Finite Group Databasebeta

← C7×#10076 #13187292

Subgroups — C7×#10076 #13187292

Order 1792.

654 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#500C211yesmaximal, characteristic, complemented
2C2×C7×#474C211yesmaximal, characteristic
3C7×#2542C211yesmaximal, characteristic
4C7×#1981C211yesmaximal, characteristic, complemented
5C2×C7×#332C211yesmaximal, characteristic, complemented
6C2×C7×D4⋊C8C211yesmaximal, characteristic
7C7×#1279C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C22⋊Q8—22no—
11C2×C4×C7×D4C2211yescharacteristic
12C7×#500—22no—
13C7×#500—22no—
14C7×#474—22no—
15C7×#474—22no—
16C2×C7×#107—22no—
17C2×C7×#107—21no—
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#103—22no—
21C2×C7×#102—21no—
22C2×C4×C7×C8C2211yescharacteristic
23C7×#399—22no—
24C7×#390—21no—
25C7×#384—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10076C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8—21no—
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—
45C7×C22⋊Q8—44no—
46C7×C22⋊Q8—44no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D412yes—
57C2×C7×C4⋊C4D412yes—
58C2×C7×C22⋊C4—24no—
59C2×C7×C22⋊C4—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—24no—
63C7×#107—28no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8D412yes—
68C7×#103—44no—
69C7×#102—24no—
70C4×C7×C8—21no—
71C4×C7×C8D412yes—
72C7×#95—21no—
73C2×#500C2×C711yescharacteristic, complemented
74C2×#474C2×C711yescharacteristic
75#2542C2×C711yescharacteristic
76#1981C2×C711yescharacteristic, complemented
77C2×#332C2×C711yescharacteristic, complemented
78C2×D4⋊C8C2×C711yescharacteristic
79#1279C2×C711yescharacteristic, complemented
80C24×C7—21no—
81C2×C7×Q8—22no—
82C2×C7×Q8—41no—
83C2×C7×Q8—41no—
84C2×C7×Q8—41no—
85C2×C7×Q8—42no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×Q8—42no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4Q1612yes—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—21no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7—41no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C2×C7×C8—24no—
107C2×C7×C8—22no—
108C2×C7×C8—22no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C2×C7×C8—42no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—48no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4D814yes—
126C7×C22⋊C4—48no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C42×C7—41no—
130C42×C7—41no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7#4111yescharacteristic
135C2×C4⋊Q8C22×C711yescharacteristic
136C2×C4⋊Q8C22×C711yescharacteristic
137C2×C22⋊Q8—22no—
138C2×C4×D4C22×C711yescharacteristic
139#500—22no—
140#500—22no—
141#474—22no—
142#474—22no—
143C2×#107—22no—
144C2×#107—21no—
145C2×C4⋊C8C22×C711yescharacteristic
146C2×C4⋊C8C22×C711yescharacteristic
147C2×#103—22no—
148C2×#102—21no—
149C2×C4×C8C22×C711yescharacteristic
150#399—22no—
151#390—21no—
152#384—21no—
153C4×C4⋊C4C22×C711yescharacteristic
154#332—22no—
155D4⋊C8—22no—
156C23×C7—22no—
157C23×C7—41no—
158C23×C7—42no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7#12011yescharacteristic
162C7×Q8—44no—
163C7×Q8—82no—
164C7×Q8—81no—
165C7×Q8—81no—
166C7×Q8—81no—
167C7×Q8—81no—
168C7×D4—22no—
169C7×D4—22no—
170C7×D4—41no—
171C7×D4—41no—
172C2×C4×C7—22no—
173C2×C4×C7—24no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—44no—
180C2×C4×C7—42no—
181C2×C4×C7—82no—
182C2×C4×C7—82no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12011yescharacteristic
187C2×C4×C7#12111yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7#12111yescharacteristic
190C2×C4×C7#12711yescharacteristic
191C2×C4×C7C2×D812yes—
192C2×C4×C7C2×Q1611yescharacteristic
193C2×C4×C7#13511yescharacteristic
194C2×C4×C7AGL(1,ℤ/8)12yes—
