Finite Group Databasebeta

← C7×#10036 #13187252

Subgroups — C7×#10036 #13187252

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×#3162C211yesmaximal, characteristic, complemented
2C7×#3101C212yesmaximal
3C7×#1955C211yesmaximal, characteristic
4C7×#1481C212yesmaximal, complemented
5C7×#1275C211yesmaximal, characteristic, complemented
6C7×#533C2212yes—
7C2×C4×C7×D4C2211yescharacteristic
8C7×#501—22no—
9C7×#501—22no—
10C7×#483—22no—
11C7×#476—22no—
12C7×#461—22no—
13C7×#450—22no—
14C4×C7×SD16—24no—
15C7×#431C2212yes—
16C7×#427—22no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C22⋊C8—22no—
19C2×C7×#97C2211yescharacteristic
20C7×Q8⋊C8—24no—
21#10036C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×D4D411yescharacteristic
23C23×C4×C7—21no—
24C2×C7×SD16—24no—
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×D4—22no—
35C4×C7×D4—22no—
36C7×#117—22no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C42×C7C2311yescharacteristic
40C7×#107—22no—
41C7×#107—22no—
42C7×#106—24no—
43C7×#106—24no—
44C7×C4⋊C8—22no—
45C7×C4⋊C8—22no—
46C7×#103—24no—
47C7×#102—24no—
48C7×C22⋊C8—22no—
49C7×C22⋊C8—44no—
50C7×#97—22no—
51C4×C7×C8D414yes—
52#3162C2×C711yescharacteristic, complemented
53#3101C2×C712yes—
54#1955C2×C711yescharacteristic
55#1481C2×C712yescomplemented
56#1275C2×C711yescharacteristic, complemented
57C24×C7—21no—
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×D4—22no—
61C2×C7×D4—42no—
62C22×C4×C7—21no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7#4111yescharacteristic
70C7×SD16—48no—
71C2×C7×C8—22no—
72C2×C7×C8—22no—
73C2×C7×C8—24no—
74C2×C7×C8—41no—
75C2×C7×C8—42no—
76C2×C7×C8SD1614yescomplemented
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—42no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C42×C7—42no—
87C42×C7C2×D412yes—
88C42×C7C2×D412yes—
89#533C22×C712yes—
90C2×C4×D4C22×C711yescharacteristic
91#501—22no—
92#501—22no—
93#483—22no—
94#476—22no—
95#461—22no—
96#450—22no—
97C4×SD16—24no—
98#431C22×C712yes—
99#427—22no—
100C2×C4⋊C8C22×C711yescharacteristic
101C2×C22⋊C8—22no—
102C2×#97C22×C711yescharacteristic
103Q8⋊C8—24no—
104C23×C7—22no—
105C23×C7—41no—
106C23×C7—42no—
107C23×C7—41no—
108C23×C7—42no—
109C23×C7#12111yescharacteristic
110C7×Q8—44no—
111C7×Q8—44no—
112C7×D4—24no—
113C7×D4—42no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—42no—
120C2×C4×C7—82no—
121C2×C4×C7—81no—
122C2×C4×C7—81no—
123C2×C4×C7#12012yes—
124C2×C4×C7#12011yescharacteristic
125C2×C4×C7#12112yes—
126C2×C4×C7#12711yescharacteristic
127C2×C4×C7C2×SD1612yes—
128C2×C4×C7#13712yes—
129C7×C8—24no—
130C7×C8—44no—
131C7×C8—81no—
132C22×D4C7×D411yescharacteristic
133C23×C4—21no—
134C2×SD16—24no—
135C22×C8—21no—
136C22×C8C7×D412yes—
137C4⋊Q8C7×D412yes—
138C4⋊Q8C7×D412yes—
139#125—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×D4—22no—
145C4×D4—22no—
146#117—22no—
147C2×C4⋊C4C7×D411yescharacteristic
148C2×C22⋊C4—21no—
149C2×C42C23×C711yescharacteristic
150#107—22no—
151#107—22no—
152#106—24no—
153#106—24no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156#103—24no—
157#102—24no—
158C22⋊C8—22no—
159C22⋊C8—44no—
160#97—22no—
161C4×C8C7×D414yes—
162C22×C7—21no—
163C22×C7—21no—
164C22×C7—44no—
165C22×C7—42no—
166C22×C7—82no—
167C22×C7—81no—
168C22×C7—81no—
169C22×C7#39211yescharacteristic
170C22×C7#46112yes—
171C4×C7—22no—
172C4×C7—22no—
173C4×C7—82no—
174C4×C7—81no—
175C4×C7—82no—
176C4×C7#44812yes—
177C4×C7#49211yescharacteristic
178C4×C7#49711yescharacteristic
179C24—21no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×D4—22no—
183C2×D4—42no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4C2×C7×D411yescharacteristic
190C22×C4C2×C7×D411yescharacteristic
191C22×C4C7×#4111yescharacteristic
192SD16—48no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8C7×SD1614yescomplemented
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—42no—
209C42C2×C7×D412yes—
210C42C2×C7×D412yes—
211C2×C7—22no—
212C2×C7—82no—
213C2×C7#201011yescharacteristic
214C2×C7#201311yescharacteristic
215C2×C7#201711yescharacteristic
216C23—22no—
217C23—41no—
218C23—42no—
219C23—41no—
220C23—42no—
221C23C7×#12111yescharacteristic
222Q8—44no—
223Q8—44no—
224D4—24no—
225D4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—82no—
235C2×C4C7×#12012yes—
236C2×C4C7×#12011yescharacteristic
237C2×C4C7×#12112yes—
238C2×C4C7×#12711yescharacteristic
239C2×C4C2×C7×SD1612yes—
240C2×C4C7×#13712yes—
241C8—24no—
242C8—44no—
243C8—81no—
244C7#1003611yescharacteristic, direct product factor, Sylow
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22C7×#39211yescharacteristic
253C22C7×#46112yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—81no—
258C4—82no—
259C4C7×#44812yes—
260C4C7×#49211yescharacteristic
261C4C7×#49711yescharacteristic
262C2—22no—
263C2—82no—
264C2C7×#201011yescharacteristic
265C2C7×#201311yescharacteristic
266C2C7×#201711yescharacteristic
subgroupG/H
centerC2×C4×C7#127
commutator, Frattini, cosocleC2×C42C23×C7
socleC22×C7#392
prime radicalC22×C7×D4D4
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
1. C7×#3162C2semidirect
4. C7×#1481C2semidirect
5. C7×#1275C2semidirect
21. #10036C7direct (N × H)
52. #3162C2×C7semidirect
55. #1481C2×C7semidirect
56. #1275C2×C7semidirect
76. C2×C7×C8SD16semidirect
198. C2×C8C7×SD16semidirect
244. C7#10036direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31622
C7×#31012
C7×#19552
C7×#14812
C7×#12752
#100367