Finite Group Databasebeta

← C7×#8953 #13186169

Subgroups — C7×#8953 #13186169

Order 1792.

514 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2988C211yesmaximal, characteristic
2C4×C7×#137C211yesmaximal, characteristic
3C7×#1984C211yesmaximal, characteristic, complemented
4C7×#1479C211yesmaximal, characteristic, complemented
5C7×#1476C211yesmaximal, characteristic
6C7×#1283C211yesmaximal, characteristic, complemented
7C7×#1278C211yesmaximal, characteristic, complemented
8C2×C7×#137C411yescharacteristic
9C4×C7×#41C2211yescharacteristic
10C2×C4×C7×Q8C2211yescharacteristic
11C7×C8×Q8—21no—
12C7×#441—21no—
13C7×#440—21no—
14C4×C7×Q16—21no—
15C4×C7×SD16—21no—
16C7×C8×D4—22no—
17C7×C8×D4—21no—
18C7×#431—21no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#428C411yescharacteristic
22C2×C7×C4⋊C8—21no—
23C2×C7×C4⋊C8C2211yescharacteristic
24C7×#419C411yescharacteristic
25C7×#417C411yescharacteristic
26C4×C7×M16C2211yescharacteristic
27C2×C4×C7×C8C2211yescharacteristic
28C7×#336—22no—
29C7×Q8⋊C8—21no—
30C7×Q8⋊C8—21no—
31C7×Q8⋊C8—21no—
32C7×D4⋊C8—21no—
33#8953C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41C2×C411yescharacteristic
35C22×C7×Q8C2×C411yescharacteristic
36C7×#137—22no—
37C7×#137C814yescomplemented
38C2×C7×Q16—21no—
39C2×C7×SD16—21no—
40C2×C7×M16C2×C411yescharacteristic
41C2×C7×M16C2×C411yescharacteristic
42C22×C7×C8—21no—
43C22×C7×C8—22no—
44C22×C7×C8D412yes—
45C4×C7×Q8—21no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8D411yescharacteristic
50C4×C7×Q8D411yescharacteristic
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4D411yescharacteristic
54C4×C7×D4D411yescharacteristic
55C7×#117—21no—
56C7×#117C2×C411yescharacteristic
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2×C411yescharacteristic
59C2×C42×C7—21no—
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C7×#107—21no—
63C7×#106—21no—
64C7×C4⋊C8—22no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D411yescharacteristic
69C7×C4⋊C8D411yescharacteristic
70C7×#103—21no—
71C7×#103—21no—
72C7×#103—21no—
73C7×#102—21no—
74C7×C22⋊C8—22no—
75C7×C22⋊C8—21no—
76C7×C22⋊C8—21no—
77C7×#97—21no—
78C4×C7×C8—22no—
79C4×C7×C8—21no—
80C4×C7×C8—21no—
81C4×C7×C8—21no—
82C4×C7×C8—21no—
83C4×C7×C8—21no—
84#2988C2×C711yescharacteristic
85C4×#137C2×C711yescharacteristic
86#1984C2×C711yescharacteristic, complemented
87#1479C2×C711yescharacteristic, complemented
88#1476C2×C711yescharacteristic
89#1283C2×C711yescharacteristic, complemented
90#1278C2×C711yescharacteristic, complemented
91C7×#41—21no—
92C7×#41—21no—
93C7×#41C2×C812yes—
94C7×#41M1612yes—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—21no—
98C2×C7×Q8—21no—
99C2×C7×Q8—21no—
100C2×C7×Q8C22⋊C411yescharacteristic
101C2×C7×Q8C2×C812yescomplemented
102C2×C7×D4—21no—
103C2×C7×D4C22⋊C411yescharacteristic
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7C22×C411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7#4111yescharacteristic
112C7×Q16—24no—
113C7×SD16—24no—
114C7×M16—21no—
115C7×M16—21no—
116C7×M16—21no—
117C7×M16C2×C812yescomplemented
118C2×C7×C8—21no—
119C2×C7×C8—22no—
120C2×C7×C8—21no—
121C2×C7×C8—21no—
122C2×C7×C8—22no—
123C2×C7×C8—21no—
124C2×C7×C8—41no—
125C2×C7×C8—41no—
126C2×C7×C8—42no—
