Finite Group Databasebeta

← C7×#13676 #13190892

Subgroups — C7×#13676 #13190892

Order 1792.

394 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3291C211yesmaximal, characteristic
2C7×#2198C212yesmaximal, complemented
3C7×#1630C211yesmaximal, characteristic, complemented
4C7×#1409C211yesmaximal, characteristic, complemented
5C7×#1402C212yesmaximal, complemented
6C7×#577—22no—
7C7×#559C2211yescharacteristic
8C7×#492C2211yescharacteristic
9C7×#477—21no—
10C7×#471—22no—
11C7×#458—22no—
12C7×#456—21no—
13C7×#453C2212yes—
14C7×#443—21no—
15C7×#353C2212yes—
16C7×#352—22no—
17C7×#331C2211yescharacteristic
18C7×Q8⋊C8—21no—
19#13676C711yesmaximal, characteristic, direct product factor, Sylow
20C7×2−1+4—21no—
21C7×H(5,𝔽2)D412yes—
22C7×#137—22no—
23C7×#137—42no—
24C7×AGL(1,ℤ/8)—22no—
25C7×AGL(1,ℤ/8)—42no—
26C7×#135—21no—
27C7×#135—42no—
28C2×C7×SD16—22no—
29C2×C7×SD16—42no—
30C7×#131—22no—
31C7×C4⋊Q8D411yescharacteristic
32C7×#127C2311yescharacteristic
33C7×#124—41no—
34C7×#123—22no—
35C7×#123—42no—
36C4×C7×Q8—21no—
37C4×C7×D4—41no—
38C7×#108—21no—
39C7×#107—41no—
40C7×C4⋊C8—21no—
41C7×#104—22no—
42C7×#103—41no—
43C7×#101—21no—
44C7×#100D412yes—
45C7×#99—22no—
46C7×#99—42no—
47C4×C7×C8D411yescharacteristic
48#3291C2×C711yescharacteristic
49#2198C2×C712yescomplemented
50#1630C2×C711yescharacteristic, complemented
51#1409C2×C711yescharacteristic, complemented
52#1402C2×C712yescomplemented
53C7×#41—22no—
54C7×#41—42no—
55C7×#41—42no—
56C7×#41—81no—
57C2×C7×Q8—21no—
58C2×C7×Q8—21no—
59C2×C7×Q8—42no—
60C2×C7×D4—41no—
61C2×C7×D4—42no—
62C2×C7×D4—42no—
63C2×C7×D4C2×D412yes—
64C22×C4×C7—22no—
65C7×Q16—21no—
66C7×Q16—42no—
67C7×SD16—22no—
68C7×SD16—42no—
69C7×SD16—42no—
70C7×SD16—42no—
71C7×SD16—42no—
72C7×D8—21no—
73C7×D8—42no—
74C7×M16—22no—
75C7×M16—42no—
76C2×C7×C8—21no—
77C2×C7×C8—42no—
78C2×C7×C8—41no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—82no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C7×C22⋊C4—82no—
87C42×C7—41no—
88C42×C7C2×D411yescharacteristic
89#577—22no—
90#559C22×C711yescharacteristic
91#492C22×C711yescharacteristic
92#477—21no—
93#471—22no—
94#458—22no—
95#456—21no—
96#453C22×C712yes—
97#443—21no—
98#353C22×C712yes—
99#352—22no—
100#331C22×C711yescharacteristic
101Q8⋊C8—21no—
102C23×C7—22no—
103C23×C7—42no—
104C7×Q8—22no—
105C7×Q8—42no—
106C7×Q8—41no—
107C7×Q8—41no—
108C7×Q8—81no—
109C7×D4—22no—
110C7×D4—42no—
111C7×D4—42no—
112C7×D4—81no—
113C7×D4—81no—
114C7×D4—82no—
115C7×D4—81no—
116C2×C4×C7—21no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—82no—
123C2×C4×C7—82no—
124C2×C4×C7—82no—
125C2×C4×C7#12011yescharacteristic
126C7×C8—21no—
127C7×C8—21no—
128C7×C8—42no—
129C7×C8—81no—
1302−1+4—21no—
131H(5,𝔽2)C7×D412yes—
132#137—22no—
133#137—42no—
134AGL(1,ℤ/8)—22no—
135AGL(1,ℤ/8)—42no—
136#135—21no—
137#135—42no—
138C2×SD16—22no—
139C2×SD16—42no—
140#131—22no—
141C4⋊Q8C7×D411yescharacteristic
142#127C23×C711yescharacteristic
143#124—41no—
144#123—22no—
145#123—42no—
146C4×Q8—21no—
147C4×D4—41no—
148#108—21no—
149#107—41no—
150C4⋊C8—21no—
151#104—22no—
152#103—41no—
153#101—21no—
154#100C7×D412yes—
155#99—22no—
156#99—42no—
157C4×C8C7×D411yescharacteristic
158C22×C7—42no—
159C22×C7—42no—
160C22×C7—82no—
161C22×C7—162no—
162C22×C7B(4,𝔽2)11yescharacteristic
163C4×C7—21no—
164C4×C7—21no—
165C4×C7—21no—
166C4×C7—42no—
167C4×C7—81no—
168C4×C7—81no—
169C4×C7—81no—
170C4×C7—162no—
171#41—22no—
172#41—42no—
173#41—42no—
174#41—81no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—42no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4C2×C7×D412yes—
182C22×C4—22no—
183Q16—21no—
184Q16—42no—
185SD16—22no—
186SD16—42no—
187SD16—42no—
188SD16—42no—
189SD16—42no—
190D8—21no—
191D8—42no—
192M16—22no—
193M16—42no—
194C2×C8—21no—
195C2×C8—42no—
196C2×C8—41no—
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—82no—
201C22⋊C4—22no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—41no—
206C42C2×C7×D411yescharacteristic
207C2×C7—21no—
208C2×C7—82no—
209C2×C7—82no—
210C2×C7#219611yescharacteristic
211C23—22no—
212C23—42no—
213Q8—22no—
214Q8—42no—
215Q8—41no—
216Q8—41no—
217Q8—81no—
218D4—22no—
219D4—42no—
220D4—42no—
221D4—81no—
222D4—81no—
223D4—82no—
224D4—81no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4C7×#12011yescharacteristic
235C8—21no—
236C8—21no—
237C8—42no—
238C8—81no—
239C7#1367611yescharacteristic, direct product factor, Sylow
240C22—42no—
241C22—42no—
242C22—82no—
243C22—162no—
244C22C7×B(4,𝔽2)11yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—162no—
253C2—21no—
254C2—82no—
255C2—82no—
256C2C7×#219611yescharacteristic
subgroupG/H
center, socleC2×C7#2196
commutator, Frattini, cosocle#127C23×C7
prime radicalC7×#3291C2
LCSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C22×C7 ◁ C2×C4×C7 ◁ C7×#127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2198C2semidirect
3. C7×#1630C2semidirect
4. C7×#1409C2semidirect
5. C7×#1402C2semidirect
19. #13676C7direct (N × H)
49. #2198C2×C7semidirect
50. #1630C2×C7semidirect
51. #1409C2×C7semidirect
52. #1402C2×C7semidirect
239. C7#13676direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32912
C7×#21982
C7×#16302
C7×#14092
C7×#14022
#136767