Finite Group Databasebeta

← C7×#10074 #13187290

Subgroups — C7×#10074 #13187290

Order 1792.

618 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#498C211yesmaximal, characteristic, complemented
2C2×C7×#474C211yesmaximal, characteristic
3C7×#2547C211yesmaximal, 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×#124—21no—
11C2×C4×C7×D4C2211yescharacteristic
12C7×#498—22no—
13C7×#498—22no—
14C7×#474—22no—
15C7×#474—22no—
16C2×C7×#107—21no—
17C2×C7×#106—22no—
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#103—22no—
21C2×C7×#102—21no—
22C2×C4×C7×C8C2211yescharacteristic
23C7×#397—22no—
24C7×#391—21no—
25C7×#380—22no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#332—22no—
28C7×D4⋊C8—22no—
29#10074C711yesmaximal, 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×#124—42no—
46C7×#124—42no—
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—21no—
54C2×C7×C4⋊C4D412yes—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D412yes—
57C2×C7×C22⋊C4—24no—
58C2×C7×C22⋊C4—21no—
59C2×C42×C7—21no—
60C2×C42×C7C2311yescharacteristic
61C7×#107—24no—
62C7×#106—28no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8D412yes—
67C7×#103—44no—
68C7×#102—24no—
69C4×C7×C8—21no—
70C4×C7×C8D412yes—
71C7×#95—24no—
72C7×#95—21no—
73C2×#498C2×C711yescharacteristic, complemented
74C2×#474C2×C711yescharacteristic
75#2547C2×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—41no—
86C2×C7×Q8—41no—
87C2×C7×Q8—41no—
88C2×C7×Q8—41no—
89C2×C7×Q8—42no—
90C2×C7×Q8—42no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4SD1612yes—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7—41no—
103C22×C4×C7—42no—
104C22×C4×C7—42no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C2×C7×C8—24no—
109C2×C7×C8—22no—
110C2×C7×C8—22no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C2×C7×C8—41no—
114C2×C7×C8—42no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—22no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C22⋊C4—48no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C42×C7—41no—
131C42×C7—41no—
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7#4111yescharacteristic
136C2×C4⋊Q8C22×C711yescharacteristic
137C2×C4⋊Q8C22×C711yescharacteristic
138C2×#124—21no—
139C2×C4×D4C22×C711yescharacteristic
140#498—22no—
141#498—22no—
142#474—22no—
143#474—22no—
144C2×#107—21no—
145C2×#106—22no—
146C2×C4⋊C8C22×C711yescharacteristic
147C2×C4⋊C8C22×C711yescharacteristic
148C2×#103—22no—
149C2×#102—21no—
150C2×C4×C8C22×C711yescharacteristic
151#397—22no—
152#391—21no—
153#380—22no—
154C4×C4⋊C4C22×C711yescharacteristic
155#332—22no—
156D4⋊C8—22no—
157C23×C7—22no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7—41no—
161C23×C7—42no—
162C23×C7#12011yescharacteristic
163C7×Q8—44no—
164C7×Q8—81no—
165C7×Q8—81no—
166C7×Q8—81no—
167C7×Q8—81no—
168C7×Q8—81no—
169C7×Q8—81no—
170C7×D4—22no—
171C7×D4—22no—
172C7×D4—41no—
173C7×D4—41no—
174C2×C4×C7—22no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—42no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—82no—
185C2×C4×C7—82no—
186C2×C4×C7—81no—
187C2×C4×C7—81no—
188C2×C4×C7#12011yescharacteristic
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12711yescharacteristic
194C2×C4×C7C2×SD1611yescharacteristic
195C2×C4×C7#13511yescharacteristic
196C2×C4×C7#13512yes—
197C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C7AGL(1,ℤ/8)12yes—
199C2×C4×C7#13711yescharacteristic
200C7×C8—24no—
201C7×C8—42no—
202C7×C8—81no—
203C22×Q8—21no—
204C22×Q8—21no—
205C22×D4C7×D411yescharacteristic
