Finite Group Databasebeta

← C7×#9340 #13186556

Subgroups — C7×#9340 #13186556

Order 1792.

480 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×#3217C211yesmaximal, characteristic
2C7×#2995C211yesmaximal, characteristic, complemented
3C7×#1965C211yesmaximal, characteristic, complemented
4C7×#1555C211yesmaximal, characteristic, complemented
5C7×#1529C211yesmaximal, characteristic, complemented
6C7×#1307C211yesmaximal, characteristic, complemented
7C7×#1291C211yesmaximal, characteristic, complemented
8C2×C7×#127C2211yescharacteristic
9C7×#529C2211yescharacteristic
10C7×#496—21no—
11C7×#490—21no—
12C7×#460—21no—
13C7×#459—21no—
14C7×#450C412yes—
15C7×#447C412yes—
16C7×#446—21no—
17C7×#436—21no—
18C7×#435—21no—
19C7×#435—21no—
20C7×#433—21no—
21C7×#432C2211yescharacteristic
22C7×#432C2211yescharacteristic
23C7×#432C2211yescharacteristic
24C7×#424—21no—
25C7×#423C2211yescharacteristic
26C2×C7×#100—21no—
27C2×C7×#100—21no—
28C4×C7×M16C2211yescharacteristic
29C7×#331—21no—
30C7×#331—21no—
31C7×#330—21no—
32C7×#329—21no—
33#9340C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×D4—21no—
35C22×C7×D4—21no—
36C22×C7×D4C2×C412yes—
37C2×C7×SD16—22no—
38C2×C7×D8—22no—
39C2×C7×M16—21no—
40C2×C7×M16—21no—
41C2×C7×M16D411yescharacteristic
42C2×C7×M16D411yescharacteristic
43C22×C7×C8—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127—21no—
48C7×#126—21no—
49C7×#125D411yescharacteristic
50C7×#124D411yescharacteristic
51C7×#123—21no—
52C7×C22⋊Q8C2×C411yescharacteristic
53C7×#121C2×C411yescharacteristic
54C4×C7×Q8D411yescharacteristic
55C4×C7×D4—21no—
56C4×C7×D4D411yescharacteristic
57C7×#117—21no—
58C2×C42×C7C2311yescharacteristic
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8D411yescharacteristic
66C7×C4⋊C8D411yescharacteristic
67C7×#102—22no—
68C7×#102—22no—
69C7×#100—42no—
70C7×#100—42no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×C22⋊C8C2×C412yes—
76C7×#97—21no—
77C7×#97—21no—
78C7×#97—21no—
79C7×#97—21no—
80C7×#97—21no—
81C4×C7×C8—21no—
82C4×C7×C8—21no—
83#3217C2×C711yescharacteristic
84#2995C2×C711yescharacteristic, complemented
85#1965C2×C711yescharacteristic, complemented
86#1555C2×C711yescharacteristic, complemented
87#1529C2×C711yescharacteristic, complemented
88#1307C2×C711yescharacteristic, complemented
89#1291C2×C711yescharacteristic, complemented
90C24×C7—22no—
91C2×C7×Q8C22⋊C411yescharacteristic
92C2×C7×D4—21no—
93C2×C7×D4—22no—
94C2×C7×D4—22no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
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—82no—
104C2×C7×D4—82no—
105C2×C7×D4C22⋊C411yescharacteristic
106C22×C4×C7—21no—
107C22×C4×C7C22×C411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4111yescharacteristic
110C7×SD16—82no—
111C7×D8—82no—
112C7×M16—22no—
113C7×M16—41no—
114C7×M16—41no—
115C7×M16—41no—
116C7×M16—81no—
117C2×C7×C8—22no—
118C2×C7×C8—21no—
119C2×C7×C8—21no—
120C2×C7×C8—21no—
121C2×C7×C8—21no—
122C2×C7×C8—41no—
123C2×C7×C8—41no—
124C2×C7×C8—81no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—21no—
128C7×C4⋊C4—21no—
129C7×C4⋊C4C22⋊C411yescharacteristic
130C7×C4⋊C4C22⋊C411yescharacteristic
131C7×C22⋊C4—21no—
132C7×C22⋊C4—21no—
133C7×C22⋊C4—21no—
134C7×C22⋊C4—21no—
135C7×C22⋊C4—21no—
136C42×C7—21no—
137C42×C7C2×D411yescharacteristic
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7#4111yescharacteristic
141C2×#127C22×C711yescharacteristic
142#529C22×C711yescharacteristic
143#496—21no—
144#490—21no—
145#460—21no—
146#459—21no—
147#450C4×C712yes—
148#447C4×C712yes—
149#446—21no—
150#436—21no—
151#435—21no—
152#435—21no—
153#433—21no—
154#432C22×C711yescharacteristic
155#432C22×C711yescharacteristic
156#432C22×C711yescharacteristic
157#424—21no—
158#423C22×C711yescharacteristic
159C2×#100—21no—
160C2×#100—21no—
161C4×M16C22×C711yescharacteristic
162#331—21no—
163#331—21no—
164#330—21no—
165#329—21no—
166C23×C7—21no—
167C23×C7—42no—
168C23×C7—42no—
169C23×C7—42no—
170C23×C7—82no—
171C23×C7—82no—
172C23×C7C4×D411yescharacteristic
173C7×Q8—41no—
174C7×D4—41no—
175C7×D4—42no—
176C7×D4—42no—
177C7×D4—82no—
178C7×D4—81no—
