Finite Group Databasebeta

← C7×#11791 #13189007

Subgroups — C7×#11791 #13189007

Order 1792.

572 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2409C211yesmaximal, characteristic
2C4×C7×#100C211yesmaximal, characteristic
3C7×#1465C211yesmaximal, characteristic
4C7×#1354C214yesmaximal, complemented
5C23×C7×Q8C411yescharacteristic
6C2×C7×C22⋊Q8—21no—
7C2×C4×C7×D4C2211yescharacteristic
8C2×C7×#100—21no—
9C7×#407—21no—
10C4×C7×M16—21no—
11C7×#395C2212yes—
12C7×#393C412yes—
13C7×#391—21no—
14C7×#386—21no—
15C7×#385—21no—
16C7×#380C411yescharacteristic
17C7×#343C2212yes—
18C7×#324C2212yes—
19C7×#323—21no—
20#11791C711yesmaximal, characteristic, direct product factor, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—41no—
23C22×C7×Q8—41no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C22×C7×D4D411yescharacteristic
27C23×C4×C7—21no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C2×C7×M16—22no—
31C2×C7×M16—41no—
32C7×C22⋊Q8—44no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4C2311yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C7×#100—24no—
44C7×C22⋊C8—22no—
45C7×C22⋊C8—41no—
46C7×#97—41no—
47C4×C7×C8—41no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95C2×C412yes—
51#2409C2×C711yescharacteristic
52C4×#100C2×C711yescharacteristic
53#1465C2×C711yescharacteristic
54#1354C2×C714yescomplemented
55C24×C7C22⋊C411yescharacteristic
56C24×C7C22⋊C411yescharacteristic
57C2×C7×Q8—24no—
58C2×C7×Q8—42no—
59C2×C7×Q8—44no—
60C2×C7×Q8—42no—
61C2×C7×Q8—48no—
62C2×C7×Q8—44no—
63C2×C7×Q8—44no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—41no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—22no—
75C22×C4×C7—44no—
76C22×C4×C7—42no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22×C411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C7×M16—24no—
85C7×M16—42no—
86C7×M16—82no—
87C2×C7×C8—42no—
88C2×C7×C8—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—44no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—42no—
96C42×C7—21no—
97C42×C7—21no—
98C23×Q8C4×C711yescharacteristic
99C2×C22⋊Q8—21no—
100C2×C4×D4C22×C711yescharacteristic
101C2×#100—21no—
102#407—21no—
103C4×M16—21no—
104#395C22×C712yes—
105#393C4×C712yes—
106#391—21no—
107#386—21no—
108#385—21no—
109#380C4×C711yescharacteristic
110#343C22×C712yes—
111#324C22×C712yes—
112#323—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7#10412yes—
123C23×C7C2×C22⋊C411yescharacteristic
124C7×Q8—44no—
125C7×Q8—44no—
126C7×Q8—44no—
127C7×Q8—44no—
128C7×D4—44no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—22no—
132C2×C4×C7—24no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—48no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—81no—
149C2×C4×C7#9911yescharacteristic
150C2×C4×C7#9911yescharacteristic
151C2×C4×C7#13712yes—
152C7×C8—44no—
153C7×C8—82no—
154C22×Q8—21no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—44no—
158C22×Q8C2×C4×C712yes—
159C22×D4C7×D411yescharacteristic
160C23×C4—21no—
161C23×C4C2×C4×C711yescharacteristic
162C23×C4C7×D411yescharacteristic
163C2×M16—22no—
164C2×M16—41no—
165C22⋊Q8—44no—
166C4×D4—41no—
167C4×D4—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C23×C711yescharacteristic
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4C2×C4×C711yescharacteristic
174C2×C22⋊C4C7×D411yescharacteristic
175C2×C42C7×D411yescharacteristic
176#100—24no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179#97—41no—
180C4×C8—41no—
181#95—22no—
182#95—22no—
183#95C2×C4×C712yes—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—22no—
187C22×C7—42no—
188C22×C7—42no—
189C22×C7—42no—
190C22×C7—41no—
191C22×C7—41no—
192C22×C7—41no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—81no—
196C22×C7C2×#9911yescharacteristic
197C22×C7#41911yescharacteristic
198C22×C7C2×#10411yescharacteristic
199C4×C7—21no—
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—44no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—82no—
206C4×C7AΣL(1,𝔽16)12yes—
207C24C7×C22⋊C411yescharacteristic
208C24C7×C22⋊C411yescharacteristic
209C2×Q8—24no—
210C2×Q8—44no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—48no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C7×C22⋊C411yescharacteristic
232C22×C4C7×C22⋊C411yescharacteristic
233C22×C4C22×C4×C711yescharacteristic
234C22×C4C2×C7×D411yescharacteristic
235C22×C4C2×C7×D411yescharacteristic
236M16—24no—
237M16—42no—
238M16—82no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C42—21no—
249C42—21no—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7—42no—
253C2×C7—81no—
254C2×C7#150811yescharacteristic
255C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
256C2×C7#213511yescharacteristic
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C7×#10412yes—
267C23C2×C7×C22⋊C411yescharacteristic
268Q8—44no—
269Q8—44no—
270Q8—44no—
271Q8—44no—
272D4—44no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—48no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4C7×#9911yescharacteristic
294C2×C4C7×#9911yescharacteristic
295C2×C4C7×#13712yes—
296C8—44no—
297C8—82no—
298C7#1179111yescharacteristic, direct product factor, Sylow
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×C7×#9911yescharacteristic
312C22C7×#41911yescharacteristic
313C22C2×C7×#10411yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4C7×AΣL(1,𝔽16)12yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2C7×#150811yescharacteristic
327C2C2×C7×AΣL(1,𝔽16)11yescharacteristic
328C2C7×#213511yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ 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
4. C7×#1354C2semidirect
20. #11791C7direct (N × H)
54. #1354C2×C7semidirect
298. C7#11791direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24092
C4×C7×#1002
C7×#14652
C7×#13542
#117917