Finite Group Databasebeta

← C7×#9201 #13186417

Subgroups — C7×#9201 #13186417

Order 1792.

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

normal subgroups only

This group has 612 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
1C7×#2626C211yesmaximal, characteristic
2C7×#2312C211yesmaximal, characteristic
3C7×#1520C212yesmaximal, complemented
4C2×C7×#323C211yesmaximal, characteristic
5C7×#1289C212yesmaximal, complemented
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C7×C22⋊C4C2211yescharacteristic
8C2×C7×C22⋊C8C2212yes—
9C7×#399C412yescomplemented
10C7×#396—22no—
11C7×#393C412yescomplemented
12C7×#388—24no—
13C7×#385—21no—
14C7×#380C2211yescharacteristic
15C7×#380C2212yes—
16C7×#379—21no—
17C7×#379—21no—
18C4×C7×C22⋊C4—21no—
19C7×#328—24no—
20C7×#323—22no—
21#9201C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C22×C7×Q8C2×C411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×C7×C8—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C22⋊C4—22no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4C2×C412yes—
44C2×C7×C22⋊C4D412yes—
45C2×C42×C7—22no—
46C7×C22⋊C8—22no—
47C7×C22⋊C8—22no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95C2×C412yes—
51C7×#95D414yes—
52#2626C2×C711yescharacteristic
53#2312C2×C711yescharacteristic
54#1520C2×C712yescomplemented
55C2×#323C2×C711yescharacteristic
56#1289C2×C712yescomplemented
57C24×C7—21no—
58C24×C7—21no—
59C24×C7—22no—
60C24×C7—22no—
61C24×C7—81no—
62C24×C7—81no—
63C24×C7C22⋊C412yes—
64C24×C7C2×D411yescharacteristic
65C2×C7×Q8—22no—
66C2×C7×Q8—81no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—22no—
71C22×C4×C7—21no—
72C22×C4×C7—24no—
73C22×C4×C7—81no—
74C22×C4×C7—81no—
75C22×C4×C7C22⋊C412yes—
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D412yes—
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7#4112yes—
80C2×C7×C8—42no—
81C2×C7×C8—82no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—82no—
85C7×C4⋊C4—81no—
86C7×C4⋊C4—82no—
87C7×C22⋊C4—24no—
88C7×C22⋊C4—24no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—42no—
93C7×C22⋊C4—42no—
94C7×C22⋊C4—82no—
95C7×C22⋊C4—82no—
96C7×C22⋊C4—82no—
97C7×C22⋊C4—82no—
98C42×C7—82no—
99C2×C22⋊Q8C22×C711yescharacteristic
100C22×C22⋊C4C22×C711yescharacteristic
101C2×C22⋊C8C22×C712yes—
102#399C4×C712yescomplemented
103#396—22no—
104#393C4×C712yescomplemented
105#388—24no—
106#385—21no—
107#380C22×C711yescharacteristic
108#380C22×C712yes—
109#379—21no—
110#379—21no—
111C4×C22⋊C4—21no—
112#328—24no—
113#323—22no—
114C23×C7—22no—
115C23×C7—22no—
116C23×C7—22no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7—42no—
120C23×C7—42no—
121C23×C7—42no—
122C23×C7—42no—
123C23×C7—42no—
124C23×C7—42no—
125C23×C7—81no—
126C23×C7—81no—
127C23×C7—84no—
128C23×C7—84no—
129C23×C7C2×C22⋊C411yescharacteristic
130C23×C7#12011yescharacteristic
131C23×C7#12311yescharacteristic
132C23×C7#13512yes—
133C23×C7#13712yes—
134C7×Q8—82no—
135C2×C4×C7—22no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—41no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—44no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—44no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—44no—
152C2×C4×C7—81no—
153C2×C4×C7C4×D412yes—
154C2×C4×C7#12112yes—
155C7×C8—82no—
156C25C7×D411yescharacteristic
157C22×Q8C2×C4×C711yescharacteristic
158C23×C4C7×D411yescharacteristic
159C23×C4C23×C711yescharacteristic
160C22×C8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4C2×C4×C711yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4C2×C4×C712yes—
178C2×C22⋊C4C7×D412yes—
179C2×C42—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182#95—22no—
183#95—22no—
184#95C2×C4×C712yes—
185#95C7×D414yes—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—42no—
193C22×C7—42no—
194C22×C7—44no—
195C22×C7—42no—
196C22×C7—42no—
197C22×C7—42no—
198C22×C7—82no—
199C22×C7—84no—
200C22×C7—84no—
201C22×C7#38611yescharacteristic
202C22×C7#41611yescharacteristic
203C22×C7#41911yescharacteristic
204C22×C7#45212yes—
205C22×C7#48512yes—
206C4×C7—22no—
207C4×C7—22no—
208C4×C7—42no—
209C4×C7—82no—
210C4×C7—84no—
211C4×C7—81no—
212C4×C7—81no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—81no—
218C24—81no—
219C24C7×C22⋊C412yes—
220C24C2×C7×D411yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4C7×C22⋊C412yes—
232C22×C4C22×C4×C711yescharacteristic
233C22×C4C2×C7×D412yes—
234C22×C4C2×C7×D411yescharacteristic
235C22×C4C7×#4112yes—
236C2×C8—42no—
237C2×C8—82no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C4⋊C4—81no—
242C4⋊C4—82no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—82no—
255C2×C7—21no—
256C2×C7—22no—
257C2×C7—21no—
258C2×C7—22no—
259C2×C7—82no—
260C2×C7#187311yescharacteristic
261C2×C7#191111yescharacteristic
262C2×C7#191311yescharacteristic
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—84no—
277C23—84no—
278C23C2×C7×C22⋊C411yescharacteristic
279C23C7×#12011yescharacteristic
280C23C7×#12311yescharacteristic
281C23C7×#13512yes—
282C23C7×#13712yes—
283Q8—82no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—81no—
302C2×C4C4×C7×D412yes—
303C2×C4C7×#12112yes—
304C8—82no—
305C7#920111yescharacteristic, direct product factor, Sylow
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22C7×#38611yescharacteristic
322C22C7×#41611yescharacteristic
323C22C7×#41911yescharacteristic
324C22C7×#45212yes—
325C22C7×#48512yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—82no—
330C4—84no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2C7×#187311yescharacteristic
339C2C7×#191111yescharacteristic
340C2C7×#191311yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7D4
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
3. C7×#1520C2semidirect
5. C7×#1289C2semidirect
9. C7×#399C4semidirect
11. C7×#393C4semidirect
21. #9201C7direct (N × H)
54. #1520C2×C7semidirect
56. #1289C2×C7semidirect
102. #399C4×C7semidirect
104. #393C4×C7semidirect
305. C7#9201direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26262
C7×#23122
C7×#15202
C2×C7×#3232
C7×#12892
#92017