Finite Group Databasebeta

← C7×#12263 #13189479

Subgroups — C7×#12263 #13189479

Order 1792.

438 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2760C211yesmaximal, characteristic
2C7×#2030C211yesmaximal, characteristic, complemented
3C7×#1602C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic
5C7×#1351C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic
7C7×#1317C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C7×#451—21no—
10C7×#450—21no—
11C7×#401—21no—
12C7×#393—21no—
13C7×#393C2211yescharacteristic
14C7×#392C2211yescharacteristic
15C7×#386—21no—
16C7×#386—21no—
17C7×#386—21no—
18C7×#381—21no—
19C7×#381—21no—
20C7×#381C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12263C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C23×C4×C7D411yescharacteristic
33C2×C7×Q16—41no—
34C2×C7×SD16—41no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—41no—
38C7×C22⋊Q8—41no—
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—21no—
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C7×#103—41no—
49C7×#102—41no—
50C7×C22⋊C8—21no—
51C7×C22⋊C8—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—41no—
60C7×#95—41no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2760C2×C711yescharacteristic
64#2030C2×C711yescharacteristic, complemented
65#1602C2×C711yescharacteristic, complemented
66#1596C2×C711yescharacteristic
67#1351C2×C711yescharacteristic, complemented
68#1343C2×C711yescharacteristic
69#1317C2×C711yescharacteristic, complemented
70C24×C7—21no—
71C24×C7C2×D411yescharacteristic
72C2×C7×Q8—21no—
73C2×C7×Q8—21no—
74C2×C7×Q8—41no—
75C2×C7×Q8—81no—
76C2×C7×D4—41no—
77C2×C7×D4—42no—
78C2×C7×D4—42no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—81no—
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—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—42no—
94C22×C4×C7—41no—
95C22×C4×C7—81no—
96C22×C4×C7—81no—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C7×Q16—82no—
100C7×SD16—82no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—81no—
114C7×C22⋊C4—81no—
115C7×C22⋊C4—81no—
116C7×C22⋊C4—81no—
117C7×C22⋊C4—81no—
118#543C22×C711yescharacteristic
119#451—21no—
120#450—21no—
121#401—21no—
122#393—21no—
123#393C22×C711yescharacteristic
124#392C22×C711yescharacteristic
125#386—21no—
126#386—21no—
127#386—21no—
128#381—21no—
129#381—21no—
130#381C22×C711yescharacteristic
131#342—21no—
132#342C22×C711yescharacteristic
133#328—21no—
134#328—21no—
135#323C22×C711yescharacteristic
136#323C22×C711yescharacteristic
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#12011yescharacteristic
149C23×C7#13511yescharacteristic
150C23×C7#13711yescharacteristic
151C7×Q8—42no—
152C7×Q8—81no—
153C7×Q8—81no—
154C7×Q8—81no—
155C7×D4—42no—
156C7×D4—81no—
157C7×D4—82no—
158C7×D4—82no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
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—84no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7—81no—
186C2×C4×C7—81no—
187C2×C4×C7—81no—
188C7×C8—81no—
189C7×C8—81no—
190C22×Q8C7×D411yescharacteristic
191C22×D4—21no—
192C22×D4C7×D411yescharacteristic
193C23×C4—21no—
194C23×C4C7×D411yescharacteristic
195C2×Q16—41no—
196C2×SD16—41no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4C7×D411yescharacteristic
209C2×C22⋊C4C23×C711yescharacteristic
210#103—41no—
211#102—41no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—41no—
222#95—41no—
223#95C7×D411yescharacteristic
224#95C7×D411yescharacteristic
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—42no—
232C22×C7—42no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—42no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7#45211yescharacteristic
242C22×C7#45511yescharacteristic
243C22×C7B(4,𝔽2)11yescharacteristic
244C4×C7—41no—
245C4×C7—41no—
246C4×C7—81no—
247C4×C7—81no—
248C4×C7—81no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7—81no—
253C4×C7—161no—
254C24—21no—
255C24C2×C7×D411yescharacteristic
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—41no—
259C2×Q8—81no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—81no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4C2×C7×D411yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283Q16—82no—
284SD16—82no—
285C2×C8—41no—
286C2×C8—41no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C22⋊C4—81no—
302C2×C7—21no—
303C2×C7—21no—
304C2×C7—41no—
305C2×C7—41no—
306C2×C7—82no—
307C2×C7#160611yescharacteristic
308C2×C7#219711yescharacteristic
309C2×C7#220011yescharacteristic
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23C7×#12011yescharacteristic
322C23C7×#13511yescharacteristic
323C23C7×#13711yescharacteristic
324Q8—42no—
325Q8—81no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330D4—82no—
331D4—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C8—81no—
362C8—81no—
363C7#1226311yescharacteristic, direct product factor, Sylow
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C7×#45211yescharacteristic
381C22C7×#45511yescharacteristic
382C22C7×B(4,𝔽2)11yescharacteristic
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—161no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C7×#160611yescharacteristic
399C2C7×#219711yescharacteristic
400C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2030C2semidirect
3. C7×#1602C2semidirect
5. C7×#1351C2semidirect
7. C7×#1317C2semidirect
27. #12263C7direct (N × H)
64. #2030C2×C7semidirect
65. #1602C2×C7semidirect
67. #1351C2×C7semidirect
69. #1317C2×C7semidirect
363. C7#12263direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27602
C7×#20302
C7×#16022
C7×#15962
C7×#13512
C7×#13432
C7×#13172
#122637