206C23×C4—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8C7×D412yes—
218#124—42no—
219#124—42no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C7×D411yescharacteristic
228C2×C4⋊C4C7×D412yes—
229C2×C4⋊C4C7×D412yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—21no—
232C2×C42—21no—
233C2×C42C23×C711yescharacteristic
234#107—24no—
235#106—28no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8C7×D412yes—
240#103—44no—
241#102—24no—
242C4×C8—21no—
243C4×C8C7×D412yes—
244#95—24no—
245#95—21no—
246C22×C7—44no—
247C22×C7—42no—
248C22×C7—82no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7#39211yescharacteristic
252C22×C7#44912yes—
253C22×C7#45011yescharacteristic
254C22×C7#45211yescharacteristic
255C22×C7#45612yes—
256C4×C7—21no—
257C4×C7—21no—
258C4×C7—82no—
259C4×C7—82no—
260C4×C7—81no—
261C4×C7—82no—
262C4×C7#44811yescharacteristic
263C4×C7#44911yescharacteristic
264C4×C7#46312yes—
265C4×C7#46511yescharacteristic
266C4×C7#46811yescharacteristic
267C4×C7#49511yescharacteristic
268C4×C7#49611yescharacteristic
269C24—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4C7×SD1612yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4C2×C7×D411yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297C2×C8—24no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C22⋊C4—48no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42—41no—
320C42—41no—
321C42C2×C7×D411yescharacteristic
322C42C2×C7×D411yescharacteristic
323C42C2×C7×D411yescharacteristic
324C42C7×#4111yescharacteristic
325C2×C7—82no—
326C2×C7#165212yes—
327C2×C7#165412yes—
328C2×C7#200011yescharacteristic
329C2×C7#200511yescharacteristic
330C2×C7#201211yescharacteristic
331C23—22no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23C7×#12011yescharacteristic
337Q8—44no—
338Q8—81no—
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343Q8—81no—
344D4—22no—
345D4—22no—
346D4—41no—
347D4—41no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4C7×#12011yescharacteristic
363C2×C4C7×#12011yescharacteristic
364C2×C4C7×#12111yescharacteristic
365C2×C4C7×#12111yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12711yescharacteristic
368C2×C4C2×C7×SD1611yescharacteristic
369C2×C4C7×#13512yes—
370C2×C4C7×#13511yescharacteristic
371C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C7×AGL(1,ℤ/8)12yes—
373C2×C4C7×#13711yescharacteristic
374C8—24no—
375C8—42no—
376C8—81no—
377C7#1007411yescharacteristic, direct product factor, Sylow
378C22—42no—
379C22—44no—
380C22—81no—
381C22—81no—
382C22—82no—
383C22C7×#39211yescharacteristic
384C22C7×#44912yes—
385C22C7×#45011yescharacteristic
386C22C7×#45211yescharacteristic
387C22C7×#45612yes—
388C4—21no—
389C4—21no—
390C4—82no—
391C4—81no—
392C4—82no—
393C4—82no—
394C4C7×#44811yescharacteristic
395C4C7×#44911yescharacteristic
396C4C7×#46312yes—
397C4C7×#46511yescharacteristic
398C4C7×#46811yescharacteristic
399C4C7×#49511yescharacteristic
400C4C7×#49611yescharacteristic
401C2—82no—
402C2C7×#165212yes—
403C2C7×#165412yes—
404C2C7×#200011yescharacteristic
405C2C7×#200511yescharacteristic
406C2C7×#201211yescharacteristic
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×#1981C2semidirect
5. C2×C7×#332C2semidirect
7. C7×#1279C2semidirect
29. #10074C7direct (N × H)
73. C2×#498C2×C7semidirect
76. #1981C2×C7semidirect
77. C2×#332C2×C7semidirect
79. #1279C2×C7semidirect
377. C7#10074direct (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×#4742
C7×#25472
C7×#19812
C2×C7×#3322
C2×C7×D4⋊C82
C7×#12792
#100747