Finite Group Databasebeta

← C7×#10031 #13187247

Subgroups — C7×#10031 #13187247

Order 1792.

468 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3113C212yesmaximal
3C7×#1985C211yesmaximal, characteristic
4C7×#1531C212yesmaximal
5C7×#1290C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#483—22no—
9C7×#477—22no—
10C7×#475—22no—
11C7×#467—22no—
12C7×#462—22no—
13C7×#462—22no—
14C7×#451—22no—
15C7×#441—24no—
16C7×#432C2212yes—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#97C2211yescharacteristic
19C7×#336—22no—
20C7×#330—24no—
21#10031C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D411yescharacteristic
23C2×C7×Q16—24no—
24C2×C7×Q16—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×#106—24no—
43C7×C4⋊C8—22no—
44C7×C4⋊C8—22no—
45C7×#103—24no—
46C7×#103—24no—
47C7×#103—24no—
48C7×C22⋊C8—22no—
49C7×#97—22no—
50C7×#97D414yes—
51#3249C2×C711yescharacteristic
52#3113C2×C712yes—
53#1985C2×C711yescharacteristic
54#1531C2×C712yes—
55#1290C2×C711yescharacteristic
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—42no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7C2×D411yescharacteristic
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7#4111yescharacteristic
65C7×Q16—48no—
66C7×Q16—82no—
67C2×C7×C8—22no—
68C2×C7×C8—24no—
69C2×C7×C8—22no—
70C2×C7×C8—41no—
71C2×C7×C8—41no—
72C2×C7×C8—42no—
73C7×C4⋊C4—22no—
74C7×C4⋊C4—22no—
75C7×C4⋊C4—22no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C22⋊C4—42no—
82C42×C7—41no—
83C42×C7—42no—
84C42×C7—41no—
85C42×C7C2×D412yes—
86C42×C7C2×D412yes—
87#533C22×C712yes—
88C2×C4×Q8C22×C711yescharacteristic
89#483—22no—
90#477—22no—
91#475—22no—
92#467—22no—
93#462—22no—
94#462—22no—
95#451—22no—
96#441—24no—
97#432C22×C712yes—
98C2×C4⋊C8C22×C711yescharacteristic
99C2×#97C22×C711yescharacteristic
100#336—22no—
101#330—24no—
102C23×C7#12111yescharacteristic
103C7×Q8—24no—
104C7×Q8—42no—
105C7×Q8—44no—
106C7×Q8—42no—
107C7×Q8—42no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—41no—
113C2×C4×C7—41no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7#12012yes—
120C2×C4×C7#12011yescharacteristic
121C2×C4×C7#12112yes—
122C2×C4×C7#12711yescharacteristic
123C7×C8—42no—
124C7×C8—44no—
125C7×C8—42no—
126C22×Q8C7×D411yescharacteristic
127C2×Q16—24no—
128C2×Q16—22no—
129C22×C8—21no—
130C22×C8C7×D412yes—
131C4⋊Q8C7×D412yes—
132C4⋊Q8C7×D412yes—
133#125—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140#117—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4C7×D411yescharacteristic
143C2×C42—21no—
144C2×C42C23×C711yescharacteristic
145#107—22no—
146#106—24no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149#103—24no—
150#103—24no—
151#103—24no—
152C22⋊C8—22no—
153#97—22no—
154#97C7×D414yes—
155C22×C7—21no—
156C22×C7—21no—
157C22×C7#39211yescharacteristic
158C22×C7#47312yes—
159C4×C7—21no—
160C4×C7—22no—
161C4×C7—22no—
162C4×C7—42no—
163C4×C7—82no—
164C4×C7—82no—
165C4×C7—81no—
166C4×C7#45511yescharacteristic
167C4×C7#45611yescharacteristic
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—42no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4C2×C7×D411yescharacteristic
175C22×C4C2×C7×D411yescharacteristic
176C22×C4C7×#4111yescharacteristic
177Q16—48no—
178Q16—82no—
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—22no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C22⋊C4—42no—
194C42—41no—
195C42—42no—
196C42—41no—
197C42C2×C7×D412yes—
198C42C2×C7×D412yes—
199C2×C7—22no—
200C2×C7#200711yescharacteristic
201C2×C7#201011yescharacteristic
202C2×C7#202011yescharacteristic
203C23C7×#12111yescharacteristic
204Q8—24no—
205Q8—42no—
206Q8—44no—
207Q8—42no—
208Q8—42no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4C7×#12012yes—
221C2×C4C7×#12011yescharacteristic
222C2×C4C7×#12112yes—
223C2×C4C7×#12711yescharacteristic
224C8—42no—
225C8—44no—
226C8—42no—
227C7#1003111yescharacteristic, direct product factor, Sylow
228C22—21no—
229C22—21no—
230C22C7×#39211yescharacteristic
231C22C7×#47312yes—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—82no—
237C4—82no—
238C4—81no—
239C4C7×#45511yescharacteristic
240C4C7×#45611yescharacteristic
241C2—22no—
242C2C7×#200711yescharacteristic
243C2C7×#201011yescharacteristic
244C2C7×#202011yescharacteristic
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. #10031C7direct (N × H)
227. C7#10031direct (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×#31132
C7×#19852
C7×#15312
C7×#12902
#100317