharacteristic
245C22×C7#45512yes—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—82no—
249C4×C7—82no—
250C4×C7—81no—
251C4×C7—82no—
252C4×C7#45111yescharacteristic
253C4×C7#45211yescharacteristic
254C4×C7#46312yes—
255C4×C7#46711yescharacteristic
256C4×C7#47011yescharacteristic
257C4×C7#49511yescharacteristic
258C4×C7#49611yescharacteristic
259C24—21no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—44no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4C7×Q1612yes—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4C2×C7×D411yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283C22×C4C2×C7×D411yescharacteristic
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—84no—
306C42—41no—
307C42—41no—
308C42C2×C7×D411yescharacteristic
309C42C2×C7×D411yescharacteristic
310C42C2×C7×D411yescharacteristic
311C42C7×#4111yescharacteristic
312C2×C7—82no—
313C2×C7#164412yes—
314C2×C7#164612yes—
315C2×C7#200111yescharacteristic
316C2×C7#200411yescharacteristic
317C2×C7#201211yescharacteristic
318C23—22no—
319C23—41no—
320C23—42no—
321C23—41no—
322C23—42no—
323C23C7×#12011yescharacteristic
324Q8—44no—
325Q8—44no—
326Q8—82no—
327Q8—81no—
328Q8—81no—
329D4—22no—
330D4—22no—
331D4—41no—
332D4—41no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4C7×#12011yescharacteristic
347C2×C4C7×#12011yescharacteristic
348C2×C4C7×#12111yescharacteristic
349C2×C4C7×#12111yescharacteristic
350C2×C4C7×#12111yescharacteristic
351C2×C4C7×#12711yescharacteristic
352C2×C4C2×C7×Q1611yescharacteristic
353C2×C4C7×#13511yescharacteristic
354C2×C4C7×#13512yes—
355C2×C4C7×AGL(1,ℤ/8)12yes—
356C2×C4C7×#13711yescharacteristic
357C2×C4C7×#13711yescharacteristic
358C8—42no—
359C8—42no—
360C8—81no—
361C7#1007211yescharacteristic, direct product factor, Sylow
362C22—42no—
363C22—44no—
364C22—81no—
365C22—81no—
366C22—82no—
367C22C7×#39211yescharacteristic
368C22C7×#44912yes—
369C22C7×#45111yescharacteristic
370C22C7×#45211yescharacteristic
371C22C7×#45512yes—
372C4—21no—
373C4—21no—
374C4—82no—
375C4—81no—
376C4—82no—
377C4—82no—
378C4C7×#45111yescharacteristic
379C4C7×#45211yescharacteristic
380C4C7×#46312yes—
381C4C7×#46711yescharacteristic
382C4C7×#47011yescharacteristic
383C4C7×#49511yescharacteristic
384C4C7×#49611yescharacteristic
385C2—82no—
386C2C7×#164412yes—
387C2C7×#164612yes—
388C2C7×#200111yescharacteristic
389C2C7×#200411yescharacteristic
390C2C7×#201211yescharacteristic
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×#1981C2semidirect
5. C2×C7×#333C2semidirect
7. C7×#1279C2semidirect
29. #10072C7direct (N × H)
71. C2×#487C2×C7semidirect
74. #1981C2×C7semidirect
75. C2×#333C2×C7semidirect
77. #1279C2×C7semidirect
361. C7#10072direct (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×#4742
C7×#25632
C7×#19812
C2×C7×#3332
C2×C7×D4⋊C82
C7×#12792
#100727