Finite Group Databasebeta

← C7×#11813 #13189029

Subgroups — C7×#11813 #13189029

Order 1792.

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

normal subgroups only

This group has 382 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×#2628C211yesmaximal, characteristic
2C7×#1820C211yesmaximal, characteristic
3C7×#1464C211yesmaximal, characteristic
4C7×#1352C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×#117C2211yescharacteristic
7C7×#432—21no—
8C7×#400C412yescomplemented
9C7×#399C2212yes—
10C7×#395—21no—
11C7×#393—21no—
12C7×#389—21no—
13C7×#382C411yescharacteristic, complemented
14C7×#381—21no—
15C7×#380—21no—
16C7×#342—21no—
17C7×#342C2212yes—
18C7×#324—21no—
19C7×#323C2212yes—
20#11813C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C23×C4×C7D411yescharacteristic
23C7×C4⋊Q8—41no—
24C7×C4⋊Q8—41no—
25C7×#117—41no—
26C7×#117—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4C2×C412yes—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4D411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C7×C22⋊C4C2311yescharacteristic
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C7×C4⋊C8—41no—
40C7×C22⋊C8—22no—
41C7×#97—41no—
42C7×#95—22no—
43C7×#95—21no—
44C7×#95—22no—
45C7×#95—41no—
46C7×#95—41no—
47C7×#95C2×C412yes—
48#2628C2×C711yescharacteristic
49#1820C2×C711yescharacteristic
50#1464C2×C711yescharacteristic
51#1352C2×C714yes—
52C24×C7C2×D411yescharacteristic
53C2×C7×Q8—42no—
54C2×C7×Q8—82no—
55C22×C4×C7—22no—
56C22×C4×C7—21no—
57C22×C4×C7—21no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—41no—
61C22×C4×C7—42no—
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22×C411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C2×C7×C8—42no—
69C7×C4⋊C4—41no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—82no—
76C7×C22⋊C4—41no—
77C7×C22⋊C4—41no—
78C7×C22⋊C4—42no—
79C7×C22⋊C4—42no—
80C7×C22⋊C4—82no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7—21no—
84C42×C7—21no—
85C2×C4⋊Q8C4×C711yescharacteristic, complemented
86C2×#117C22×C711yescharacteristic
87#432—21no—
88#400C4×C712yescomplemented
89#399C22×C712yes—
90#395—21no—
91#393—21no—
92#389—21no—
93#382C4×C711yescharacteristic, complemented
94#381—21no—
95#380—21no—
96#342—21no—
97#342C22×C712yes—
98#324—21no—
99#323C22×C712yes—
100C23×C7—21no—
101C23×C7—21no—
102C23×C7—41no—
103C23×C7—41no—
104C23×C7C2×C22⋊C411yescharacteristic
105C23×C7#13712yes—
106C7×Q8—84no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—42no—
128C2×C4×C7—82no—
129C2×C4×C7—82no—
130C2×C4×C7—82no—
131C2×C4×C7#9911yescharacteristic
132C2×C4×C7#9911yescharacteristic
133C7×C8—82no—
134C22×Q8—21no—
135C23×C4C7×D411yescharacteristic
136C4⋊Q8—41no—
137C4⋊Q8—41no—
138#117—41no—
139#117—41no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C2×C4×C712yes—
144C2×C4⋊C4C2×C4×C711yescharacteristic
145C2×C4⋊C4C7×D411yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4C7×D411yescharacteristic
149C2×C22⋊C4C23×C711yescharacteristic
150C2×C42C2×C4×C711yescharacteristic
151C2×C42C7×D411yescharacteristic
152C4⋊C8—41no—
153C22⋊C8—22no—
154#97—41no—
155#95—22no—
156#95—21no—
157#95—22no—
158#95—41no—
159#95—41no—
160#95C2×C4×C712yes—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7—22no—
164C22×C7—42no—
165C22×C7—41no—
166C22×C7—41no—
167C22×C7—81no—
168C22×C7C2×#9911yescharacteristic
169C22×C7#41911yescharacteristic
170C22×C7#42111yescharacteristic
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—41no—
174C4×C7—41no—
175C4×C7—41no—
176C4×C7—41no—
177C4×C7—41no—
178C4×C7—82no—
179C4×C7—82no—
180C4×C7—162no—
181C24C2×C7×D411yescharacteristic
182C2×Q8—42no—
183C2×Q8—82no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4C7×C22⋊C411yescharacteristic
192C22×C4C7×C22⋊C411yescharacteristic
193C22×C4C7×C22⋊C411yescharacteristic
194C22×C4C7×C22⋊C411yescharacteristic
195C22×C4C22×C4×C711yescharacteristic
196C22×C4C2×C7×D411yescharacteristic
197C2×C8—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—82no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—82no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C2×C7—21no—
215C2×C7—21no—
216C2×C7—42no—
217C2×C7#151111yescharacteristic
218C2×C7#212311yescharacteristic
219C2×C7#212811yescharacteristic
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23C2×C7×C22⋊C411yescharacteristic
225C23C7×#13712yes—
226Q8—84no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4C7×#9911yescharacteristic
252C2×C4C7×#9911yescharacteristic
253C8—82no—
254C7#1181311yescharacteristic, direct product factor, Sylow
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22C2×C7×#9911yescharacteristic
263C22C7×#41911yescharacteristic
264C22C7×#42111yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—162no—
275C2—21no—
276C2—21no—
277C2—42no—
278C2C7×#151111yescharacteristic
279C2C7×#212311yescharacteristic
280C2C7×#212811yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C7×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C7×C4⋊Q8C4semidirect
8. C7×#400C4semidirect
13. C7×#382C4semidirect
20. #11813C7direct (N × H)
85. C2×C4⋊Q8C4×C7semidirect
88. #400C4×C7semidirect
93. #382C4×C7semidirect
254. C7#11813direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26282
C7×#18202
C7×#14642
C7×#13522
#118137