Finite Group Databasebeta

← C7×#11843 #13189059

Subgroups — C7×#11843 #13189059

Order 1792.

620 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2539C211yesmaximal, characteristic
2C7×#1819C211yesmaximal, characteristic, complemented
3C2×C7×#323C211yesmaximal, characteristic, complemented
4C7×#1352C212yesmaximal, complemented
5C7×#1343C212yesmaximal, complemented
6C2×C7×#123—21no—
7C2×C7×#121C411yescharacteristic, complemented
8C2×C7×#120—21no—
9C22×C7×C22⋊C4C2211yescharacteristic
10C2×C7×C22⋊C8—21no—
11C7×#399C411yescharacteristic, complemented
12C7×#399C2211yescharacteristic
13C7×#392C411yescharacteristic, complemented
14C7×#392C2211yescharacteristic
15C7×#390—21no—
16C7×#390—21no—
17C7×#379—21no—
18C4×C7×C22⋊C4C411yescharacteristic, complemented
19C7×#342—21no—
20C7×#342C2212yes—
21C7×#323—21no—
22C7×#323C2212yes—
23#11843C711yesmaximal, characteristic, direct product factor, Sylow
24C25×C7D411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C22×C7×C8—41no—
32C7×#123—44no—
33C7×#121—41no—
34C7×#121—41no—
35C7×#120—44no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4—41no—
48C2×C7×C22⋊C4—41no—
49C2×C7×C22⋊C4—41no—
50C2×C7×C22⋊C4C2×C411yescharacteristic
51C2×C7×C22⋊C4D411yescharacteristic
52C2×C7×C22⋊C4D411yescharacteristic
53C2×C7×C22⋊C4C2311yescharacteristic
54C2×C42×C7—21no—
55C7×C22⋊C8—22no—
56C7×C22⋊C8—41no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#2539C2×C711yescharacteristic
60#1819C2×C711yescharacteristic, complemented
61C2×#323C2×C711yescharacteristic, complemented
62#1352C2×C712yescomplemented
63#1343C2×C712yescomplemented
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—22no—
67C24×C7—21no—
68C24×C7—41no—
69C24×C7—42no—
70C24×C7—81no—
71C24×C7C22⋊C411yescharacteristic
72C24×C7C22⋊C411yescharacteristic
73C24×C7C2×D411yescharacteristic
74C2×C7×D4—42no—
75C2×C7×D4—42no—
76C2×C7×D4—41no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—42no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—42no—
83C2×C7×D4—81no—
84C2×C7×D4—81no—
85C2×C7×D4—81no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7—81no—
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22×C411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C2×C7×C8—42no—
101C2×C7×C8—82no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—42no—
105C7×C22⋊C4—24no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—82no—
118C7×C22⋊C4—82no—
119C42×C7—81no—
120C2×#123—21no—
121C2×#121C4×C711yescharacteristic, complemented
122C2×#120—21no—
123C22×C22⋊C4C22×C711yescharacteristic
124C2×C22⋊C8—21no—
125#399C4×C711yescharacteristic, complemented
126#399C22×C711yescharacteristic
127#392C4×C711yescharacteristic, complemented
128#392C22×C711yescharacteristic
129#390—21no—
130#390—21no—
131#379—21no—
132C4×C22⋊C4C4×C711yescharacteristic, complemented
133#342—21no—
134#342C22×C712yes—
135#323—21no—
136#323C22×C712yes—
137C23×C7—22no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—41no—
146C23×C7—42no—
147C23×C7—42no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—42no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—41no—
161C23×C7—81no—
162C23×C7—81no—
163C23×C7—84no—
164C23×C7—84no—
165C23×C7#9911yescharacteristic
166C23×C7#9911yescharacteristic
167C23×C7C2×C22⋊C411yescharacteristic
168C23×C7#13512yes—
169C7×D4—84no—
170C7×D4—84no—
171C7×D4—82no—
172C7×D4—82no—
173C7×D4—82no—
174C2×C4×C7—22no—
175C2×C4×C7—22no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7—81no—
195C2×C4×C7—82no—
196C2×C4×C7—82no—
197C2×C4×C7—82no—
198C7×C8—82no—
199C25C7×D411yescharacteristic
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C2×C4×C711yescharacteristic
