Finite Group Databasebeta

← C7×#12226 #13189442

Subgroups — C7×#12226 #13189442

Order 1792.

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

normal subgroups only

This group has 510 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×#2455C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic, complemented
3C7×#1614C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C2×C7×#123—21no—
10C2×C4×C7×D4—21no—
11C7×#484—21no—
12C7×#450—21no—
13C7×#399C2211yescharacteristic
14C7×#394—21no—
15C7×#392C2211yescharacteristic
16C7×#389—21no—
17C7×#388—21no—
18C7×#385—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12226C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7D411yescharacteristic
32C2×C7×SD16—41no—
33C7×#123—44no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—41no—
37C7×C22⋊Q8—41no—
38C4×C7×D4—44no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—41no—
44C2×C7×C4⋊C4D411yescharacteristic
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⋊C4—21no—
51C2×C7×C22⋊C4—21no—
52C2×C7×C22⋊C4D411yescharacteristic
53C2×C7×C22⋊C4C2311yescharacteristic
54C2×C42×C7—21no—
55C7×#107—41no—
56C7×#103—41no—
57C7×#102—41no—
58C7×C22⋊C8—21no—
59C7×C22⋊C8—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—41no—
63C7×#95D411yescharacteristic
64C7×#95D411yescharacteristic
65#2455C2×C711yescharacteristic
66#2062C2×C711yescharacteristic, complemented
67#1614C2×C711yescharacteristic, complemented
68#1596C2×C711yescharacteristic
69#1352C2×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×D4—42no—
77C2×C7×D4—41no—
78C2×C7×D4—42no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—42no—
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—42no—
90C22×C4×C7—42no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—42no—
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—42no—
104C7×C4⋊C4—42no—
105C7×C4⋊C4—42no—
106C7×C4⋊C4—42no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—82no—
110C7×C22⋊C4—24no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—21no—
113C7×C22⋊C4—21no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C42×C7—42no—
123#543C22×C711yescharacteristic
124C2×#123—21no—
125C2×C4×D4—21no—
126#484—21no—
127#450—21no—
128#399C22×C711yescharacteristic
129#394—21no—
130#392C22×C711yescharacteristic
131#389—21no—
132#388—21no—
133#385—21no—
134#385—21no—
135#385C22×C711yescharacteristic
136#342—21no—
137#342C22×C711yescharacteristic
138#328—21no—
139#328—21no—
140#323C22×C711yescharacteristic
141#323C22×C711yescharacteristic
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—22no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7#12011yescharacteristic
154C23×C7#13511yescharacteristic
155C23×C7#13711yescharacteristic
156C7×Q8—81no—
157C7×Q8—81no—
158C7×D4—42no—
159C7×D4—44no—
160C7×D4—44no—
161C7×D4—81no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—22no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—41no—
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—84no—
187C2×C4×C7—81no—
188C2×C4×C7—81no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C7×C8—81no—
192C7×C8—81no—
193C22×D4—21no—
194C22×D4C7×D411yescharacteristic
195C23×C4—21no—
196C23×C4C7×D411yescharacteristic
197C2×SD16—41no—
198#123—44no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C4×D4—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C7×D411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4C7×D411yescharacteristic
218C2×C22⋊C4C23×C711yescharacteristic
219C2×C42—21no—
220#107—41no—
221#103—41no—
222#102—41no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225#95—21no—
226#95—21no—
227#95—41no—
228#95C7×D411yescharacteristic
229#95C7×D411yescharacteristic
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—42no—
237C22×C7—42no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—42no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7#45211yescharacteristic
247C22×C7#45611yescharacteristic
248C22×C7B(4,𝔽2)11yescharacteristic
249C4×C7—42no—
250C4×C7—42no—
251C4×C7—41no—
252C4×C7—42no—
253C4×C7—41no—
254C4×C7—81no—
255C4×C7—82no—
256C4×C7—81no—
257C4×C7—81no—
258C4×C7—81no—
259C24—21no—
260C24C2×C7×D411yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—81no—
282C22×C4C2×C7×D411yescharacteristic
283C22×C4C2×C7×D411yescharacteristic
284SD16—82no—
285C2×C8—41no—
286C2×C8—41no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—82no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—42no—
310C2×C7—21no—
311C2×C7—21no—
312C2×C7—41no—
313C2×C7—41no—
314C2×C7—82no—
315C2×C7#160311yescharacteristic
316C2×C7#219611yescharacteristic
317C2×C7#219811yescharacteristic
318C23—21no—
319C23—21no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C7×#12011yescharacteristic
330C23C7×#13511yescharacteristic
331C23C7×#13711yescharacteristic
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—44no—
336D4—44no—
337D4—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
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—41no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C8—81no—
368C8—81no—
369C7#1222611yescharacteristic, direct product factor, Sylow
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C7×#45211yescharacteristic
387C22C7×#45611yescharacteristic
388C22C7×B(4,𝔽2)11yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—82no—
404C2C7×#160311yescharacteristic
405C2C7×#219611yescharacteristic
406C2C7×#219811yescharacteristic
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×#1614C2semidirect
5. C7×#1352C2semidirect
7. C7×#1326C2semidirect
27. #12226C7direct (N × H)
66. #2062C2×C7semidirect
67. #1614C2×C7semidirect
69. #1352C2×C7semidirect
71. #1326C2×C7semidirect
369. C7#12226direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24552
C7×#20622
C7×#16142
C7×#15962
C7×#13522
C7×#13432
C7×#13262
#122267