Finite Group Databasebeta

← C7×#9444 #13186660

Subgroups — C7×#9444 #13186660

Order 1792.

474 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2921C211yesmaximal, characteristic
2C7×#2499C211yesmaximal, characteristic
3C7×#1820C211yesmaximal, characteristic
4C7×#1511C212yesmaximal
5C7×#1284C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C2×C7×#117C2211yescharacteristic
8C7×#433—24no—
9C7×#423—21no—
10C7×#407C2212yes—
11C4×C7×M16—21no—
12C7×#397—21no—
13C7×#397—21no—
14C7×#391C411yescharacteristic, complemented
15C7×#391C411yescharacteristic, complemented
16C7×#387—21no—
17C7×#387—21no—
18C7×#384C411yescharacteristic, complemented
19C7×#382C411yescharacteristic, complemented
20C7×#381C2212yes—
21C4×C7×C22⋊C4C2211yescharacteristic
22C7×#342—21no—
23C7×#342—21no—
24C7×#328—24no—
25#9444C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C23×C4×C7C2311yescharacteristic
28C2×C7×M16—22no—
29C7×C22⋊Q8—22no—
30C7×C22⋊Q8—22no—
31C7×#117—22no—
32C7×#117—22no—
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⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7C2×C411yescharacteristic
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×C42×C7D411yescharacteristic
48C7×C4⋊C8—24no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8D414yes—
51C7×#97—22no—
52C4×C7×C8—22no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—22no—
61C7×#95—22no—
62#2921C2×C711yescharacteristic
63#2499C2×C711yescharacteristic
64#1820C2×C711yescharacteristic
65#1511C2×C712yes—
66#1284C2×C712yes—
67C24×C7C2×D411yescharacteristic
68C2×C7×Q8—22no—
69C2×C7×Q8—41no—
70C2×C7×Q8—41no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—41no—
76C22×C4×C7—41no—
77C22×C4×C7—42no—
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22×C411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D412yes—
85C22×C4×C7#4112yes—
86C7×M16—42no—
87C7×M16—42no—
88C2×C7×C8—24no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—22no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C42×C7—21no—
103C42×C7—21no—
104C42×C7—21no—
105C42×C7—21no—
106C42×C7—41no—
107C42×C7—41no—
108C2×C22⋊Q8C22×C711yescharacteristic
109C2×#117C22×C711yescharacteristic
110#433—24no—
111#423—21no—
112#407C22×C712yes—
113C4×M16—21no—
114#397—21no—
115#397—21no—
116#391C4×C711yescharacteristic, complemented
117#391C4×C711yescharacteristic, complemented
118#387—21no—
119#387—21no—
120#384C4×C711yescharacteristic, complemented
121#382C4×C711yescharacteristic, complemented
122#381C22×C712yes—
123C4×C22⋊C4C22×C711yescharacteristic
124#342—21no—
125#342—21no—
126#328—24no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7C2×C22⋊C411yescharacteristic
132C23×C7#12112yes—
133C7×Q8—82no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—22no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
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—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—82no—
161C2×C4×C7—82no—
162C2×C4×C7—82no—
163C2×C4×C7C4×D412yes—
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12311yescharacteristic
166C7×C8—44no—
167C22×Q8C2×C4×C711yescharacteristic
168C23×C4C23×C711yescharacteristic
169C2×M16—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C2×C4×C711yescharacteristic
178C2×C4⋊C4C2×C4×C711yescharacteristic
179C2×C4⋊C4C2×C4×C711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4C7×D411yescharacteristic
184C2×C22⋊C4C7×D411yescharacteristic
185C2×C42C2×C4×C711yescharacteristic
186C2×C42C2×C4×C711yescharacteristic
187C2×C42C7×D411yescharacteristic
188C2×C42C7×D411yescharacteristic
189C4⋊C8—24no—
190C22⋊C8—22no—
191C22⋊C8C7×D414yes—
192#97—22no—
193C4×C8—22no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—22no—
202#95—22no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—81no—
209C22×C7—81no—
210C22×C7#38611yescharacteristic
211C22×C7#41011yescharacteristic
212C22×C7#41011yescharacteristic
213C4×C7—21no—
214C4×C7—22no—
215C4×C7—21no—
216C4×C7—42no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—81no—
222C4×C7—81no—
223C4×C7—81no—
224C4×C7—162no—
225C24C2×C7×D411yescharacteristic
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4C7×C22⋊C411yescharacteristic
237C22×C4C7×C22⋊C411yescharacteristic
238C22×C4C7×C22⋊C411yescharacteristic
239C22×C4C7×C22⋊C411yescharacteristic
240C22×C4C22×C4×C711yescharacteristic
241C22×C4C2×C7×D411yescharacteristic
242C22×C4C2×C7×D412yes—
243C22×C4C7×#4112yes—
244M16—42no—
245M16—42no—
246C2×C8—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—41no—
265C42—41no—
266C2×C7—21no—
267C2×C7—21no—
268C2×C7—42no—
269C2×C7#188411yescharacteristic
270C2×C7#189911yescharacteristic
271C2×C7#190911yescharacteristic
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×C7×C22⋊C411yescharacteristic
277C23C7×#12112yes—
278Q8—82no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
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—41no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4C4×C7×D412yes—
309C2×C4C7×#12011yescharacteristic
310C2×C4C7×#12311yescharacteristic
311C8—44no—
312C7#944411yescharacteristic, direct product factor, Sylow
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22C7×#38611yescharacteristic
321C22C7×#41011yescharacteristic
322C22C7×#41011yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—162no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C7×#188411yescharacteristic
339C2C7×#189911yescharacteristic
340C2C7×#190911yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
14. C7×#391C4semidirect
15. C7×#391C4semidirect
18. C7×#384C4semidirect
19. C7×#382C4semidirect
25. #9444C7direct (N × H)
116. #391C4×C7semidirect
117. #391C4×C7semidirect
120. #384C4×C7semidirect
121. #382C4×C7semidirect
312. C7#9444direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29212
C7×#24992
C7×#18202
C7×#15112
C7×#12842
#94447