Finite Group Databasebeta

← C7×#10051 #13187267

Subgroups — C7×#10051 #13187267

Order 1792.

670 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#486C211yesmaximal, characteristic
2C2×C7×#460C211yesmaximal, characteristic
3C7×#2546C211yesmaximal, characteristic, complemented
4C7×#1982C211yesmaximal, characteristic
5C2×C7×#332C211yesmaximal, characteristic, complemented
6C2×C7×Q8⋊C8C211yesmaximal, characteristic, complemented
7C7×#1279C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C2×C7×C4⋊Q8—21no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×#127C2211yescharacteristic
12C2×C4×C7×Q8C2211yescharacteristic
13C7×#486—22no—
14C7×#486—22no—
15C7×#460—22no—
16C7×#460—22no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#102—21no—
20C2×C7×#102—22no—
21C2×C7×#102—22no—
22C2×C4×C7×C8C2211yescharacteristic
23C7×#398—22no—
24C7×#391—21no—
25C7×#389—22no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#332—22no—
28C7×Q8⋊C8—22no—
29#10051C711yesmaximal, 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—24no—
36C2×C7×SD16—42no—
37C2×C7×SD16—42no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C22×C7×C8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—42no—
46C7×C4⋊Q8—42no—
47C7×#127—21no—
48C7×#127—21no—
49C7×#127—21no—
50C7×#127D412yes—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C2×C7×C4⋊C4—24no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D412yes—
60C2×C7×C4⋊C4D412yes—
61C2×C42×C7—21no—
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8D412yes—
68C7×#102—28no—
69C7×#102—24no—
70C7×#102—44no—
71C4×C7×C8—21no—
72C4×C7×C8D412yes—
73C7×#95—24no—
74C7×#95—21no—
75C2×#486C2×C711yescharacteristic
76C2×#460C2×C711yescharacteristic
77#2546C2×C711yescharacteristic, complemented
78#1982C2×C711yescharacteristic
79C2×#332C2×C711yescharacteristic, complemented
80C2×Q8⋊C8C2×C711yescharacteristic, complemented
81#1279C2×C711yescharacteristic, complemented
82C24×C7—41no—
83C2×C7×Q8—21no—
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×Q8—41no—
93C2×C7×Q8—41no—
94C2×C7×Q8D812yes—
95C2×C7×D4—22no—
96C2×C7×D4—22no—
97C2×C7×D4—22no—
98C2×C7×D4—44no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—42no—
104C2×C7×D4—41no—
105C2×C7×D4—41no—
106C2×C7×D4—41no—
107C2×C7×D4—42no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—22no—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×D411yescharacteristic
115C7×SD16—48no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—42no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C2×C7×C8—41no—
122C2×C7×C8—42no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—48no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C7×C4⋊C4—41no—
133C7×C4⋊C4SD1614yescomplemented
134C42×C7—41no—
135C42×C7—41no—
136C42×C7—41no—
137C42×C7—41no—
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7C2×D411yescharacteristic
141C42×C7#4111yescharacteristic
142C22×SD16—21no—
143C2×C4⋊Q8—21no—
144C2×C4⋊Q8C22×C711yescharacteristic
145C2×#127C22×C711yescharacteristic
146C2×C4×Q8C22×C711yescharacteristic
147#486—22no—
148#486—22no—
149#460—22no—
150#460—22no—
151C2×C4⋊C8C22×C711yescharacteristic
152C2×C4⋊C8C22×C711yescharacteristic
153C2×#102—22no—
154C2×#102—21no—
155C2×#102—22no—
156C2×C4×C8C22×C711yescharacteristic
157#398—22no—
158#391—21no—
159#389—22no—
160C4×C4⋊C4C22×C711yescharacteristic
161#332—22no—
162Q8⋊C8—22no—
163C23×C7—42no—
164C23×C7—81no—
165C23×C7—82no—
166C23×C7—81no—
167C23×C7—82no—
168C23×C7#12011yescharacteristic
169C7×Q8—22no—
170C7×Q8—22no—
171C7×Q8—41no—
172C7×Q8—41no—
173C7×Q8—81no—
174C7×Q8—81no—
175C7×Q8—81no—
176C7×Q8—81no—
177C7×D4—44no—
178C7×D4—42no—
179C7×D4—42no—
180C7×D4—44no—
181C7×D4—82no—
182C7×D4—81no—
183C7×D4—81no—
184C7×D4—81no—
185C7×D4—81no—
186C2×C4×C7—22no—
187C2×C4×C7—22no—
188C2×C4×C7—24no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—42no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7#12011yescharacteristic
199C2×C4×C7#12011yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12111yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×C4×C7#12711yescharacteristic
204C2×C4×C7C2×D811yescharacteristic
205C2×C4×C7C2×SD1612yes—
206C2×C4×C7#13511yescharacteristic
