Finite Group Databasebeta

← C7×#11875 #13189091

Subgroups — C7×#11875 #13189091

Order 1792.

536 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2535C211yesmaximal, characteristic
2C7×#1799C211yesmaximal, characteristic, complemented
3C7×#1459C211yesmaximal, characteristic, complemented
4C7×#1344C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C7×#121—21no—
7C2×C4×C7×D4C411yescharacteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#99—21no—
10C2×C7×C22⋊C8—21no—
11C7×#397C411yescharacteristic, complemented
12C7×#397C2211yescharacteristic
13C7×#392C411yescharacteristic, complemented
14C7×#392C2211yescharacteristic
15C7×#386—21no—
16C7×#386—21no—
17C7×#382C411yescharacteristic, complemented
18C7×#380—21no—
19C7×#380—21no—
20C7×#342C2212yes—
21C7×#324—21no—
22C7×#323C2212yes—
23#11875C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C22×C7×D4D411yescharacteristic
28C23×C4×C7—21no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C22×C7×C8—41no—
32C7×#121—42no—
33C7×#121—42no—
34C4×C7×D4—41no—
35C4×C7×D4—41no—
36C4×C7×D4—41no—
37C4×C7×D4—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4C2×C411yescharacteristic
49C2×C7×C22⋊C4C2311yescharacteristic
50C2×C42×C7C2×C411yescharacteristic
51C7×#99—42no—
52C7×C22⋊C8—22no—
53C7×C22⋊C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—41no—
58C7×#95C2×C411yescharacteristic
59C7×#95C2×C411yescharacteristic
60#2535C2×C711yescharacteristic
61#1799C2×C711yescharacteristic, complemented
62#1459C2×C711yescharacteristic, complemented
63#1344C2×C712yescomplemented
64#1343C2×C712yescomplemented
65C24×C7—21no—
66C24×C7C22⋊C411yescharacteristic
67C24×C7C2×D411yescharacteristic
68C2×C7×D4—41no—
69C2×C7×D4—41no—
70C2×C7×D4—41no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—42no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—81no—
78C2×C7×D4—81no—
79C2×C7×D4D812yes—
80C2×C7×D4SD1612yes—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—42no—
89C22×C4×C7—42no—
90C22×C4×C7—42no—
91C22×C4×C7—42no—
92C22×C4×C7—81no—
93C22×C4×C7—81no—
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22×C411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C2×C7×C8—42no—
100C2×C7×C8—82no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—82no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—81no—
115C7×C22⋊C4—81no—
116C42×C7—21no—
117C42×C7—21no—
118C2×#121—21no—
119C2×C4×D4C4×C711yescharacteristic, complemented
120C2×C4×D4C22×C711yescharacteristic
121C2×#99—21no—
122C2×C22⋊C8—21no—
123#397C4×C711yescharacteristic, complemented
124#397C22×C711yescharacteristic
125#392C4×C711yescharacteristic, complemented
126#392C22×C711yescharacteristic
127#386—21no—
128#386—21no—
129#382C4×C711yescharacteristic, complemented
130#380—21no—
131#380—21no—
132#342C22×C712yes—
133#324—21no—
134#323C22×C712yes—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—22no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7#10212yes—
151C23×C7C2×C22⋊C411yescharacteristic
152C23×C7C2×D811yescharacteristic
153C23×C7C2×SD1611yescharacteristic
154C7×D4—42no—
155C7×D4—42no—
156C7×D4—42no—
157C7×D4—42no—
158C7×D4—84no—
159C7×D4—82no—
160C7×D4—82no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—41no—
167C2×C4×C7—42no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7—84no—
184C2×C4×C7—82no—
185C2×C4×C7—82no—
186C2×C4×C7—82no—
187C2×C4×C7#9911yescharacteristic
188C2×C4×C7#9911yescharacteristic
189C2×C4×C7#10212yes—
190C7×C8—82no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2×C4×C711yescharacteristic
194C22×D4C7×D411yescharacteristic
195C23×C4—21no—
196C23×C4C7×D411yescharacteristic
197C23×C4C7×D411yescharacteristic
198C22×C8—41no—
199#121—42no—
200#121—42no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4C2×C4×C711yescharacteristic
208C2×C4⋊C4C7×D411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4C2×C4×C711yescharacteristic
216C2×C22⋊C4C23×C711yescharacteristic
217C2×C42C2×C4×C711yescharacteristic
218#99—42no—
219C22⋊C8—22no—
220C22⋊C8—41no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—41no—
225#95C2×C4×C711yescharacteristic
226#95C2×C4×C711yescharacteristic
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—21no—
230C22×C7—22no—
231C22×C7—21no—
232C22×C7—42no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—42no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7C2×#9911yescharacteristic
249C22×C7C2×#10211yescharacteristic
250C22×C7#42111yescharacteristic
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—22no—
254C4×C7—41no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7—81no—
258C4×C7—81no—
259C4×C7—81no—
260C4×C7—162no—
261C24—21no—
262C24C7×C22⋊C411yescharacteristic
263C24C2×C7×D411yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4C7×D812yes—
276C2×D4C7×SD1612yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C7×C22⋊C411yescharacteristic
291C22×C4C7×C22⋊C411yescharacteristic
292C22×C4C7×C22⋊C411yescharacteristic
293C22×C4C22×C4×C711yescharacteristic
294C22×C4C2×C7×D411yescharacteristic
295C2×C8—42no—
296C2×C8—82no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—82no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C42—21no—
313C42—21no—
314C2×C7—21no—
315C2×C7—21no—
316C2×C7—41no—
317C2×C7—42no—
318C2×C7—41no—
319C2×C7—82no—
320C2×C7#151911yescharacteristic
321C2×C7#212011yescharacteristic
322C2×C7#212711yescharacteristic
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23C7×#10212yes—
339C23C2×C7×C22⋊C411yescharacteristic
340C23C2×C7×D811yescharacteristic
341C23C2×C7×SD1611yescharacteristic
342D4—42no—
343D4—42no—
344D4—42no—
345D4—42no—
346D4—84no—
347D4—82no—
348D4—82no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4C7×#9911yescharacteristic
376C2×C4C7×#9911yescharacteristic
377C2×C4C7×#10212yes—
378C8—82no—
379C7#1187511yescharacteristic, direct product factor, Sylow
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×C7×#9911yescharacteristic
402C22C2×C7×#10211yescharacteristic
403C22C7×#42111yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—162no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—82no—
420C2C7×#151911yescharacteristic
421C2C7×#212011yescharacteristic
422C2C7×#212711yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1799C2semidirect
3. C7×#1459C2semidirect
4. C7×#1344C2semidirect
5. C7×#1343C2semidirect
7. C2×C4×C7×D4C4semidirect
11. C7×#397C4semidirect
13. C7×#392C4semidirect
17. C7×#382C4semidirect
23. #11875C7direct (N × H)
61. #1799C2×C7semidirect
62. #1459C2×C7semidirect
63. #1344C2×C7semidirect
64. #1343C2×C7semidirect
119. C2×C4×D4C4×C7semidirect
123. #397C4×C7semidirect
125. #392C4×C7semidirect
129. #382C4×C7semidirect
379. C7#11875direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25352
C7×#17992
C7×#14592
C7×#13442
C7×#13432
#118757