Finite Group Databasebeta

← C7×#10195 #13187411

Subgroups — C7×#10195 #13187411

Order 1792.

554 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3249C211yesmaximal, characteristic, complemented
2C7×#3208C211yesmaximal, characteristic
3C7×#3138C211yesmaximal, characteristic
4C7×#1986C211yesmaximal, characteristic, complemented
5C7×#1982C211yesmaximal, characteristic
6C7×#1503C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C7×#533C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C7×#492—21no—
12C7×#490—22no—
13C7×#486—21no—
14C7×#486—21no—
15C7×#486—21no—
16C7×#477—21no—
17C7×#477—21no—
18C7×#467—22no—
19C7×#462—21no—
20C7×#462—21no—
21C7×#460—22no—
22C7×#450—22no—
23C7×#431C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C7×#418—22no—
26C2×C7×#102—22no—
27C4×C7×M16C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C7×#339—22no—
30C7×#328—22no—
31#10195C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C2×C7×Q16—22no—
36C2×C7×SD16—24no—
37C2×C7×M16D412yes—
38C22×C7×C8D412yes—
39C7×C4⋊Q8D411yescharacteristic
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×#127—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#127D412yes—
47C7×#125—21no—
48C7×#125—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C7×C22⋊Q8—21no—
52C7×C22⋊Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C7×#117—21no—
58C7×#117—21no—
59C2×C7×C4⋊C4—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—22no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×#103—22no—
67C7×#103—22no—
68C7×#102—24no—
69C7×#102—24no—
70C7×#102—24no—
71C7×#102—44no—
72C7×C22⋊C8D412yes—
73C7×#97—21no—
74C4×C7×C8—21no—
75C4×C7×C8—21no—
76C7×#95—21no—
77#3249C2×C711yescharacteristic, complemented
78#3208C2×C711yescharacteristic
79#3138C2×C711yescharacteristic
80#1986C2×C711yescharacteristic, complemented
81#1982C2×C711yescharacteristic
82#1503C2×C711yescharacteristic, complemented
83#1283C2×C711yescharacteristic, complemented
84C24×C7—41no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×D4—22no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—42no—
93C2×C7×D4—42no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—42no—
98C2×C7×D4—42no—
99C2×C7×D4—82no—
100C22×C4×C7—41no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C7×Q16—82no—
105C7×SD16—84no—
106C7×M16—22no—
107C7×M16—22no—
108C2×C7×C8—22no—
109C2×C7×C8—22no—
110C2×C7×C8—22no—
111C2×C7×C8—42no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C42×C7—41no—
129C42×C7—41no—
130C42×C7—42no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7#4111yescharacteristic
135#533C22×C711yescharacteristic
136#533C22×C711yescharacteristic
137C2×#127C22×C711yescharacteristic
138#492—21no—
139#490—22no—
140#486—21no—
141#486—21no—
142#486—21no—
143#477—21no—
144#477—21no—
145#467—22no—
146#462—21no—
147#462—21no—
148#460—22no—
149#450—22no—
150#431C22×C711yescharacteristic
151C2×C4⋊C8C22×C711yescharacteristic
152#418—22no—
153C2×#102—22no—
154C4×M16C22×C711yescharacteristic
155C4×C4⋊C4C22×C711yescharacteristic
156#339—22no—
157#328—22no—
158C23×C7—42no—
159C23×C7—82no—
160C23×C7—82no—
161C23×C7—81no—
162C23×C7—81no—
163C23×C7#12711yescharacteristic
164C7×Q8—42no—
165C7×Q8—41no—
166C7×Q8—41no—
167C7×Q8—41no—
168C7×Q8—41no—
169C7×Q8—81no—
170C7×D4—44no—
171C7×D4—44no—
172C7×D4—44no—
173C7×D4—82no—
174C7×D4—81no—
175C7×D4—81no—
176C7×D4—82no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—81no—
186C2×C4×C7—81no—
187C2×C4×C7#12011yescharacteristic
188C2×C4×C7#12011yescharacteristic
