Finite Group Databasebeta

← C7×#10268 #13187484

Subgroups — C7×#10268 #13187484

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#492C211yesmaximal, characteristic
2C2×C7×#475C211yesmaximal, characteristic, complemented
3C2×C7×#460C211yesmaximal, characteristic
4C7×#1984C211yesmaximal, characteristic, complemented
5C2×C7×#332C211yesmaximal, characteristic, complemented
6C2×C7×#331C211yesmaximal, characteristic
7C7×#1277C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C22×C7×SD16—22no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×#127C2211yescharacteristic
12C2×C4×C7×Q8C2211yescharacteristic
13C7×#492—22no—
14C7×#492—22no—
15C7×#475—22no—
16C7×#475—22no—
17C7×#460—22no—
18C7×#460—22no—
19C2×C7×#106—21no—
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×#103—21no—
24C2×C7×#102—21no—
25C2×C4×C7×C8C2211yescharacteristic
26C7×#336—22no—
27C7×#332—22no—
28C7×#331—22no—
29#10268C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8D411yescharacteristic
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C2×C7×SD16—28no—
36C2×C7×SD16—24no—
37C2×C7×SD16—44no—
38C2×C7×SD16—42no—
39C2×C7×SD16—42no—
40C2×C7×SD16—44no—
41C22×C7×C8—21no—
42C22×C7×C8—21no—
43C22×C7×C8—21no—
44C22×C7×C8D412yes—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8D412yes—
49C7×#127—21no—
50C7×#127—21no—
51C7×#127—21no—
52C7×#127D412yes—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4—21no—
60C2×C7×C4⋊C4D411yescharacteristic
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63C7×#106—24no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×C4⋊C8D412yes—
70C7×#103—24no—
71C7×#102—24no—
72C4×C7×C8—21no—
73C4×C7×C8—21no—
74C2×#492C2×C711yescharacteristic
75C2×#475C2×C711yescharacteristic, complemented
76C2×#460C2×C711yescharacteristic
77#1984C2×C711yescharacteristic, complemented
78C2×#332C2×C711yescharacteristic, complemented
79C2×#331C2×C711yescharacteristic
80#1277C2×C711yescharacteristic, complemented
81C24×C7—41no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×Q8—21no—
85C2×C7×Q8—22no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—41no—
89C2×C7×Q8—42no—
90C2×C7×Q8—42no—
91C2×C7×Q8SD1612yescomplemented
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—42no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—42no—
103C2×C7×D4—41no—
104C2×C7×D4—82no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—41no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4111yescharacteristic
111C7×SD16—416no—
112C7×SD16—48no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—41no—
118C2×C7×C8—41no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C2×C7×C8—41no—
122C2×C7×C8—42no—
123C2×C7×C8SD1614yescomplemented
124C7×C4⋊C4—21no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C22×SD16—21no—
139C22×SD16—22no—
140C2×C4⋊Q8C22×C711yescharacteristic
141C2×#127C22×C711yescharacteristic
142C2×C4×Q8C22×C711yescharacteristic
143#492—22no—
144#492—22no—
145#475—22no—
146#475—22no—
147#460—22no—
148#460—22no—
149C2×#106—21no—
150C2×C4⋊C8C22×C711yescharacteristic
151C2×C4⋊C8C22×C711yescharacteristic
152C2×C4⋊C8C22×C711yescharacteristic
153C2×#103—21no—
154C2×#102—21no—
155C2×C4×C8C22×C711yescharacteristic
156#336—22no—
157#332—22no—
158#331—22no—
159C23×C7—42no—
160C23×C7—82no—
161C23×C7—81no—
162C23×C7—81no—
163C23×C7—82no—
164C23×C7#12111yescharacteristic
165C7×Q8—22no—
166C7×Q8—22no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—44no—
170C7×Q8—81no—
171C7×Q8—81no—
172C7×D4—42no—
173C7×D4—44no—
174C7×D4—42no—
175C7×D4—82no—
176C7×D4—82no—
177C7×D4—81no—
178C7×D4—81no—
179C7×D4—81no—
180C7×D4—81no—
181C2×C4×C7—22no—
182C2×C4×C7—22no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—82no—
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12711yescharacteristic
195C2×C4×C7C2×SD1611yescharacteristic
196C2×C4×C7C2×SD1612yes—
197C2×C4×C7#13511yescharacteristic
198C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C7AGL(1,ℤ/8)12yes—
200C2×C4×C7#13711yescharacteristic
201C7×C8—24no—
202C7×C8—42no—
203C7×C8—42no—
