Finite Group Databasebeta

← C7×#13295 #13190511

Subgroups — C7×#13295 #13190511

Order 1792.

488 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2487C211yesmaximal, characteristic
2C7×#1800C211yesmaximal, characteristic
3C4×C7×#99C211yesmaximal, characteristic
4C7×#1390C214yesmaximal
5C2×C4×C7×D4C2211yescharacteristic
6C22×C7×C4⋊C4C411yescharacteristic, complemented
7C7×#423—21no—
8C2×C7×#100C2212yes—
9C2×C7×#99C2212yes—
10C7×#400—21no—
11C7×#399—21no—
12C7×#389—21no—
13C7×#386C2212yes—
14C7×#385C412yescomplemented
15C7×#380—21no—
16C7×#380C411yescharacteristic, complemented
17C4×C7×C22⋊C4—21no—
18C7×#343—21no—
19C7×#342—21no—
20#13295C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×D4C2311yescharacteristic
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7C2×C412yes—
24C23×C4×C7D411yescharacteristic
25C2×C7×M16—22no—
26C4×C7×D4—41no—
27C4×C7×D4—41no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—41no—
33C2×C7×C4⋊C4—41no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4C2×C412yes—
38C2×C7×C22⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7—41no—
41C2×C42×C7D411yescharacteristic
42C7×C4⋊C8—41no—
43C7×C4⋊C8—41no—
44C7×#100—24no—
45C7×#99—22no—
46C7×#99Q814yes—
47C7×#95—22no—
48C7×#95—21no—
49C7×#95—22no—
50C7×#95—41no—
51#2487C2×C711yescharacteristic
52#1800C2×C711yescharacteristic
53C4×#99C2×C711yescharacteristic
54#1390C2×C714yes—
55C24×C7C22×C411yescharacteristic
56C24×C7C2×D411yescharacteristic
57C2×C7×D4—41no—
58C2×C7×D4—41no—
59C2×C7×D4C2×Q812yes—
60C2×C7×D4#4112yes—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7—82no—
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C7×M16—42no—
78C7×M16—42no—
79C2×C7×C8—42no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C4⋊C4—82no—
86C7×C22⋊C4—24no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—42no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—82no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—82no—
95C2×C4×D4C22×C711yescharacteristic
96C22×C4⋊C4C4×C711yescharacteristic, complemented
97#423—21no—
98C2×#100C22×C712yes—
99C2×#99C22×C712yes—
100#400—21no—
101#399—21no—
102#389—21no—
103#386C22×C712yes—
104#385C4×C712yescomplemented
105#380—21no—
106#380C4×C711yescharacteristic, complemented
107C4×C22⋊C4—21no—
108#343—21no—
109#342—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—41no—
115C23×C7—41no—
116C23×C7—41no—
117C23×C7—41no—
118C23×C7C2×C22⋊C411yescharacteristic
119C23×C7C4×Q812yes—
120C23×C7C22⋊Q812yes—
121C7×D4—42no—
122C7×D4—42no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—44no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—81no—
143C2×C4×C7—82no—
144C2×C4×C7—82no—
145C2×C4×C7—84no—
146C2×C4×C7#9911yescharacteristic
147C2×C4×C7#9911yescharacteristic
148C2×C4×C7#12411yescharacteristic
149C2×C4×C7C4⋊Q811yescharacteristic
150C7×C8—82no—
151C22×D4C23×C711yescharacteristic
152C23×C4C2×C4×C711yescharacteristic
153C23×C4C2×C4×C712yes—
154C23×C4C7×D411yescharacteristic
155C2×M16—22no—
156C4×D4—41no—
157C4×D4—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C7×D411yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4C2×C4×C712yes—
168C2×C22⋊C4C2×C4×C711yescharacteristic
169C2×C22⋊C4C7×D411yescharacteristic
170C2×C42—41no—
171C2×C42C7×D411yescharacteristic
172C4⋊C8—41no—
173C4⋊C8—41no—
174#100—24no—
175#99—22no—
176#99C7×Q814yes—
177#95—22no—
178#95—21no—
179#95—22no—
180#95—41no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—22no—
184C22×C7—22no—
185C22×C7—42no—
186C22×C7—42no—
187C22×C7—42no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—81no—
193C22×C7#39111yescharacteristic
194C22×C7C2×#9911yescharacteristic
195C22×C7#41011yescharacteristic
196C4×C7—21no—
197C4×C7—21no—
198C4×C7—21no—
199C4×C7—41no—
200C4×C7—41no—
201C4×C7—41no—
202C4×C7—82no—
203C4×C7—84no—
204C4×C7—82no—
205C4×C7#35212yes—
206C24C22×C4×C711yescharacteristic
207C24C2×C7×D411yescharacteristic
208C2×D4—41no—
209C2×D4—41no—
210C2×D4C2×C7×Q812yes—
211C2×D4C7×#4112yes—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—82no—
223C22×C4C7×C22⋊C411yescharacteristic
224C22×C4C7×C22⋊C411yescharacteristic
225C22×C4C7×C22⋊C411yescharacteristic
226C22×C4C7×C22⋊C411yescharacteristic
227C22×C4C2×C7×D411yescharacteristic
228M16—42no—
229M16—42no—
230C2×C8—42no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—82no—
237C22⋊C4—24no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C42—21no—
244C42—21no—
245C42—82no—
246C2×C7—21no—
247C2×C7—21no—
248C2×C7—42no—
249C2×C7—42no—
250C2×C7#199611yescharacteristic
251C2×C7C2×#35211yescharacteristic
252C2×C7#212211yescharacteristic
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23C2×C7×C22⋊C411yescharacteristic
262C23C4×C7×Q812yes—
263C23C7×C22⋊Q812yes—
264D4—42no—
265D4—42no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—84no—
289C2×C4C7×#9911yescharacteristic
290C2×C4C7×#9911yescharacteristic
291C2×C4C7×#12411yescharacteristic
292C2×C4C7×C4⋊Q811yescharacteristic
293C8—82no—
294C7#1329511yescharacteristic, direct product factor, Sylow
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22C7×#39111yescharacteristic
308C22C2×C7×#9911yescharacteristic
309C22C7×#41011yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—82no—
317C4—84no—
318C4—82no—
319C4C7×#35212yes—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2C7×#199611yescharacteristic
325C2C2×C7×#35211yescharacteristic
326C2C7×#212211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC24C22×C4×C7
Frattini, cosocleC22×D4C23×C7
prime radicalC22×C7×D4C23
LCSG ▷ C24 ▷ 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. C22×C7×C4⋊C4C4semidirect
14. C7×#385C4semidirect
16. C7×#380C4semidirect
20. #13295C7direct (N × H)
96. C22×C4⋊C4C4×C7semidirect
104. #385C4×C7semidirect
106. #380C4×C7semidirect
294. C7#13295direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24872
C7×#18002
C4×C7×#992
C7×#13902
#132957