Finite Group Databasebeta

← C7×#12181 #13189397

Subgroups — C7×#12181 #13189397

Order 1792.

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

normal subgroups only

This group has 372 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×#2571C211yesmaximal, characteristic, complemented
2C7×#2039C211yesmaximal, characteristic
3C7×#1616C212yesmaximal
4C7×#1353C212yesmaximal, complemented
5C7×#1325C211yesmaximal, characteristic, complemented
6C7×#537C2211yescharacteristic
7C7×#481—22no—
8C2×C7×#104—21no—
9C7×#417—21no—
10C7×#400—21no—
11C7×#400C2212yes—
12C7×#389—22no—
13C7×#384C2211yescharacteristic
14C7×#382—22no—
15C4×C7×C4⋊C4—21no—
16C7×#343C2211yescharacteristic
17C7×#327—22no—
18C7×#324C2212yes—
19#12181C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—21no—
21C2×C7×M16—21no—
22C7×C22⋊Q8—41no—
23C7×#121—22no—
24C7×#121—41no—
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4—21no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4D412yes—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C4⋊C4C2311yescharacteristic
32C2×C42×C7—22no—
33C2×C42×C7D411yescharacteristic
34C7×#106—42no—
35C7×#104—42no—
36C7×#103—41no—
37C7×#102—42no—
38C7×#102—41no—
39C7×C22⋊C8—22no—
40C7×#95—21no—
41C7×#95—22no—
42C7×#95—22no—
43C7×#95—22no—
44C7×#95D412yes—
45#2571C2×C711yescharacteristic, complemented
46#2039C2×C711yescharacteristic
47#1616C2×C712yes—
48#1353C2×C712yescomplemented
49#1325C2×C711yescharacteristic, complemented
50C7×#41—41no—
51C7×#41—41no—
52C7×#41—41no—
53C7×#41—41no—
54C2×C7×Q8—21no—
55C2×C7×D4—21no—
56C2×C7×D4—22no—
57C2×C7×D4—41no—
58C22×C4×C7—21no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—21no—
62C22×C4×C7—41no—
63C22×C4×C7C2×D411yescharacteristic
64C22×C4×C7C2×D412yes—
65C7×M16—41no—
66C7×M16—41no—
67C2×C7×C8—42no—
68C2×C7×C8—41no—
69C7×C4⋊C4—22no—
70C7×C4⋊C4—41no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—42no—
73C7×C4⋊C4—42no—
74C7×C4⋊C4—44no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—81no—
77C7×C22⋊C4—42no—
78C42×C7—21no—
79C42×C7—44no—
80C42×C7SD1612yes—
81#537C22×C711yescharacteristic
82#481—22no—
83C2×#104—21no—
84#417—21no—
85#400—21no—
86#400C22×C712yes—
87#389—22no—
88#384C22×C711yescharacteristic
89#382—22no—
90C4×C4⋊C4—21no—
91#343C22×C711yescharacteristic
92#327—22no—
93#324C22×C712yes—
94C23×C7—41no—
95C23×C7#12011yescharacteristic
96C7×Q8—41no—
97C7×Q8—41no—
98C7×D4—41no—
99C7×D4—41no—
100C7×D4—82no—
101C7×D4—81no—
102C7×D4—81no—
103C2×C4×C7—22no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—22no—
109C2×C4×C7—42no—
110C2×C4×C7—42no—
111C2×C4×C7—42no—
112C2×C4×C7—42no—
113C2×C4×C7—42no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—41no—
117C2×C4×C7—41no—
118C2×C4×C7—42no—
119C2×C4×C7—81no—
120C2×C4×C7—81no—
121C2×C4×C7C2×SD1611yescharacteristic
122C2×C4×C7#13711yescharacteristic
123C7×C8—81no—
124C7×C8—82no—
125C2×#41—21no—
126C2×M16—21no—
127C22⋊Q8—41no—
128#121—22no—
129#121—41no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C7×D411yescharacteristic
135C2×C4⋊C4C7×D412yes—
136C2×C4⋊C4C23×C711yescharacteristic
137C2×C42—22no—
138C2×C42C7×D411yescharacteristic
139#106—42no—
140#104—42no—
141#103—41no—
142#102—41no—
143#102—42no—
144C22⋊C8—22no—
145#95—21no—
146#95—22no—
147#95—22no—
148#95—22no—
149#95C7×D412yes—
150C22×C7—21no—
151C22×C7—21no—
152C22×C7—81no—
153C22×C7—81no—
154C22×C7—81no—
155C22×C7#44811yescharacteristic
156C22×C7#45411yescharacteristic
157C22×C7B(4,𝔽2)11yescharacteristic
158C4×C7—24no—
159C4×C7—21no—
160C4×C7—21no—
161C4×C7—42no—
162C4×C7—82no—
163C4×C7—81no—
164C4×C7—81no—
165C4×C7—82no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×D4—21no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4C2×C7×D411yescharacteristic
180C22×C4C2×C7×D412yes—
181M16—41no—
182M16—41no—
183C2×C8—42no—
184C2×C8—41no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—42no—
192C4⋊C4—81no—
193C22⋊C4—42no—
194C42—21no—
195C42—44no—
196C42C7×SD1612yes—
197C2×C7—21no—
198C2×C7—21no—
199C2×C7—161no—
200C2×C7#159911yescharacteristic
201C2×C7#219811yescharacteristic
202C2×C7#220411yescharacteristic
203C23—41no—
204C23C7×#12011yescharacteristic
205Q8—41no—
206Q8—41no—
207D4—41no—
208D4—41no—
209D4—82no—
210D4—81no—
211D4—81no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4C2×C7×SD1611yescharacteristic
231C2×C4C7×#13711yescharacteristic
232C8—81no—
233C8—82no—
234C7#1218111yescharacteristic, direct product factor, Sylow
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22C7×#44811yescharacteristic
241C22C7×#45411yescharacteristic
242C22C7×B(4,𝔽2)11yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—81no—
250C4—82no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2C7×#159911yescharacteristic
255C2C7×#219811yescharacteristic
256C2C7×#220411yescharacteristic
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×#2571C2semidirect
4. C7×#1353C2semidirect
5. C7×#1325C2semidirect
19. #12181C7direct (N × H)
45. #2571C2×C7semidirect
48. #1353C2×C7semidirect
49. #1325C2×C7semidirect
234. C7#12181direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25712
C7×#20392
C7×#16162
C7×#13532
C7×#13252
#121817