Finite Group Databasebeta

← C7×#10017 #13187233

Subgroups — C7×#10017 #13187233

Order 1792.

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

normal subgroups only

This group has 528 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×#3104C212yesmaximal
2C7×#2399C211yesmaximal, characteristic
3C7×#1995C211yesmaximal, characteristic
4C7×#1487C212yesmaximal
5C7×#1274C211yesmaximal, characteristic
6C7×#533C2211yescharacteristic
7C2×C4×C7×Q8C2212yes—
8C7×#484—22no—
9C7×#477—22no—
10C7×#462—22no—
11C7×#451—22no—
12C4×C7×Q16—24no—
13C7×#431C2212yes—
14C7×#423C2211yescharacteristic
15C7×#398—22no—
16C7×#395—22no—
17C4×C7×C4⋊C4—22no—
18C7×#376C2211yescharacteristic
19C7×#339—22no—
20C7×Q8⋊C8—24no—
21#10017C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8D412yes—
23C2×C7×Q16—24no—
24C2×C7×M16—21no—
25C7×C4⋊Q8D412yes—
26C7×#125—21no—
27C7×C22⋊Q8—22no—
28C4×C7×Q8—22no—
29C4×C7×Q8—22no—
30C4×C7×Q8—22no—
31C4×C7×Q8—22no—
32C4×C7×Q8—22no—
33C7×#117—21no—
34C2×C7×C4⋊C4—24no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4D412yes—
37C2×C42×C7—22no—
38C2×C42×C7D412yes—
39C2×C42×C7C2311yescharacteristic
40C7×#107—24no—
41C7×C4⋊C8—22no—
42C7×C4⋊C8—22no—
43C7×#103—24no—
44C7×#103—24no—
45C7×C22⋊C8—22no—
46C4×C7×C8D414yes—
47C7×#95—22no—
48C7×#95—22no—
49C7×#95—22no—
50#3104C2×C712yes—
51#2399C2×C711yescharacteristic
52#1995C2×C711yescharacteristic
53#1487C2×C712yes—
54#1274C2×C711yescharacteristic
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—44no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7C2×D412yes—
63C22×C4×C7C2×D411yescharacteristic
64C7×Q16—48no—
65C7×M16—42no—
66C2×C7×C8—24no—
67C2×C7×C8—41no—
68C7×C4⋊C4—22no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—22no—
71C7×C4⋊C4—22no—
72C7×C4⋊C4—48no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C22⋊C4—41no—
77C42×C7—22no—
78C42×C7—42no—
79C42×C7—42no—
80C42×C7—41no—
81C42×C7Q1614yescomplemented
82C42×C7C2×D411yescharacteristic
83C42×C7C2×D412yes—
84C42×C7#4111yescharacteristic
85#533C22×C711yescharacteristic
86C2×C4×Q8C22×C712yes—
87#484—22no—
88#477—22no—
89#462—22no—
90#451—22no—
91C4×Q16—24no—
92#431C22×C712yes—
93#423C22×C711yescharacteristic
94#398—22no—
95#395—22no—
96C4×C4⋊C4—22no—
97#376C22×C711yescharacteristic
98#339—22no—
99Q8⋊C8—24no—
100C23×C7#12011yescharacteristic
101C7×Q8—24no—
102C7×Q8—24no—
103C7×Q8—42no—
104C7×Q8—42no—
105C7×Q8—44no—
106C2×C4×C7—24no—
107C2×C4×C7—24no—
108C2×C4×C7—42no—
109C2×C4×C7—44no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—44no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7#12012yes—
116C2×C4×C7#12112yes—
117C2×C4×C7#12111yescharacteristic
118C2×C4×C7#12711yescharacteristic
119C2×C4×C7C2×Q1612yes—
120C2×C4×C7#13712yes—
121C7×C8—44no—
122C7×C8—81no—
123C22×Q8C7×D412yes—
124C2×Q16—24no—
125C2×M16—21no—
126C4⋊Q8C7×D412yes—
127#125—21no—
128C22⋊Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134#117—21no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4C7×D412yes—
138C2×C42—22no—
139C2×C42C7×D412yes—
140C2×C42C23×C711yescharacteristic
141#107—24no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144#103—24no—
145#103—24no—
146C22⋊C8—22no—
147C4×C8C7×D414yes—
148#95—22no—
149#95—22no—
150#95—22no—
151C22×C7—22no—
152C22×C7#39211yescharacteristic
153C22×C7#45112yes—
154C4×C7—22no—
155C4×C7—22no—
156C4×C7—44no—
157C4×C7—44no—
158C4×C7—82no—
159C4×C7—82no—
160C4×C7#46212yes—
161C4×C7#49411yescharacteristic
162C4×C7#49711yescharacteristic
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—44no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×C7×D412yes—
171C22×C4C2×C7×D411yescharacteristic
172Q16—48no—
173M16—42no—
174C2×C8—24no—
175C2×C8—41no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—48no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C22⋊C4—41no—
185C42—22no—
186C42—42no—
187C42—42no—
188C42—41no—
189C42C7×Q1614yescomplemented
190C42C2×C7×D411yescharacteristic
191C42C2×C7×D412yes—
192C42C7×#4111yescharacteristic
193C2×C7—22no—
194C2×C7#201012yes—
195C2×C7#201611yescharacteristic
196C23C7×#12011yescharacteristic
197Q8—24no—
198Q8—24no—
199Q8—42no—
200Q8—42no—
201Q8—44no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4C7×#12012yes—
212C2×C4C7×#12112yes—
213C2×C4C7×#12111yescharacteristic
214C2×C4C7×#12711yescharacteristic
215C2×C4C2×C7×Q1612yes—
216C2×C4C7×#13712yes—
217C8—44no—
218C8—81no—
219C7#1001711yescharacteristic, direct product factor, Sylow
220C22—22no—
221C22C7×#39211yescharacteristic
222C22C7×#45112yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—82no—
229C4C7×#46212yes—
230C4C7×#49411yescharacteristic
231C4C7×#49711yescharacteristic
232C2—22no—
233C2C7×#201012yes—
234C2C7×#201611yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC23×C7#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
21. #10017C7direct (N × H)
81. C42×C7Q16semidirect
189. C42C7×Q16semidirect
219. C7#10017direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31042
C7×#23992
C7×#19952
C7×#14872
C7×#12742
#100177