Finite Group Databasebeta

← C7×#9991 #13187207

Subgroups — C7×#9991 #13187207

Order 1792.

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

normal subgroups only

This group has 698 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×#3195C211yesmaximal, characteristic
2C2×C7×#451C212yesmaximal
3C7×#1784C211yesmaximal, characteristic
4C7×#1472C212yesmaximal
5C7×#1298C211yesmaximal, characteristic
6C23×C7×Q8C2211yescharacteristic
7C22×C7×Q16—22no—
8C2×C7×C22⋊Q8C2212yes—
9C7×#483—22no—
10C7×#470—22no—
11C7×#451—22no—
12C7×#451—24no—
13C7×#451—24no—
14C2×C7×#103—22no—
15C2×C7×#101—22no—
16C7×#407C2211yescharacteristic
17C7×#407C2211yescharacteristic
18C2×C7×C22⋊C8C2212yes—
19C7×#324—24no—
20#9991C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8D412yes—
26C22×C7×Q8D412yes—
27C23×C4×C7—21no—
28C23×C4×C7C2311yescharacteristic
29C2×C7×Q16—28no—
30C2×C7×Q16—22no—
31C2×C7×Q16—48no—
32C2×C7×M16—21no—
33C2×C7×M16D412yes—
34C22×C7×C8—22no—
35C7×C22⋊Q8—22no—
36C7×C22⋊Q8—22no—
37C7×C22⋊Q8—22no—
38C7×C22⋊Q8—22no—
39C2×C7×C4⋊C4—22no—
40C2×C7×C4⋊C4D412yes—
41C2×C7×C22⋊C4—22no—
42C7×#106—22no—
43C7×#103—24no—
44C7×#103—28no—
45C7×#101—44no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8D414yes—
50#3195C2×C711yescharacteristic
51C2×#451C2×C712yes—
52#1784C2×C711yescharacteristic
53#1472C2×C712yes—
54#1298C2×C711yescharacteristic
55C24×C7#4111yescharacteristic
56C2×C7×Q8—24no—
57C2×C7×Q8—28no—
58C2×C7×Q8—24no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8—44no—
62C2×C7×Q8—44no—
63C2×C7×Q8—42no—
64C2×C7×Q8—44no—
65C2×C7×Q8—48no—
66C2×C7×Q8—41no—
67C2×C7×Q8—41no—
68C22×C4×C7—22no—
69C22×C4×C7—22no—
70C22×C4×C7—22no—
71C22×C4×C7—42no—
72C22×C4×C7—44no—
73C22×C4×C7—42no—
74C22×C4×C7C2×D412yes—
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D412yes—
78C7×Q16—416no—
79C7×Q16—82no—
80C7×M16—42no—
81C7×M16—42no—
82C2×C7×C8—22no—
83C2×C7×C8—24no—
84C2×C7×C8—44no—
85C2×C7×C8—41no—
86C7×C4⋊C4—22no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—42no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—42no—
92C23×Q8C22×C711yescharacteristic
93C22×Q16—22no—
94C2×C22⋊Q8C22×C712yes—
95#483—22no—
96#470—22no—
97#451—22no—
98#451—24no—
99#451—24no—
100C2×#103—22no—
101C2×#101—22no—
102#407C22×C711yescharacteristic
103#407C22×C711yescharacteristic
104C2×C22⋊C8C22×C712yes—
105#324—24no—
106C23×C7—24no—
107C23×C7—41no—
108C23×C7#12111yescharacteristic
109C23×C7#12112yes—
110C7×Q8—28no—
111C7×Q8—44no—
112C7×Q8—44no—
113C7×Q8—44no—
114C7×Q8—44no—
115C7×Q8—44no—
116C2×C4×C7—24no—
117C2×C4×C7—22no—
118C2×C4×C7—24no—
119C2×C4×C7—41no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—48no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—44no—
127C2×C4×C7—82no—
128C2×C4×C7#12012yes—
129C2×C4×C7#12011yescharacteristic
130C2×C4×C7#12711yescharacteristic
131C7×C8—44no—
132C7×C8—42no—
133C7×C8—81no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—44no—
138C22×Q8C7×D412yes—
139C22×Q8C7×D412yes—
140C23×C4—21no—
141C23×C4C23×C711yescharacteristic
142C2×Q16—28no—
143C2×Q16—22no—
144C2×Q16—48no—
145C2×M16—21no—
146C2×M16C7×D412yes—
147C22×C8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C7×D412yes—
154C2×C22⋊C4—22no—
155#106—22no—
156#103—28no—
157#103—24no—
158#101—44no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8C7×D414yes—
163C22×C7—22no—
164C22×C7—22no—
165C22×C7—22no—
166C22×C7—21no—
167C22×C7—44no—
168C22×C7#39211yescharacteristic
169C22×C7#47314yes—
170C4×C7—22no—
171C4×C7—21no—
172C4×C7—44no—
173C4×C7—82no—
174C4×C7—82no—
175C4×C7B(4,𝔽2)12yes—
176C24C7×#4111yescharacteristic
177C2×Q8—24no—
178C2×Q8—28no—
179C2×Q8—24no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—44no—
183C2×Q8—44no—
184C2×Q8—42no—
185C2×Q8—44no—
186C2×Q8—48no—
187C2×Q8—41no—
188C2×Q8—41no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4C2×C7×D412yes—
196C22×C4C2×C7×D411yescharacteristic
197C22×C4C2×C7×D411yescharacteristic
198C22×C4C2×C7×D412yes—
199Q16—416no—
200Q16—82no—
201M16—42no—
202M16—42no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—44no—
206C2×C8—41no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C2×C7—24no—
214C2×C7—41no—
215C2×C7#202012yes—
216C2×C7#202111yescharacteristic
217C23—24no—
218C23—41no—
219C23C7×#12111yescharacteristic
220C23C7×#12112yes—
221Q8—28no—
222Q8—44no—
223Q8—44no—
224Q8—44no—
225Q8—44no—
226Q8—44no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—48no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—82no—
239C2×C4C7×#12012yes—
240C2×C4C7×#12011yescharacteristic
241C2×C4C7×#12711yescharacteristic
242C8—44no—
243C8—42no—
244C8—81no—
245C7#999111yescharacteristic, direct product factor, Sylow
246C22—22no—
247C22—22no—
248C22—22no—
249C22—21no—
250C22—44no—
251C22C7×#39211yescharacteristic
252C22C7×#47314yes—
253C4—22no—
254C4—21no—
255C4—44no—
256C4—82no—
257C4—82no—
258C4C7×B(4,𝔽2)12yes—
259C2—24no—
260C2—41no—
261C2C7×#202012yes—
262C2C7×#202111yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
20. #9991C7direct (N × H)
245. C7#9991direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31952
C2×C7×#4512
C7×#17842
C7×#14722
C7×#12982
#99917