Finite Group Databasebeta

← C7×#12801 #13190017

Subgroups — C7×#12801 #13190017

Order 1792.

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

normal subgroups only

This group has 500 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×#124C211yesmaximal, characteristic
2C2×C7×#355C212yesmaximal
3C7×#1801C212yesmaximal, complemented
4C7×#1394C212yesmaximal, complemented
5C2×C7×#124C2211yescharacteristic
6C2×C4×C7×Q8C2211yescharacteristic
7C2×C4×C7×D4C411yescharacteristic
8C7×#423—22no—
9C2×C7×#101C2214yes—
10C7×#391C2211yescharacteristic
11C7×#390C411yescharacteristic
12C7×#388—22no—
13C7×#382C411yescharacteristic
14C4×C7×C22⋊C4—22no—
15C43×C7C411yescharacteristic
16C7×#355—24no—
17C7×#343—22no—
18#12801C711yesmaximal, characteristic, direct product factor, Sylow
19C22×C7×Q8C2311yescharacteristic
20C22×C7×D4C2×C411yescharacteristic
21C23×C4×C7—21no—
22C2×C7×M16—24no—
23C7×#124—22no—
24C7×#124D412yes—
25C7×#124Q812yes—
26C4×C7×Q8—41no—
27C4×C7×Q8—41no—
28C4×C7×D4—41no—
29C4×C7×D4—41no—
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4D412yes—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—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×#124C2×C711yescharacteristic
49C2×#355C2×C712yes—
50#1801C2×C712yescomplemented
51#1394C2×C712yescomplemented
52C24×C7—21no—
53C2×C7×Q8—41no—
54C2×C7×Q8—41no—
55C2×C7×Q8C2×D411yescharacteristic
56C2×C7×Q8C2×Q811yescharacteristic
57C2×C7×Q8#4112yes—
58C2×C7×D4—21no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4C4⋊C412yes—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—44no—
66C22×C4×C7—41no—
67C22×C4×C7—41no—
68C22×C4×C7C22⋊C412yes—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D412yes—
73C7×M16—44no—
74C7×M16—44no—
75C2×C7×C8—44no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—41no—
79C7×C22⋊C4—24no—
80C7×C22⋊C4—24no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—42no—
83C42×C7—22no—
84C42×C7—22no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—21no—
88C42×C7—41no—
89C42×C7—42no—
90C42×C7—42no—
91C42×C7C4⋊C412yes—
92C2×#124C22×C711yescharacteristic
93C2×C4×Q8C22×C711yescharacteristic
94C2×C4×D4C4×C711yescharacteristic
95#423—22no—
96C2×#101C22×C714yes—
97#391C22×C711yescharacteristic
98#390C4×C711yescharacteristic
99#388—22no—
100#382C4×C711yescharacteristic
101C4×C22⋊C4—22no—
102C43C4×C711yescharacteristic
103#355—24no—
104#343—22no—
105C23×C7—22no—
106C23×C7—42no—
107C23×C7—42no—
108C23×C7—41no—
109C23×C7—41no—
110C23×C7C2×C22⋊C411yescharacteristic
111C7×Q8—41no—
112C7×Q8—41no—
113C7×Q8—42no—
114C7×D4—42no—
115C7×D4—42no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—21no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—41no—
133C2×C4×C7—82no—
134C2×C4×C7—82no—
135C2×C4×C7#9911yescharacteristic
136C2×C4×C7#9911yescharacteristic
137C2×C4×C7C2×C4⋊C411yescharacteristic
138C2×C4×C7#11711yescharacteristic
139C2×C4×C7#12112yes—
140C2×C4×C7C22⋊Q812yes—
141C7×C8—84no—
142C22×Q8C23×C711yescharacteristic
143C22×D4C2×C4×C711yescharacteristic
144C23×C4—21no—
145C2×M16—24no—
146#124—22no—
147#124C7×D412yes—
148#124C7×Q812yes—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4C2×C4×C711yescharacteristic
154C2×C4⋊C4C2×C4×C711yescharacteristic
155C2×C4⋊C4C7×D412yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42C2×C4×C711yescharacteristic
162C2×C42C2×C4×C711yescharacteristic
163C2×C42C2×C4×C711yescharacteristic
164C2×C42C7×D412yes—
165C4⋊C8—42no—
166C4⋊C8—42no—
167#101—24no—
168#101—24no—
169#95—22no—
170#95—22no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7—44no—
174C22×C7—42no—
175C22×C7—82no—
176C22×C7—81no—
177C22×C7—81no—
178C22×C7#38011yescharacteristic
179C22×C7C2×#9911yescharacteristic
180C22×C7#41011yescharacteristic
181C4×C7—22no—
182C4×C7—21no—
183C4×C7—21no—
184C4×C7—22no—
185C4×C7—21no—
186C4×C7—42no—
187C4×C7—42no—
188C4×C7—41no—
189C4×C7—42no—
190C4×C7—81no—
191C4×C7#35512yescomplemented
192C24—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C2×C7×D411yescharacteristic
196C2×Q8C2×C7×Q811yescharacteristic
197C2×Q8C7×#4112yes—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4C7×C4⋊C412yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—44no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4C7×C22⋊C411yescharacteristic
209C22×C4C7×C22⋊C412yes—
210C22×C4C7×C22⋊C411yescharacteristic
211C22×C4C22×C4×C711yescharacteristic
212C22×C4C2×C7×D412yes—
213M16—44no—
214M16—44no—
215C2×C8—44no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—42no—
222C22⋊C4—44no—
223C42—22no—
224C42—22no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—41no—
229C42—42no—
230C42—42no—
231C42C7×C4⋊C412yes—
232C2×C7—21no—
233C2×C7—21no—
234C2×C7—82no—
235C2×C7#179911yescharacteristic
236C2×C7C2×#35511yescharacteristic
237C2×C7#213211yescharacteristic
238C23—22no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23C2×C7×C22⋊C411yescharacteristic
244Q8—41no—
245Q8—41no—
246Q8—42no—
247D4—42no—
248D4—42no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4C7×#9911yescharacteristic
269C2×C4C7×#9911yescharacteristic
270C2×C4C2×C7×C4⋊C411yescharacteristic
271C2×C4C7×#11711yescharacteristic
272C2×C4C7×#12112yes—
273C2×C4C7×C22⋊Q812yes—
274C8—84no—
275C7#1280111yescharacteristic, direct product factor, Sylow
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22C7×#38011yescharacteristic
284C22C2×C7×#9911yescharacteristic
285C22C7×#41011yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—21no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—81no—
296C4C7×#35512yescomplemented
297C2—21no—
298C2—21no—
299C2—82no—
300C2C7×#179911yescharacteristic
301C2C2×C7×#35511yescharacteristic
302C2C7×#213211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×Q8C23×C7
prime radicalC22×C7×D4C2×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
3. C7×#1801C2semidirect
4. C7×#1394C2semidirect
18. #12801C7direct (N × H)
50. #1801C2×C7semidirect
51. #1394C2×C7semidirect
191. C4×C7#355semidirect
275. C7#12801direct (N × H)
296. C4C7×#355semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1242
C2×C7×#3552
C7×#18012
C7×#13942
#128017