Finite Group Databasebeta

← C7×#8492 #13185708

Subgroups — C7×#8492 #13185708

Order 1792.

692 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3550C211yesmaximal, characteristic
2C7×#1963C211yesmaximal, characteristic
3C7×#1559C212yesmaximal, complemented
4C7×#1536C211yesmaximal, characteristic
5C7×#1506C212yesmaximal, complemented
6C7×#563—22no—
7C7×#559C412yescomplemented
8C7×Q82C412yescomplemented
9C7×#556—22no—
10C7×D4×Q8—22no—
11C7×#542—24no—
12C7×#536—22no—
13C7×#533C2212yes—
14C7×#532C2211yescharacteristic
15C7×#432C2212yes—
16C7×#423C2211yescharacteristic
17C2×C7×#104—22no—
18C7×#376C2211yescharacteristic
19C7×#333—22no—
20C7×#332—22no—
21C7×#329—22no—
22C7×#328—24no—
23#8492C711yesmaximal, characteristic, direct product factor, Sylow
24C2×C7×#41D412yes—
25C22×C7×Q8—22no—
26C2×C7×M16—21no—
27C7×C4⋊Q8—24no—
28C7×C4⋊Q8—24no—
29C7×C4⋊Q8C2×C411yescharacteristic
30C7×C4⋊Q8D412yes—
31C7×C4⋊Q8D412yes—
32C7×#127C2×C411yescharacteristic
33C7×#126—28no—
34C7×#125—22no—
35C7×#124—24no—
36C7×#124—24no—
37C7×#124D412yes—
38C7×#123—24no—
39C7×#123—24no—
40C7×C22⋊Q8—24no—
41C7×C22⋊Q8—28no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8D414yes—
44C7×#121—22no—
45C4×C7×Q8—24no—
46C4×C7×Q8—22no—
47C4×C7×Q8C2×C414yes—
48C4×C7×D4—24no—
49C4×C7×D4—22no—
50C7×#117—22no—
51C7×#117—22no—
52C2×C42×C7—21no—
53C2×C42×C7C2311yescharacteristic
54C7×C4⋊C8—22no—
55C7×C4⋊C8—22no—
56C7×#104—44no—
57C7×C22⋊C8—22no—
58C7×#97—22no—
59C7×#95—22no—
60#3550C2×C711yescharacteristic
61#1963C2×C711yescharacteristic
62#1559C2×C712yescomplemented
63#1536C2×C711yescharacteristic
64#1506C2×C712yescomplemented
65C7×#41—42no—
66C7×#41—42no—
67C2×C7×Q8—24no—
68C2×C7×Q8—22no—
69C2×C7×Q8—82no—
70C2×C7×Q8—82no—
71C2×C7×Q8C22⋊C414yes—
72C2×C7×Q8C22⋊C412yes—
73C2×C7×D4—22no—
74C2×C7×D4—22no—
75C2×C7×D4C22⋊C412yes—
76C22×C4×C7—24no—
77C22×C4×C7—22no—
78C22×C4×C7—41no—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D412yes—
81C7×M16—42no—
82C2×C7×C8—42no—
83C2×C7×C8—41no—
84C7×C4⋊C4—28no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—24no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—22no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—24no—
91C7×C4⋊C4C22⋊C414yes—
92C7×C22⋊C4—24no—
93C7×C22⋊C4—28no—
94C7×C22⋊C4—24no—
95C7×C22⋊C4—24no—
96C42×C7—24no—
97C42×C7—24no—
98C42×C7—22no—
99C42×C7—41no—
100C42×C7C22×C411yescharacteristic
101C42×C7C2×D411yescharacteristic
102C42×C7C2×D412yes—
103#563—22no—
104#559C4×C712yescomplemented
105Q82C4×C712yescomplemented
106#556—22no—
107D4×Q8—22no—
108#542—24no—
109#536—22no—
110#533C22×C712yes—
111#532C22×C711yescharacteristic
112#432C22×C712yes—
113#423C22×C711yescharacteristic
114C2×#104—22no—
115#376C22×C711yescharacteristic
116#333—22no—
117#332—22no—
118#329—22no—
119#328—24no—
120C23×C7—22no—
121C23×C7#12011yescharacteristic
122C7×Q8—44no—
123C7×Q8—42no—
124C7×Q8—42no—
125C7×Q8—42no—
126C7×Q8—84no—
127C7×D4—42no—
128C7×D4—44no—
129C7×D4—82no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—24no—
133C2×C4×C7—22no—
