Finite Group Databasebeta

← C7×#10331 #13187547

Subgroups — C7×#10331 #13187547

Order 1792.

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

normal subgroups only

This group has 672 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×#3193C211yesmaximal, characteristic
2C7×#2502C211yesmaximal, characteristic, complemented
3C7×#2501C211yesmaximal, characteristic
4C7×#1821C211yesmaximal, characteristic, complemented
5C7×#1508C211yesmaximal, characteristic, complemented
6C7×#1507C211yesmaximal, characteristic
7C7×#1297C211yesmaximal, characteristic, complemented
8C2×C7×#123—22no—
9C2×C7×C22⋊Q8C2211yescharacteristic
10C2×C7×#121C2211yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C7×#484—21no—
13C7×#480—21no—
14C7×#468—22no—
15C7×#450—21no—
16C7×#448—21no—
17C7×#407C2211yescharacteristic
18C7×#407C2211yescharacteristic
19C7×#399—22no—
20C7×#398—22no—
21C7×#391—21no—
22C7×#390—21no—
23C7×#385—22no—
24C7×#384—21no—
25C7×#384—21no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C4×C7×C22⋊C4C2211yescharacteristic
28C7×#343—22no—
29C7×#328—22no—
30C7×#327—22no—
31#10331C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7D412yes—
36C23×C4×C7C2311yescharacteristic
37C2×C7×SD16—22no—
38C2×C7×M16—21no—
39C2×C7×M16D412yes—
40C7×#123—48no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C7×#121—21no—
46C7×#121—21no—
47C7×#121—21no—
48C7×#121—21no—
49C2×C7×C4⋊C4—24no—
50C2×C7×C4⋊C4—21no—
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⋊C4—24no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—24no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C42×C7—21no—
66C2×C42×C7—21no—
67C7×#107—22no—
68C7×#103—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#3193C2×C711yescharacteristic
78#2502C2×C711yescharacteristic, complemented
79#2501C2×C711yescharacteristic
80#1821C2×C711yescharacteristic, complemented
81#1508C2×C711yescharacteristic, complemented
82#1507C2×C711yescharacteristic
83#1297C2×C711yescharacteristic, complemented
84C24×C7—21no—
85C24×C7C2×D411yescharacteristic
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×D4—21no—
90C2×C7×D4—21no—
91C2×C7×D4—42no—
92C2×C7×D4—44no—
93C2×C7×D4—41no—
94C2×C7×D4—41no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—22no—
103C22×C4×C7—42no—
104C22×C4×C7—44no—
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7#4111yescharacteristic
111C7×SD16—82no—
112C7×M16—42no—
113C7×M16—42no—
114C2×C7×C8—22no—
115C2×C7×C8—41no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—48no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4—41no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—84no—
128C7×C22⋊C4—24no—
129C7×C22⋊C4—24no—
130C7×C22⋊C4—48no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C42×C7—42no—
136C42×C7—42no—
137C2×#123—22no—
138C2×C22⋊Q8C22×C711yescharacteristic
139C2×#121C22×C711yescharacteristic
140C22×C4⋊C4C22×C711yescharacteristic
141#484—21no—
142#480—21no—
143#468—22no—
144#450—21no—
145#448—21no—
146#407C22×C711yescharacteristic
147#407C22×C711yescharacteristic
148#399—22no—
149#398—22no—
150#391—21no—
151#390—21no—
152#385—22no—
153#384—21no—
154#384—21no—
155C4×C22⋊C4C22×C711yescharacteristic
156C4×C22⋊C4C22×C711yescharacteristic
157#343—22no—
158#328—22no—
159#327—22no—
160C23×C7—22no—
161C23×C7—22no—
162C23×C7—41no—
163C23×C7—42no—
164C23×C7—42no—
165C23×C7—41no—
166C23×C7—42no—
167C23×C7#12011yescharacteristic
168C23×C7#12011yescharacteristic
169C23×C7#12111yescharacteristic
170C7×Q8—42no—
171C7×Q8—81no—
172C7×D4—42no—
173C7×D4—81no—
174C7×D4—82no—
175C7×D4—82no—
176C2×C4×C7—22no—
177C2×C4×C7—22no—
