Finite Group Databasebeta

← C7×#7826 #13185042

Subgroups — C7×#7826 #13185042

Order 1792.

462 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2683C211yesmaximal, characteristic
2C7×#2100C212yesmaximal, complemented
3C7×#2094C211yesmaximal, characteristic
4C7×#1271C212yesmaximal, complemented
5C7×#1270C211yesmaximal, characteristic
6C2×C7×C22⋊Q8—22no—
7C2×C7×#106—21no—
8C2×C7×#102—21no—
9C7×#396—22no—
10C7×#395—21no—
11C7×#390C2211yescharacteristic
12C7×#389C2212yes—
13C7×#386—21no—
14C7×#384C2211yescharacteristic
15C7×#340—22no—
16C7×#340—22no—
17C7×#336C2212yes—
18C7×#336C2211yescharacteristic
19#7826C711yesmaximal, characteristic, direct product factor, Sylow
20C22×C7×Q8—21no—
21C22×C7×D4D411yescharacteristic
22C23×C4×C7—21no—
23C22×C7×C8—21no—
24C22×C7×C8—22no—
25C7×C22⋊Q8—44no—
26C7×C22⋊Q8—44no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4D412yes—
32C2×C7×C4⋊C4D412yes—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—21no—
37C2×C42×C7C2311yescharacteristic
38C7×#106—42no—
39C7×#102—42no—
40C7×#95—21no—
41C7×#95—22no—
42#2683C2×C711yescharacteristic
43#2100C2×C712yescomplemented
44#2094C2×C711yescharacteristic
45#1271C2×C712yescomplemented
46#1270C2×C711yescharacteristic
47C24×C7—21no—
48C2×C7×Q8—44no—
49C2×C7×Q8—81no—
50C2×C7×D4—21no—
51C2×C7×D4—41no—
52C2×C7×D4—41no—
53C2×C7×D4Q1612yes—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—21no—
57C22×C4×C7—44no—
58C22×C4×C7—41no—
59C22×C4×C7—81no—
60C22×C4×C7C2×D412yes—
61C22×C4×C7C2×D411yescharacteristic
62C2×C7×C8—22no—
63C2×C7×C8—41no—
64C2×C7×C8—41no—
65C2×C7×C8—42no—
66C2×C7×C8—42no—
67C2×C7×C8—42no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—44no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—44no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4SD1614yes—
78C7×C22⋊C4—44no—
79C7×C22⋊C4—42no—
80C7×C22⋊C4—42no—
81C7×C22⋊C4—81no—
82C42×C7—41no—
83C2×C22⋊Q8—22no—
84C2×#106—21no—
85C2×#102—21no—
86#396—22no—
87#395—21no—
88#390C22×C711yescharacteristic
89#389C22×C712yes—
90#386—21no—
91#384C22×C711yescharacteristic
92#340—22no—
93#340—22no—
94#336C22×C711yescharacteristic
95#336C22×C712yes—
96C23×C7—22no—
97C23×C7—42no—
98C23×C7—41no—
99C23×C7—42no—
100C23×C7—41no—
101C23×C7#12011yescharacteristic
102C7×Q8—84no—
103C7×D4—42no—
104C7×D4—81no—
105C2×C4×C7—21no—
106C2×C4×C7—22no—
107C2×C4×C7—22no—
108C2×C4×C7—21no—
109C2×C4×C7—24no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—41no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—84no—
118C2×C4×C7—81no—
119C2×C4×C7—82no—
120C2×C4×C7C2×SD1612yes—
121C2×C4×C7C2×Q1611yescharacteristic
122C2×C4×C7AGL(1,ℤ/8)12yes—
123C2×C4×C7#13711yescharacteristic
124C7×C8—42no—
125C7×C8—82no—
126C22×Q8—21no—
127C22×D4C7×D411yescharacteristic
128C23×C4—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C7×D412yes—
138C2×C4⋊C4C7×D412yes—
139C2×C4⋊C4C7×D411yescharacteristic
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—21no—
143C2×C42C23×C711yescharacteristic
144#106—42no—
145#102—42no—
146#95—21no—
147#95—22no—
148C22×C7—42no—
149C22×C7—44no—
150C22×C7—82no—
151C22×C7—81no—
152C22×C7—81no—
153C22×C7#45012yes—
154C22×C7#45111yescharacteristic
155C22×C7#45311yescharacteristic
156C22×C7#45412yes—
157C22×C7#45811yescharacteristic
158C4×C7—21no—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—44no—
163C4×C7—82no—
164C4×C7—81no—
165C4×C7—161no—
166C24—21no—
167C2×Q8—44no—
168C2×Q8—81no—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4C7×Q1612yes—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—44no—
177C22×C4—41no—
178C22×C4—81no—
179C22×C4C2×C7×D412yes—
180C22×C4C2×C7×D411yescharacteristic
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4C7×SD1614yes—
197C22⋊C4—44no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—81no—
201C42—41no—
202C2×C7—82no—
203C2×C7#161411yescharacteristic
204C2×C7#161612yes—
205C2×C7#162911yescharacteristic
206C2×C7#163712yes—
207C2×C7#165711yescharacteristic
208C23—22no—
209C23—42no—
210C23—41no—
211C23—42no—
212C23—41no—
213C23C7×#12011yescharacteristic
214Q8—84no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—84no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4C2×C7×SD1612yes—
233C2×C4C2×C7×Q1611yescharacteristic
234C2×C4C7×AGL(1,ℤ/8)12yes—
235C2×C4C7×#13711yescharacteristic
236C8—42no—
237C8—82no—
238C7#782611yescharacteristic, direct product factor, Sylow
239C22—42no—
240C22—44no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22C7×#45012yes—
245C22C7×#45111yescharacteristic
246C22C7×#45311yescharacteristic
247C22C7×#45412yes—
248C22C7×#45811yescharacteristic
249C4—21no—
250C4—21no—
251C4—22no—
252C4—22no—
253C4—44no—
254C4—82no—
255C4—81no—
256C4—161no—
257C2—82no—
258C2C7×#161411yescharacteristic
259C2C7×#161612yes—
260C2C7×#162911yescharacteristic
261C2C7×#163712yes—
262C2C7×#165711yescharacteristic
subgroupG/H
center, socleC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2100C2semidirect
4. C7×#1271C2semidirect
19. #7826C7direct (N × H)
43. #2100C2×C7semidirect
45. #1271C2×C7semidirect
238. C7#7826direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26832
C7×#21002
C7×#20942
C7×#12712
C7×#12702
#78267