Finite Group Databasebeta

← C7×#18732 #13195948

Subgroups — C7×#18732 #13195948

Order 1792.

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

normal subgroups only

This group has 808 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×#3556C211yesmaximal, characteristic
2C7×#3230C212yesmaximal
3C7×#3139C211yesmaximal, characteristic
4C7×#1636C218yesmaximal, complemented
5C7×#1544C211yesmaximal, characteristic
6C7×#1485C212yesmaximal
7C7×#564—22no—
8C7×#557—22no—
9C7×#557—22no—
10C7×#557C2214yes—
11C7×#541—24no—
12C7×#541—22no—
13C7×#533C2212yes—
14C7×#533C2211yescharacteristic
15C2×C7×C4⋊Q8C2211yescharacteristic
16C7×#500—22no—
17C7×#491—22no—
18C7×#487—22no—
19C7×#487C2214yes—
20C7×#484—24no—
21C7×#477C2218yes—
22C7×#431C2212yes—
23C7×#419—22no—
24C4×C7×M16C2211yescharacteristic
25C7×#332C2214yes—
26C7×Q8⋊C8C2218yes—
27#18732C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8—21no—
29C2×C7×M16—21no—
30C7×C4⋊Q8—22no—
31C7×C4⋊Q8—24no—
32C7×C4⋊Q8—22no—
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8D412yes—
35C7×C4⋊Q8D412yes—
36C7×C4⋊Q8C2312yes—
37C7×C4⋊Q8C2312yes—
38C7×#126—28no—
39C7×#125—22no—
40C7×#125—28no—
41C7×#125—28no—
42C7×#125—24no—
43C7×#125—28no—
44C7×#125—22no—
45C7×#125D412yes—
46C7×C22⋊Q8—22no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8D414yes—
49C4×C7×Q8—24no—
50C4×C7×Q8—24no—
51C4×C7×Q8—24no—
52C4×C7×Q8—22no—
53C4×C7×Q8—22no—
54C4×C7×Q8C2314yes—
55C7×#117—22no—
56C7×#117D412yes—
57C2×C7×C4⋊C4—22no—
58C2×C42×C7C2311yescharacteristic
59C7×#107—28no—
60C7×C4⋊C8C2314yes—
61C7×#103—28no—
62C7×#103—28no—
63C7×C22⋊C8—22no—
64C7×#97—21no—
65C4×C7×C8—22no—
66C4×C7×C8C2312yes—
67#3556C2×C711yescharacteristic
68#3230C2×C712yes—
69#3139C2×C711yescharacteristic
70#1636C2×C718yescomplemented
71#1544C2×C711yescharacteristic
72#1485C2×C712yes—
73C2×C7×Q8—22no—
74C2×C7×Q8—22no—
75C2×C7×Q8—22no—
76C2×C7×Q8—44no—
77C2×C7×Q8—41no—
78C2×C7×Q8C2×D414yes—
79C22×C4×C7—41no—
80C22×C4×C7C2×D412yes—
81C22×C4×C7C2×D411yescharacteristic
82C7×M16—42no—
83C2×C7×C8—24no—
84C2×C7×C8—22no—
85C7×C4⋊C4—28no—
86C7×C4⋊C4—28no—
87C7×C4⋊C4—28no—
88C7×C4⋊C4—22no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—24no—
92C7×C4⋊C4—24no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—24no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4C2×D414yes—
98C7×C4⋊C4C2×D414yes—
99C7×C22⋊C4—24no—
100C7×C22⋊C4—22no—
101C42×C7—28no—
102C42×C7—22no—
103C42×C7—24no—
104C42×C7C2×D412yes—
105C42×C7C2×D411yescharacteristic
106C42×C7C2411yescharacteristic
107#564—22no—
108#557—22no—
109#557—22no—
110#557C22×C714yes—
111#541—24no—
112#541—22no—
113#533C22×C712yes—
114#533C22×C711yescharacteristic
115C2×C4⋊Q8C22×C711yescharacteristic
116#500—22no—
117#491—22no—
118#487—22no—
119#487C22×C714yes—
120#484—24no—
121#477C22×C718yes—
122#431C22×C712yes—
123#419—22no—
