Finite Group Databasebeta

← C7×#18728 #13195944

Subgroups — C7×#18728 #13195944

Order 1792.

920 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3548C211yesmaximal, characteristic
2C7×#3227C212yesmaximal, complemented
3C7×#3139C211yesmaximal, characteristic, complemented
4C7×#1635C218yesmaximal, complemented
5C7×#1544C211yesmaximal, characteristic, complemented
6C7×#1484C212yesmaximal, complemented
7C7×#562—22no—
8C7×#554—22no—
9C7×#550—22no—
10C7×#547C2214yes—
11C7×#540—22no—
12C7×#537—24no—
13C7×#533C2211yescharacteristic
14C7×#532C2212yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C7×#498—22no—
17C7×#491—22no—
18C7×#487C2214yes—
19C7×#486—22no—
20C7×#481—24no—
21C7×#476C2218yes—
22C7×#431C2212yes—
23C7×#419—22no—
24C4×C7×M16C2211yescharacteristic
25C7×#332C2214yes—
26C7×D4⋊C8C2218yes—
27#18728C711yesmaximal, characteristic, direct product factor, Sylow
28C2×C7×#41—24no—
29C2×C7×#41—22no—
30C22×C7×Q8—21no—
31C2×C7×M16—21no—
32C7×C4⋊Q8—22no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8D412yes—
35C7×C4⋊Q8C2312yes—
36C7×C4⋊Q8C2312yes—
37C7×#127—22no—
38C7×#127D412yes—
39C7×#125—22no—
40C7×#125D412yes—
41C7×#124—22no—
42C7×#123—28no—
43C7×C22⋊Q8—28no—
44C7×C22⋊Q8—22no—
45C7×C22⋊Q8—24no—
46C7×#121—28no—
47C7×#121—28no—
48C7×#121—24no—
49C7×#121—22no—
50C7×#121D414yes—
51C4×C7×Q8—22no—
52C4×C7×D4—24no—
53C4×C7×D4—22no—
54C4×C7×D4C2314yes—
55C7×#117D412yes—
56C2×C7×C4⋊C4—24no—
57C2×C7×C4⋊C4—22no—
58C2×C42×C7C2311yescharacteristic
59C7×#106—28no—
60C7×C4⋊C8C2314yes—
61C7×#103—28no—
62C7×#102—28no—
63C7×C22⋊C8—22no—
64C7×#97—21no—
65C4×C7×C8—22no—
66C4×C7×C8C2312yes—
67#3548C2×C711yescharacteristic
68#3227C2×C712yescomplemented
69#3139C2×C711yescharacteristic, complemented
70#1635C2×C718yescomplemented
71#1544C2×C711yescharacteristic, complemented
72#1484C2×C712yescomplemented
73C7×#41—84no—
74C7×#41—88no—
75C2×C7×Q8—22no—
76C2×C7×Q8—22no—
77C2×C7×Q8—22no—
78C2×C7×Q8—44no—
79C2×C7×Q8—41no—
80C2×C7×D4—28no—
81C2×C7×D4—24no—
82C2×C7×D4—22no—
83C2×C7×D4—22no—
84C2×C7×D4C2×D414yes—
85C22×C4×C7—28no—
86C22×C4×C7—24no—
87C22×C4×C7—41no—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C7×M16—42no—
91C2×C7×C8—24no—
92C2×C7×C8—22no—
93C7×C4⋊C4—28no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—24no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—24no—
98C7×C4⋊C4—42no—
99C7×C4⋊C4C2×D414yes—
100C7×C4⋊C4C2×D414yes—
101C7×C22⋊C4—28no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—24no—
104C7×C22⋊C4—24no—
105C42×C7—22no—
106C42×C7C2×D412yes—
107C42×C7C2×D411yescharacteristic
108C42×C7C2411yescharacteristic
109#562—22no—
110#554—22no—
111#550—22no—
112#547C22×C714yes—
113#540—22no—
114#537—24no—
115#533C22×C711yescharacteristic
116#532C22×C712yes—
117C2×C4⋊Q8C22×C711yescharacteristic
118#498—22no—
119#491—22no—
120#487C22×C714yes—
121#486—22no—
122#481—24no—
123#476C22×C718yes—
124#431C22×C712yes—
125#419—22no—
126C4×M16C22×C711yescharacteristic
127#332C22×C714yes—
128D4⋊C8C22×C718yes—
129C23×C7—24no—
130C23×C7#12011yescharacteristic
131C7×Q8—44no—
132C7×Q8—44no—
133C7×Q8—82no—
134C7×Q8—82no—
135C7×D4—28no—
136C7×D4—44no—
137C7×D4—82no—
138C7×D4—84no—
139C7×D4—88no—
