Finite Group Databasebeta

← C7×#10055 #13187271

Subgroups — C7×#10055 #13187271

Order 1792.

702 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
1C2×C7×#492C211yesmaximal, characteristic
2C2×C7×#477C211yesmaximal, characteristic, complemented
3C7×#2548C211yesmaximal, characteristic
4C7×#1987C211yesmaximal, characteristic, complemented
5C2×C7×#331C211yesmaximal, characteristic
6C2×C7×Q8⋊C8C211yesmaximal, characteristic, complemented
7C7×#1279C211yesmaximal, characteristic, complemented
8C22×C7×SD16—22no—
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C2×C7×#124—21no—
12C2×C4×C7×Q8C2211yescharacteristic
13C7×#492—22no—
14C7×#492—22no—
15C7×#477—22no—
16C7×#477—22no—
17C2×C7×#107—21no—
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#103—21no—
21C2×C4×C7×C8C2211yescharacteristic
22C7×#394—22no—
23C7×#390—21no—
24C7×#390—22no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C7×#340—22no—
27C7×#331—22no—
28C7×Q8⋊C8—22no—
29#10055C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×Q8D411yescharacteristic
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4D412yes—
35C2×C7×SD16—28no—
36C2×C7×SD16—48no—
37C22×C7×C8—21no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8D412yes—
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127—21no—
48C7×#124—42no—
49C7×#124—42no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4D412yes—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—24no—
61C2×C42×C7—21no—
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64C7×#107—24no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8D412yes—
69C7×#103—24no—
70C4×C7×C8—21no—
71C4×C7×C8D412yes—
72C7×#95—21no—
73C2×#492C2×C711yescharacteristic
74C2×#477C2×C711yescharacteristic, complemented
75#2548C2×C711yescharacteristic
76#1987C2×C711yescharacteristic, complemented
77C2×#331C2×C711yescharacteristic
78C2×Q8⋊C8C2×C711yescharacteristic, complemented
79#1279C2×C711yescharacteristic, complemented
80C24×C7—22no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—22no—
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—44no—
88C2×C7×Q8—41no—
89C2×C7×Q8SD1612yescomplemented
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—44no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C2×C7×D4—42no—
100C2×C7×D4—41no—
101C2×C7×D4—41no—
102C2×C7×D4—42no—
103C22×C4×C7—21no—
104C22×C4×C7—22no—
105C22×C4×C7—21no—
106C22×C4×C7—41no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C7×SD16—416no—
111C2×C7×C8—24no—
112C2×C7×C8—22no—
113C2×C7×C8—22no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C2×C7×C8—42no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—22no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C22⋊C4—48no—
129C7×C22⋊C4—84no—
130C42×C7—41no—
131C42×C7—41no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7C2×D411yescharacteristic
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7#4111yescharacteristic
138C22×SD16—22no—
139C2×C4⋊Q8C22×C711yescharacteristic
140C2×#127C22×C711yescharacteristic
141C2×#124—21no—
142C2×C4×Q8C22×C711yescharacteristic
143#492—22no—
144#492—22no—
145#477—22no—
146#477—22no—
147C2×#107—21no—
148C2×C4⋊C8C22×C711yescharacteristic
149C2×C4⋊C8C22×C711yescharacteristic
150C2×#103—21no—
151C2×C4×C8C22×C711yescharacteristic
152#394—22no—
153#390—22no—
154#390—21no—
155C4×C4⋊C4C22×C711yescharacteristic
156#340—22no—
157#331—22no—
158Q8⋊C8—22no—
159C23×C7—42no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7—44no—
163C23×C7—44no—
164C23×C7#12011yescharacteristic
165C7×Q8—22no—
166C7×Q8—22no—
167C7×Q8—44no—
168C7×Q8—41no—
169C7×Q8—41no—
170C7×Q8—82no—
171C7×D4—44no—
172C7×D4—82no—
173C7×D4—81no—
174C7×D4—81no—
175C7×D4—82no—
176C7×D4—81no—
177C7×D4—81no—
178C7×D4—81no—
179C7×D4—81no—
180C2×C4×C7—22no—
181C2×C4×C7—22no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—82no—
191C2×C4×C7#12011yescharacteristic
192C2×C4×C7#12011yescharacteristic
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12711yescharacteristic
197C2×C4×C7C2×SD1611yescharacteristic
198C2×C4×C7#13511yescharacteristic
