Finite Group Databasebeta

← C7×#10193 #13187409

Subgroups — C7×#10193 #13187409

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3246C211yesmaximal, characteristic, complemented
2C7×#3205C211yesmaximal, characteristic
3C7×#3138C211yesmaximal, characteristic, complemented
4C7×#1982C211yesmaximal, characteristic, complemented
5C7×#1980C211yesmaximal, characteristic, complemented
6C7×#1503C211yesmaximal, characteristic, complemented
7C7×#1283C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C7×#532C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C7×#493—21no—
12C7×#490—22no—
13C7×#486—21no—
14C7×#486—21no—
15C7×#486—21no—
16C7×#476—21no—
17C7×#475—21no—
18C7×#465—22no—
19C7×#461—21no—
20C7×#460—21no—
21C7×#459—22no—
22C7×#447—22no—
23C7×#431C2211yescharacteristic
24C2×C7×C4⋊C8C2211yescharacteristic
25C2×C7×#104—22no—
26C7×#418—22no—
27C2×C7×#102—22no—
28C4×C7×M16C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C7×#328—22no—
31#10193C711yesmaximal, characteristic, direct product factor, Sylow
32C2×C7×#41—21no—
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C22×C7×D4—21no—
36C2×C7×SD16—22no—
37C2×C7×D8—24no—
38C2×C7×M16D412yes—
39C22×C7×C8D412yes—
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×#127—21no—
44C7×#127—21no—
45C7×#127—21no—
46C7×#127D412yes—
47C7×#127D411yescharacteristic
48C7×#125—21no—
49C7×#124—21no—
50C7×C22⋊Q8—21no—
51C7×C22⋊Q8—21no—
52C7×#121—21no—
53C7×#121—21no—
54C4×C7×Q8—21no—
55C4×C7×Q8—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C7×#117—21no—
59C2×C7×C4⋊C4—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#106—22no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×#104—44no—
67C7×#103—22no—
68C7×#102—22no—
69C7×#102—24no—
70C7×#102—24no—
71C7×#102—24no—
72C7×#102—44no—
73C7×C22⋊C8D412yes—
74C7×#97—21no—
75C4×C7×C8—21no—
76C4×C7×C8—21no—
77C7×#95—21no—
78#3246C2×C711yescharacteristic, complemented
79#3205C2×C711yescharacteristic
80#3138C2×C711yescharacteristic, complemented
81#1982C2×C711yescharacteristic, complemented
82#1980C2×C711yescharacteristic, complemented
83#1503C2×C711yescharacteristic, complemented
84#1283C2×C711yescharacteristic, complemented
85C24×C7—41no—
86C7×#41—42no—
87C7×#41—42no—
88C2×C7×Q8—21no—
89C2×C7×Q8—21no—
90C2×C7×Q8—21no—
91C2×C7×D4—22no—
92C2×C7×D4—21no—
93C2×C7×D4—22no—
94C2×C7×D4—21no—
95C2×C7×D4—21no—
96C2×C7×D4—22no—
97C2×C7×D4—42no—
98C2×C7×D4—42no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—42no—
104C2×C7×D4—42no—
105C2×C7×D4—82no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C7×SD16—82no—
112C7×D8—84no—
113C7×M16—22no—
114C7×M16—22no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—42no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C42×C7—41no—
133C42×C7—42no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7#4111yescharacteristic
138#533C22×C711yescharacteristic
139#532C22×C711yescharacteristic
140C2×#127C22×C711yescharacteristic
141#493—21no—
142#490—22no—
143#486—21no—
144#486—21no—
145#486—21no—
146#476—21no—
147#475—21no—
148#465—22no—
149#461—21no—
150#460—21no—
151#459—22no—
152#447—22no—
153#431C22×C711yescharacteristic
154C2×C4⋊C8C22×C711yescharacteristic
155C2×#104—22no—
156#418—22no—
157C2×#102—22no—
158C4×M16C22×C711yescharacteristic
159C4×C4⋊C4C22×C711yescharacteristic
160#328—22no—
161C23×C7—41no—
162C23×C7—42no—
163C23×C7—82no—
164C23×C7—82no—
165C23×C7—81no—
166C23×C7—81no—
167C23×C7#12711yescharacteristic
168C7×Q8—41no—
169C7×Q8—41no—
170C7×Q8—42no—
171C7×Q8—81no—
172C7×D4—42no—
173C7×D4—44no—
174C7×D4—44no—
175C7×D4—44no—
176C7×D4—41no—
177C7×D4—41no—
178C7×D4—42no—
179C7×D4—82no—
180C7×D4—81no—
181C7×D4—81no—
182C7×D4—81no—
183C7×D4—81no—
184C7×D4—82no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7—82no—
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12011yescharacteristic
