Finite Group Databasebeta

← C7×#12115 #13189331

Subgroups — C7×#12115 #13189331

Order 1792.

422 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3195C211yesmaximal, characteristic, complemented
2C7×#2191C212yesmaximal, complemented
3C7×#2010C211yesmaximal, characteristic
4C7×#1340C212yesmaximal, complemented
5C7×#1320C211yesmaximal, characteristic, complemented
6C7×#575C2211yescharacteristic
7C7×#525—21no—
8C2×C7×C22⋊Q8—21no—
9C7×#510—22no—
10C7×#483—22no—
11C7×#470—21no—
12C7×#470C2212yes—
13C7×#451—22no—
14C7×#421—21no—
15C7×#407C2211yescharacteristic
16C7×#360—22no—
17C7×#349C2212yes—
18C7×H(3,ℤ/4)C2211yescharacteristic
19#12115C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—21no—
21C22×C7×Q8—21no—
22C23×C4×C7D411yescharacteristic
23C7×#137—41no—
24C7×AGL(1,ℤ/8)—41no—
25C7×#135—22no—
26C2×C7×Q16—21no—
27C2×C7×Q16D412yescomplemented
28C2×C7×M16D411yescharacteristic
29C2×C7×M16C2311yescharacteristic
30C7×#123—41no—
31C7×C22⋊Q8—22no—
32C7×C22⋊Q8—22no—
33C7×C22⋊Q8—41no—
34C7×C22⋊Q8—41no—
35C7×C22⋊Q8—41no—
36C7×#121—41no—
37C7×#120—41no—
38C4×C7×D4—42no—
39C7×#117—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—41no—
42C2×C7×C22⋊C4—41no—
43C7×Q32—42no—
44C7×SD32—42no—
45C7×M32—22no—
46C7×#106—21no—
47C7×#106D412yescomplemented
48C7×#104—42no—
49C7×#103—22no—
50C7×#103—22no—
51C7×C22⋊C8—22no—
52#3195C2×C711yescharacteristic, complemented
53#2191C2×C712yescomplemented
54#2010C2×C711yescharacteristic
55#1340C2×C712yescomplemented
56#1320C2×C711yescharacteristic, complemented
57C24×C7D811yescharacteristic
58C7×#41—41no—
59C7×#41—41no—
60C7×#41—41no—
61C7×#41—41no—
62C2×C7×Q8—21no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8—81no—
66C2×C7×Q8—81no—
67C2×C7×D4—21no—
68C2×C7×D4—42no—
69C2×C7×D4—41no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7—41no—
77C22×C4×C7D811yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C7×Q16—22no—
80C7×Q16—42no—
81C7×Q16—81no—
82C7×SD16—42no—
83C7×D8—22no—
84C7×M16—41no—
85C7×M16—41no—
86C2×C7×C8—21no—
87C2×C7×C8C2×D412yes—
88C7×C4⋊C4—22no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—42no—
93C7×C4⋊C4—42no—
94C7×C22⋊C4—42no—
95C7×C22⋊C4—42no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—42no—
98C7×C22⋊C4—41no—
99C42×C7—22no—
100C7×C16—42no—
101#575C22×C711yescharacteristic
102#525—21no—
103C2×C22⋊Q8—21no—
104#510—22no—
105#483—22no—
106#470—21no—
107#470C22×C712yes—
108#451—22no—
109#421—21no—
110#407C22×C711yescharacteristic
111#360—22no—
112#349C22×C712yes—
113H(3,ℤ/4)C22×C711yescharacteristic
114C23×C7—21no—
115C23×C7—41no—
116C23×C7—42no—
117C23×C7—41no—
118C23×C7C2×D811yescharacteristic
119C7×Q8—41no—
120C7×Q8—42no—
121C7×Q8—41no—
122C7×Q8—42no—
123C7×Q8—82no—
124C7×D4—41no—
125C7×D4—41no—
126C7×D4—81no—
127C7×D4—84no—
128C7×D4—81no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—42no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7—82no—
142C2×C4×C7—82no—
143C2×C4×C7—81no—
144C2×C4×C7#12011yescharacteristic
145C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
