Finite Group Databasebeta

← C7×#12180 #13189396

Subgroups — C7×#12180 #13189396

Order 1792.

472 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2551C211yesmaximal, characteristic, complemented
2C7×#2039C211yesmaximal, characteristic
3C7×#1604C212yesmaximal
4C7×#1354C212yesmaximal, complemented
5C7×#1325C211yesmaximal, characteristic, complemented
6C7×#537C2211yescharacteristic
7C2×C7×C4⋊Q8—22no—
8C7×#448—22no—
9C2×C7×#104—21no—
10C7×#417—21no—
11C7×#395—21no—
12C7×#395C2212yes—
13C7×#389—22no—
14C7×#384C2211yescharacteristic
15C4×C7×C4⋊C4—21no—
16C7×#343C2211yescharacteristic
17C7×#327—22no—
18C7×#324C2212yes—
19#12180C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—21no—
21C22×C7×Q8—21no—
22C22×C7×Q8D412yes—
23C2×C7×SD16—42no—
24C2×C7×M16—21no—
25C7×C4⋊Q8—48no—
26C7×C22⋊Q8—41no—
27C7×#121—22no—
28C7×#121—41no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C4⋊C4C2311yescharacteristic
36C2×C42×C7—22no—
37C2×C42×C7D411yescharacteristic
38C7×#104—42no—
39C7×#103—42no—
40C7×#103—41no—
41C7×#102—41no—
42C7×C22⋊C8—22no—
43C7×#95—21no—
44C7×#95—22no—
45C7×#95D412yes—
46#2551C2×C711yescharacteristic, complemented
47#2039C2×C711yescharacteristic
48#1604C2×C712yes—
49#1354C2×C712yescomplemented
50#1325C2×C711yescharacteristic, complemented
51C7×#41—41no—
52C7×#41—41no—
53C7×#41—41no—
54C7×#41—41no—
55C2×C7×Q8—21no—
56C2×C7×Q8—44no—
57C2×C7×Q8—44no—
58C2×C7×Q8—42no—
59C2×C7×Q8—81no—
60C2×C7×D4—21no—
61C2×C7×D4—22no—
62C2×C7×D4—41no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—41no—
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D412yes—
70C7×SD16—84no—
71C7×M16—41no—
72C7×M16—41no—
73C2×C7×C8—42no—
74C2×C7×C8—41no—
75C7×C4⋊C4—24no—
76C7×C4⋊C4—24no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—41no—
79C7×C4⋊C4—41no—
80C7×C4⋊C4—44no—
81C7×C4⋊C4—44no—
82C7×C4⋊C4—44no—
83C7×C22⋊C4—42no—
84C42×C7—21no—
85C42×C7—44no—
86C42×C7SD1612yescomplemented
87#537C22×C711yescharacteristic
88C2×C4⋊Q8—22no—
89#448—22no—
90C2×#104—21no—
91#417—21no—
92#395—21no—
93#395C22×C712yes—
94#389—22no—
95#384C22×C711yescharacteristic
96C4×C4⋊C4—21no—
97#343C22×C711yescharacteristic
98#327—22no—
99#324C22×C712yes—
100C23×C7—41no—
101C23×C7#12011yescharacteristic
102C7×Q8—41no—
103C7×Q8—41no—
104C7×Q8—44no—
105C7×Q8—82no—
106C7×Q8—84no—
107C7×D4—41no—
108C7×D4—41no—
109C7×D4—82no—
110C7×D4—81no—
111C7×D4—81no—
112C2×C4×C7—22no—
113C2×C4×C7—22no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—24no—
118C2×C4×C7—24no—
119C2×C4×C7—22no—
120C2×C4×C7—42no—
121C2×C4×C7—42no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—41no—
127C2×C4×C7—41no—
128C2×C4×C7—81no—
129C2×C4×C7—81no—
130C2×C4×C7C2×SD1611yescharacteristic
131C2×C4×C7#13711yescharacteristic
132C7×C8—81no—
133C7×C8—82no—
134C2×#41—21no—
135C22×Q8—21no—
136C22×Q8C7×D412yes—
137C2×SD16—42no—
138C2×M16—21no—
139C4⋊Q8—48no—
140C22⋊Q8—41no—
141#121—22no—
142#121—41no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C7×D411yescharacteristic
149C2×C4⋊C4C23×C711yescharacteristic
150C2×C42—22no—
151C2×C42C7×D411yescharacteristic
152#104—42no—
153#103—42no—
154#103—41no—
155#102—41no—
156C22⋊C8—22no—
157#95—21no—
158#95—22no—
159#95C7×D412yes—
160C22×C7—21no—
161C22×C7—21no—
162C22×C7—81no—
163C22×C7—81no—
164C22×C7—81no—
165C22×C7#44811yescharacteristic
166C22×C7#45411yescharacteristic
167C22×C7B(4,𝔽2)11yescharacteristic
168C4×C7—24no—
169C4×C7—21no—
170C4×C7—21no—
171C4×C7—44no—
172C4×C7—42no—
173C4×C7—44no—
174C4×C7—81no—
175C4×C7—81no—
176#41—41no—
177#41—41no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—42no—
184C2×Q8—81no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4—41no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—41no—
193C22×C4C2×C7×D411yescharacteristic
194C22×C4C2×C7×D412yes—
195SD16—84no—
196M16—41no—
197M16—41no—
198C2×C8—42no—
199C2×C8—41no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C22⋊C4—42no—
209C42—21no—
210C42—44no—
211C42C7×SD1612yescomplemented
212C2×C7—21no—
213C2×C7—21no—
214C2×C7—161no—
215C2×C7#159911yescharacteristic
216C2×C7#219611yescharacteristic
217C2×C7#220511yescharacteristic
218C23—41no—
219C23C7×#12011yescharacteristic
220Q8—41no—
221Q8—41no—
222Q8—44no—
223Q8—82no—
224Q8—84no—
225D4—41no—
226D4—41no—
227D4—82no—
228D4—81no—
229D4—81no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4C2×C7×SD1611yescharacteristic
249C2×C4C7×#13711yescharacteristic
250C8—81no—
251C8—82no—
252C7#1218011yescharacteristic, direct product factor, Sylow
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22C7×#44811yescharacteristic
259C22C7×#45411yescharacteristic
260C22C7×B(4,𝔽2)11yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—44no—
265C4—42no—
266C4—44no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2C7×#159911yescharacteristic
273C2C7×#219611yescharacteristic
274C2C7×#220511yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC7×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2551C2semidirect
4. C7×#1354C2semidirect
5. C7×#1325C2semidirect
19. #12180C7direct (N × H)
46. #2551C2×C7semidirect
49. #1354C2×C7semidirect
50. #1325C2×C7semidirect
86. C42×C7SD16semidirect
211. C42C7×SD16semidirect
252. C7#12180direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25512
C7×#20392
C7×#16042
C7×#13542
C7×#13252
#121807