Finite Group Databasebeta

← C7×#12245 #13189461

Subgroups — C7×#12245 #13189461

Order 1792.

458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2712C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic, complemented
3C7×#1608C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic
5C7×#1344C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C2×C7×#124—21no—
10C7×#483—21no—
11C7×#450—21no—
12C7×#397C2211yescharacteristic
13C7×#394—21no—
14C7×#393—21no—
15C7×#392C2211yescharacteristic
16C7×#391—21no—
17C7×#388—21no—
18C7×#386—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12245C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8—21no—
29C22×C7×D4—21no—
30C22×C7×D4D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C2×C7×SD16—41no—
33C7×#124—42no—
34C7×#124—42no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—41no—
38C7×C22⋊Q8—41no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—41no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4D411yescharacteristic
51C2×C7×C22⋊C4C2311yescharacteristic
52C2×C42×C7—21no—
53C7×#106—41no—
54C7×#103—41no—
55C7×#102—41no—
56C7×C22⋊C8—21no—
57C7×C22⋊C8—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—41no—
63C7×#95D411yescharacteristic
64C7×#95D411yescharacteristic
65#2712C2×C711yescharacteristic
66#2062C2×C711yescharacteristic, complemented
67#1608C2×C711yescharacteristic, complemented
68#1596C2×C711yescharacteristic
69#1344C2×C711yescharacteristic, complemented
70#1343C2×C711yescharacteristic
71#1326C2×C711yescharacteristic, complemented
72C24×C7—21no—
73C24×C7C2×D411yescharacteristic
74C2×C7×Q8—21no—
75C2×C7×Q8—21no—
76C2×C7×Q8—42no—
77C2×C7×Q8—42no—
78C2×C7×Q8—81no—
79C2×C7×D4—41no—
80C2×C7×D4—42no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—81no—
84C2×C7×D4—81no—
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—21no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—81no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C7×SD16—82no—
98C2×C7×C8—41no—
99C2×C7×C8—41no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—81no—
106C7×C4⋊C4—82no—
107C7×C22⋊C4—21no—
108C7×C22⋊C4—22no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—42no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—81no—
119C7×C22⋊C4—81no—
120C7×C22⋊C4SD1612yescomplemented
121C42×C7—42no—
122#543C22×C711yescharacteristic
123C2×#124—21no—
124#483—21no—
125#450—21no—
126#397C22×C711yescharacteristic
127#394—21no—
128#393—21no—
129#392C22×C711yescharacteristic
130#391—21no—
131#388—21no—
132#386—21no—
133#385—21no—
134#385C22×C711yescharacteristic
135#342—21no—
136#342C22×C711yescharacteristic
137#328—21no—
138#328—21no—
139#323C22×C711yescharacteristic
140#323C22×C711yescharacteristic
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7#12011yescharacteristic
153C23×C7C2×SD1611yescharacteristic
154C23×C7AGL(1,ℤ/8)11yescharacteristic
155C7×Q8—84no—
156C7×Q8—81no—
157C7×Q8—81no—
158C7×D4—42no—
159C7×D4—82no—
160C7×D4—82no—
161C7×D4—81no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—42no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—42no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—81no—
186C2×C4×C7—81no—
187C2×C4×C7—81no—
188C2×C4×C7—81no—
189C2×C4×C7—81no—
190C7×C8—81no—
191C7×C8—81no—
192C22×Q8—21no—
193C22×D4—21no—
194C22×D4C7×D411yescharacteristic
195C23×C4C7×D411yescharacteristic
196C2×SD16—41no—
197#124—42no—
198#124—42no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C7×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C7×D411yescharacteristic
215C2×C22⋊C4C23×C711yescharacteristic
216C2×C42—21no—
217#106—41no—
218#103—41no—
219#102—41no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—41no—
227#95C7×D411yescharacteristic
228#95C7×D411yescharacteristic
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7#45011yescharacteristic
246C22×C7#45611yescharacteristic
247C22×C7B(4,𝔽2)11yescharacteristic
248C4×C7—42no—
249C4×C7—42no—
250C4×C7—41no—
251C4×C7—41no—
252C4×C7—81no—
253C4×C7—82no—
254C4×C7—81no—
255C4×C7—81no—
256C4×C7—81no—
257C4×C7—81no—
258C24—21no—
259C24C2×C7×D411yescharacteristic
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—81no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4C2×C7×D411yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283SD16—82no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—81no—
292C4⋊C4—82no—
293C22⋊C4—21no—
294C22⋊C4—22no—
295C22⋊C4—41no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4C7×SD1612yescomplemented
307C42—42no—
308C2×C7—21no—
309C2×C7—21no—
310C2×C7—41no—
311C2×C7—41no—
312C2×C7—161no—
313C2×C7#160411yescharacteristic
314C2×C7#219811yescharacteristic
315C2×C7#219911yescharacteristic
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23C7×#12011yescharacteristic
328C23C2×C7×SD1611yescharacteristic
329C23C7×AGL(1,ℤ/8)11yescharacteristic
330Q8—84no—
331Q8—81no—
332Q8—81no—
333D4—42no—
334D4—82no—
335D4—82no—
336D4—81no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C8—81no—
366C8—81no—
367C7#1224511yescharacteristic, direct product factor, Sylow
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22C7×#45011yescharacteristic
385C22C7×#45611yescharacteristic
386C22C7×B(4,𝔽2)11yescharacteristic
387C4—42no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2C7×#160411yescharacteristic
403C2C7×#219811yescharacteristic
404C2C7×#219911yescharacteristic
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×#2062C2semidirect
3. C7×#1608C2semidirect
5. C7×#1344C2semidirect
7. C7×#1326C2semidirect
27. #12245C7direct (N × H)
66. #2062C2×C7semidirect
67. #1608C2×C7semidirect
69. #1344C2×C7semidirect
71. #1326C2×C7semidirect
120. C7×C22⋊C4SD16semidirect
306. C22⋊C4C7×SD16semidirect
367. C7#12245direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27122
C7×#20622
C7×#16082
C7×#15962
C7×#13442
C7×#13432
C7×#13262
#122457