127C2×C7×C8—41no—
128C2×C7×C8—82no—
129C7×C4⋊C4—21no—
130C7×C4⋊C4—21no—
131C7×C4⋊C4—21no—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—41no—
134C7×C4⋊C4—41no—
135C7×C4⋊C4C22⋊C411yescharacteristic
136C7×C4⋊C4C22⋊C411yescharacteristic
137C7×C22⋊C4—21no—
138C7×C22⋊C4—21no—
139C7×C22⋊C4—21no—
140C42×C7—21no—
141C42×C7—21no—
142C42×C7—21no—
143C42×C7—41no—
144C42×C7—41no—
145C42×C7C2×D411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7C2×D411yescharacteristic
148C42×C7#4111yescharacteristic
149C2×#137C4×C711yescharacteristic
150C4×#41C22×C711yescharacteristic
151C2×C4×Q8C22×C711yescharacteristic
152C8×Q8—21no—
153#441—21no—
154#440—21no—
155C4×Q16—21no—
156C4×SD16—21no—
157C8×D4—22no—
158C8×D4—21no—
159#431—21no—
160#431C22×C711yescharacteristic
161#431C22×C711yescharacteristic
162#428C4×C711yescharacteristic
163C2×C4⋊C8—21no—
164C2×C4⋊C8C22×C711yescharacteristic
165#419C4×C711yescharacteristic
166#417C4×C711yescharacteristic
167C4×M16C22×C711yescharacteristic
168C2×C4×C8C22×C711yescharacteristic
169#336—22no—
170Q8⋊C8—21no—
171Q8⋊C8—21no—
172Q8⋊C8—21no—
173D4⋊C8—21no—
174C23×C7—21no—
175C23×C7C4×D411yescharacteristic
176C7×Q8—21no—
177C7×Q8—22no—
178C7×Q8—22no—
179C7×Q8—41no—
180C7×Q8—41no—
181C7×Q8C22⋊C812yes—
182C7×D4—22no—
183C7×D4—22no—
184C7×D4—21no—
185C7×D4C22⋊C812yes—
186C2×C4×C7—21no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—22no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—42no—
200C2×C4×C7—41no—
201C2×C4×C7—81no—
202C2×C4×C7C2×C22⋊C411yescharacteristic
203C2×C4×C7C4×D411yescharacteristic
204C2×C4×C7#12011yescharacteristic
205C2×C4×C7#12111yescharacteristic
206C2×C4×C7#12111yescharacteristic
207C2×C4×C7#12311yescharacteristic
208C2×C4×C7C22×C811yescharacteristic
209C2×C4×C7C2×M1611yescharacteristic
210C2×C4×C7#13111yescharacteristic
211C2×C4×C7#13111yescharacteristic
212C7×C8—22no—
213C7×C8—42no—
214C7×C8—41no—
215C7×C8—42no—
216C7×C8—41no—
217C7×C8—41no—
218C7×C8—81no—
219C2×#41C2×C4×C711yescharacteristic
220C22×Q8C2×C4×C711yescharacteristic
221#137—22no—
222#137C7×C814yescomplemented
223C2×Q16—21no—
224C2×SD16—21no—
225C2×M16C2×C4×C711yescharacteristic
226C2×M16C2×C4×C711yescharacteristic
227C22×C8—21no—
228C22×C8—22no—
229C22×C8C7×D412yes—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C7×D411yescharacteristic
235C4×Q8C7×D411yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C7×D411yescharacteristic
239C4×D4C7×D411yescharacteristic
240#117—21no—
241#117C2×C4×C711yescharacteristic
242C2×C4⋊C4—21no—
243C2×C4⋊C4C2×C4×C711yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C23×C711yescharacteristic
247#107—21no—
248#106—21no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8C7×D411yescharacteristic
254C4⋊C8C7×D411yescharacteristic
255#103—21no—
256#103—21no—
257#103—21no—
258#102—21no—
259C22⋊C8—22no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262#97—21no—
263C4×C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—22no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7#38611yescharacteristic
275C22×C7C8×D411yescharacteristic
276C22×C7#43511yescharacteristic
277C4×C7—22no—
278C4×C7—21no—
279C4×C7—21no—
280C4×C7—21no—
281C4×C7—42no—
282C4×C7—41no—
283C4×C7—41no—
284C4×C7—81no—