204C7×C8—81no—
205C22×Q8—21no—
206C22×Q8C7×D411yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C2×SD16—24no—
211C2×SD16—28no—
212C2×SD16—44no—
213C2×SD16—42no—
214C2×SD16—42no—
215C2×SD16—44no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8C7×D412yes—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8C7×D412yes—
224#127—21no—
225#127—21no—
226#127—21no—
227#127C7×D412yes—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C42—21no—
237C2×C42C23×C711yescharacteristic
238#106—24no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C7×D412yes—
244C4⋊C8C7×D412yes—
245#103—24no—
246#102—24no—
247C4×C8—21no—
248C4×C8—21no—
249C22×C7—82no—
250C22×C7—82no—
251C22×C7—82no—
252C22×C7—162no—
253C22×C7—162no—
254C22×C7#39211yescharacteristic
255C22×C7#46012yes—
256C22×C7#46511yescharacteristic
257C22×C7#46811yescharacteristic
258C22×C7#47112yes—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—42no—
262C4×C7—82no—
263C4×C7—81no—
264C4×C7#44811yescharacteristic
265C4×C7#44911yescharacteristic
266C4×C7#45011yescharacteristic
267C4×C7#45012yes—
268C4×C7#45211yescharacteristic
269C4×C7#49211yescharacteristic
270C4×C7#49611yescharacteristic
271C24—41no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C7×SD1612yescomplemented
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—82no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4C2×C7×D411yescharacteristic
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C7×#4111yescharacteristic
301SD16—416no—
302SD16—48no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8C7×SD1614yescomplemented
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C42—41no—
323C42—41no—
324C42C2×C7×D411yescharacteristic
325C42C2×C7×D411yescharacteristic
326C42C2×C7×D411yescharacteristic
327C42C2×C7×D411yescharacteristic
328C2×C7—162no—
329C2×C7#166612yes—
330C2×C7#168612yes—
331C2×C7#200011yescharacteristic
332C2×C7#201411yescharacteristic
333C2×C7#201911yescharacteristic
334C23—42no—
335C23—81no—
336C23—82no—
337C23—82no—
338C23—81no—
339C23C7×#12111yescharacteristic
340Q8—22no—
341Q8—22no—
342Q8—41no—
343Q8—41no—
344Q8—44no—
345Q8—81no—
346Q8—81no—
347D4—42no—
348D4—44no—
349D4—42no—
350D4—82no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4C7×#12011yescharacteristic
365C2×C4C7×#12011yescharacteristic
366C2×C4C7×#12011yescharacteristic
367C2×C4C7×#12111yescharacteristic
368C2×C4C7×#12111yescharacteristic
369C2×C4C7×#12711yescharacteristic
370C2×C4C2×C7×SD1612yes—
371C2×C4C2×C7×SD1611yescharacteristic
372C2×C4C7×#13511yescharacteristic
373C2×C4C7×AGL(1,ℤ/8)12yes—
374C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
375C2×C4C7×#13711yescharacteristic
376C8—24no—
377C8—42no—
378C8—42no—
379C8—81no—
380C7#1026811yescharacteristic, direct product factor, Sylow
381C22—82no—
382C22—82no—
383C22—82no—
384C22—162no—
385C22—162no—
386C22C7×#39211yescharacteristic
387C22C7×#46012yes—
388C22C7×#46511yescharacteristic
389C22C7×#46811yescharacteristic
390C22C7×#47112yes—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4C7×#44811yescharacteristic
397C4C7×#44911yescharacteristic
398C4C7×#45011yescharacteristic
399C4C7×#45012yes—
400C4C7×#45211yescharacteristic
401C4C7×#49211yescharacteristic
402C4C7×#49611yescharacteristic
403C2—162no—
404C2C7×#166612yes—
405C2C7×#168612yes—
406C2C7×#200011yescharacteristic
407C2C7×#201411yescharacteristic
408C2C7×#201911yescharacteristic
subgroupG/H
center, socleC23×C7#121
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
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
2. C2×C7×#475C2semidirect
4. C7×#1984C2semidirect
5. C2×C7×#332C2semidirect
7. C7×#1277C2semidirect
29. #10268C7direct (N × H)
75. C2×#475C2×C7semidirect
77. #1984C2×C7semidirect
78. C2×#332C2×C7semidirect
80. #1277C2×C7semidirect
91. C2×C7×Q8SD16semidirect
123. C2×C7×C8SD16semidirect
281. C2×Q8C7×SD16semidirect
313. C2×C8C7×SD16semidirect
380. C7#10268direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4922
C2×C7×#4752
C2×C7×#4602
C7×#19842
C2×C7×#3322
C2×C7×#3312
C7×#12772
#102687