146C7×C8—22no—
147C7×C8—41no—
148C2×#41—21no—
149C22×Q8—21no—
150C23×C4C7×D411yescharacteristic
151#137—41no—
152AGL(1,ℤ/8)—41no—
153#135—22no—
154C2×Q16—21no—
155C2×Q16C7×D412yescomplemented
156C2×M16C7×D411yescharacteristic
157C2×M16C23×C711yescharacteristic
158#123—41no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C22⋊Q8—41no—
163C22⋊Q8—41no—
164#121—41no—
165#120—41no—
166C4×D4—42no—
167#117—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—41no—
171Q32—42no—
172SD32—42no—
173M32—22no—
174#106—21no—
175#106C7×D412yescomplemented
176#104—42no—
177#103—22no—
178#103—22no—
179C22⋊C8—22no—
180C22×C7—22no—
181C22×C7—21no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—42no—
185C22×C7—41no—
186C22×C7—81no—
187C22×C7—82no—
188C22×C7—81no—
189C22×C7#44711yescharacteristic
190C4×C7—21no—
191C4×C7—41no—
192C4×C7—82no—
193C4×C7—82no—
194C4×C7—81no—
195C4×C7—82no—
196C4×C7#45311yescharacteristic
197C4×C7B(4,𝔽2)11yescharacteristic
198C24C7×D811yescharacteristic
199#41—41no—
200#41—41no—
201#41—41no—
202#41—41no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—81no—
207C2×Q8—81no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4C7×D811yescharacteristic
219C22×C4C2×C7×D411yescharacteristic
220Q16—22no—
221Q16—42no—
222Q16—81no—
223SD16—42no—
224D8—22no—
225M16—41no—
226M16—41no—
227C2×C8—21no—
228C2×C8C2×C7×D412yes—
229C4⋊C4—22no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—41no—
240C42—22no—
241C16—42no—
242C2×C7—21no—
243C2×C7—41no—
244C2×C7—42no—
245C2×C7—161no—
246C2×C7#159511yescharacteristic
247C23—21no—
248C23—41no—
249C23—42no—
250C23—41no—
251C23C2×C7×D811yescharacteristic
252Q8—41no—
253Q8—42no—
254Q8—41no—
255Q8—42no—
256Q8—82no—
257D4—41no—
258D4—41no—
259D4—81no—
260D4—84no—
261D4—81no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4C7×#12011yescharacteristic
278C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
279C8—22no—
280C8—41no—
281C7#1211511yescharacteristic, direct product factor, Sylow
282C22—22no—
283C22—21no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—41no—
288C22—81no—
289C22—82no—
290C22—81no—
291C22C7×#44711yescharacteristic
292C4—21no—
293C4—41no—
294C4—82no—
295C4—82no—
296C4—81no—
297C4—82no—
298C4C7×#45311yescharacteristic
299C4C7×B(4,𝔽2)11yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—161no—
304C2C7×#159511yescharacteristic
subgroupG/H
center, socleC2×C7#1595
commutator, Frattini, cosocleC2×M16C23×C7
prime radicalC7×#2010C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C2×C7×M16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3195C2semidirect
2. C7×#2191C2semidirect
4. C7×#1340C2semidirect
5. C7×#1320C2semidirect
19. #12115C7direct (N × H)
27. C2×C7×Q16D4semidirect
47. C7×#106D4semidirect
52. #3195C2×C7semidirect
53. #2191C2×C7semidirect
55. #1340C2×C7semidirect
56. #1320C2×C7semidirect
155. C2×Q16C7×D4semidirect
175. #106C7×D4semidirect
281. C7#12115direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31952
C7×#21912
C7×#20102
C7×#13402
C7×#13202
#121157