195C2×C4×C7#13711yescharacteristic
196C2×C4×C7#13711yescharacteristic
197C7×C8—24no—
198C7×C8—42no—
199C7×C8—81no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×D4C7×D411yescharacteristic
203C23×C4—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8C7×D412yes—
215C22⋊Q8—44no—
216C22⋊Q8—44no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C4⋊C4C7×D412yes—
227C2×C4⋊C4C7×D412yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42C23×C711yescharacteristic
232#107—24no—
233#107—28no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8C7×D412yes—
238#103—44no—
239#102—24no—
240C4×C8—21no—
241C4×C8C7×D412yes—
242#95—21no—
243C22×C7—44no—
244C22×C7—42no—
245C22×C7—82no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7#39211yescharacteristic
249C22×C7#44712yes—
250C22×C7#45111yescharacteristic
251C22×C7#45211yescharacteristic
252C22×C7#45612yes—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—44no—
256C4×C7—81no—
257C4×C7—82no—
258C4×C7—82no—
259C4×C7#45111yescharacteristic
260C4×C7#45211yescharacteristic
261C4×C7#45912yes—
262C4×C7#46711yescharacteristic
263C4×C7#47011yescharacteristic
264C4×C7#49311yescharacteristic
265C4×C7#49611yescharacteristic
266C24—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4C7×Q1612yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291C22×C4C2×C7×D411yescharacteristic
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—48no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C7×D814yes—
312C22⋊C4—48no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—41no—
317C42C2×C7×D411yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C2×C7×D411yescharacteristic
320C42C7×#4111yescharacteristic
321C2×C7—82no—
322C2×C7#164712yes—
323C2×C7#164912yes—
324C2×C7#200111yescharacteristic
325C2×C7#200511yescharacteristic
326C2×C7#201111yescharacteristic
327C23—22no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23C7×#12011yescharacteristic
333Q8—44no—
334Q8—81no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—82no—
339D4—22no—
340D4—22no—
341D4—41no—
342D4—41no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4C7×#12011yescharacteristic
357C2×C4C7×#12011yescharacteristic
358C2×C4C7×#12111yescharacteristic
359C2×C4C7×#12111yescharacteristic
360C2×C4C7×#12111yescharacteristic
361C2×C4C7×#12711yescharacteristic
362C2×C4C2×C7×D812yes—
363C2×C4C2×C7×Q1611yescharacteristic
364C2×C4C7×#13511yescharacteristic
365C2×C4C7×AGL(1,ℤ/8)12yes—
366C2×C4C7×#13711yescharacteristic
367C2×C4C7×#13711yescharacteristic
368C8—24no—
369C8—42no—
370C8—81no—
371C7#1007611yescharacteristic, direct product factor, Sylow
372C22—42no—
373C22—44no—
374C22—81no—
375C22—81no—
376C22—82no—
377C22C7×#39211yescharacteristic
378C22C7×#44712yes—
379C22C7×#45111yescharacteristic
380C22C7×#45211yescharacteristic
381C22C7×#45612yes—
382C4—21no—
383C4—21no—
384C4—44no—
385C4—81no—
386C4—82no—
387C4—82no—
388C4C7×#45111yescharacteristic
389C4C7×#45211yescharacteristic
390C4C7×#45912yes—
391C4C7×#46711yescharacteristic
392C4C7×#47011yescharacteristic
393C4C7×#49311yescharacteristic
394C4C7×#49611yescharacteristic
395C2—82no—
396C2C7×#164712yes—
397C2C7×#164912yes—
398C2C7×#200111yescharacteristic
399C2C7×#200511yescharacteristic
400C2C7×#201111yescharacteristic
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×#500C2semidirect
4. C7×#1981C2semidirect
5. C2×C7×#332C2semidirect
7. C7×#1279C2semidirect
29. #10076C7direct (N × H)
73. C2×#500C2×C7semidirect
76. #1981C2×C7semidirect
77. C2×#332C2×C7semidirect
79. #1279C2×C7semidirect
371. C7#10076direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5002
C2×C7×#4742
C7×#25422
C7×#19812
C2×C7×#3322
C2×C7×D4⋊C82
C7×#12792
#100767