Finite Group Databasebeta

← C7×#10068 #13187284

Subgroups — C7×#10068 #13187284

Order 1792.

670 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#498C211yesmaximal, characteristic, complemented
2C2×C7×#461C211yesmaximal, characteristic
3C7×#2542C211yesmaximal, characteristic
4C7×#1988C211yesmaximal, characteristic, complemented
5C2×C7×#332C211yesmaximal, characteristic, complemented
6C2×C7×D4⋊C8C211yesmaximal, characteristic
7C7×#1279C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×C22⋊Q8—22no—
12C2×C4×C7×D4C2211yescharacteristic
13C7×#498—22no—
14C7×#498—22no—
15C7×#461—22no—
16C7×#461—22no—
17C2×C7×#106—22no—
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#103—21no—
21C2×C4×C7×C8C2211yescharacteristic
22C7×#399—22no—
23C7×#390—21no—
24C7×#384—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C7×#340—22no—
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10068C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8—21no—
32C22×C7×D4D411yescharacteristic
33C23×C4×C7—21no—
34C2×C7×SD16—24no—
35C2×C7×SD16—42no—
36C2×C7×SD16—42no—
37C22×C7×C8—21no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8D412yes—
48C7×C22⋊Q8—44no—
49C7×C22⋊Q8—44no—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4D412yes—
61C2×C7×C22⋊C4—24no—
62C2×C7×C22⋊C4—21no—
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65C7×#106—28no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8D412yes—
70C7×#103—24no—
71C4×C7×C8—21no—
72C4×C7×C8D412yes—
73C7×#95—21no—
74C2×#498C2×C711yescharacteristic, complemented
75C2×#461C2×C711yescharacteristic
76#2542C2×C711yescharacteristic
77#1988C2×C711yescharacteristic, complemented
78C2×#332C2×C711yescharacteristic, complemented
79C2×D4⋊C8C2×C711yescharacteristic
80#1279C2×C711yescharacteristic, complemented
81C24×C7—21no—
82C2×C7×Q8—22no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C2×C7×Q8—41no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—21no—
97C2×C7×D4SD1612yes—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—41no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7—42no—
105C22×C4×C7—42no—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C7×SD16—48no—
110C2×C7×C8—24no—
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—41no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—42no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—48no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4SD1614yescomplemented
130C7×C22⋊C4—48no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7#4111yescharacteristic
139C22×SD16—21no—
140C2×C4⋊Q8C22×C711yescharacteristic
141C2×C4⋊Q8C22×C711yescharacteristic
142C2×C22⋊Q8—22no—
143C2×C4×D4C22×C711yescharacteristic
144#498—22no—
145#498—22no—
146#461—22no—
147#461—22no—
148C2×#106—22no—
149C2×C4⋊C8C22×C711yescharacteristic
150C2×C4⋊C8C22×C711yescharacteristic
151C2×#103—21no—
152C2×C4×C8C22×C711yescharacteristic
153#399—22no—
154#390—21no—
155#384—21no—
156C4×C4⋊C4C22×C711yescharacteristic
157#340—22no—
158#332—22no—
159D4⋊C8—22no—
160C23×C7—22no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—42no—
164C23×C7—42no—
165C23×C7#12011yescharacteristic
166C7×Q8—42no—
167C7×Q8—42no—
168C7×Q8—82no—
169C7×Q8—81no—
170C7×Q8—81no—
171C7×Q8—81no—
172C7×Q8—81no—
173C7×D4—22no—
174C7×D4—22no—
175C7×D4—41no—
176C7×D4—41no—
177C2×C4×C7—22no—
178C2×C4×C7—24no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—44no—
185C2×C4×C7—42no—
186C2×C4×C7—81no—
187C2×C4×C7—81no—
188C2×C4×C7—82no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12711yescharacteristic
197C2×C4×C7C2×SD1611yescharacteristic
198C2×C4×C7C2×SD1612yes—
199C2×C4×C7#13511yescharacteristic
200C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C7AGL(1,ℤ/8)12yes—
202C2×C4×C7#13711yescharacteristic
203C7×C8—24no—
204C7×C8—42no—