124C4×M16C22×C711yescharacteristic
125#332C22×C714yes—
126Q8⋊C8C22×C718yes—
127C23×C7#12011yescharacteristic
128C7×Q8—28no—
129C7×Q8—44no—
130C7×Q8—44no—
131C7×Q8—44no—
132C7×Q8—82no—
133C2×C4×C7—24no—
134C2×C4×C7—22no—
135C2×C4×C7—24no—
136C2×C4×C7—24no—
137C2×C4×C7—24no—
138C2×C4×C7—42no—
139C2×C4×C7—82no—
140C2×C4×C7#12012yes—
141C2×C4×C7#12011yescharacteristic
142C2×C4×C7C22×D411yescharacteristic
143C2×C4×C7C22×D412yes—
144C7×C8—42no—
145C7×C8—44no—
146C22×Q8—21no—
147C2×M16—21no—
148C4⋊Q8—22no—
149C4⋊Q8—24no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8C7×D412yes—
153C4⋊Q8C7×D412yes—
154C4⋊Q8C23×C712yes—
155C4⋊Q8C23×C712yes—
156#126—28no—
157#125—22no—
158#125—28no—
159#125—28no—
160#125—24no—
161#125—28no—
162#125—22no—
163#125C7×D412yes—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8C7×D414yes—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8C23×C714yes—
173#117—22no—
174#117C7×D412yes—
175C2×C4⋊C4—22no—
176C2×C42C23×C711yescharacteristic
177#107—28no—
178C4⋊C8C23×C714yes—
179#103—28no—
180#103—28no—
181C22⋊C8—22no—
182#97—21no—
183C4×C8—22no—
184C4×C8C23×C712yes—
185C22×C7—22no—
186C22×C7—21no—
187C22×C7C2×#12011yescharacteristic
188C4×C7—22no—
189C4×C7—21no—
190C4×C7—44no—
191C4×C7—44no—
192C4×C7—44no—
193C4×C7—82no—
194C4×C7#45412yes—
195C4×C7#57814yes—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—44no—
200C2×Q8—41no—
201C2×Q8C2×C7×D414yes—
202C22×C4—41no—
203C22×C4C2×C7×D412yes—
204C22×C4C2×C7×D411yescharacteristic
205M16—42no—
206C2×C8—24no—
207C2×C8—22no—
208C4⋊C4—28no—
209C4⋊C4—28no—
210C4⋊C4—28no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—42no—
220C4⋊C4C2×C7×D414yes—
221C4⋊C4C2×C7×D414yes—
222C22⋊C4—24no—
223C22⋊C4—22no—
224C42—28no—
225C42—22no—
226C42—24no—
227C42C2×C7×D412yes—
228C42C2×C7×D411yescharacteristic
229C42C24×C711yescharacteristic
230C2×C7—41no—
231C2×C7#301112yes—
232C2×C7C2×#45411yescharacteristic
233C23C7×#12011yescharacteristic
234Q8—28no—
235Q8—44no—
236Q8—44no—
237Q8—44no—
238Q8—82no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4C7×#12012yes—
247C2×C4C7×#12011yescharacteristic
248C2×C4C22×C7×D411yescharacteristic
249C2×C4C22×C7×D412yes—
250C8—42no—
251C8—44no—
252C7#1873211yescharacteristic, direct product factor, Sylow
253C22—22no—
254C22—21no—
255C22C2×C7×#12011yescharacteristic
256C4—22no—
257C4—21no—
258C4—44no—
259C4—44no—
260C4—44no—
261C4—82no—
262C4C7×#45412yes—
263C4C7×#57814yes—
264C2—41no—
265C2C7×#301112yes—
266C2C2×C7×#45411yescharacteristic
subgroupG/H
center, socleC22×C7C2×#120
commutator, Frattini, cosocleC42C24×C7
prime radicalC23×C7#120
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
4. C7×#1636C2semidirect
27. #18732C7direct (N × H)
70. #1636C2×C7semidirect
252. C7#18732direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#35562
C7×#32302
C7×#31392
C7×#16362
C7×#15442
C7×#14852
#187327