Finite Group Databasebeta

← C7×#10179 #13187395

Subgroups — C7×#10179 #13187395

Order 1792.

560 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3208C211yesmaximal, characteristic
2C7×#3078C211yesmaximal, characteristic, complemented
3C2×C7×#460C211yesmaximal, characteristic
4C7×#1545C211yesmaximal, characteristic
5C7×#1481C211yesmaximal, characteristic, complemented
6C7×#1461C211yesmaximal, characteristic, complemented
7C7×#1301C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C7×#533C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C7×#492—21no—
13C7×#486—21no—
14C7×#479—22no—
15C7×#477—21no—
16C7×#475—21no—
17C7×#460—22no—
18C7×#460—22no—
19C7×#460—22no—
20C7×#450—22no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#428—21no—
24C7×#423C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C7×#419—21no—
27C2×C7×#102—21no—
28C7×#331—22no—
29C7×Q8⋊C8—22no—
30C7×#324—22no—
31#10179C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4—21no—
36C2×C7×SD16—24no—
37C2×C7×SD16—24no—
38C2×C7×SD16—42no—
39C2×C7×SD16—42no—
40C2×C7×M16—21no—
41C22×C7×C8—21no—
42C7×C4⋊Q8D411yescharacteristic
43C7×C4⋊Q8D411yescharacteristic
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127D412yes—
48C7×#125—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C7×#117—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—22no—
63C7×#106—22no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×C4⋊C8D412yes—
70C7×#103—22no—
71C7×#103—22no—
72C7×#102—24no—
73C7×#102—24no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8D412yes—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78#3208C2×C711yescharacteristic
79#3078C2×C711yescharacteristic, complemented
80C2×#460C2×C711yescharacteristic
81#1545C2×C711yescharacteristic
82#1481C2×C711yescharacteristic, complemented
83#1461C2×C711yescharacteristic, complemented
84#1301C2×C711yescharacteristic, complemented
85C24×C7—41no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8SD1612yescomplemented
94C2×C7×D4—22no—
95C2×C7×D4—22no—
96C2×C7×D4—22no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—82no—
105C2×C7×D4—82no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4111yescharacteristic
111C7×SD16—48no—
112C7×SD16—84no—
113C7×M16—41no—
114C7×M16—41no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—41no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C22×SD16—21no—
140#533C22×C711yescharacteristic
141C2×#127C22×C711yescharacteristic
142C2×C4×Q8C22×C711yescharacteristic
143#492—21no—
144#486—21no—
145#479—22no—
146#477—21no—
147#475—21no—
148#460—22no—
149#460—22no—
150#460—22no—
151#450—22no—
152#431C22×C711yescharacteristic
153#431C22×C711yescharacteristic
154#428—21no—
155#423C22×C711yescharacteristic
156C2×C4⋊C8C22×C711yescharacteristic
157#419—21no—
158C2×#102—21no—
159#331—22no—
160Q8⋊C8—22no—
161#324—22no—
162C23×C7—42no—
163C23×C7—82no—
164C23×C7—82no—
165C23×C7—81no—
166C23×C7—81no—
167C23×C7#12111yescharacteristic
168C7×Q8—22no—
169C7×Q8—22no—
170C7×Q8—42no—
171C7×Q8—41no—
172C7×Q8—41no—
173C7×Q8—42no—
174C7×D4—44no—
175C7×D4—42no—
176C7×D4—42no—
177C7×D4—82no—
178C7×D4—82no—
179C7×D4—81no—
180C7×D4—81no—
181C7×D4—82no—
182C2×C4×C7—22no—
183C2×C4×C7—22no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—81no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12011yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12711yescharacteristic
197C2×C4×C7C2×SD1611yescharacteristic
198C2×C4×C7#13511yescharacteristic
199C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
200C2×C4×C7#13711yescharacteristic
201C7×C8—42no—
202C7×C8—42no—
203C7×C8—42no—
204C7×C8—81no—
205C22×Q8C7×D411yescharacteristic
206C22×D4—21no—
