Finite Group Databasebeta

← C7×#10016 #13187232

Subgroups — C7×#10016 #13187232

Order 1792.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3104C211yesmaximal, characteristic, complemented
2C7×#3101C211yesmaximal, characteristic
3C7×#2398C211yesmaximal, characteristic
4C7×#1995C211yesmaximal, characteristic, complemented
5C7×#1487C211yesmaximal, characteristic, complemented
6C7×#1486C211yesmaximal, characteristic
7C7×#1274C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#484—21no—
12C7×#483—21no—
13C7×#477—21no—
14C7×#476—21no—
15C7×#462—21no—
16C7×#461—21no—
17C7×#451—21no—
18C7×#450—21no—
19C4×C7×Q16—22no—
20C4×C7×SD16—22no—
21C7×#431C2211yescharacteristic
22C7×#431C2211yescharacteristic
23C7×#423C2211yescharacteristic
24C7×#400—21no—
25C7×#397—21no—
26C7×#394—21no—
27C7×#393—21no—
28C4×C7×C22⋊C4—22no—
29C7×#376C2211yescharacteristic
30C7×#339—22no—
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#10016C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C2×C7×Q16—22no—
38C2×C7×SD16—22no—
39C2×C7×M16—21no—
40C7×C4⋊Q8D411yescharacteristic
41C7×C4⋊Q8D411yescharacteristic
42C7×#125—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C4×C7×Q8—21no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C7×#117—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—22no—
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7—21no—
63C2×C42×C7D412yes—
64C2×C42×C7C2311yescharacteristic
65C7×#107—22no—
66C7×#106—22no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8—21no—
71C7×#103—22no—
72C7×#103—22no—
73C7×#103—22no—
74C7×#102—22no—
75C7×C22⋊C8—21no—
76C7×C22⋊C8—21no—
77C4×C7×C8D412yes—
78C4×C7×C8D412yes—
79C7×#95—22no—
80C7×#95—22no—
81C7×#95—21no—
82C7×#95—21no—
83#3104C2×C711yescharacteristic, complemented
84#3101C2×C711yescharacteristic
85#2398C2×C711yescharacteristic
86#1995C2×C711yescharacteristic, complemented
87#1487C2×C711yescharacteristic, complemented
88#1486C2×C711yescharacteristic
89#1274C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—42no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—42no—
99C22×C4×C7—21no—
100C22×C4×C7—22no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—41no—
104C22×C4×C7—41no—
105C22×C4×C7—42no—
106C22×C4×C7—42no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C7×Q16—44no—
111C7×SD16—44no—
112C7×M16—42no—
113C2×C7×C8—22no—
114C2×C7×C8—22no—
115C2×C7×C8—41no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C22⋊C4—44no—
129C7×C22⋊C4—44no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C42×C7—22no—
134C42×C7—22no—
135C42×C7—41no—
136C42×C7—41no—
137C42×C7—41no—
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140C42×C7C2×D411yescharacteristic
141C42×C7#4111yescharacteristic
142#533C22×C711yescharacteristic
143C2×C4×Q8C22×C711yescharacteristic
144C2×C4×D4C22×C711yescharacteristic
145#484—21no—
146#483—21no—
147#477—21no—
148#476—21no—
149#462—21no—
150#461—21no—
151#451—21no—
152#450—21no—
153C4×Q16—22no—
154C4×SD16—22no—
155#431C22×C711yescharacteristic
156#431C22×C711yescharacteristic
157#423C22×C711yescharacteristic
158#400—21no—
159#397—21no—
160#394—21no—
161#393—21no—
162C4×C22⋊C4—22no—
163#376C22×C711yescharacteristic
164#339—22no—
165Q8⋊C8—22no—
166D4⋊C8—22no—
167C23×C7—22no—
168C23×C7—42no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—42no—
172C23×C7#12011yescharacteristic
173C7×Q8—22no—
174C7×Q8—22no—
175C7×Q8—41no—
176C7×Q8—41no—
177C7×Q8—42no—
178C7×Q8—42no—
179C7×D4—22no—
180C7×D4—22no—
181C7×D4—41no—
