Finite Group Databasebeta

← C7×#10227 #13187443

Subgroups — C7×#10227 #13187443

Order 1792.

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

normal subgroups only

This group has 466 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×#3229C212yesmaximal
2C7×#3162C211yesmaximal, characteristic
3C7×#1960C211yesmaximal, characteristic
4C7×#1518C212yesmaximal
5C7×#1281C211yesmaximal, characteristic
6C7×#533C2212yes—
7C2×C7×C4⋊Q8C2211yescharacteristic
8C7×#501—22no—
9C7×#501—22no—
10C7×#498—22no—
11C7×#487—22no—
12C7×#483—24no—
13C7×#475—24no—
14C7×#431C2212yes—
15C7×#427—22no—
16C2×C7×#103—22no—
17C2×C7×#97C2211yescharacteristic
18C7×#382C2211yescharacteristic
19C7×#328—24no—
20#10227C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×C8D412yes—
23C7×C4⋊Q8—22no—
24C7×C4⋊Q8—21no—
25C7×C4⋊Q8D412yes—
26C7×C4⋊Q8D412yes—
27C7×C4⋊Q8D412yes—
28C7×#125—22no—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C4×C7×Q8—22no—
32C4×C7×Q8—22no—
33C7×#117—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—21no—
36C2×C42×C7C2311yescharacteristic
37C7×#107—24no—
38C7×#106—28no—
39C7×#106—22no—
40C7×#106—22no—
41C7×C4⋊C8—22no—
42C7×#103—28no—
43C7×#103—44no—
44C7×C22⋊C8D414yes—
45C7×#97—22no—
46C4×C7×C8—22no—
47C7×#95—22no—
48#3229C2×C712yes—
49#3162C2×C711yescharacteristic
50#1960C2×C711yescharacteristic
51#1518C2×C712yes—
52#1281C2×C711yescharacteristic
53C2×C7×Q8—22no—
54C2×C7×Q8—22no—
55C2×C7×Q8—22no—
56C2×C7×Q8—41no—
57C2×C7×Q8—42no—
58C2×C7×Q8—42no—
59C22×C4×C7—41no—
60C22×C4×C7—41no—
61C22×C4×C7C2×D412yes—
62C22×C4×C7C2×D411yescharacteristic
63C2×C7×C8—22no—
64C2×C7×C8—22no—
65C2×C7×C8—24no—
66C2×C7×C8SD1614yes—
67C7×C4⋊C4—24no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C22⋊C4—42no—
77C42×C7—42no—
78C42×C7C2×D412yes—
79C42×C7C2×D411yescharacteristic
80C42×C7#4111yescharacteristic
81#533C22×C712yes—
82C2×C4⋊Q8C22×C711yescharacteristic
83#501—22no—
84#501—22no—
85#498—22no—
86#487—22no—
87#483—24no—
88#475—24no—
89#431C22×C712yes—
90#427—22no—
91C2×#103—22no—
92C2×#97C22×C711yescharacteristic
93#382C22×C711yescharacteristic
94#328—24no—
95C23×C7#12711yescharacteristic
96C7×Q8—44no—
97C7×Q8—44no—
98C7×Q8—82no—
99C7×Q8—82no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—42no—
104C2×C4×C7—81no—
105C2×C4×C7—81no—
106C2×C4×C7—82no—
107C2×C4×C7—81no—
108C2×C4×C7#12012yes—
109C2×C4×C7#12011yescharacteristic
110C2×C4×C7#12112yes—
111C2×C4×C7#12111yescharacteristic
112C2×C4×C7C2×SD1612yes—
113C2×C4×C7#13712yes—
114C7×C8—24no—
115C7×C8—44no—
116C22×Q8—21no—
117C22×C8C7×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—21no—
120C4⋊Q8C7×D412yes—
121C4⋊Q8C7×D412yes—
122C4⋊Q8C7×D412yes—
123#125—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128#117—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C42C23×C711yescharacteristic
132#107—24no—
133#106—22no—
134#106—22no—
135#106—28no—
136C4⋊C8—22no—
137#103—28no—
138#103—44no—
139C22⋊C8C7×D414yes—
140#97—22no—
141C4×C8—22no—
142#95—22no—
143C22×C7—22no—
144C22×C7#39211yescharacteristic
145C22×C7#49211yescharacteristic
146C22×C7#49711yescharacteristic
147C4×C7—22no—
148C4×C7—22no—
149C4×C7—82no—
150C4×C7—82no—
151C4×C7—82no—
152C4×C7—161no—
153C4×C7#44812yes—
154C4×C7#46112yes—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—42no—
160C2×Q8—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4C2×C7×D412yes—
164C22×C4C2×C7×D411yescharacteristic
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—24no—
168C2×C8C7×SD1614yes—
169C4⋊C4—24no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C22⋊C4—42no—
179C42—42no—
180C42C2×C7×D412yes—
181C42C2×C7×D411yescharacteristic
182C42C7×#4111yescharacteristic
183C2×C7—21no—
184C2×C7—21no—
185C2×C7#200912yes—
186C2×C7#201311yescharacteristic
187C23C7×#12711yescharacteristic
188Q8—44no—
189Q8—44no—
190Q8—82no—
191Q8—82no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—82no—
199C2×C4—81no—
200C2×C4C7×#12012yes—
201C2×C4C7×#12011yescharacteristic
202C2×C4C7×#12112yes—
203C2×C4C7×#12111yescharacteristic
204C2×C4C2×C7×SD1612yes—
205C2×C4C7×#13712yes—
206C8—24no—
207C8—44no—
208C7#1022711yescharacteristic, direct product factor, Sylow
209C22—22no—
210C22C7×#39211yescharacteristic
211C22C7×#49211yescharacteristic
212C22C7×#49711yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4—161no—
219C4C7×#44812yes—
220C4C7×#46112yes—
221C2—21no—
222C2—21no—
223C2C7×#200912yes—
224C2C7×#201311yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#127
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
20. #10227C7direct (N × H)
208. C7#10227direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32292
C7×#31622
C7×#19602
C7×#15182
C7×#12812
#102277