Finite Group Databasebeta

← C7×#10217 #13187433

Subgroups — C7×#10217 #13187433

Order 1792.

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3255C211yesmaximal, characteristic
2C7×#3237C212yesmaximal
3C7×#1995C211yesmaximal, characteristic
4C7×#1506C212yesmaximal
5C7×#1307C211yesmaximal, characteristic
6C7×#533C2211yescharacteristic
7C7×#533C2212yes—
8C7×#501—22no—
9C7×#491—22no—
10C7×#485—24no—
11C7×#477—22no—
12C7×#475—22no—
13C7×#475—22no—
14C7×#470—22no—
15C7×#462—22no—
16C7×#432C2212yes—
17C7×#423C2211yescharacteristic
18C7×#376C2211yescharacteristic
19C7×#339—22no—
20C7×#328—24no—
21#10217C711yesmaximal, characteristic, direct product factor, Sylow
22C2×C7×Q16—22no—
23C2×C7×M16D412yes—
24C7×C4⋊Q8D412yes—
25C7×C4⋊Q8D412yes—
26C7×C4⋊Q8D412yes—
27C7×#125—22no—
28C7×#125—21no—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C7×C22⋊Q8—22no—
32C4×C7×Q8—22no—
33C4×C7×Q8—22no—
34C4×C7×Q8—22no—
35C7×#117—22no—
36C7×#117—21no—
37C2×C42×C7—21no—
38C2×C42×C7C2311yescharacteristic
39C7×#107—24no—
40C7×#106—22no—
41C7×#106—24no—
42C7×C4⋊C8—22no—
43C7×C4⋊C8—22no—
44C7×#103—24no—
45C7×#103—24no—
46C7×#103—24no—
47C7×C22⋊C8D414yes—
48C7×#97—22no—
49C7×#95—22no—
50#3255C2×C711yescharacteristic
51#3237C2×C712yes—
52#1995C2×C711yescharacteristic
53#1506C2×C712yes—
54#1307C2×C711yescharacteristic
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C22×C4×C7—41no—
59C22×C4×C7C2×D411yescharacteristic
60C22×C4×C7C2×D412yes—
61C7×Q16—82no—
62C7×M16—42no—
63C2×C7×C8—24no—
64C2×C7×C8—22no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C22⋊C4—42no—
75C7×C22⋊C4—41no—
76C42×C7—22no—
77C42×C7—41no—
78C42×C7—41no—
79C42×C7—42no—
80C42×C7C2×D411yescharacteristic
81C42×C7C2×D412yes—
82C42×C7#4111yescharacteristic
83#533C22×C711yescharacteristic
84#533C22×C712yes—
85#501—22no—
86#491—22no—
87#485—24no—
88#477—22no—
89#475—22no—
90#475—22no—
91#470—22no—
92#462—22no—
93#432C22×C712yes—
94#423C22×C711yescharacteristic
95#376C22×C711yescharacteristic
96#339—22no—
97#328—24no—
98C23×C7#12711yescharacteristic
99C7×Q8—44no—
100C7×Q8—42no—
101C7×Q8—42no—
102C7×Q8—82no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7—82no—
111C2×C4×C7#12012yes—
112C2×C4×C7#12011yescharacteristic
113C2×C4×C7#12112yes—
114C2×C4×C7#12111yescharacteristic
115C7×C8—44no—
116C7×C8—42no—
117C2×Q16—22no—
118C2×M16C7×D412yes—
119C4⋊Q8C7×D412yes—
120C4⋊Q8C7×D412yes—
121C4⋊Q8C7×D412yes—
122#125—22no—
123#125—21no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130#117—21no—
131#117—22no—
132C2×C42—21no—
133C2×C42C23×C711yescharacteristic
134#107—24no—
135#106—22no—
136#106—24no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139#103—24no—
140#103—24no—
141#103—24no—
142C22⋊C8C7×D414yes—
143#97—22no—
144#95—22no—
145C22×C7—21no—
146C22×C7—22no—
147C22×C7#39211yescharacteristic
148C4×C7—21no—
149C4×C7—22no—
150C4×C7—22no—
151C4×C7—82no—
152C4×C7—82no—
153C4×C7—82no—
154C4×C7—82no—
155C4×C7#45511yescharacteristic
156C4×C7#45611yescharacteristic
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C22×C4—41no—
161C22×C4C2×C7×D411yescharacteristic
162C22×C4C2×C7×D412yes—
163Q16—82no—
164M16—42no—
165C2×C8—24no—
166C2×C8—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C22⋊C4—41no—
177C22⋊C4—42no—
178C42—22no—
179C42—41no—
180C42—41no—
181C42—42no—
182C42C2×C7×D411yescharacteristic
183C42C2×C7×D412yes—
184C42C7×#4111yescharacteristic
185C2×C7—41no—
186C2×C7#200711yescharacteristic
187C2×C7#200912yes—
188C23C7×#12711yescharacteristic
189Q8—44no—
190Q8—42no—
191Q8—42no—
192Q8—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—82no—
201C2×C4C7×#12012yes—
202C2×C4C7×#12011yescharacteristic
203C2×C4C7×#12112yes—
204C2×C4C7×#12111yescharacteristic
205C8—44no—
206C8—42no—
207C7#1021711yescharacteristic, direct product factor, Sylow
208C22—21no—
209C22—22no—
210C22C7×#39211yescharacteristic
211C4—21no—
212C4—22no—
213C4—22no—
214C4—82no—
215C4—82no—
216C4—82no—
217C4—82no—
218C4C7×#45511yescharacteristic
219C4C7×#45611yescharacteristic
220C2—41no—
221C2C7×#200711yescharacteristic
222C2C7×#200912yes—
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
21. #10217C7direct (N × H)
207. C7#10217direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32552
C7×#32372
C7×#19952
C7×#15062
C7×#13072
#102177