207C22×D4—21no—
208C22×D4—21no—
209C2×SD16—24no—
210C2×SD16—24no—
211C2×SD16—42no—
212C2×SD16—42no—
213C2×M16—21no—
214C22×C8—21no—
215C4⋊Q8C7×D411yescharacteristic
216C4⋊Q8C7×D411yescharacteristic
217#127—21no—
218#127—21no—
219#127—21no—
220#127C7×D412yes—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230#117—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C7×D411yescharacteristic
233C2×C42—21no—
234C2×C42C23×C711yescharacteristic
235#107—22no—
236#106—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8C7×D412yes—
242C4⋊C8C7×D412yes—
243#103—22no—
244#103—22no—
245#102—24no—
246#102—24no—
247C22⋊C8—21no—
248C22⋊C8C7×D412yes—
249C4×C8—21no—
250C4×C8—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—82no—
254C22×C7—82no—
255C22×C7—82no—
256C22×C7—162no—
257C22×C7—162no—
258C22×C7#39211yescharacteristic
259C22×C7#46511yescharacteristic
260C22×C7#46811yescharacteristic
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—42no—
265C4×C7—81no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7#44811yescharacteristic
269C4×C7#44911yescharacteristic
270C4×C7#45011yescharacteristic
271C4×C7#45211yescharacteristic
272C4×C7#47112yes—
273C24—41no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C7×SD1612yescomplemented
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—82no—
293C2×D4—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4C2×C7×D411yescharacteristic
297C22×C4C2×C7×D411yescharacteristic
298C22×C4C7×#4111yescharacteristic
299SD16—48no—
300SD16—84no—
301M16—41no—
302M16—41no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C22⋊C4—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42C2×C7×D411yescharacteristic
324C42C2×C7×D411yescharacteristic
325C42C2×C7×D411yescharacteristic
326C42C2×C7×D411yescharacteristic
327C2×C7—21no—
328C2×C7—21no—
329C2×C7—162no—
330C2×C7#200011yescharacteristic
331C2×C7#200811yescharacteristic
332C2×C7#201911yescharacteristic
333C23—42no—
334C23—82no—
335C23—82no—
336C23—81no—
337C23—81no—
338C23C7×#12111yescharacteristic
339Q8—22no—
340Q8—22no—
341Q8—42no—
342Q8—41no—
343Q8—41no—
344Q8—42no—
345D4—44no—
346D4—42no—
347D4—42no—
348D4—82no—
349D4—82no—
350D4—82no—
351D4—81no—
352D4—81no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4C7×#12011yescharacteristic
363C2×C4C7×#12011yescharacteristic
364C2×C4C7×#12011yescharacteristic
365C2×C4C7×#12111yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12711yescharacteristic
368C2×C4C2×C7×SD1611yescharacteristic
369C2×C4C7×#13511yescharacteristic
370C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C7×#13711yescharacteristic
372C8—42no—
373C8—42no—
374C8—42no—
375C8—81no—
376C7#1017911yescharacteristic, direct product factor, Sylow
377C22—21no—
378C22—21no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—162no—
383C22—162no—
384C22C7×#39211yescharacteristic
385C22C7×#46511yescharacteristic
386C22C7×#46811yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4C7×#44811yescharacteristic
395C4C7×#44911yescharacteristic
396C4C7×#45011yescharacteristic
397C4C7×#45211yescharacteristic
398C4C7×#47112yes—
399C2—21no—
400C2—21no—
401C2—162no—
402C2C7×#200011yescharacteristic
403C2C7×#200811yescharacteristic
404C2C7×#201911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C7×#3078C2semidirect
5. C7×#1481C2semidirect
6. C7×#1461C2semidirect
7. C7×#1301C2semidirect
31. #10179C7direct (N × H)
79. #3078C2×C7semidirect
82. #1481C2×C7semidirect
83. #1461C2×C7semidirect
84. #1301C2×C7semidirect
93. C2×C7×Q8SD16semidirect
281. C2×Q8C7×SD16semidirect
376. C7#10179direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32082
C7×#30782
C2×C7×#4602
C7×#15452
C7×#14812
C7×#14612
C7×#13012
#101797