Finite Group Databasebeta

← C7×#12250 #13189466

Subgroups — C7×#12250 #13189466

Order 1792.

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

normal subgroups only

This group has 454 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×#2819C211yesmaximal, characteristic
2C7×#2061C211yesmaximal, characteristic, complemented
3C7×#1599C212yesmaximal, complemented
4C7×#1345C212yesmaximal, complemented
5C7×#1324C211yesmaximal, characteristic, complemented
6C7×#544C2211yescharacteristic
7C2×C7×#124—21no—
8C7×#450—22no—
9C7×#428—21no—
10C7×#394—21no—
11C7×#394C2212yes—
12C7×#393—22no—
13C7×#392—22no—
14C7×#382C2211yescharacteristic
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17C7×#328—22no—
18C7×#324C2212yes—
19#12250C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×D4—21no—
22C22×C7×D4—21no—
23C22×C7×D4D412yes—
24C2×C7×SD16—42no—
25C2×C7×M16—21no—
26C7×C4⋊Q8—41no—
27C7×#125—41no—
28C7×#124—42no—
29C7×#124—42no—
30C7×C22⋊Q8—22no—
31C7×#117—21no—
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C4⋊C4C2311yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C42×C7D411yescharacteristic
39C7×#107—41no—
40C7×#106—41no—
41C7×#102—42no—
42C7×C22⋊C8—22no—
43C7×#95—21no—
44C7×#95—22no—
45C7×#95D412yes—
46#2819C2×C711yescharacteristic
47#2061C2×C711yescharacteristic, complemented
48#1599C2×C712yescomplemented
49#1345C2×C712yescomplemented
50#1324C2×C711yescharacteristic, complemented
51C24×C7—22no—
52C2×C7×Q8—22no—
53C2×C7×Q8—42no—
54C2×C7×Q8—81no—
55C2×C7×Q8—81no—
56C2×C7×D4—42no—
57C2×C7×D4—42no—
58C2×C7×D4—82no—
59C2×C7×D4—81no—
60C2×C7×D4—81no—
61C2×C7×D4—82no—
62C2×C7×D4—81no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D412yes—
68C7×SD16—84no—
69C7×M16—22no—
70C7×M16—41no—
71C2×C7×C8—41no—
72C2×C7×C8—42no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4—21no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—42no—
79C7×C22⋊C4—44no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—42no—
82C7×C22⋊C4—42no—
83C7×C22⋊C4—82no—
84C7×C22⋊C4—82no—
85C42×C7—21no—
86C42×C7—41no—
87C42×C7SD1612yescomplemented
88#544C22×C711yescharacteristic
89C2×#124—21no—
90#450—22no—
91#428—21no—
92#394—21no—
93#394C22×C712yes—
94#393—22no—
95#392—22no—
96#382C22×C711yescharacteristic
97#343C22×C711yescharacteristic
98#339—21no—
99#328—22no—
100#324C22×C712yes—
101C23×C7—42no—
102C23×C7—42no—
103C23×C7—42no—
104C23×C7—42no—
105C23×C7—42no—
106C23×C7—42no—
107C23×C7—42no—
108C23×C7#12011yescharacteristic
109C7×Q8—82no—
110C7×Q8—84no—
111C7×D4—44no—
112C7×D4—82no—
113C7×D4—84no—
114C7×D4—84no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7C2×SD1611yescharacteristic
130C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
131C7×C8—42no—
132C7×C8—82no—
133C22×Q8—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4C7×D412yes—
137C2×SD16—42no—
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141#124—42no—
142#124—42no—
143C22⋊Q8—22no—
144#117—21no—
145C2×C4⋊C4C7×D411yescharacteristic
146C2×C4⋊C4C23×C711yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C42C7×D411yescharacteristic
152#107—41no—
153#106—41no—
154#102—42no—
155C22⋊C8—22no—
156#95—21no—
157#95—22no—
158#95C7×D412yes—
159C22×C7—21no—
160C22×C7—21no—
161C22×C7—82no—
162C22×C7—82no—
163C22×C7—82no—
164C22×C7—82no—
165C22×C7—82no—
166C22×C7—82no—
167C22×C7—82no—
168C22×C7#45011yescharacteristic
169C22×C7#45611yescharacteristic
170C22×C7B(4,𝔽2)11yescharacteristic
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—42no—
174C4×C7—42no—
175C4×C7—82no—
176C4×C7—81no—
177C4×C7—81no—
178C4×C7—81no—
179C24—22no—
180C2×Q8—22no—
181C2×Q8—42no—
182C2×Q8—81no—
183C2×Q8—81no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—82no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—82no—
190C2×D4—81no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4C2×C7×D411yescharacteristic
195C22×C4C2×C7×D412yes—
196SD16—84no—
197M16—22no—
198M16—41no—
199C2×C8—41no—
200C2×C8—42no—
201C4⋊C4—22no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C22⋊C4—44no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—82no—
212C22⋊C4—82no—
213C42—21no—
214C42—41no—
215C42C7×SD1612yescomplemented
216C2×C7—21no—
217C2×C7—21no—
218C2×C7—162no—
219C2×C7#160411yescharacteristic
220C2×C7#219911yescharacteristic
221C2×C7#220211yescharacteristic
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23C7×#12011yescharacteristic
230Q8—82no—
231Q8—84no—
232D4—44no—
233D4—82no—
234D4—84no—
235D4—84no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4C2×C7×SD1611yescharacteristic
251C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
252C8—42no—
253C8—82no—
254C7#1225011yescharacteristic, direct product factor, Sylow
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22C7×#45011yescharacteristic
265C22C7×#45611yescharacteristic
266C22C7×B(4,𝔽2)11yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—81no—
274C4—81no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2C7×#160411yescharacteristic
279C2C7×#219911yescharacteristic
280C2C7×#220211yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC7×#2819C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2061C2semidirect
3. C7×#1599C2semidirect
4. C7×#1345C2semidirect
5. C7×#1324C2semidirect
19. #12250C7direct (N × H)
47. #2061C2×C7semidirect
48. #1599C2×C7semidirect
49. #1345C2×C7semidirect
50. #1324C2×C7semidirect
87. C42×C7SD16semidirect
215. C42C7×SD16semidirect
254. C7#12250direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28192
C7×#20612
C7×#15992
C7×#13452
C7×#13242
#122507