Finite Group Databasebeta

← C7×#10204 #13187420

Subgroups — C7×#10204 #13187420

Order 1792.

598 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
1C7×#3245C211yesmaximal, characteristic
2C7×#3226C211yesmaximal, characteristic, complemented
3C7×#3152C211yesmaximal, characteristic, complemented
4C7×#1988C211yesmaximal, characteristic, complemented
5C7×#1980C211yesmaximal, characteristic, complemented
6C7×#1502C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C2×C7×#137—22no—
9C7×#532C2211yescharacteristic
10C7×#532C2211yescharacteristic
11C2×C7×C4⋊Q8C2211yescharacteristic
12C7×#500—21no—
13C7×#497—22no—
14C7×#494—21no—
15C7×#492—21no—
16C7×#486—21no—
17C7×#480—22no—
18C7×#474—22no—
19C7×#474—21no—
20C7×#474—21no—
21C7×#466—22no—
22C7×#459—21no—
23C7×#459—21no—
24C7×#431C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#104—22no—
27C4×C7×M16C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C7×#340—22no—
30C7×#327—22no—
31#10204C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C2×C7×#41—21no—
34C22×C7×Q8—21no—
35C7×#137—44no—
36C7×#137—44no—
37C2×C7×Q16—24no—
38C2×C7×SD16—24no—
39C2×C7×D8—22no—
40C2×C7×M16D412yes—
41C22×C7×C8D412yes—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8D411yescharacteristic
46C7×C4⋊Q8D412yes—
47C7×C4⋊Q8D411yescharacteristic
48C7×#127D411yescharacteristic
49C7×#127D411yescharacteristic
50C7×#124—21no—
51C7×#124—21no—
52C7×#121—21no—
53C7×#121—21no—
54C7×#121—21no—
55C7×#121—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C4×C7×D4—21no—
60C2×C7×C4⋊C4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4—21no—
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65C7×#107—22no—
66C7×#107—24no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×#104—44no—
71C7×#102—22no—
72C7×#102—22no—
73C7×#102—24no—
74C7×C22⋊C8D412yes—
75C7×#97—21no—
76C4×C7×C8—21no—
77C4×C7×C8—21no—
78C7×#95—21no—
79#3245C2×C711yescharacteristic
80#3226C2×C711yescharacteristic, complemented
81#3152C2×C711yescharacteristic, complemented
82#1988C2×C711yescharacteristic, complemented
83#1980C2×C711yescharacteristic, complemented
84#1502C2×C711yescharacteristic, complemented
85#1283C2×C711yescharacteristic, complemented
86C7×#41—42no—
87C7×#41—42no—
88C7×#41—42no—
89C7×#41—42no—
90C2×C7×Q8—21no—
91C2×C7×Q8—22no—
92C2×C7×Q8—21no—
93C2×C7×Q8—41no—
94C2×C7×Q8—42no—
95C2×C7×Q8—42no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C22×C4×C7—41no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C7×Q16—48no—
112C7×SD16—48no—
113C7×D8—82no—
114C7×M16—22no—
115C7×M16D814yescomplemented
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—22no—
119C2×C7×C8—42no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C42×C7—42no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7#4111yescharacteristic
139C2×#137—22no—
140#532C22×C711yescharacteristic
141#532C22×C711yescharacteristic
142C2×C4⋊Q8C22×C711yescharacteristic
143#500—21no—
144#497—22no—
145#494—21no—
146#492—21no—
147#486—21no—
148#480—22no—
149#474—22no—
150#474—21no—
151#474—21no—
152#466—22no—
153#459—21no—
154#459—21no—
155#431C22×C711yescharacteristic
156C2×C4⋊C8C22×C711yescharacteristic
157C2×#104—22no—
158C4×M16C22×C711yescharacteristic
159C4×C4⋊C4C22×C711yescharacteristic
160#340—22no—
161#327—22no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7#12711yescharacteristic
165C7×Q8—44no—
166C7×Q8—42no—
167C7×Q8—42no—
168C7×Q8—82no—
169C7×D4—42no—
170C7×D4—42no—
171C7×D4—41no—
172C7×D4—41no—
173C7×D4—41no—
174C7×D4—41no—
175C7×D4—42no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—81no—
179C7×D4—81no—
180C7×D4—81no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—81no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—82no—
