Finite Group Databasebeta

← C7×#10328 #13187544

Subgroups — C7×#10328 #13187544

Order 1792.

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3192C211yesmaximal, characteristic
2C7×#2504C211yesmaximal, characteristic, complemented
3C7×#2501C211yesmaximal, characteristic, complemented
4C7×#1821C211yesmaximal, characteristic, complemented
5C7×#1507C211yesmaximal, characteristic, complemented
6C7×#1507C211yesmaximal, characteristic, complemented
7C7×#1297C211yesmaximal, characteristic, complemented
8C2×C7×#123—22no—
9C2×C7×#121C2211yescharacteristic
10C2×C7×#121C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C7×#481—21no—
13C7×#481—21no—
14C7×#469—22no—
15C7×#447—21no—
16C7×#447—21no—
17C7×#407C2211yescharacteristic
18C7×#407C2211yescharacteristic
19C7×#399—22no—
20C7×#397—22no—
21C7×#390—21no—
22C7×#390—21no—
23C7×#386—22no—
24C7×#384—21no—
25C7×#382—21no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C4×C7×C22⋊C4C2211yescharacteristic
28C7×#343—22no—
29C7×#327—22no—
30C7×#327—22no—
31#10328C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×D4—21no—
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7D412yes—
37C23×C4×C7C2311yescharacteristic
38C2×C7×D8—22no—
39C2×C7×M16—21no—
40C2×C7×M16D412yes—
41C7×#123—48no—
42C7×#121—21no—
43C7×#121—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C7×#121—21no—
48C7×#121—21no—
49C7×#121—21no—
50C2×C7×C4⋊C4—24no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—24no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—24no—
65C2×C42×C7—21no—
66C2×C42×C7—21no—
67C7×#106—22no—
68C7×#102—22no—
69C7×#102—22no—
70C7×C22⋊C8—21no—
71C7×C22⋊C8—21no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×#95—24no—
75C7×#95D412yes—
76C7×#95D412yes—
77#3192C2×C711yescharacteristic
78#2504C2×C711yescharacteristic, complemented
79#2501C2×C711yescharacteristic, complemented
80#1821C2×C711yescharacteristic, complemented
81#1507C2×C711yescharacteristic, complemented
82#1507C2×C711yescharacteristic, complemented
83#1297C2×C711yescharacteristic, complemented
84C24×C7—21no—
85C24×C7—21no—
86C24×C7C2×D411yescharacteristic
87C2×C7×D4—21no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—42no—
92C2×C7×D4—42no—
93C2×C7×D4—44no—
94C2×C7×D4—44no—
95C2×C7×D4—41no—
96C2×C7×D4—41no—
97C2×C7×D4—41no—
98C2×C7×D4—41no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—42no—
107C22×C4×C7—44no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7#4111yescharacteristic
114C7×D8—82no—
115C7×M16—42no—
116C7×M16—42no—
117C2×C7×C8—22no—
118C2×C7×C8—41no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—48no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C22⋊C4—24no—
129C7×C22⋊C4—48no—
130C7×C22⋊C4—42no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—84no—
136C42×C7—42no—
137C42×C7—42no—
138C2×#123—22no—
139C2×#121C22×C711yescharacteristic
140C2×#121C22×C711yescharacteristic
141C22×C4⋊C4C22×C711yescharacteristic
142#481—21no—
143#481—21no—
144#469—22no—
145#447—21no—
146#447—21no—
147#407C22×C711yescharacteristic
148#407C22×C711yescharacteristic
149#399—22no—
150#397—22no—
151#390—21no—
152#390—21no—
153#386—22no—
154#384—21no—
155#382—21no—
156C4×C22⋊C4C22×C711yescharacteristic
157C4×C22⋊C4C22×C711yescharacteristic
158#343—22no—
159#327—22no—
160#327—22no—
161C23×C7—22no—
162C23×C7—22no—
163C23×C7—41no—
164C23×C7—42no—
165C23×C7—42no—
166C23×C7—42no—
167C23×C7—42no—
168C23×C7—41no—
169C23×C7—41no—
170C23×C7—42no—
171C23×C7—42no—
172C23×C7#12011yescharacteristic
173C23×C7#12011yescharacteristic
174C23×C7#12111yescharacteristic
175C7×D4—42no—
176C7×D4—42no—
177C7×D4—81no—
178C7×D4—81no—
179C7×D4—82no—
180C7×D4—82no—
181C7×D4—82no—
182C7×D4—82no—
183C2×C4×C7—22no—
184C2×C4×C7—22no—
185C2×C4×C7—22no—