134C2×C4×C7—24no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—84no—
140C2×C4×C7—82no—
141C2×C4×C7C2×C22⋊C412yes—
142C2×C4×C7C2×C22⋊C411yescharacteristic
143C2×C4×C7#12012yes—
144C2×C4×C7#12011yescharacteristic
145C7×C8—82no—
146C7×C8—81no—
147C2×#41C7×D412yes—
148C22×Q8—22no—
149C2×M16—21no—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8C2×C4×C711yescharacteristic
153C4⋊Q8C7×D412yes—
154C4⋊Q8C7×D412yes—
155#127C2×C4×C711yescharacteristic
156#126—28no—
157#125—22no—
158#124—24no—
159#124—24no—
160#124C7×D412yes—
161#123—24no—
162#123—24no—
163C22⋊Q8—24no—
164C22⋊Q8—28no—
165C22⋊Q8—22no—
166C22⋊Q8C7×D414yes—
167#121—22no—
168C4×Q8—24no—
169C4×Q8—22no—
170C4×Q8C2×C4×C714yes—
171C4×D4—24no—
172C4×D4—22no—
173#117—22no—
174#117—22no—
175C2×C42—21no—
176C2×C42C23×C711yescharacteristic
177C4⋊C8—22no—
178C4⋊C8—22no—
179#104—44no—
180C22⋊C8—22no—
181#97—22no—
182#95—22no—
183C22×C7—21no—
184C22×C7—22no—
185C22×C7—42no—
186C22×C7—44no—
187C22×C7#37911yescharacteristic
188C4×C7—21no—
189C4×C7—22no—
190C4×C7—22no—
191C4×C7—42no—
192C4×C7—44no—
193C4×C7—44no—
194C4×C7—82no—
195C4×C7#45311yescharacteristic
196C4×C7#45511yescharacteristic
197#41—42no—
198#41—42no—
199C2×Q8—24no—
200C2×Q8—22no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8C7×C22⋊C414yes—
204C2×Q8C7×C22⋊C412yes—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4C7×C22⋊C412yes—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4C2×C7×D411yescharacteristic
212C22×C4C2×C7×D412yes—
213M16—42no—
214C2×C8—42no—
215C2×C8—41no—
216C4⋊C4—28no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4C7×C22⋊C414yes—
224C22⋊C4—24no—
225C22⋊C4—28no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C42—24no—
229C42—24no—
230C42—22no—
231C42—41no—
232C42C22×C4×C711yescharacteristic
233C42C2×C7×D411yescharacteristic
234C42C2×C7×D412yes—
235C2×C7—41no—
236C2×C7—82no—
237C2×C7#179211yescharacteristic
238C2×C7#179612yes—
239C23—22no—
240C23C7×#12011yescharacteristic
241Q8—44no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—84no—
246D4—42no—
247D4—44no—
248D4—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4—82no—
260C2×C4C2×C7×C22⋊C412yes—
261C2×C4C2×C7×C22⋊C411yescharacteristic
262C2×C4C7×#12012yes—
263C2×C4C7×#12011yescharacteristic
264C8—82no—
265C8—81no—
266C7#849211yescharacteristic, direct product factor, Sylow
267C22—21no—
268C22—22no—
269C22—42no—
270C22—44no—
271C22C7×#37911yescharacteristic
272C4—21no—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—44no—
277C4—44no—
278C4—82no—
279C4C7×#45311yescharacteristic
280C4C7×#45511yescharacteristic
281C2—41no—
282C2—82no—
283C2C7×#179211yescharacteristic
284C2C7×#179612yes—
subgroupG/H
center, socleC22×C7#379
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1559C2semidirect
5. C7×#1506C2semidirect
7. C7×#559C4semidirect
8. C7×Q82C4semidirect
23. #8492C7direct (N × H)
62. #1559C2×C7semidirect
64. #1506C2×C7semidirect
104. #559C4×C7semidirect
105. Q82C4×C7semidirect
266. C7#8492direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35502
C7×#19632
C7×#15592
C7×#15362
C7×#15062
#84927