178C2×C4×C7—22no—
179C2×C4×C7—24no—
180C2×C4×C7—24no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—44no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—42no—
196C2×C4×C7—44no—
197C2×C4×C7—82no—
198C2×C4×C7#12011yescharacteristic
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12711yescharacteristic
202C7×C8—42no—
203C7×C8—81no—
204C22×Q8C7×D411yescharacteristic
205C22×D4—21no—
206C22×D4C7×D411yescharacteristic
207C23×C4C7×D412yes—
208C23×C4C23×C711yescharacteristic
209C2×SD16—22no—
210C2×M16—21no—
211C2×M16C7×D412yes—
212#123—48no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217#121—21no—
218#121—21no—
219#121—21no—
220#121—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4C7×D411yescharacteristic
231C2×C4⋊C4C7×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—21no—
239#107—22no—
240#103—22no—
241#102—22no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246#95—24no—
247#95C7×D412yes—
248#95C7×D412yes—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—44no—
257C22×C7—84no—
258C22×C7—81no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7#39211yescharacteristic
262C22×C7#45312yes—
263C22×C7#45612yes—
264C4×C7—21no—
265C4×C7—21no—
266C4×C7—21no—
267C4×C7—44no—
268C4×C7—44no—
269C4×C7—81no—
270C4×C7—82no—
271C4×C7—81no—
272C4×C7—81no—
273C4×C7#45812yes—
274C24—21no—
275C24C2×C7×D411yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—41no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4C2×C7×D411yescharacteristic
296C22×C4C2×C7×D411yescharacteristic
297C22×C4C2×C7×D411yescharacteristic
298C22×C4C2×C7×D411yescharacteristic
299C22×C4C2×C7×D411yescharacteristic
300C22×C4C7×#4111yescharacteristic
301SD16—82no—
302M16—42no—
303M16—42no—
304C2×C8—22no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—48no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—84no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—48no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—42no—
326C42—42no—
327C2×C7—22no—
328C2×C7—22no—
329C2×C7—41no—
330C2×C7—161no—
331C2×C7#200211yescharacteristic
332C2×C7#200511yescharacteristic
333C2×C7#202211yescharacteristic
334C23—22no—
335C23—22no—
336C23—41no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—42no—
341C23C7×#12011yescharacteristic
342C23C7×#12011yescharacteristic
343C23C7×#12111yescharacteristic
344Q8—42no—
345Q8—81no—
346D4—42no—
347D4—81no—
348D4—82no—
349D4—82no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—82no—
372C2×C4C7×#12011yescharacteristic
373C2×C4C7×#12111yescharacteristic
374C2×C4C7×#12111yescharacteristic
375C2×C4C7×#12711yescharacteristic
376C8—42no—
377C8—81no—
378C7#1033111yescharacteristic, direct product factor, Sylow
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22—84no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22C7×#39211yescharacteristic
392C22C7×#45312yes—
393C22C7×#45612yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—44no—
398C4—44no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4C7×#45812yes—
404C2—22no—
405C2—22no—
406C2—41no—
407C2—161no—
408C2C7×#200211yescharacteristic
409C2C7×#200511yescharacteristic
410C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
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×#2502C2semidirect
4. C7×#1821C2semidirect
5. C7×#1508C2semidirect
7. C7×#1297C2semidirect
31. #10331C7direct (N × H)
78. #2502C2×C7semidirect
80. #1821C2×C7semidirect
81. #1508C2×C7semidirect
83. #1297C2×C7semidirect
378. C7#10331direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31932
C7×#25022
C7×#25012
C7×#18212
C7×#15082
C7×#15072
C7×#12972
#103317