Finite Group Databasebeta

← C7×#10071 #13187287

Subgroups — C7×#10071 #13187287

Order 1792.

654 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#486C211yesmaximal, characteristic
2C2×C7×#475C211yesmaximal, characteristic, complemented
3C7×#2548C211yesmaximal, characteristic
4C7×#1981C211yesmaximal, characteristic, complemented
5C2×C7×#331C211yesmaximal, characteristic
6C2×C7×Q8⋊C8C211yesmaximal, characteristic, complemented
7C7×#1279C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C7×#127C2211yescharacteristic
10C2×C7×#124—21no—
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#486—22no—
13C7×#486—22no—
14C7×#475—22no—
15C7×#475—22no—
16C2×C7×#106—21no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#103—21no—
20C2×C7×#103—22no—
21C2×C7×#102—22no—
22C2×C4×C7×C8C2211yescharacteristic
23C7×#394—22no—
24C7×#390—21no—
25C7×#390—22no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#331—22no—
28C7×Q8⋊C8—22no—
29#10071C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8D411yescharacteristic
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4D412yes—
35C22×C7×C8—21no—
36C22×C7×C8—21no—
37C22×C7×C8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8D412yes—
42C7×#127—21no—
43C7×#127—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#124—42no—
47C7×#124—42no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D412yes—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—24no—
59C2×C42×C7—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#106—24no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8D412yes—
67C7×#103—24no—
68C7×#103—44no—
69C7×#102—28no—
70C4×C7×C8—21no—
71C4×C7×C8D412yes—
72C7×#95—21no—
73C2×#486C2×C711yescharacteristic
74C2×#475C2×C711yescharacteristic, complemented
75#2548C2×C711yescharacteristic
76#1981C2×C711yescharacteristic, complemented
77C2×#331C2×C711yescharacteristic
78C2×Q8⋊C8C2×C711yescharacteristic, complemented
79#1279C2×C711yescharacteristic, complemented
80C24×C7—22no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—41no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C2×C7×Q8SD1612yes—
91C2×C7×D4—22no—
92C2×C7×D4—22no—
93C2×C7×D4—44no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—42no—
101C2×C7×D4—41no—
102C2×C7×D4—42no—
103C2×C7×D4—41no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—21no—
107C22×C4×C7—41no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—42no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C2×C7×C8—42no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C22⋊C4—48no—
129C7×C22⋊C4—84no—
130C42×C7—41no—
131C42×C7—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7#4111yescharacteristic
138C2×C4⋊Q8C22×C711yescharacteristic
139C2×#127C22×C711yescharacteristic
140C2×#124—21no—
141C2×C4×Q8C22×C711yescharacteristic
142#486—22no—
143#486—22no—
144#475—22no—
145#475—22no—
146C2×#106—21no—
147C2×C4⋊C8C22×C711yescharacteristic
148C2×C4⋊C8C22×C711yescharacteristic
149C2×#103—21no—
150C2×#103—22no—
151C2×#102—22no—
152C2×C4×C8C22×C711yescharacteristic
153#394—22no—
154#390—22no—
155#390—21no—
156C4×C4⋊C4C22×C711yescharacteristic
157#331—22no—
158Q8⋊C8—22no—
159C23×C7—42no—
160C23×C7—44no—
161C23×C7—42no—
162C23×C7—42no—
163C23×C7—44no—
164C23×C7#12011yescharacteristic
165C7×Q8—22no—
166C7×Q8—22no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—44no—
170C7×Q8—81no—
171C7×Q8—81no—
172C7×D4—44no—
173C7×D4—82no—
174C7×D4—81no—
175C7×D4—81no—
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—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—82no—
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12011yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12711yescharacteristic
198C2×C4×C7C2×SD1611yescharacteristic
199C2×C4×C7#13511yescharacteristic
200C2×C4×C7#13512yes—
201C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C7AGL(1,ℤ/8)12yes—
