Finite Group Databasebeta

← C7×#10334 #13187550

Subgroups — C7×#10334 #13187550

Order 1792.

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

normal subgroups only

This group has 754 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
2C7×#2434C212yesmaximal
3C7×#1784C211yesmaximal, characteristic
4C7×#1508C212yesmaximal
5C7×#1297C211yesmaximal, characteristic
6C23×C7×Q8C2211yescharacteristic
7C2×C7×C4⋊Q8—22no—
8C2×C7×C22⋊Q8C2212yes—
9C7×#483—22no—
10C7×#470—22no—
11C7×#451—22no—
12C2×C7×#101—22no—
13C7×#407C2211yescharacteristic
14C7×#407C2211yescharacteristic
15C7×#395—24no—
16C7×#391—22no—
17C7×#385—24no—
18C4×C7×C22⋊C4C2212yes—
19C7×#328—24no—
20#10334C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—22no—
22C22×C7×Q8—21no—
23C22×C7×Q8—22no—
24C22×C7×Q8—21no—
25C22×C7×Q8—44no—
26C22×C7×Q8D412yes—
27C23×C4×C7D412yes—
28C23×C4×C7C2311yescharacteristic
29C2×C7×Q16—22no—
30C2×C7×M16—21no—
31C2×C7×M16D412yes—
32C7×C4⋊Q8—48no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—22no—
35C7×C22⋊Q8—22no—
36C7×C22⋊Q8—22no—
37C2×C7×C4⋊C4—28no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4D412yes—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C42×C7—22no—
43C7×#106—22no—
44C7×#103—24no—
45C7×#101—44no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×#95—28no—
49C7×#95D414yes—
50#3195C2×C711yescharacteristic
51#2434C2×C712yes—
52#1784C2×C711yescharacteristic
53#1508C2×C712yes—
54#1297C2×C711yescharacteristic
55C24×C7C2×D411yescharacteristic
56C2×C7×Q8—24no—
57C2×C7×Q8—28no—
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—44no—
61C2×C7×Q8—42no—
62C2×C7×Q8—48no—
63C2×C7×Q8—44no—
64C2×C7×Q8—44no—
65C2×C7×Q8—41no—
66C2×C7×Q8—42no—
67C2×C7×Q8—41no—
68C22×C4×C7—24no—
69C22×C4×C7—24no—
70C22×C4×C7—22no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—44no—
74C22×C4×C7—42no—
75C22×C4×C7C2×D412yes—
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D412yes—
78C22×C4×C7#4111yescharacteristic
79C7×Q16—82no—
80C7×M16—42no—
81C7×M16—42no—
82C2×C7×C8—22no—
83C2×C7×C8—41no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—416no—
87C7×C4⋊C4—42no—
88C7×C4⋊C4—42no—
89C7×C22⋊C4—28no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—42no—
92C42×C7—44no—
93C23×Q8C22×C711yescharacteristic
94C2×C4⋊Q8—22no—
95C2×C22⋊Q8C22×C712yes—
96#483—22no—
97#470—22no—
98#451—22no—
99C2×#101—22no—
100#407C22×C711yescharacteristic
101#407C22×C711yescharacteristic
102#395—24no—
103#391—22no—
104#385—24no—
105C4×C22⋊C4C22×C712yes—
106#328—24no—
107C23×C7—24no—
108C23×C7—41no—
109C23×C7#12012yes—
110C23×C7#12111yescharacteristic
111C7×Q8—44no—
112C7×Q8—48no—
113C7×Q8—44no—
114C7×Q8—44no—
115C7×Q8—82no—
116C2×C4×C7—24no—
117C2×C4×C7—22no—
118C2×C4×C7—28no—
119C2×C4×C7—28no—
120C2×C4×C7—24no—
121C2×C4×C7—42no—
122C2×C4×C7—41no—
123C2×C4×C7—44no—
124C2×C4×C7—44no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—44no—
128C2×C4×C7—44no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—44no—
132C2×C4×C7#12011yescharacteristic
133C2×C4×C7#12112yes—
134C2×C4×C7#12711yescharacteristic
135C7×C8—42no—
136C7×C8—81no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—44no—
142C22×Q8C7×D412yes—
143C23×C4C7×D412yes—
144C23×C4C23×C711yescharacteristic
145C2×Q16—22no—
146C2×M16—21no—
147C2×M16C7×D412yes—
148C4⋊Q8—48no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C7×D412yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C42—22no—
159#106—22no—
160#103—24no—
161#101—44no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164#95—28no—
165#95C7×D414yes—
166C22×C7—22no—
167C22×C7—22no—
168C22×C7—22no—
169C22×C7—21no—
170C22×C7—44no—
171C22×C7#39211yescharacteristic
172C22×C7#45614yes—
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—48no—
176C4×C7—44no—
177C4×C7—82no—
178C4×C7—82no—
179C4×C7B(4,𝔽2)12yes—
180C24C2×C7×D411yescharacteristic
181C2×Q8—24no—
182C2×Q8—28no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—44no—
186C2×Q8—42no—
187C2×Q8—48no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—41no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4C2×C7×D412yes—
201C22×C4C2×C7×D411yescharacteristic
202C22×C4C2×C7×D412yes—
203C22×C4C7×#4111yescharacteristic
204Q16—82no—
205M16—42no—
206M16—42no—
207C2×C8—22no—
208C2×C8—41no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—416no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C22⋊C4—28no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C42—44no—
218C2×C7—24no—
219C2×C7—41no—
220C2×C7#200512yes—
221C2×C7#202111yescharacteristic
222C23—24no—
223C23—41no—
224C23C7×#12012yes—
225C23C7×#12111yescharacteristic
226Q8—44no—
227Q8—48no—
228Q8—44no—
229Q8—44no—
230Q8—82no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—28no—
234C2×C4—28no—
235C2×C4—24no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—44no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4C7×#12011yescharacteristic
248C2×C4C7×#12112yes—
249C2×C4C7×#12711yescharacteristic
250C8—42no—
251C8—81no—
252C7#1033411yescharacteristic, direct product factor, Sylow
253C22—22no—
254C22—22no—
255C22—22no—
256C22—21no—
257C22—44no—
258C22C7×#39211yescharacteristic
259C22C7×#45614yes—
260C4—21no—
261C4—22no—
262C4—48no—
263C4—44no—
264C4—82no—
265C4—82no—
266C4C7×B(4,𝔽2)12yes—
267C2—24no—
268C2—41no—
269C2C7×#200512yes—
270C2C7×#202111yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
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. #10334C7direct (N × H)
252. C7#10334direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31952
C7×#24342
C7×#17842
C7×#15082
C7×#12972
#103347