Finite Group Databasebeta

← C7×#10259 #13187475

Subgroups — C7×#10259 #13187475

Order 1792.

474 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3249C211yesmaximal, characteristic
2C7×#3078C212yesmaximal
3C7×#1984C211yesmaximal, characteristic
4C7×#1493C212yesmaximal
5C7×#1292C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#479—24no—
9C7×#477—22no—
10C7×#477—22no—
11C7×#475—22no—
12C7×#467—22no—
13C7×#462—22no—
14C7×#428—22no—
15C7×#423C2212yes—
16C2×C7×C4⋊C8C2211yescharacteristic
17C7×#419—22no—
18C2×C4×C7×C8C2211yescharacteristic
19C7×#336—22no—
20C7×Q8⋊C8—24no—
21#10259C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D411yescharacteristic
23C2×C7×Q16—22no—
24C2×C7×M16—22no—
25C22×C7×C8—21no—
26C22×C7×C8D412yes—
27C7×C4⋊Q8D412yes—
28C7×C4⋊Q8D412yes—
29C7×#125—22no—
30C7×C22⋊Q8—22no—
31C7×C22⋊Q8—22no—
32C4×C7×Q8—22no—
33C4×C7×Q8—22no—
34C4×C7×Q8—22no—
35C4×C7×Q8—22no—
36C7×#117—22no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C42×C7—21no—
40C2×C42×C7C2311yescharacteristic
41C7×#107—22no—
42C7×#107—24no—
43C7×#106—24no—
44C7×C4⋊C8—22no—
45C7×C4⋊C8—22no—
46C7×C4⋊C8D414yes—
47C7×#103—24no—
48C7×#103—24no—
49C7×#103—24no—
50C4×C7×C8—22no—
51#3249C2×C711yescharacteristic
52#3078C2×C712yes—
53#1984C2×C711yescharacteristic
54#1493C2×C712yes—
55#1292C2×C711yescharacteristic
56C2×C7×Q8—21no—
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8Q1612yescomplemented
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7#4111yescharacteristic
67C7×Q16—82no—
68C7×M16—42no—
69C7×M16—42no—
70C2×C7×C8—22no—
71C2×C7×C8—22no—
72C2×C7×C8—24no—
73C2×C7×C8—41no—
74C2×C7×C8—41no—
75C2×C7×C8—42no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—42no—
84C7×C22⋊C4—42no—
85C42×C7—41no—
86C42×C7—42no—
87C42×C7—41no—
88C42×C7C2×D412yes—
89C42×C7C2×D412yes—
90#533C22×C712yes—
91C2×C4×Q8C22×C711yescharacteristic
92#479—24no—
93#477—22no—
94#477—22no—
95#475—22no—
96#467—22no—
97#462—22no—
98#428—22no—
99#423C22×C712yes—
100C2×C4⋊C8C22×C711yescharacteristic
101#419—22no—
102C2×C4×C8C22×C711yescharacteristic
103#336—22no—
104Q8⋊C8—24no—
105C23×C7#12111yescharacteristic
106C7×Q8—24no—
107C7×Q8—44no—
108C7×Q8—42no—
109C7×Q8—42no—
110C7×Q8—42no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—22no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7—42no—
121C2×C4×C7#12012yes—
122C2×C4×C7#12011yescharacteristic
123C2×C4×C7#12112yes—
124C2×C4×C7#12711yescharacteristic
125C2×C4×C7C2×Q1611yescharacteristic
126C2×C4×C7#13511yescharacteristic
127C2×C4×C7#13712yes—
128C7×C8—42no—
129C7×C8—44no—
130C7×C8—42no—
131C22×Q8C7×D411yescharacteristic
132C2×Q16—22no—
133C2×M16—22no—
134C22×C8—21no—
135C22×C8C7×D412yes—
136C4⋊Q8C7×D412yes—
137C4⋊Q8C7×D412yes—
138#125—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145#117—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C7×D411yescharacteristic
148C2×C42—21no—
149C2×C42C23×C711yescharacteristic
150#107—24no—
151#107—22no—
152#106—24no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8C7×D414yes—
156#103—24no—
157#103—24no—
158#103—24no—
159C4×C8—22no—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7#39211yescharacteristic
163C22×C7#46711yescharacteristic
164C22×C7#47011yescharacteristic
165C4×C7—21no—
166C4×C7—22no—
167C4×C7—42no—
168C4×C7—82no—
169C4×C7—81no—
170C4×C7—82no—
171C4×C7#45112yes—
172C4×C7#45212yes—
173C4×C7#45511yescharacteristic
174C4×C7#45611yescharacteristic
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8C7×Q1612yescomplemented
181C22×C4—21no—
182C22×C4—21no—
183C22×C4C2×C7×D411yescharacteristic
184C22×C4C2×C7×D411yescharacteristic
185C22×C4C7×#4111yescharacteristic
186Q16—82no—
187M16—42no—
188M16—42no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—24no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C22⋊C4—42no—
204C42—41no—
205C42—42no—
206C42—41no—
207C42C2×C7×D412yes—
208C42C2×C7×D412yes—
209C2×C7—21no—
210C2×C7—21no—
211C2×C7#200111yescharacteristic
212C2×C7#200711yescharacteristic
213C2×C7#201711yescharacteristic
214C23C7×#12111yescharacteristic
215Q8—24no—
216Q8—44no—
217Q8—42no—
218Q8—42no—
219Q8—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4C7×#12012yes—
231C2×C4C7×#12011yescharacteristic
232C2×C4C7×#12112yes—
233C2×C4C7×#12711yescharacteristic
234C2×C4C2×C7×Q1611yescharacteristic
235C2×C4C7×#13511yescharacteristic
236C2×C4C7×#13712yes—
237C8—42no—
238C8—44no—
239C8—42no—
240C7#1025911yescharacteristic, direct product factor, Sylow
241C22—21no—
242C22—21no—
243C22C7×#39211yescharacteristic
244C22C7×#46711yescharacteristic
245C22C7×#47011yescharacteristic
246C4—21no—
247C4—22no—
248C4—42no—
249C4—82no—
250C4—81no—
251C4—82no—
252C4C7×#45112yes—
253C4C7×#45212yes—
254C4C7×#45511yescharacteristic
255C4C7×#45611yescharacteristic
256C2—21no—
257C2—21no—
258C2C7×#200111yescharacteristic
259C2C7×#200711yescharacteristic
260C2C7×#201711yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
21. #10259C7direct (N × H)
61. C2×C7×Q8Q16semidirect
180. C2×Q8C7×Q16semidirect
240. C7#10259direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32492
C7×#30782
C7×#19842
C7×#14932
C7×#12922
#102597