186C2×C4×C7—24no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—44no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—44no—
202C2×C4×C7—82no—
203C2×C4×C7#12011yescharacteristic
204C2×C4×C7#12111yescharacteristic
205C2×C4×C7#12111yescharacteristic
206C2×C4×C7#12711yescharacteristic
207C7×C8—42no—
208C7×C8—81no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C7×D411yescharacteristic
212C22×D4C7×D411yescharacteristic
213C23×C4C7×D412yes—
214C23×C4C23×C711yescharacteristic
215C2×D8—22no—
216C2×M16—21no—
217C2×M16C7×D412yes—
218#123—48no—
219#121—21no—
220#121—21no—
221#121—21no—
222#121—21no—
223#121—21no—
224#121—21no—
225#121—21no—
226#121—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4C7×D411yescharacteristic
235C2×C4⋊C4C7×D411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—24no—
242C2×C42—21no—
243C2×C42—21no—
244#106—22no—
245#102—22no—
246#102—22no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251#95—24no—
252#95C7×D412yes—
253#95C7×D412yes—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—44no—
262C22×C7—84no—
263C22×C7—84no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7—81no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7#39211yescharacteristic
271C22×C7#45312yes—
272C22×C7#45412yes—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—21no—
276C4×C7—44no—
277C4×C7—82no—
278C4×C7—82no—
279C4×C7—81no—
280C4×C7—81no—
281C4×C7#45812yes—
282C24—21no—
283C24—21no—
284C24C2×C7×D411yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4C2×C7×D411yescharacteristic
307C22×C4C2×C7×D411yescharacteristic
308C22×C4C2×C7×D411yescharacteristic
309C22×C4C2×C7×D411yescharacteristic
310C22×C4C2×C7×D411yescharacteristic
311C22×C4C7×#4111yescharacteristic
312D8—82no—
313M16—42no—
314M16—42no—
315C2×C8—22no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—48no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—24no—
327C22⋊C4—48no—
328C22⋊C4—42no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—84no—
334C42—42no—
335C42—42no—
336C2×C7—22no—
337C2×C7—22no—
338C2×C7—41no—
339C2×C7—161no—
340C2×C7—161no—
341C2×C7#200211yescharacteristic
342C2×C7#200311yescharacteristic
343C2×C7#202211yescharacteristic
344C23—22no—
345C23—22no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23C7×#12011yescharacteristic
356C23C7×#12011yescharacteristic
357C23C7×#12111yescharacteristic
358D4—42no—
359D4—42no—
360D4—81no—
361D4—81no—
362D4—82no—
363D4—82no—
364D4—82no—
365D4—82no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4C7×#12011yescharacteristic
387C2×C4C7×#12111yescharacteristic
388C2×C4C7×#12111yescharacteristic
389C2×C4C7×#12711yescharacteristic
390C8—42no—
391C8—81no—
392C7#1032811yescharacteristic, direct product factor, Sylow
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22—84no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22C7×#39211yescharacteristic
410C22C7×#45312yes—
411C22C7×#45412yes—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—44no—
416C4—82no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4C7×#45812yes—
421C2—22no—
422C2—22no—
423C2—41no—
424C2—161no—
425C2—161no—
426C2C7×#200211yescharacteristic
427C2C7×#200311yescharacteristic
428C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC7×#3192C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2504C2semidirect
3. C7×#2501C2semidirect
4. C7×#1821C2semidirect
5. C7×#1507C2semidirect
6. C7×#1507C2semidirect
7. C7×#1297C2semidirect
31. #10328C7direct (N × H)
78. #2504C2×C7semidirect
79. #2501C2×C7semidirect
80. #1821C2×C7semidirect
81. #1507C2×C7semidirect
82. #1507C2×C7semidirect
83. #1297C2×C7semidirect
392. C7#10328direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31922
C7×#25042
C7×#25012
C7×#18212
C7×#15072
C7×#12972
#103287