Finite Group Databasebeta

← C7×#12227 #13189443

Subgroups — C7×#12227 #13189443

Order 1792.

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

normal subgroups only

This group has 520 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×#2463C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic, complemented
3C7×#1602C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic
5C7×#1351C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C2×C7×C22⋊Q8—21no—
10C2×C4×C7×D4—21no—
11C7×#451—21no—
12C7×#450—21no—
13C7×#394—21no—
14C7×#393C2211yescharacteristic
15C7×#392C2211yescharacteristic
16C7×#388—21no—
17C7×#387—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#12227C711yesmaximal, 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—
39C7×C22⋊Q8—44no—
40C4×C7×D4—44no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—41no—
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×#103—41no—
56C7×#102—41no—
57C7×C22⋊C8—21no—
58C7×C22⋊C8—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—41no—
63C7×#95D411yescharacteristic
64C7×#95D411yescharacteristic
65#2463C2×C711yescharacteristic
66#2062C2×C711yescharacteristic, complemented
67#1602C2×C711yescharacteristic, complemented
68#1596C2×C711yescharacteristic
69#1351C2×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—41no—
78C2×C7×D4—42no—
79C2×C7×D4—41no—
80C2×C7×D4—42no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—42no—
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—42no—
92C22×C4×C7—42no—
93C22×C4×C7—42no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
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—24no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—21no—
106C7×C4⋊C4—42no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—82no—
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—42no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C42×C7—42no—
124#543C22×C711yescharacteristic
125C2×C22⋊Q8—21no—
126C2×C4×D4—21no—
127#451—21no—
128#450—21no—
129#394—21no—
130#393C22×C711yescharacteristic
131#392C22×C711yescharacteristic
132#388—21no—
133#387—21no—
134#386—21no—
135#385—21no—
136#385C22×C711yescharacteristic
137#342—21no—
138#342C22×C711yescharacteristic
139#328—21no—
140#328—21no—
141#323C22×C711yescharacteristic
142#323C22×C711yescharacteristic
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—22no—
146C23×C7—41no—
147C23×C7—41no—
148C23×C7—41no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7#12011yescharacteristic
155C23×C7#13511yescharacteristic
156C23×C7#13711yescharacteristic
157C7×Q8—42no—
158C7×Q8—81no—
159C7×Q8—81no—
160C7×Q8—81no—
161C7×D4—42no—
162C7×D4—44no—
163C7×D4—44no—
164C7×D4—81no—
165C2×C4×C7—22no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—41no—
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—42no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—81no—
189C2×C4×C7—84no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C7×C8—81no—
195C7×C8—81no—
196C22×Q8C7×D411yescharacteristic
197C22×D4—21no—
198C22×D4C7×D411yescharacteristic
199C23×C4—21no—
200C23×C4C7×D411yescharacteristic
201C2×Q16—41no—
202C2×SD16—41no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—44no—
208C4×D4—44no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4C7×D411yescharacteristic
221C2×C22⋊C4C23×C711yescharacteristic
222C2×C42—21no—
223#103—41no—
224#102—41no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—41no—
231#95C7×D411yescharacteristic
232#95C7×D411yescharacteristic
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—42no—
240C22×C7—42no—
241C22×C7—41no—
242C22×C7—41no—
243C22×C7—42no—
244C22×C7—81no—
245C22×C7—81no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7—81no—
249C22×C7#45211yescharacteristic
250C22×C7#45611yescharacteristic
251C22×C7B(4,𝔽2)11yescharacteristic
252C4×C7—42no—
253C4×C7—41no—
254C4×C7—42no—
255C4×C7—42no—
256C4×C7—41no—
257C4×C7—81no—
258C4×C7—81no—
259C4×C7—82no—
260C4×C7—81no—
261C4×C7—81no—
262C24—21no—
263C24C2×C7×D411yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4C2×C7×D411yescharacteristic
288C22×C4C2×C7×D411yescharacteristic
289Q16—82no—
290SD16—82no—
291C2×C8—41no—
292C2×C8—41no—
293C4⋊C4—24no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—82no—
301C22⋊C4—22no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—42no—
314C2×C7—21no—
315C2×C7—21no—
316C2×C7—41no—
317C2×C7—41no—
318C2×C7—82no—
319C2×C7#160311yescharacteristic
320C2×C7#219611yescharacteristic
321C2×C7#219711yescharacteristic
322C23—21no—
323C23—21no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23C7×#12011yescharacteristic
334C23C7×#13511yescharacteristic
335C23C7×#13711yescharacteristic
336Q8—42no—
337Q8—81no—
338Q8—81no—
339Q8—81no—
340D4—42no—
341D4—44no—
342D4—44no—
343D4—81no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—84no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C8—81no—
374C8—81no—
375C7#1222711yescharacteristic, direct product factor, Sylow
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22C7×#45211yescharacteristic
393C22C7×#45611yescharacteristic
394C22C7×B(4,𝔽2)11yescharacteristic
395C4—42no—
396C4—41no—
397C4—42no—
398C4—42no—
399C4—41no—
400C4—81no—
401C4—81no—
402C4—82no—
403C4—81no—
404C4—81no—
405C2—21no—
406C2—21no—
407C2—41no—
408C2—41no—
409C2—82no—
410C2C7×#160311yescharacteristic
411C2C7×#219611yescharacteristic
412C2C7×#219711yescharacteristic
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×#1602C2semidirect
5. C7×#1351C2semidirect
7. C7×#1326C2semidirect
27. #12227C7direct (N × H)
66. #2062C2×C7semidirect
67. #1602C2×C7semidirect
69. #1351C2×C7semidirect
71. #1326C2×C7semidirect
375. C7#12227direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24632
C7×#20622
C7×#16022
C7×#15962
C7×#13512
C7×#13432
C7×#13262
#122277