Finite Group Databasebeta

← C7×#10049 #13187265

Subgroups — C7×#10049 #13187265

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#492C211yesmaximal, characteristic
2C2×C7×#460C211yesmaximal, characteristic
3C7×#2567C211yesmaximal, characteristic, complemented
4C7×#1982C211yesmaximal, characteristic
5C2×C7×#333C211yesmaximal, characteristic, complemented
6C2×C7×Q8⋊C8C211yesmaximal, characteristic, complemented
7C7×#1279C211yesmaximal, characteristic, complemented
8C22×C7×SD16—21no—
9C22×C7×SD16—22no—
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×#127C2211yescharacteristic
12C2×C4×C7×Q8C2211yescharacteristic
13C7×#492—22no—
14C7×#492—22no—
15C7×#460—22no—
16C7×#460—22no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×#102—21no—
20C2×C7×#102—22no—
21C2×C4×C7×C8C2211yescharacteristic
22C7×#395—22no—
23C7×#391—21no—
24C7×#389—22no—
25C7×#384—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C7×#333—22no—
28C7×Q8⋊C8—22no—
29#10049C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8D411yescharacteristic
31C22×C7×Q8D412yes—
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4—21no—
35C2×C7×SD16—28no—
36C2×C7×SD16—24no—
37C2×C7×SD16—48no—
38C2×C7×SD16—42no—
39C2×C7×SD16—42no—
40C22×C7×C8—21no—
41C22×C7×C8—21no—
42C22×C7×C8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×#127—21no—
48C7×#127—21no—
49C7×#127—21no—
50C7×#127D412yes—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C4×C7×Q8—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—24no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4D412yes—
61C2×C42×C7—21no—
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8D412yes—
68C7×#102—24no—
69C7×#102—44no—
70C4×C7×C8—21no—
71C4×C7×C8D412yes—
72C7×#95—24no—
73C7×#95—21no—
74C2×#492C2×C711yescharacteristic
75C2×#460C2×C711yescharacteristic
76#2567C2×C711yescharacteristic, complemented
77#1982C2×C711yescharacteristic
78C2×#333C2×C711yescharacteristic, complemented
79C2×Q8⋊C8C2×C711yescharacteristic, complemented
80#1279C2×C711yescharacteristic, complemented
81C24×C7—41no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×Q8—22no—
85C2×C7×Q8—21no—
86C2×C7×Q8—22no—
87C2×C7×Q8—21no—
88C2×C7×Q8—21no—
89C2×C7×Q8—44no—
90C2×C7×Q8SD1612yescomplemented
91C2×C7×D4—22no—
92C2×C7×D4—22no—
93C2×C7×D4—22no—
94C2×C7×D4—44no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—42no—
99C2×C7×D4—41no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—41no—
103C2×C7×D4—42no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C7×SD16—416no—
112C7×SD16—48no—
113C2×C7×C8—24no—
114C2×C7×C8—22no—
115C2×C7×C8—22no—
116C2×C7×C8—41no—
117C2×C7×C8—41no—
118C2×C7×C8—41no—
119C2×C7×C8—42no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—84no—
130C7×C4⋊C4SD1614yescomplemented
131C42×C7—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7#4111yescharacteristic
139C22×SD16—22no—
140C22×SD16—21no—
141C2×C4⋊Q8C22×C711yescharacteristic
142C2×#127C22×C711yescharacteristic
143C2×C4×Q8C22×C711yescharacteristic
144#492—22no—
145#492—22no—
146#460—22no—
147#460—22no—
148C2×C4⋊C8C22×C711yescharacteristic
149C2×C4⋊C8C22×C711yescharacteristic
150C2×#102—21no—
151C2×#102—22no—
152C2×C4×C8C22×C711yescharacteristic
153#395—22no—
154#391—21no—
155#389—22no—
156#384—21no—
157C4×C4⋊C4C22×C711yescharacteristic
158#333—22no—
159Q8⋊C8—22no—
160C23×C7—42no—
161C23×C7—81no—
162C23×C7—82no—
163C23×C7—81no—
164C23×C7—82no—
165C23×C7#12011yescharacteristic
166C7×Q8—22no—
167C7×Q8—22no—
168C7×Q8—44no—
169C7×Q8—41no—
170C7×Q8—41no—
171C7×Q8—82no—
172C7×D4—44no—
173C7×D4—42no—
174C7×D4—42no—
175C7×D4—44no—
176C7×D4—82no—
177C7×D4—81no—
178C7×D4—81no—
179C7×D4—81no—
180C7×D4—81no—
181C2×C4×C7—22no—
182C2×C4×C7—24no—
183C2×C4×C7—22no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—41no—
187C2×C4×C7—42no—
188C2×C4×C7—41no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—44no—
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12011yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12711yescharacteristic
198C2×C4×C7C2×SD1611yescharacteristic
199C2×C4×C7C2×SD1612yes—
200C2×C4×C7#13511yescharacteristic