203C2×C4×C7#13711yescharacteristic
204C7×C8—42no—
205C7×C8—42no—
206C7×C8—81no—
207C22×Q8—21no—
208C22×Q8C7×D411yescharacteristic
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C7×D412yes—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8C7×D412yes—
219#127—21no—
220#127—21no—
221#127—21no—
222#127—21no—
223#124—42no—
224#124—42no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C7×D411yescharacteristic
233C2×C4⋊C4C7×D412yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C23×C711yescharacteristic
239#106—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8C7×D412yes—
244#103—24no—
245#103—44no—
246#102—28no—
247C4×C8—21no—
248C4×C8C7×D412yes—
249#95—21no—
250C22×C7—84no—
251C22×C7—84no—
252C22×C7—82no—
253C22×C7—82no—
254C22×C7—82no—
255C22×C7#39211yescharacteristic
256C22×C7#44912yes—
257C22×C7#45011yescharacteristic
258C22×C7#45211yescharacteristic
259C22×C7#45512yes—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—42no—
263C4×C7—82no—
264C4×C7—81no—
265C4×C7—82no—
266C4×C7#44811yescharacteristic
267C4×C7#44911yescharacteristic
268C4×C7#46312yes—
269C4×C7#46511yescharacteristic
270C4×C7#46811yescharacteristic
271C4×C7#49511yescharacteristic
272C4×C7#49611yescharacteristic
273C24—22no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8C7×SD1612yes—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—44no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4C2×C7×D411yescharacteristic
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C2×C7×D411yescharacteristic
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—48no—
322C22⋊C4—84no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42C2×C7×D411yescharacteristic
328C42C2×C7×D411yescharacteristic
329C42C2×C7×D411yescharacteristic
330C42C7×#4111yescharacteristic
331C2×C7—162no—
332C2×C7#164012yes—
333C2×C7#164212yes—
334C2×C7#200011yescharacteristic
335C2×C7#200411yescharacteristic
336C2×C7#201211yescharacteristic
337C23—42no—
338C23—44no—
339C23—42no—
340C23—42no—
341C23—44no—
342C23C7×#12011yescharacteristic
343Q8—22no—
344Q8—22no—
345Q8—41no—
346Q8—41no—
347Q8—44no—
348Q8—81no—
349Q8—81no—
350D4—44no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—82no—
358D4—81no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4C7×#12011yescharacteristic
371C2×C4C7×#12011yescharacteristic
372C2×C4C7×#12111yescharacteristic
373C2×C4C7×#12111yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12711yescharacteristic
376C2×C4C2×C7×SD1611yescharacteristic
377C2×C4C7×#13512yes—
378C2×C4C7×#13511yescharacteristic
379C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C7×AGL(1,ℤ/8)12yes—
381C2×C4C7×#13711yescharacteristic
382C8—42no—
383C8—42no—
384C8—81no—
385C7#1007111yescharacteristic, direct product factor, Sylow
386C22—82no—
387C22—82no—
388C22—82no—
389C22—84no—
390C22—84no—
391C22C7×#39211yescharacteristic
392C22C7×#44912yes—
393C22C7×#45011yescharacteristic
394C22C7×#45211yescharacteristic
395C22C7×#45512yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—82no—
401C4—82no—
402C4C7×#44811yescharacteristic
403C4C7×#44911yescharacteristic
404C4C7×#46312yes—
405C4C7×#46511yescharacteristic
406C4C7×#46811yescharacteristic
407C4C7×#49511yescharacteristic
408C4C7×#49611yescharacteristic
409C2—162no—
410C2C7×#164012yes—
411C2C7×#164212yes—
412C2C7×#200011yescharacteristic
413C2C7×#200411yescharacteristic
414C2C7×#201211yescharacteristic
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
2. C2×C7×#475C2semidirect
4. C7×#1981C2semidirect
6. C2×C7×Q8⋊C8C2semidirect
7. C7×#1279C2semidirect
29. #10071C7direct (N × H)
74. C2×#475C2×C7semidirect
76. #1981C2×C7semidirect
78. C2×Q8⋊C8C2×C7semidirect
79. #1279C2×C7semidirect
385. C7#10071direct (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×#4752
C7×#25482
C7×#19812
C2×C7×#3312
C2×C7×Q8⋊C82
C7×#12792
#100717