140C2×C4×C7—24no—
141C2×C4×C7—22no—
142C2×C4×C7—24no—
143C2×C4×C7—24no—
144C2×C4×C7—48no—
145C2×C4×C7—42no—
146C2×C4×C7—88no—
147C2×C4×C7—82no—
148C2×C4×C7#12012yes—
149C2×C4×C7#12011yescharacteristic
150C2×C4×C7C22×D411yescharacteristic
151C2×C4×C7C22×D412yes—
152C7×C8—42no—
153C7×C8—44no—
154C2×#41—24no—
155C2×#41—22no—
156C22×Q8—21no—
157C2×M16—21no—
158C4⋊Q8—22no—
159C4⋊Q8—21no—
160C4⋊Q8C7×D412yes—
161C4⋊Q8C23×C712yes—
162C4⋊Q8C23×C712yes—
163#127—22no—
164#127C7×D412yes—
165#125—22no—
166#125C7×D412yes—
167#124—22no—
168#123—28no—
169C22⋊Q8—28no—
170C22⋊Q8—22no—
171C22⋊Q8—24no—
172#121—28no—
173#121—28no—
174#121—24no—
175#121—22no—
176#121C7×D414yes—
177C4×Q8—22no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4C23×C714yes—
181#117C7×D412yes—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C42C23×C711yescharacteristic
185#106—28no—
186C4⋊C8C23×C714yes—
187#103—28no—
188#102—28no—
189C22⋊C8—22no—
190#97—21no—
191C4×C8—22no—
192C4×C8C23×C712yes—
193C22×C7—22no—
194C22×C7—21no—
195C22×C7—44no—
196C22×C7—44no—
197C22×C7—44no—
198C22×C7C2×#12011yescharacteristic
199C4×C7—22no—
200C4×C7—21no—
201C4×C7—44no—
202C4×C7—44no—
203C4×C7—82no—
204C4×C7#45412yes—
205C4×C7#57714yes—
206#41—84no—
207#41—88no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×D4—28no—
214C2×D4—24no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4C2×C7×D414yes—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4—41no—
221C22×C4C2×C7×D412yes—
222C22×C4C2×C7×D411yescharacteristic
223M16—42no—
224C2×C8—24no—
225C2×C8—22no—
226C4⋊C4—28no—
227C4⋊C4—22no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4C2×C7×D414yes—
233C4⋊C4C2×C7×D414yes—
234C22⋊C4—28no—
235C22⋊C4—22no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C42—22no—
239C42C2×C7×D412yes—
240C42C2×C7×D411yescharacteristic
241C42C24×C711yescharacteristic
242C2×C7—41no—
243C2×C7—84no—
244C2×C7#300912yes—
245C2×C7C2×#45411yescharacteristic
246C23—24no—
247C23C7×#12011yescharacteristic
248Q8—44no—
249Q8—44no—
250Q8—82no—
251Q8—82no—
252D4—28no—
253D4—44no—
254D4—82no—
255D4—84no—
256D4—88no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—48no—
262C2×C4—42no—
263C2×C4—88no—
264C2×C4—82no—
265C2×C4C7×#12012yes—
266C2×C4C7×#12011yescharacteristic
267C2×C4C22×C7×D411yescharacteristic
268C2×C4C22×C7×D412yes—
269C8—42no—
270C8—44no—
271C7#1872811yescharacteristic, direct product factor, Sylow
272C22—22no—
273C22—21no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22C2×C7×#12011yescharacteristic
278C4—22no—
279C4—21no—
280C4—44no—
281C4—44no—
282C4—82no—
283C4C7×#45412yes—
284C4C7×#57714yes—
285C2—41no—
286C2—84no—
287C2C7×#300912yes—
288C2C2×C7×#45411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC7×#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#3227C2semidirect
3. C7×#3139C2semidirect
4. C7×#1635C2semidirect
5. C7×#1544C2semidirect
6. C7×#1484C2semidirect
27. #18728C7direct (N × H)
68. #3227C2×C7semidirect
69. #3139C2×C7semidirect
70. #1635C2×C7semidirect
71. #1544C2×C7semidirect
72. #1484C2×C7semidirect
271. C7#18728direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35482
C7×#32272
C7×#31392
C7×#16352
C7×#15442
C7×#14842
#187287