179C7×D4—81no—
180C7×D4—82no—
181C7×D4—82no—
182C7×D4—81no—
183C7×D4—81no—
184C7×D4—81no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7C2×C22⋊C411yescharacteristic
195C2×C4×C7C4×D411yescharacteristic
196C2×C4×C7#12011yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12311yescharacteristic
200C7×C8—42no—
201C7×C8—42no—
202C7×C8—41no—
203C7×C8—41no—
204C7×C8—41no—
205C7×C8—41no—
206C7×C8—81no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C2×C4×C712yes—
210C2×SD16—22no—
211C2×D8—22no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16C7×D411yescharacteristic
215C2×M16C7×D411yescharacteristic
216C22×C8—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220#127—21no—
221#126—21no—
222#125C7×D411yescharacteristic
223#124C7×D411yescharacteristic
224#123—21no—
225C22⋊Q8C2×C4×C711yescharacteristic
226#121C2×C4×C711yescharacteristic
227C4×Q8C7×D411yescharacteristic
228C4×D4—21no—
229C4×D4C7×D411yescharacteristic
230#117—21no—
231C2×C42C23×C711yescharacteristic
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8C7×D411yescharacteristic
239C4⋊C8C7×D411yescharacteristic
240#102—22no—
241#102—22no—
242#100—42no—
243#100—42no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8C2×C4×C712yes—
249#97—21no—
250#97—21no—
251#97—21no—
252#97—21no—
253#97—21no—
254C4×C8—21no—
255C4×C8—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—41no—
262C22×C7—82no—
263C22×C7—82no—
264C22×C7—82no—
265C22×C7—162no—
266C22×C7—162no—
267C22×C7#38611yescharacteristic
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—41no—
274C4×C7—41no—
275C4×C7—41no—
276C4×C7#45311yescharacteristic
277C4×C7#45411yescharacteristic
278C24—22no—
279C2×Q8C7×C22⋊C411yescharacteristic
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4C7×C22⋊C411yescharacteristic
294C22×C4—21no—
295C22×C4C22×C4×C711yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297C22×C4C7×#4111yescharacteristic
298SD16—82no—
299D8—82no—
300M16—22no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—81no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—81no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4C7×C22⋊C411yescharacteristic
318C4⋊C4C7×C22⋊C411yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C42—21no—
325C42C2×C7×D411yescharacteristic
326C42C2×C7×D411yescharacteristic
327C42C2×C7×D411yescharacteristic
328C42C7×#4111yescharacteristic
329C2×C7—41no—
330C2×C7—81no—
331C2×C7—162no—
332C2×C7#188311yescharacteristic
333C2×C7#190311yescharacteristic
334C2×C7#191411yescharacteristic
335C23—21no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—82no—
340C23—82no—
341C23C4×C7×D411yescharacteristic
342Q8—41no—
343D4—41no—
344D4—42no—
345D4—42no—
346D4—82no—
347D4—81no—
348D4—81no—
349D4—82no—
350D4—82no—
351D4—81no—
352D4—81no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4C2×C7×C22⋊C411yescharacteristic
364C2×C4C4×C7×D411yescharacteristic
365C2×C4C7×#12011yescharacteristic
366C2×C4C7×#12111yescharacteristic
367C2×C4C7×#12111yescharacteristic
368C2×C4C7×#12311yescharacteristic
369C8—42no—
370C8—42no—
371C8—41no—
372C8—41no—
373C8—41no—
374C8—41no—
375C8—81no—
376C7#934011yescharacteristic, direct product factor, Sylow
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—162no—
387C22—162no—
388C22C7×#38611yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4C7×#45311yescharacteristic
398C4C7×#45411yescharacteristic
399C2—41no—
400C2—81no—
401C2—162no—
402C2C7×#188311yescharacteristic
403C2C7×#190311yescharacteristic
404C2C7×#191411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3217C2
LCSG ▷ C22×C4 ▷ 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×#2995C2semidirect
3. C7×#1965C2semidirect
4. C7×#1555C2semidirect
5. C7×#1529C2semidirect
6. C7×#1307C2semidirect
7. C7×#1291C2semidirect
33. #9340C7direct (N × H)
84. #2995C2×C7semidirect
85. #1965C2×C7semidirect
86. #1555C2×C7semidirect
87. #1529C2×C7semidirect
88. #1307C2×C7semidirect
89. #1291C2×C7semidirect
376. C7#9340direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32172
C7×#29952
C7×#19652
C7×#15552
C7×#15292
C7×#13072
C7×#12912
#93407