Finite Group Databasebeta

← C7×#11778 #13188994

Subgroups — C7×#11778 #13188994

Order 1792.

446 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2536C211yesmaximal, characteristic
2C4×C7×#101C211yesmaximal, characteristic, complemented
3C7×#1467C211yesmaximal, characteristic, complemented
4C7×#1346C212yesmaximal, complemented
5C7×#1345C212yesmaximal
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×Q8C2211yescharacteristic
8C2×C4×C7×D4C411yescharacteristic
9C7×#432—21no—
10C2×C7×#101—21no—
11C4×C7×M16—21no—
12C7×#398C411yescharacteristic
13C7×#398C2211yescharacteristic
14C7×#394C411yescharacteristic
15C7×#394C2211yescharacteristic
16C7×#388—21no—
17C7×#388—21no—
18C7×#382C411yescharacteristic
19C7×#380—21no—
20C7×#343C2212yes—
21C7×#324—21no—
22C7×#324C2212yes—
23#11778C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8D411yescharacteristic
25C22×C7×D4C2×C411yescharacteristic
26C23×C4×C7—21no—
27C2×C7×M16—22no—
28C7×C22⋊Q8—44no—
29C4×C7×Q8—41no—
30C4×C7×Q8—41no—
31C4×C7×D4—41no—
32C4×C7×D4—41no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×C42×C7D411yescharacteristic
48C7×C4⋊C8—41no—
49C7×#101—24no—
50C7×C22⋊C8—22no—
51C7×#97—41no—
52C7×#97—41no—
53C4×C7×C8—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95C2×C411yescharacteristic
58C7×#95C2×C411yescharacteristic
59#2536C2×C711yescharacteristic
60C4×#101C2×C711yescharacteristic, complemented
61#1467C2×C711yescharacteristic, complemented
62#1346C2×C712yescomplemented
63#1345C2×C712yes—
64C24×C7—21no—
65C2×C7×Q8—22no—
66C2×C7×Q8—22no—
67C2×C7×Q8—41no—
68C2×C7×D4—41no—
69C2×C7×D4—42no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7—42no—
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C7×M16—24no—
84C7×M16—42no—
85C2×C7×C8—42no—
86C2×C7×C8—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—42no—
100C42×C7—21no—
101C42×C7—21no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—21no—
106C2×C22⋊Q8—21no—
107C2×C4×Q8C22×C711yescharacteristic
108C2×C4×D4C4×C711yescharacteristic
109#432—21no—
110C2×#101—21no—
111C4×M16—21no—
112#398C4×C711yescharacteristic
113#398C22×C711yescharacteristic
114#394C4×C711yescharacteristic
115#394C22×C711yescharacteristic
116#388—21no—
117#388—21no—
118#382C4×C711yescharacteristic
119#380—21no—
120#343C22×C712yes—
121#324—21no—
122#324C22×C712yes—
123C23×C7—22no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7C2×C22⋊C411yescharacteristic
131C7×Q8—44no—
132C7×D4—42no—
133C7×D4—42no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—84no—
158C2×C4×C7#9911yescharacteristic
159C2×C4×C7#9911yescharacteristic
160C2×C4×C7#10412yes—
161C2×C4×C7AGL(1,ℤ/8)12yes—
162C7×C8—44no—
163C7×C8—82no—
164C22×Q8C7×D411yescharacteristic
165C22×D4C2×C4×C711yescharacteristic
166C23×C4—21no—
167C2×M16—22no—
168C22⋊Q8—44no—
169C4×Q8—41no—
170C4×Q8—41no—
171C4×D4—41no—
172C4×D4—41no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×C4×C711yescharacteristic
177C2×C4⋊C4C2×C4×C711yescharacteristic
178C2×C4⋊C4C7×D411yescharacteristic
179C2×C4⋊C4C23×C711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42C2×C4×C711yescharacteristic
186C2×C42C7×D411yescharacteristic
187C2×C42C7×D411yescharacteristic
188C4⋊C8—41no—
189#101—24no—
190C22⋊C8—22no—
191#97—41no—
192#97—41no—
193C4×C8—41no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95C2×C4×C711yescharacteristic
198#95C2×C4×C711yescharacteristic
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—42no—
202C22×C7—42no—
203C22×C7—42no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7—81no—
207C22×C7—81no—
208C22×C7C2×#9911yescharacteristic
209C22×C7#41811yescharacteristic
210C22×C7C2×#10411yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—22no—
214C4×C7—22no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—81no—
220C4×C7—81no—
221C4×C7—81no—
222C24—21no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×D4—41no—
227C2×D4—42no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C7×C22⋊C411yescharacteristic
235C22×C4C7×C22⋊C411yescharacteristic
236C22×C4C7×C22⋊C411yescharacteristic
237C22×C4C7×C22⋊C411yescharacteristic
238C22×C4C22×C4×C711yescharacteristic
239C22×C4C2×C7×D411yescharacteristic
240C22×C4C2×C7×D411yescharacteristic
241M16—24no—
242M16—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—42no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C2×C7—21no—
265C2×C7—21no—
266C2×C7—82no—
267C2×C7#150711yescharacteristic
268C2×C7#212011yescharacteristic
269C2×C7#213311yescharacteristic
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C7×C22⋊C411yescharacteristic
278Q8—44no—
279D4—42no—
280D4—42no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—84no—
305C2×C4C7×#9911yescharacteristic
306C2×C4C7×#9911yescharacteristic
307C2×C4C7×#10412yes—
308C2×C4C7×AGL(1,ℤ/8)12yes—
309C8—44no—
310C8—82no—
311C7#1177811yescharacteristic, direct product factor, Sylow
312C22—21no—
313C22—21no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×C7×#9911yescharacteristic
322C22C7×#41811yescharacteristic
323C22C2×C7×#10411yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2C7×#150711yescharacteristic
339C2C7×#212011yescharacteristic
340C2C7×#213311yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
2. C4×C7×#101C2semidirect
3. C7×#1467C2semidirect
4. C7×#1346C2semidirect
23. #11778C7direct (N × H)
60. C4×#101C2×C7semidirect
61. #1467C2×C7semidirect
62. #1346C2×C7semidirect
311. C7#11778direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25362
C4×C7×#1012
C7×#14672
C7×#13462
C7×#13452
#117787