197C2×C4×C7#12011yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#13512yes—
202C2×C4×C7AGL(1,ℤ/8)12yes—
203C7×C8—42no—
204C7×C8—42no—
205C7×C8—42no—
206C2×#41—21no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C2×SD16—22no—
211C2×D8—24no—
212C2×M16C7×D412yes—
213C22×C8C7×D412yes—
214C4⋊Q8C7×D411yescharacteristic
215C4⋊Q8C7×D411yescharacteristic
216C4⋊Q8C7×D411yescharacteristic
217#127—21no—
218#127—21no—
219#127—21no—
220#127C7×D412yes—
221#127C7×D411yescharacteristic
222#125—21no—
223#124—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226#121—21no—
227#121—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×D4—21no—
231C4×D4—21no—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C42—21no—
235C2×C42C23×C711yescharacteristic
236#106—22no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240#104—44no—
241#103—22no—
242#102—24no—
243#102—24no—
244#102—24no—
245#102—22no—
246#102—44no—
247C22⋊C8C7×D412yes—
248#97—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—82no—
255C22×C7—81no—
256C22×C7—82no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7—82no—
260C22×C7—162no—
261C22×C7—162no—
262C22×C7#39211yescharacteristic
263C22×C7#49511yescharacteristic
264C22×C7#49611yescharacteristic
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—21no—
268C4×C7—82no—
269C4×C7—81no—
270C4×C7—81no—
271C4×C7—81no—
272C4×C7#44912yes—
273C4×C7#45311yescharacteristic
274C4×C7#45411yescharacteristic
275C4×C7#46312yes—
276C24—41no—
277#41—42no—
278#41—42no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—82no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C2×C7×D411yescharacteristic
301C22×C4C2×C7×D411yescharacteristic
302SD16—82no—
303D8—84no—
304M16—22no—
305M16—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—42no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—41no—
324C42—42no—
325C42C2×C7×D411yescharacteristic
326C42C2×C7×D411yescharacteristic
327C42C2×C7×D411yescharacteristic
328C42C7×#4111yescharacteristic
329C2×C7—21no—
330C2×C7—21no—
331C2×C7—161no—
332C2×C7—162no—
333C2×C7#200611yescharacteristic
334C2×C7#200811yescharacteristic
335C2×C7#201211yescharacteristic
336C23—41no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23—81no—
341C23—81no—
342C23C7×#12711yescharacteristic
343Q8—41no—
344Q8—41no—
345Q8—42no—
346Q8—81no—
347D4—42no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—41no—
352D4—41no—
353D4—42no—
354D4—82no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4C7×#12011yescharacteristic
371C2×C4C7×#12011yescharacteristic
372C2×C4C7×#12011yescharacteristic
373C2×C4C7×#12111yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12111yescharacteristic
376C2×C4C7×#13512yes—
377C2×C4C7×AGL(1,ℤ/8)12yes—
378C8—42no—
379C8—42no—
380C8—42no—
381C7#1019311yescharacteristic, direct product factor, Sylow
382C22—21no—
383C22—21no—
384C22—82no—
385C22—81no—
386C22—82no—
387C22—81no—
388C22—81no—
389C22—82no—
390C22—162no—
391C22—162no—
392C22C7×#39211yescharacteristic
393C22C7×#49511yescharacteristic
394C22C7×#49611yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4C7×#44912yes—
403C4C7×#45311yescharacteristic
404C4C7×#45411yescharacteristic
405C4C7×#46312yes—
406C2—21no—
407C2—21no—
408C2—162no—
409C2—161no—
410C2C7×#200611yescharacteristic
411C2C7×#200811yescharacteristic
412C2C7×#201211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC7×#3205C2
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×#3246C2semidirect
3. C7×#3138C2semidirect
4. C7×#1982C2semidirect
5. C7×#1980C2semidirect
6. C7×#1503C2semidirect
7. C7×#1283C2semidirect
31. #10193C7direct (N × H)
78. #3246C2×C7semidirect
80. #3138C2×C7semidirect
81. #1982C2×C7semidirect
82. #1980C2×C7semidirect
83. #1503C2×C7semidirect
84. #1283C2×C7semidirect
381. C7#10193direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32462
C7×#32052
C7×#31382
C7×#19822
C7×#19802
C7×#15032
C7×#12832
#101937