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#13512yes—
194C2×C4×C7#13712yes—
195C7×C8—42no—
196C7×C8—42no—
197C7×C8—42no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C2×Q16—22no—
202C2×SD16—24no—
203C2×M16C7×D412yes—
204C22×C8C7×D412yes—
205C4⋊Q8C7×D411yescharacteristic
206C4⋊Q8C7×D411yescharacteristic
207C4⋊Q8C7×D411yescharacteristic
208C4⋊Q8C7×D411yescharacteristic
209#127—21no—
210#127—21no—
211#127—21no—
212#127C7×D412yes—
213#125—21no—
214#125—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223#117—21no—
224#117—21no—
225C2×C4⋊C4—21no—
226C2×C42—21no—
227C2×C42C23×C711yescharacteristic
228#107—22no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232#103—22no—
233#103—22no—
234#102—24no—
235#102—24no—
236#102—24no—
237#102—44no—
238C22⋊C8C7×D412yes—
239#97—21no—
240C4×C8—21no—
241C4×C8—21no—
242#95—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—82no—
246C22×C7—82no—
247C22×C7—82no—
248C22×C7—162no—
249C22×C7—162no—
250C22×C7#39211yescharacteristic
251C22×C7#49511yescharacteristic
252C22×C7#49711yescharacteristic
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—82no—
257C4×C7—81no—
258C4×C7—81no—
259C4×C7—81no—
260C4×C7—81no—
261C4×C7#45212yes—
262C4×C7#45311yescharacteristic
263C4×C7#45411yescharacteristic
264C4×C7#46412yes—
265C24—41no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—82no—
281C22×C4—41no—
282C22×C4C2×C7×D411yescharacteristic
283C22×C4C2×C7×D411yescharacteristic
284C22×C4C2×C7×D411yescharacteristic
285Q16—82no—
286SD16—84no—
287M16—22no—
288M16—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—42no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—41no—
310C42—41no—
311C42—42no—
312C42C2×C7×D411yescharacteristic
313C42C2×C7×D411yescharacteristic
314C42C2×C7×D411yescharacteristic
315C42C7×#4111yescharacteristic
316C2×C7—21no—
317C2×C7—21no—
318C2×C7—162no—
319C2×C7#200611yescharacteristic
320C2×C7#200811yescharacteristic
321C2×C7#201511yescharacteristic
322C23—42no—
323C23—82no—
324C23—82no—
325C23—81no—
326C23—81no—
327C23C7×#12711yescharacteristic
328Q8—42no—
329Q8—41no—
330Q8—41no—
331Q8—41no—
332Q8—41no—
333Q8—81no—
334D4—44no—
335D4—44no—
336D4—44no—
337D4—82no—
338D4—81no—
339D4—81no—
340D4—82no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4C7×#12011yescharacteristic
352C2×C4C7×#12011yescharacteristic
353C2×C4C7×#12011yescharacteristic
354C2×C4C7×#12111yescharacteristic
355C2×C4C7×#12111yescharacteristic
356C2×C4C7×#12111yescharacteristic
357C2×C4C7×#13512yes—
358C2×C4C7×#13712yes—
359C8—42no—
360C8—42no—
361C8—42no—
362C7#1019511yescharacteristic, direct product factor, Sylow
363C22—21no—
364C22—21no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—162no—
369C22—162no—
370C22C7×#39211yescharacteristic
371C22C7×#49511yescharacteristic
372C22C7×#49711yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4C7×#45212yes—
382C4C7×#45311yescharacteristic
383C4C7×#45411yescharacteristic
384C4C7×#46412yes—
385C2—21no—
386C2—21no—
387C2—162no—
388C2C7×#200611yescharacteristic
389C2C7×#200811yescharacteristic
390C2C7×#201511yescharacteristic
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
1. C7×#3249C2semidirect
4. C7×#1986C2semidirect
6. C7×#1503C2semidirect
7. C7×#1283C2semidirect
31. #10195C7direct (N × H)
77. #3249C2×C7semidirect
80. #1986C2×C7semidirect
82. #1503C2×C7semidirect
83. #1283C2×C7semidirect
362. C7#10195direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32492
C7×#32082
C7×#31382
C7×#19862
C7×#19822
C7×#15032
C7×#12832
#101957