193C2×C4×C7—82no—
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12011yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7C2×D812yes—
201C2×C4×C7AGL(1,ℤ/8)12yes—
202C7×C8—24no—
203C7×C8—42no—
204C7×C8—42no—
205C2×#41—21no—
206C2×#41—21no—
207C22×Q8—21no—
208#137—44no—
209#137—44no—
210C2×Q16—24no—
211C2×SD16—24no—
212C2×D8—22no—
213C2×M16C7×D412yes—
214C22×C8C7×D412yes—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8C7×D412yes—
219C4⋊Q8C7×D411yescharacteristic
220C4⋊Q8C7×D411yescharacteristic
221#127C7×D411yescharacteristic
222#127C7×D411yescharacteristic
223#124—21no—
224#124—21no—
225#121—21no—
226#121—21no—
227#121—21no—
228#121—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C42—21no—
237C2×C42C23×C711yescharacteristic
238#107—22no—
239#107—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243#104—44no—
244#102—22no—
245#102—22no—
246#102—24no—
247C22⋊C8C7×D412yes—
248#97—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—81no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7—81no—
260C22×C7#39211yescharacteristic
261C22×C7#49311yescharacteristic
262C22×C7#49611yescharacteristic
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—21no—
266C4×C7—82no—
267C4×C7—81no—
268C4×C7—81no—
269C4×C7—82no—
270C4×C7#44712yes—
271C4×C7#45511yescharacteristic
272C4×C7#45611yescharacteristic
273C4×C7#45912yes—
274#41—42no—
275#41—42no—
276#41—42no—
277#41—42no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C2×C7×D411yescharacteristic
297C22×C4C2×C7×D411yescharacteristic
298C22×C4C2×C7×D411yescharacteristic
299Q16—48no—
300SD16—48no—
301D8—82no—
302M16—22no—
303M16C7×D814yescomplemented
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C42—42no—
323C42C2×C7×D411yescharacteristic
324C42C2×C7×D411yescharacteristic
325C42C2×C7×D411yescharacteristic
326C42C7×#4111yescharacteristic
327C2×C7—21no—
328C2×C7—21no—
329C2×C7—161no—
330C2×C7—161no—
331C2×C7#200711yescharacteristic
332C2×C7#200911yescharacteristic
333C2×C7#201111yescharacteristic
334C23—41no—
335C23—41no—
336C23C7×#12711yescharacteristic
337Q8—44no—
338Q8—42no—
339Q8—42no—
340Q8—82no—
341D4—42no—
342D4—42no—
343D4—41no—
344D4—41no—
345D4—41no—
346D4—41no—
347D4—42no—
348D4—81no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4C7×#12011yescharacteristic
367C2×C4C7×#12011yescharacteristic
368C2×C4C7×#12011yescharacteristic
369C2×C4C7×#12111yescharacteristic
370C2×C4C7×#12111yescharacteristic
371C2×C4C7×#12111yescharacteristic
372C2×C4C2×C7×D812yes—
373C2×C4C7×AGL(1,ℤ/8)12yes—
374C8—24no—
375C8—42no—
376C8—42no—
377C7#1020411yescharacteristic, direct product factor, Sylow
378C22—21no—
379C22—21no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C7×#39211yescharacteristic
387C22C7×#49311yescharacteristic
388C22C7×#49611yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—82no—
396C4C7×#44712yes—
397C4C7×#45511yescharacteristic
398C4C7×#45611yescharacteristic
399C4C7×#45912yes—
400C2—21no—
401C2—21no—
402C2—161no—
403C2—161no—
404C2C7×#200711yescharacteristic
405C2C7×#200911yescharacteristic
406C2C7×#201111yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3245C2
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×#3226C2semidirect
3. C7×#3152C2semidirect
4. C7×#1988C2semidirect
5. C7×#1980C2semidirect
6. C7×#1502C2semidirect
7. C7×#1283C2semidirect
31. #10204C7direct (N × H)
80. #3226C2×C7semidirect
81. #3152C2×C7semidirect
82. #1988C2×C7semidirect
83. #1980C2×C7semidirect
84. #1502C2×C7semidirect
85. #1283C2×C7semidirect
115. C7×M16D8semidirect
303. M16C7×D8semidirect
377. C7#10204direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32452
C7×#32262
C7×#31522
C7×#19882
C7×#19802
C7×#15022
C7×#12832
#102047