199C2×C4×C7#13512yes—
200C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
201C2×C4×C7#13711yescharacteristic
202C2×C4×C7#13712yes—
203C7×C8—24no—
204C7×C8—42no—
205C7×C8—81no—
206C22×Q8—21no—
207C22×Q8C7×D411yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C7×D412yes—
211C2×SD16—28no—
212C2×SD16—48no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8C7×D412yes—
220#127—21no—
221#127—21no—
222#127—21no—
223#127—21no—
224#124—42no—
225#124—42no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C7×D411yescharacteristic
234C2×C4⋊C4C7×D412yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C23×C711yescharacteristic
240#107—24no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8C7×D412yes—
245#103—24no—
246C4×C8—21no—
247C4×C8C7×D412yes—
248#95—21no—
249C22×C7—84no—
250C22×C7—84no—
251C22×C7—82no—
252C22×C7—82no—
253C22×C7—82no—
254C22×C7#39211yescharacteristic
255C22×C7#45011yescharacteristic
256C22×C7#45211yescharacteristic
257C22×C7#45212yes—
258C22×C7#45312yes—
259C4×C7—21no—
260C4×C7—21no—
261C4×C7—42no—
262C4×C7—82no—
263C4×C7—81no—
264C4×C7—82no—
265C4×C7#44811yescharacteristic
266C4×C7#44911yescharacteristic
267C4×C7#46412yes—
268C4×C7#46511yescharacteristic
269C4×C7#46811yescharacteristic
270C4×C7#49511yescharacteristic
271C4×C7#49711yescharacteristic
272C24—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—44no—
280C2×Q8—41no—
281C2×Q8C7×SD1612yescomplemented
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—44no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C2×C7×D411yescharacteristic
301C22×C4C2×C7×D411yescharacteristic
302SD16—416no—
303C2×C8—24no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—48no—
321C22⋊C4—84no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42C2×C7×D411yescharacteristic
327C42C2×C7×D411yescharacteristic
328C42C2×C7×D411yescharacteristic
329C42C7×#4111yescharacteristic
330C2×C7—162no—
331C2×C7#163012yes—
332C2×C7#163112yes—
333C2×C7#200011yescharacteristic
334C2×C7#200211yescharacteristic
335C2×C7#201511yescharacteristic
336C23—42no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—44no—
341C23C7×#12011yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—44no—
345Q8—41no—
346Q8—41no—
347Q8—82no—
348D4—44no—
349D4—82no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—82no—
356D4—81no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4C7×#12011yescharacteristic
369C2×C4C7×#12011yescharacteristic
370C2×C4C7×#12111yescharacteristic
371C2×C4C7×#12111yescharacteristic
372C2×C4C7×#12111yescharacteristic
373C2×C4C7×#12711yescharacteristic
374C2×C4C2×C7×SD1611yescharacteristic
375C2×C4C7×#13512yes—
376C2×C4C7×#13511yescharacteristic
377C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
378C2×C4C7×#13711yescharacteristic
379C2×C4C7×#13712yes—
380C8—24no—
381C8—42no—
382C8—81no—
383C7#1005511yescharacteristic, direct product factor, Sylow
384C22—82no—
385C22—82no—
386C22—82no—
387C22—84no—
388C22—84no—
389C22C7×#39211yescharacteristic
390C22C7×#45011yescharacteristic
391C22C7×#45211yescharacteristic
392C22C7×#45212yes—
393C22C7×#45312yes—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—82no—
399C4—82no—
400C4C7×#44811yescharacteristic
401C4C7×#44911yescharacteristic
402C4C7×#46412yes—
403C4C7×#46511yescharacteristic
404C4C7×#46811yescharacteristic
405C4C7×#49511yescharacteristic
406C4C7×#49711yescharacteristic
407C2—162no—
408C2C7×#163012yes—
409C2C7×#163112yes—
410C2C7×#200011yescharacteristic
411C2C7×#200211yescharacteristic
412C2C7×#201511yescharacteristic
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
2. C2×C7×#477C2semidirect
4. C7×#1987C2semidirect
6. C2×C7×Q8⋊C8C2semidirect
7. C7×#1279C2semidirect
29. #10055C7direct (N × H)
74. C2×#477C2×C7semidirect
76. #1987C2×C7semidirect
78. C2×Q8⋊C8C2×C7semidirect
79. #1279C2×C7semidirect
89. C2×C7×Q8SD16semidirect
281. C2×Q8C7×SD16semidirect
383. C7#10055direct (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×#4772
C7×#25482
C7×#19872
C2×C7×#3312
C2×C7×Q8⋊C82
C7×#12792
#100557