Finite Group Databasebeta

← C7×#12802 #13190018

Subgroups — C7×#12802 #13190018

Order 1792.

468 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×C7×C4⋊Q8C211yesmaximal, characteristic
2C2×C7×#356C212yesmaximal
3C7×#1801C212yesmaximal
4C7×#1394C212yesmaximal
5C2×C7×C4⋊Q8C2211yescharacteristic
6C2×C4×C7×Q8C411yescharacteristic
7C2×C4×C7×Q8C2211yescharacteristic
8C7×#423—22no—
9C2×C7×#101C2214yes—
10C7×#391C411yescharacteristic
11C7×#391C2211yescharacteristic
12C7×#389—22no—
13C7×#384C411yescharacteristic
14C4×C7×C4⋊C4—22no—
15C43×C7C411yescharacteristic
16C7×#356—24no—
17C7×#343—22no—
18#12802C711yesmaximal, characteristic, direct product factor, Sylow
19C22×C7×Q8C2×C411yescharacteristic
20C22×C7×Q8C2311yescharacteristic
21C2×C7×M16—24no—
22C7×C4⋊Q8—22no—
23C7×C4⋊Q8D412yes—
24C7×C4⋊Q8Q812yes—
25C4×C7×Q8—41no—
26C4×C7×Q8—41no—
27C4×C7×Q8—41no—
28C4×C7×Q8—41no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D412yes—
35C2×C42×C7—21no—
36C2×C42×C7—21no—
37C2×C42×C7—21no—
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7C2×C411yescharacteristic
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7D412yes—
42C7×C4⋊C8—42no—
43C7×C4⋊C8—42no—
44C7×#101—24no—
45C7×#101—24no—
46C7×#95—22no—
47C7×#95—22no—
48C4×C4⋊Q8C2×C711yescharacteristic
49C2×#356C2×C712yes—
50#1801C2×C712yes—
51#1394C2×C712yes—
52C2×C7×Q8—21no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8C4⋊C412yes—
58C2×C7×Q8C2×D411yescharacteristic
59C2×C7×Q8C2×Q811yescharacteristic
60C2×C7×Q8#4112yes—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7C22⋊C412yes—
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D412yes—
70C7×M16—44no—
71C7×M16—44no—
72C2×C7×C8—44no—
73C7×C4⋊C4—24no—
74C7×C4⋊C4—24no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—41no—
78C7×C4⋊C4—44no—
79C7×C4⋊C4—42no—
80C42×C7—24no—
81C42×C7—22no—
82C42×C7—22no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—41no—
87C42×C7—42no—
88C42×C7—41no—
89C42×C7—41no—
90C42×C7C4⋊C412yes—
91C2×C4⋊Q8C22×C711yescharacteristic
92C2×C4×Q8C4×C711yescharacteristic
93C2×C4×Q8C22×C711yescharacteristic
94#423—22no—
95C2×#101C22×C714yes—
96#391C4×C711yescharacteristic
97#391C22×C711yescharacteristic
98#389—22no—
99#384C4×C711yescharacteristic
100C4×C4⋊C4—22no—
101C43C4×C711yescharacteristic
102#356—24no—
103#343—22no—
104C23×C7C2×C22⋊C411yescharacteristic
105C7×Q8—41no—
106C7×Q8—41no—
107C7×Q8—42no—
108C7×Q8—42no—
109C7×Q8—42no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—22no—
119C2×C4×C7—22no—
120C2×C4×C7—21no—
121C2×C4×C7—22no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—41no—
130C2×C4×C7#9911yescharacteristic
131C2×C4×C7#9911yescharacteristic
132C2×C4×C7C2×C4⋊C411yescharacteristic
133C2×C4×C7#11711yescharacteristic
134C2×C4×C7#12112yes—
135C2×C4×C7C22⋊Q812yes—
136C7×C8—84no—
137C22×Q8C2×C4×C711yescharacteristic
138C22×Q8C23×C711yescharacteristic
139C2×M16—24no—
140C4⋊Q8—22no—
141C4⋊Q8C7×D412yes—
142C4⋊Q8C7×Q812yes—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C4×Q8—41no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×C4×C711yescharacteristic
151C2×C4⋊C4C2×C4×C711yescharacteristic
152C2×C4⋊C4C7×D412yes—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42C2×C4×C711yescharacteristic
157C2×C42C2×C4×C711yescharacteristic
158C2×C42C2×C4×C711yescharacteristic
159C2×C42C7×D412yes—
160C4⋊C8—42no—
161C4⋊C8—42no—
162#101—24no—
163#101—24no—
164#95—22no—
165#95—22no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7#38011yescharacteristic
169C22×C7C2×#9911yescharacteristic
170C22×C7#41011yescharacteristic
171C4×C7—24no—
172C4×C7—22no—
173C4×C7—21no—
174C4×C7—21no—
175C4×C7—22no—
176C4×C7—21no—
177C4×C7—42no—
178C4×C7—42no—
179C4×C7—41no—
180C4×C7—42no—
181C4×C7—81no—
182C4×C7#35612yescomplemented
183C2×Q8—21no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8C7×C4⋊C412yes—
189C2×Q8C2×C7×D411yescharacteristic
190C2×Q8C2×C7×Q811yescharacteristic
191C2×Q8C7×#4112yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4C7×C22⋊C411yescharacteristic
197C22×C4C7×C22⋊C412yes—
198C22×C4C7×C22⋊C411yescharacteristic
199C22×C4C22×C4×C711yescharacteristic
200C22×C4C2×C7×D412yes—
201M16—44no—
202M16—44no—
203C2×C8—44no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C42—24no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—41no—
218C42—42no—
219C42—41no—
220C42—41no—
221C42C7×C4⋊C412yes—
222C2×C7—21no—
223C2×C7—21no—
224C2×C7#179911yescharacteristic
225C2×C7C2×#35611yescharacteristic
226C2×C7#213211yescharacteristic
227C23C2×C7×C22⋊C411yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231Q8—42no—
232Q8—42no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4C7×#9911yescharacteristic
254C2×C4C7×#9911yescharacteristic
255C2×C4C2×C7×C4⋊C411yescharacteristic
256C2×C4C7×#11711yescharacteristic
257C2×C4C7×#12112yes—
258C2×C4C7×C22⋊Q812yes—
259C8—84no—
260C7#1280211yescharacteristic, direct product factor, Sylow
261C22—21no—
262C22—21no—
263C22C7×#38011yescharacteristic
264C22C2×C7×#9911yescharacteristic
265C22C7×#41011yescharacteristic
266C4—24no—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—22no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4C7×#35612yescomplemented
278C2—21no—
279C2—21no—
280C2C7×#179911yescharacteristic
281C2C2×C7×#35611yescharacteristic
282C2C7×#213211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×Q8C23×C7
prime radicalC23×C7C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
18. #12802C7direct (N × H)
182. C4×C7#356semidirect
260. C7#12802direct (N × H)
277. C4C7×#356semidirect

Maximal subgroups

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

subgroupindex
C4×C7×C4⋊Q82
C2×C7×#3562
C7×#18012
C7×#13942
#128027