207C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
208C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
209C2×C4×C7#13712yes—
210C7×C8—42no—
211C7×C8—42no—
212C7×C8—81no—
213C22×Q8—21no—
214C22×Q8C7×D411yescharacteristic
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—21no—
218C2×SD16—24no—
219C2×SD16—42no—
220C2×SD16—42no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230#127—21no—
231#127—21no—
232#127—21no—
233#127C7×D412yes—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C7×D411yescharacteristic
242C2×C4⋊C4C7×D412yes—
243C2×C4⋊C4C7×D412yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C23×C711yescharacteristic
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C7×D412yes—
251#102—24no—
252#102—28no—
253#102—44no—
254C4×C8—21no—
255C4×C8C7×D412yes—
256#95—24no—
257#95—21no—
258C22×C7—82no—
259C22×C7—82no—
260C22×C7—82no—
261C22×C7—162no—
262C22×C7—162no—
263C22×C7#39211yescharacteristic
264C22×C7#44711yescharacteristic
265C22×C7#44812yes—
266C22×C7#44911yescharacteristic
267C22×C7#45412yes—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—44no—
271C4×C7—42no—
272C4×C7—82no—
273C4×C7—81no—
274C4×C7#44711yescharacteristic
275C4×C7#44911yescharacteristic
276C4×C7#46112yes—
277C4×C7#46611yescharacteristic
278C4×C7#46911yescharacteristic
279C4×C7#49211yescharacteristic
280C4×C7#49711yescharacteristic
281C24—41no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8C7×D812yes—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—44no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4C2×C7×D411yescharacteristic
312C22×C4C2×C7×D411yescharacteristic
313C22×C4C2×C7×D411yescharacteristic
314SD16—48no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—48no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4C7×SD1614yescomplemented
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42C2×C7×D411yescharacteristic
338C42C2×C7×D411yescharacteristic
339C42C2×C7×D411yescharacteristic
340C42C7×#4111yescharacteristic
341C2×C7—162no—
342C2×C7#162612yes—
343C2×C7#162712yes—
344C2×C7#199911yescharacteristic
345C2×C7#200311yescharacteristic
346C2×C7#201311yescharacteristic
347C23—42no—
348C23—81no—
349C23—82no—
350C23—81no—
351C23—82no—
352C23C7×#12011yescharacteristic
353Q8—22no—
354Q8—22no—
355Q8—41no—
356Q8—41no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361D4—44no—
362D4—42no—
363D4—42no—
364D4—44no—
365D4—82no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4C7×#12011yescharacteristic
383C2×C4C7×#12011yescharacteristic
384C2×C4C7×#12111yescharacteristic
385C2×C4C7×#12111yescharacteristic
386C2×C4C7×#12111yescharacteristic
387C2×C4C7×#12711yescharacteristic
388C2×C4C2×C7×D811yescharacteristic
389C2×C4C2×C7×SD1612yes—
390C2×C4C7×#13511yescharacteristic
391C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
392C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
393C2×C4C7×#13712yes—
394C8—42no—
395C8—42no—
396C8—81no—
397C7#1005111yescharacteristic, direct product factor, Sylow
398C22—82no—
399C22—82no—
400C22—82no—
401C22—162no—
402C22—162no—
403C22C7×#39211yescharacteristic
404C22C7×#44711yescharacteristic
405C22C7×#44812yes—
406C22C7×#44911yescharacteristic
407C22C7×#45412yes—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—44no—
412C4—82no—
413C4—81no—
414C4C7×#44711yescharacteristic
415C4C7×#44911yescharacteristic
416C4C7×#46112yes—
417C4C7×#46611yescharacteristic
418C4C7×#46911yescharacteristic
419C4C7×#49211yescharacteristic
420C4C7×#49711yescharacteristic
421C2—162no—
422C2C7×#162612yes—
423C2C7×#162712yes—
424C2C7×#199911yescharacteristic
425C2C7×#200311yescharacteristic
426C2C7×#201311yescharacteristic
subgroupG/H
center, socleC23×C7#120
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
3. C7×#2546C2semidirect
5. C2×C7×#332C2semidirect
6. C2×C7×Q8⋊C8C2semidirect
7. C7×#1279C2semidirect
29. #10051C7direct (N × H)
77. #2546C2×C7semidirect
79. C2×#332C2×C7semidirect
80. C2×Q8⋊C8C2×C7semidirect
81. #1279C2×C7semidirect
133. C7×C4⋊C4SD16semidirect
332. C4⋊C4C7×SD16semidirect
397. C7#10051direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4862
C2×C7×#4602
C7×#25462
C7×#19822
C2×C7×#3322
C2×C7×Q8⋊C82
C7×#12792
#100517