201C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C7#13711yescharacteristic
203C2×C4×C7#13712yes—
204C7×C8—24no—
205C7×C8—42no—
206C7×C8—81no—
207C22×Q8C7×D411yescharacteristic
208C22×Q8C7×D412yes—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C2×SD16—24no—
213C2×SD16—28no—
214C2×SD16—48no—
215C2×SD16—42no—
216C2×SD16—42no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224#127—21no—
225#127—21no—
226#127—21no—
227#127C7×D412yes—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4C7×D411yescharacteristic
237C2×C4⋊C4C7×D412yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42C23×C711yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8C7×D412yes—
245#102—24no—
246#102—44no—
247C4×C8—21no—
248C4×C8C7×D412yes—
249#95—24no—
250#95—21no—
251C22×C7—82no—
252C22×C7—82no—
253C22×C7—82no—
254C22×C7—162no—
255C22×C7—162no—
256C22×C7#39211yescharacteristic
257C22×C7#44812yes—
258C22×C7#44811yescharacteristic
259C22×C7#44911yescharacteristic
260C22×C7#45312yes—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—44no—
264C4×C7—42no—
265C4×C7—81no—
266C4×C7—82no—
267C4×C7#45011yescharacteristic
268C4×C7#45211yescharacteristic
269C4×C7#46112yes—
270C4×C7#46511yescharacteristic
271C4×C7#46811yescharacteristic
272C4×C7#49211yescharacteristic
273C4×C7#49711yescharacteristic
274C24—41no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—44no—
283C2×Q8C7×SD1612yescomplemented
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4C2×C7×D411yescharacteristic
302C22×C4C2×C7×D411yescharacteristic
303C22×C4C2×C7×D411yescharacteristic
304SD16—416no—
305SD16—48no—
306C2×C8—24no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—84no—
323C4⋊C4C7×SD1614yescomplemented
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42C2×C7×D411yescharacteristic
329C42C2×C7×D411yescharacteristic
330C42C2×C7×D411yescharacteristic
331C42C7×#4111yescharacteristic
332C2×C7—162no—
333C2×C7#162212yes—
334C2×C7#162312yes—
335C2×C7#200011yescharacteristic
336C2×C7#200211yescharacteristic
337C2×C7#201311yescharacteristic
338C23—42no—
339C23—81no—
340C23—82no—
341C23—81no—
342C23—82no—
343C23C7×#12011yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—44no—
347Q8—41no—
348Q8—41no—
349Q8—82no—
350D4—44no—
351D4—42no—
352D4—42no—
353D4—44no—
354D4—82no—
355D4—81no—
356D4—81no—
357D4—81no—
358D4—81no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4C7×#12011yescharacteristic
371C2×C4C7×#12011yescharacteristic
372C2×C4C7×#12111yescharacteristic
373C2×C4C7×#12111yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12711yescharacteristic
376C2×C4C2×C7×SD1612yes—
377C2×C4C2×C7×SD1611yescharacteristic
378C2×C4C7×#13511yescharacteristic
379C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C7×#13711yescharacteristic
381C2×C4C7×#13712yes—
382C8—24no—
383C8—42no—
384C8—81no—
385C7#1004911yescharacteristic, direct product factor, Sylow
386C22—82no—
387C22—82no—
388C22—82no—
389C22—162no—
390C22—162no—
391C22C7×#39211yescharacteristic
392C22C7×#44811yescharacteristic
393C22C7×#44812yes—
394C22C7×#44911yescharacteristic
395C22C7×#45312yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—44no—
400C4—82no—
401C4—81no—
402C4C7×#45011yescharacteristic
403C4C7×#45211yescharacteristic
404C4C7×#46112yes—
405C4C7×#46511yescharacteristic
406C4C7×#46811yescharacteristic
407C4C7×#49211yescharacteristic
408C4C7×#49711yescharacteristic
409C2—162no—
410C2C7×#162212yes—
411C2C7×#162312yes—
412C2C7×#200011yescharacteristic
413C2C7×#200211yescharacteristic
414C2C7×#201311yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#2567C2semidirect
5. C2×C7×#333C2semidirect
6. C2×C7×Q8⋊C8C2semidirect
7. C7×#1279C2semidirect
29. #10049C7direct (N × H)
76. #2567C2×C7semidirect
78. C2×#333C2×C7semidirect
79. C2×Q8⋊C8C2×C7semidirect
80. #1279C2×C7semidirect
90. C2×C7×Q8SD16semidirect
130. C7×C4⋊C4SD16semidirect
283. C2×Q8C7×SD16semidirect
323. C4⋊C4C7×SD16semidirect
385. C7#10049direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4922
C2×C7×#4602
C7×#25672
C7×#19822
C2×C7×#3332
C2×C7×Q8⋊C82
C7×#12792
#100497