285C4×C7C2×C22⋊C811yescharacteristic
286C4×C7#40811yescharacteristic
287C4×C7C8×D411yescharacteristic
288C4×C7#43511yescharacteristic
289C4×C7#45511yescharacteristic
290C4×C7#45611yescharacteristic
291#41—21no—
292#41—21no—
293#41C2×C7×C812yes—
294#41C7×M1612yes—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8C7×C22⋊C411yescharacteristic
301C2×Q8C2×C7×C812yescomplemented
302C2×D4—21no—
303C2×D4C7×C22⋊C411yescharacteristic
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4C22×C4×C711yescharacteristic
310C22×C4C2×C7×D411yescharacteristic
311C22×C4C7×#4111yescharacteristic
312Q16—24no—
313SD16—24no—
314M16—21no—
315M16—21no—
316M16—21no—
317M16C2×C7×C812yescomplemented
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—82no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4C7×C22⋊C411yescharacteristic
336C4⋊C4C7×C22⋊C411yescharacteristic
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—41no—
344C42—41no—
345C42C2×C7×D411yescharacteristic
346C42C2×C7×D411yescharacteristic
347C42C2×C7×D411yescharacteristic
348C42C7×#4111yescharacteristic
349C2×C7—21no—
350C2×C7—21no—
351C2×C7—42no—
352C2×C7#186911yescharacteristic
353C2×C7#188111yescharacteristic
354C2×C7#188211yescharacteristic
355C23—21no—
356C23C4×C7×D411yescharacteristic
357Q8—21no—
358Q8—22no—
359Q8—22no—
360Q8—41no—
361Q8—41no—
362Q8C7×C22⋊C812yes—
363D4—22no—
364D4—22no—
365D4—21no—
366D4C7×C22⋊C812yes—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4C2×C7×C22⋊C411yescharacteristic
384C2×C4C4×C7×D411yescharacteristic
385C2×C4C7×#12011yescharacteristic
386C2×C4C7×#12111yescharacteristic
387C2×C4C7×#12111yescharacteristic
388C2×C4C7×#12311yescharacteristic
389C2×C4C22×C7×C811yescharacteristic
390C2×C4C2×C7×M1611yescharacteristic
391C2×C4C7×#13111yescharacteristic
392C2×C4C7×#13111yescharacteristic
393C8—22no—
394C8—42no—
395C8—41no—
396C8—42no—
397C8—41no—
398C8—41no—
399C8—81no—
400C7#895311yescharacteristic, direct product factor, Sylow
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22C7×#38611yescharacteristic
407C22C7×C8×D411yescharacteristic
408C22C7×#43511yescharacteristic
409C4—22no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—41no—
415C4—41no—
416C4—81no—
417C4C2×C7×C22⋊C811yescharacteristic
418C4C7×#40811yescharacteristic
419C4C7×C8×D411yescharacteristic
420C4C7×#43511yescharacteristic
421C4C7×#45511yescharacteristic
422C4C7×#45611yescharacteristic
423C2—21no—
424C2—21no—
425C2—42no—
426C2C7×#186911yescharacteristic
427C2C7×#188111yescharacteristic
428C2C7×#188211yescharacteristic
subgroupG/H
centerC2×C4×C7#120
commutatorC2×C4C22×C7×C8
Frattini, cosocleC2×C42C23×C7
socleC22×C7#386
prime radicalC2×C7×#41C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1984C2semidirect
4. C7×#1479C2semidirect
6. C7×#1283C2semidirect
7. C7×#1278C2semidirect
33. #8953C7direct (N × H)
37. C7×#137C8semidirect
86. #1984C2×C7semidirect
87. #1479C2×C7semidirect
89. #1283C2×C7semidirect
90. #1278C2×C7semidirect
101. C2×C7×Q8C2×C8semidirect
117. C7×M16C2×C8semidirect
222. #137C7×C8semidirect
301. C2×Q8C2×C7×C8semidirect
317. M16C2×C7×C8semidirect
400. C7#8953direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29882
C4×C7×#1372
C7×#19842
C7×#14792
C7×#14762
C7×#12832
C7×#12782
#89537