Finite Group Databasebeta

← C7×#12798 #13190014

Subgroups — C7×#12798 #13190014

Order 1792.

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

normal subgroups only

This group has 544 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×#354C212yesmaximal
3C7×#1799C212yesmaximal
4C7×#1393C212yesmaximal
5C2×C7×#124C2211yescharacteristic
6C2×C4×C7×Q8C411yescharacteristic, complemented
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#99C2214yes—
9C7×#391C411yescharacteristic, complemented
10C7×#390C2211yescharacteristic
11C7×#388—22no—
12C7×#382C411yescharacteristic, complemented
13C7×#380—22no—
14C4×C7×C22⋊C4—22no—
15C43×C7C411yescharacteristic, complemented
16C7×#354—24no—
17C7×#342—22no—
18#12798C711yesmaximal, characteristic, direct product factor, Sylow
19C22×C7×Q8C2×C411yescharacteristic
20C22×C7×D4C2311yescharacteristic
21C23×C4×C7D412yes—
22C7×#124—22no—
23C7×#124D412yes—
24C7×#124Q812yes—
25C4×C7×Q8—41no—
26C4×C7×Q8—41no—
27C4×C7×D4—41no—
28C4×C7×D4—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—42no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—24no—
36C2×C7×C22⋊C4D412yes—
37C2×C42×C7—21no—
38C2×C42×C7—21no—
39C2×C42×C7—21no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7C2×C411yescharacteristic
43C7×#99—24no—
44C7×#99—24no—
45C7×#95—22no—
46C7×#95—22no—
47C7×#95—42no—
48C4×#124C2×C711yescharacteristic
49C2×#354C2×C712yes—
50#1799C2×C712yes—
51#1393C2×C712yes—
52C24×C7C2×D412yes—
53C2×C7×Q8—21no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8C4⋊C412yescomplemented
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4C2×D411yescharacteristic
60C2×C7×D4C2×Q811yescharacteristic
61C2×C7×D4#4112yes—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—44no—
69C22×C4×C7—44no—
70C22×C4×C7C22⋊C412yes—
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C7×C4⋊C4—41no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—84no—
78C7×C22⋊C4—24no—
79C7×C22⋊C4—24no—
80C7×C22⋊C4—42no—
81C7×C22⋊C4—44no—
82C7×C22⋊C4—44no—
83C7×C22⋊C4—44no—
84C42×C7—21no—
85C42×C7—21no—
86C42×C7—21no—
87C42×C7—41no—
88C42×C7—42no—
89C42×C7—42no—
90C42×C7—41no—
91C42×C7—41no—
92C42×C7C4⋊C412yescomplemented
93C2×#124C22×C711yescharacteristic
94C2×C4×Q8C4×C711yescharacteristic, complemented
95C2×C4×D4C22×C711yescharacteristic
96C2×#99C22×C714yes—
97#391C4×C711yescharacteristic, complemented
98#390C22×C711yescharacteristic
99#388—22no—
100#382C4×C711yescharacteristic, complemented
101#380—22no—
102C4×C22⋊C4—22no—
103C43C4×C711yescharacteristic, complemented
104#354—24no—
105#342—22no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—42no—
109C23×C7—42no—
110C23×C7C2×C22⋊C411yescharacteristic
111C23×C7#12112yes—
112C23×C7C22⋊Q812yes—
113C7×Q8—42no—
114C7×Q8—42no—
115C7×D4—41no—
116C7×D4—41no—
117C7×D4—42no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—44no—
133C2×C4×C7—41no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—82no—
138C2×C4×C7—84no—
139C2×C4×C7—84no—
140C2×C4×C7—84no—
141C2×C4×C7#9911yescharacteristic
142C2×C4×C7#9911yescharacteristic
143C2×C4×C7C2×C4⋊C411yescharacteristic
144C2×C4×C7#11711yescharacteristic
145C22×Q8C2×C4×C711yescharacteristic
146C22×D4C23×C711yescharacteristic
147C23×C4C7×D412yes—
148#124—22no—
149#124C7×D412yes—
150#124C7×Q812yes—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×D4—41no—
154C4×D4—41no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4C2×C4×C711yescharacteristic
158C2×C4⋊C4C2×C4×C711yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C7×D412yes—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42C2×C4×C711yescharacteristic
167C2×C42C2×C4×C711yescharacteristic
168C2×C42C2×C4×C711yescharacteristic
169#99—24no—
170#99—24no—
171#95—22no—
172#95—22no—
173#95—42no—
174C22×C7—21no—
175C22×C7—21no—
176C22×C7—22no—
177C22×C7—22no—
178C22×C7—42no—
179C22×C7—42no—
180C22×C7—42no—
181C22×C7—42no—
182C22×C7—82no—
183C22×C7#38011yescharacteristic
184C22×C7C2×#9911yescharacteristic
185C22×C7#41011yescharacteristic
186C4×C7—21no—
187C4×C7—21no—
188C4×C7—21no—
189C4×C7—42no—
190C4×C7—42no—
191C4×C7—42no—
192C4×C7—42no—
193C4×C7—41no—
194C4×C7—81no—
195C4×C7—164no—
196C4×C7#35412yescomplemented
197C24C2×C7×D412yes—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8C7×C4⋊C412yescomplemented
202C2×D4—41no—
203C2×D4—41no—
204C2×D4C2×C7×D411yescharacteristic
205C2×D4C2×C7×Q811yescharacteristic
206C2×D4C7×#4112yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4C7×C22⋊C411yescharacteristic
216C22×C4C7×C22⋊C412yes—
217C22×C4C7×C22⋊C411yescharacteristic
218C22×C4C22×C4×C711yescharacteristic
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—84no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—41no—
233C42—42no—
234C42—42no—
235C42—41no—
236C42—41no—
237C42C7×C4⋊C412yescomplemented
238C2×C7—21no—
239C2×C7—21no—
240C2×C7—42no—
241C2×C7—42no—
242C2×C7#179911yescharacteristic
243C2×C7C2×#35411yescharacteristic
244C2×C7#212611yescharacteristic
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23C2×C7×C22⋊C411yescharacteristic
250C23C7×#12112yes—
251C23C7×C22⋊Q812yes—
252Q8—42no—
253Q8—42no—
254D4—41no—
255D4—41no—
256D4—42no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—44no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4C7×#9911yescharacteristic
281C2×C4C7×#9911yescharacteristic
282C2×C4C2×C7×C4⋊C411yescharacteristic
283C2×C4C7×#11711yescharacteristic
284C7#1279811yescharacteristic, direct product factor, Sylow
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22C7×#38011yescharacteristic
295C22C2×C7×#9911yescharacteristic
296C22C7×#41011yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—81no—
306C4—164no—
307C4C7×#35412yescomplemented
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2C7×#179911yescharacteristic
313C2C2×C7×#35411yescharacteristic
314C2C7×#212611yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×C7×Q8C4semidirect
9. C7×#391C4semidirect
12. C7×#382C4semidirect
15. C43×C7C4semidirect
18. #12798C7direct (N × H)
56. C2×C7×Q8C4⋊C4semidirect
92. C42×C7C4⋊C4semidirect
94. C2×C4×Q8C4×C7semidirect
97. #391C4×C7semidirect
100. #382C4×C7semidirect
103. C43C4×C7semidirect
196. C4×C7#354semidirect
201. C2×Q8C7×C4⋊C4semidirect
237. C42C7×C4⋊C4semidirect
284. C7#12798direct (N × H)
307. C4C7×#354semidirect

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×#3542
C7×#17992
C7×#13932
#127987