Finite Group Databasebeta

← C7×#12230 #13189446

Subgroups — C7×#12230 #13189446

Order 1792.

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

normal subgroups only

This group has 422 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×#2824C211yesmaximal, 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
9C7×#451—21no—
10C7×#450—21no—
11C7×#400—21no—
12C7×#397—21no—
13C7×#394—21no—
14C7×#394—21no—
15C7×#393—21no—
16C7×#393C2211yescharacteristic
17C7×#392C2211yescharacteristic
18C7×#386—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12230C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×C7×D4—21no—
30C22×C7×D4D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C2×C7×Q16—41no—
33C2×C7×SD16—41no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—21no—
36C7×C22⋊Q8—41no—
37C7×C22⋊Q8—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—41no—
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⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C7×#103—41no—
52C7×#102—41no—
53C7×C22⋊C8—21no—
54C7×C22⋊C8—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—41no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2824C2×C711yescharacteristic
64#2062C2×C711yescharacteristic, complemented
65#1602C2×C711yescharacteristic, complemented
66#1596C2×C711yescharacteristic
67#1351C2×C711yescharacteristic, complemented
68#1343C2×C711yescharacteristic
69#1326C2×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—41no—
78C2×C7×D4—41no—
79C2×C7×D4—81no—
80C2×C7×D4—81no—
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—41no—
90C22×C4×C7—41no—
91C22×C4×C7—81no—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C7×Q16—82no—
95C7×SD16—82no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—81no—
103C7×C4⋊C4—81no—
104C7×C4⋊C4—82no—
105C7×C22⋊C4—22no—
106C7×C22⋊C4—21no—
107C7×C22⋊C4—21no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—81no—
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#400—21no—
122#397—21no—
123#394—21no—
124#394—21no—
125#393—21no—
126#393C22×C711yescharacteristic
127#392C22×C711yescharacteristic
128#386—21no—
129#385—21no—
130#385C22×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—41no—
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—42no—
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—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—42no—
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—41no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7—81no—
186C2×C4×C7—81no—
187C7×C8—81no—
188C7×C8—81no—
189C22×Q8C7×D411yescharacteristic
190C22×D4—21no—
191C22×D4C7×D411yescharacteristic
192C23×C4C7×D411yescharacteristic
193C2×Q16—41no—
194C2×SD16—41no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
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⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4C7×D411yescharacteristic
211C2×C22⋊C4C23×C711yescharacteristic
212#103—41no—
213#102—41no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—41no—
222#95C7×D411yescharacteristic
223#95C7×D411yescharacteristic
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7—81no—
238C22×C7—81no—
239C22×C7—81no—
240C22×C7#45211yescharacteristic
241C22×C7#45611yescharacteristic
242C22×C7B(4,𝔽2)11yescharacteristic
243C4×C7—41no—
244C4×C7—41no—
245C4×C7—81no—
246C4×C7—81no—
247C4×C7—81no—
248C4×C7—82no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7—81no—
253C24—21no—
254C24C2×C7×D411yescharacteristic
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—41no—
258C2×Q8—81no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—81no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—81no—
275C22×C4C2×C7×D411yescharacteristic
276C22×C4C2×C7×D411yescharacteristic
277Q16—82no—
278SD16—82no—
279C2×C8—41no—
280C2×C8—41no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C4⋊C4—82no—
288C22⋊C4—22no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C2×C7—21no—
302C2×C7—21no—
303C2×C7—41no—
304C2×C7—41no—
305C2×C7—161no—
306C2×C7#160311yescharacteristic
307C2×C7#219911yescharacteristic
308C2×C7#220011yescharacteristic
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C7×#12011yescharacteristic
321C23C7×#13511yescharacteristic
322C23C7×#13711yescharacteristic
323Q8—42no—
324Q8—81no—
325Q8—81no—
326Q8—81no—
327D4—42no—
328D4—81no—
329D4—82no—
330D4—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C8—81no—
360C8—81no—
361C7#1223011yescharacteristic, direct product factor, Sylow
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22C7×#45211yescharacteristic
379C22C7×#45611yescharacteristic
380C22C7×B(4,𝔽2)11yescharacteristic
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2C7×#160311yescharacteristic
397C2C7×#219911yescharacteristic
398C2C7×#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×#2062C2semidirect
3. C7×#1602C2semidirect
5. C7×#1351C2semidirect
7. C7×#1326C2semidirect
27. #12230C7direct (N × H)
64. #2062C2×C7semidirect
65. #1602C2×C7semidirect
67. #1351C2×C7semidirect
69. #1326C2×C7semidirect
361. C7#12230direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28242
C7×#20622
C7×#16022
C7×#15962
C7×#13512
C7×#13432
C7×#13262
#122307