205C7×C8—81no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×D4C7×D411yescharacteristic
209C23×C4—21no—
210C2×SD16—24no—
211C2×SD16—42no—
212C2×SD16—42no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8C7×D412yes—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C7×D411yescharacteristic
235C2×C4⋊C4C7×D412yes—
236C2×C4⋊C4C7×D412yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—21no—
239C2×C42—21no—
240C2×C42C23×C711yescharacteristic
241#106—28no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8C7×D412yes—
246#103—24no—
247C4×C8—21no—
248C4×C8C7×D412yes—
249#95—21no—
250C22×C7—44no—
251C22×C7—42no—
252C22×C7—82no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7#39211yescharacteristic
256C22×C7#44811yescharacteristic
257C22×C7#44911yescharacteristic
258C22×C7#45012yes—
259C22×C7#45612yes—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—44no—
263C4×C7—82no—
264C4×C7—81no—
265C4×C7—82no—
266C4×C7#45011yescharacteristic
267C4×C7#45211yescharacteristic
268C4×C7#46012yes—
269C4×C7#46511yescharacteristic
270C4×C7#46811yescharacteristic
271C4×C7#49211yescharacteristic
272C4×C7#49611yescharacteristic
273C24—21no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C7×SD1612yes—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×C7×D411yescharacteristic
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C2×C7×D411yescharacteristic
301SD16—48no—
302C2×C8—24no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—48no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4C7×SD1614yescomplemented
322C22⋊C4—48no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—41no—
326C42—41no—
327C42C2×C7×D411yescharacteristic
328C42C2×C7×D411yescharacteristic
329C42C2×C7×D411yescharacteristic
330C42C7×#4111yescharacteristic
331C2×C7—82no—
332C2×C7#165112yes—
333C2×C7#165412yes—
334C2×C7#200011yescharacteristic
335C2×C7#200511yescharacteristic
336C2×C7#201411yescharacteristic
337C23—22no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23C7×#12011yescharacteristic
343Q8—42no—
344Q8—42no—
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—82no—
349Q8—81no—
350D4—22no—
351D4—22no—
352D4—41no—
353D4—41no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4C7×#12011yescharacteristic
369C2×C4C7×#12011yescharacteristic
370C2×C4C7×#12111yescharacteristic
371C2×C4C7×#12111yescharacteristic
372C2×C4C7×#12111yescharacteristic
373C2×C4C7×#12711yescharacteristic
374C2×C4C2×C7×SD1612yes—
375C2×C4C2×C7×SD1611yescharacteristic
376C2×C4C7×#13511yescharacteristic
377C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C7×AGL(1,ℤ/8)12yes—
379C2×C4C7×#13711yescharacteristic
380C8—24no—
381C8—42no—
382C8—81no—
383C7#1006811yescharacteristic, direct product factor, Sylow
384C22—42no—
385C22—44no—
386C22—81no—
387C22—81no—
388C22—82no—
389C22C7×#39211yescharacteristic
390C22C7×#44811yescharacteristic
391C22C7×#44911yescharacteristic
392C22C7×#45012yes—
393C22C7×#45612yes—
394C4—21no—
395C4—21no—
396C4—44no—
397C4—82no—
398C4—81no—
399C4—82no—
400C4C7×#45011yescharacteristic
401C4C7×#45211yescharacteristic
402C4C7×#46012yes—
403C4C7×#46511yescharacteristic
404C4C7×#46811yescharacteristic
405C4C7×#49211yescharacteristic
406C4C7×#49611yescharacteristic
407C2—82no—
408C2C7×#165112yes—
409C2C7×#165412yes—
410C2C7×#200011yescharacteristic
411C2C7×#200511yescharacteristic
412C2C7×#201411yescharacteristic
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×#498C2semidirect
4. C7×#1988C2semidirect
5. C2×C7×#332C2semidirect
7. C7×#1279C2semidirect
29. #10068C7direct (N × H)
74. C2×#498C2×C7semidirect
77. #1988C2×C7semidirect
78. C2×#332C2×C7semidirect
80. #1279C2×C7semidirect
129. C7×C4⋊C4SD16semidirect
321. C4⋊C4C7×SD16semidirect
383. C7#10068direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4982
C2×C7×#4612
C7×#25422
C7×#19882
C2×C7×#3322
C2×C7×D4⋊C82
C7×#12792
#100687