182C7×D4—41no—
183C2×C4×C7—22no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—82no—
198C2×C4×C7—81no—
199C2×C4×C7—81no—
200C2×C4×C7#12011yescharacteristic
201C2×C4×C7#12011yescharacteristic
202C2×C4×C7#12111yescharacteristic
203C2×C4×C7#12111yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C2×C4×C7#12711yescharacteristic
206C2×C4×C7#13512yes—
207C2×C4×C7#13712yes—
208C7×C8—42no—
209C7×C8—42no—
210C7×C8—81no—
211C22×Q8C7×D411yescharacteristic
212C22×D4C7×D411yescharacteristic
213C23×C4—21no—
214C2×Q16—22no—
215C2×SD16—22no—
216C2×M16—21no—
217C4⋊Q8C7×D411yescharacteristic
218C4⋊Q8C7×D411yescharacteristic
219#125—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C7×D411yescharacteristic
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C42—21no—
240C2×C42C7×D412yes—
241C2×C42C23×C711yescharacteristic
242#107—22no—
243#106—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248#103—22no—
249#103—22no—
250#103—22no—
251#102—22no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C4×C8C7×D412yes—
255C4×C8C7×D412yes—
256#95—22no—
257#95—22no—
258#95—21no—
259#95—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—44no—
263C22×C7—42no—
264C22×C7—82no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7#39211yescharacteristic
268C22×C7#45212yes—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—42no—
274C4×C7—82no—
275C4×C7—81no—
276C4×C7—81no—
277C4×C7—81no—
278C4×C7—81no—
279C4×C7#46412yes—
280C4×C7#49511yescharacteristic
281C4×C7#49711yescharacteristic
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—42no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C2×C7×D411yescharacteristic
301C22×C4C2×C7×D411yescharacteristic
302Q16—44no—
303SD16—44no—
304M16—42no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42C2×C7×D411yescharacteristic
331C42C2×C7×D411yescharacteristic
332C42C2×C7×D411yescharacteristic
333C42C7×#4111yescharacteristic
334C2×C7—22no—
335C2×C7—82no—
336C2×C7#201011yescharacteristic
337C2×C7#201011yescharacteristic
338C2×C7#201511yescharacteristic
339C23—22no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23C7×#12011yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
349Q8—42no—
350Q8—42no—
351D4—22no—
352D4—22no—
353D4—41no—
354D4—41no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4C7×#12011yescharacteristic
373C2×C4C7×#12011yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12111yescharacteristic
376C2×C4C7×#12111yescharacteristic
377C2×C4C7×#12711yescharacteristic
378C2×C4C7×#13512yes—
379C2×C4C7×#13712yes—
380C8—42no—
381C8—42no—
382C8—81no—
383C7#1001611yescharacteristic, direct product factor, Sylow
384C22—21no—
385C22—21no—
386C22—44no—
387C22—42no—
388C22—82no—
389C22—81no—
390C22—81no—
391C22C7×#39211yescharacteristic
392C22C7×#45212yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—42no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4C7×#46412yes—
404C4C7×#49511yescharacteristic
405C4C7×#49711yescharacteristic
406C2—22no—
407C2—82no—
408C2C7×#201011yescharacteristic
409C2C7×#201011yescharacteristic
410C2C7×#201511yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3104C2semidirect
4. C7×#1995C2semidirect
5. C7×#1487C2semidirect
7. C7×#1274C2semidirect
33. #10016C7direct (N × H)
83. #3104C2×C7semidirect
86. #1995C2×C7semidirect
87. #1487C2×C7semidirect
89. #1274C2×C7semidirect
383. C7#10016direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31042
C7×#31012
C7×#23982
C7×#19952
C7×#14872
C7×#14862
C7×#12742
#100167