204C23×C4C2×C4×C711yescharacteristic
205C23×C4C7×D411yescharacteristic
206C22×C8—41no—
207#123—44no—
208#121—41no—
209#121—41no—
210#120—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×C4×C711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4C2×C4×C711yescharacteristic
226C2×C22⋊C4C7×D411yescharacteristic
227C2×C22⋊C4C7×D411yescharacteristic
228C2×C22⋊C4C23×C711yescharacteristic
229C2×C42—21no—
230C22⋊C8—22no—
231C22⋊C8—41no—
232#95C2×C4×C711yescharacteristic
233#95C2×C4×C711yescharacteristic
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—22no—
241C22×C7—21no—
242C22×C7—42no—
243C22×C7—42no—
244C22×C7—42no—
245C22×C7—42no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—81no—
252C22×C7—84no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7—84no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7#35312yes—
263C22×C7C2×#9911yescharacteristic
264C22×C7#41611yescharacteristic
265C22×C7#42111yescharacteristic
266C4×C7—42no—
267C4×C7—42no—
268C4×C7—41no—
269C4×C7—41no—
270C4×C7—41no—
271C4×C7—41no—
272C4×C7—81no—
273C4×C7—162no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—41no—
279C24—42no—
280C24—81no—
281C24C7×C22⋊C411yescharacteristic
282C24C7×C22⋊C411yescharacteristic
283C24C2×C7×D411yescharacteristic
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—81no—
294C2×D4—81no—
295C2×D4—81no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—81no—
306C22×C4C7×C22⋊C411yescharacteristic
307C22×C4C7×C22⋊C411yescharacteristic
308C22×C4C22×C4×C711yescharacteristic
309C22×C4C2×C7×D411yescharacteristic
310C2×C8—42no—
311C2×C8—82no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C22⋊C4—24no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C42—81no—
330C2×C7—21no—
331C2×C7—21no—
332C2×C7—22no—
333C2×C7—42no—
334C2×C7—41no—
335C2×C7—81no—
336C2×C7—161no—
337C2×C7#151611yescharacteristic
338C2×C7#212111yescharacteristic
339C2×C7C2×#35311yescharacteristic
340C23—22no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—81no—
365C23—81no—
366C23—84no—
367C23—84no—
368C23C7×#9911yescharacteristic
369C23C7×#9911yescharacteristic
370C23C2×C7×C22⋊C411yescharacteristic
371C23C7×#13512yes—
372D4—84no—
373D4—84no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C8—82no—
402C7#1184311yescharacteristic, direct product factor, Sylow
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—84no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—84no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22C7×#35312yes—
432C22C2×C7×#9911yescharacteristic
433C22C7×#41611yescharacteristic
434C22C7×#42111yescharacteristic
435C4—42no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4—162no—
443C2—21no—
444C2—21no—
445C2—22no—
446C2—42no—
447C2—41no—
448C2—81no—
449C2—161no—
450C2C7×#151611yescharacteristic
451C2C7×#212111yescharacteristic
452C2C2×C7×#35311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#2539C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C24×C7 ◁ C22×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1819C2semidirect
3. C2×C7×#323C2semidirect
4. C7×#1352C2semidirect
5. C7×#1343C2semidirect
7. C2×C7×#121C4semidirect
11. C7×#399C4semidirect
13. C7×#392C4semidirect
18. C4×C7×C22⋊C4C4semidirect
23. #11843C7direct (N × H)
60. #1819C2×C7semidirect
61. C2×#323C2×C7semidirect
62. #1352C2×C7semidirect
63. #1343C2×C7semidirect
121. C2×#121C4×C7semidirect
125. #399C4×C7semidirect
127. #392C4×C7semidirect
132. C4×C22⋊C4C4×C7semidirect
402. C7#11843direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25392
C7×#18192
C2×C7×#3232
C7×#13522
C7×#13432
#118437