Finite Group Databasebeta

← #13249809

Subgroups — #13249809

Order 1792.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2905C211yesmaximal, characteristic
2#11739709C211yesmaximal, characteristic
3#11729325C211yesmaximal, characteristic
4#11728262C212yesmaximal, complemented
5#11725519C212yesmaximal, complemented
6C22×C7×M16—21no—
7C2×C7×#117C2211yescharacteristic
8C2×#6432C2211yescharacteristic
9C7×#424—24no—
10C7×#423—21no—
11C7×#423C2212yes—
12#89984—21no—
13#89758C412yes—
14#89751—21no—
15#89721C412yes—
16#89719—21no—
17#89708C2211yescharacteristic
18#89700—21no—
19#89697C2212yes—
20#89323—21no—
21#89320—21no—
22#89281—24no—
23#9193—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C22×Dic14—21no—
26C2×C7×M16—22no—
27C2×C7×M16—22no—
28C2×C7×M16—42no—
29C2×C7×M16—42no—
30C22×C7×C8—22no—
31C7×#117—22no—
32C7×#117—22no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7C2311yescharacteristic
39C2×C22⋊Dic7—22no—
40C2×C4⋊Dic7—21no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7C2×C412yes—
43C2×Dic7⋊C4—22no—
44#6432—22no—
45#6432—22no—
46C7×C4⋊C8—24no—
47C7×C4⋊C8—22no—
48C7×C4⋊C8D414yes—
49C7⋊#95—22no—
50C7⋊#95—22no—
51C7⋊#95—21no—
52C7⋊#95C2×C412yes—
53C22⋊(C7⋊C8)—22no—
54C4⋊(C7⋊C8)—22no—
55C4×C7⋊C8—22no—
56#2905D711yescharacteristic
57#2628—71no—
58#1549—72no—
59#1458—71no—
60#1286—72no—
61C24×C7C22⋊C411yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7SD1612yes—
70C22×C4×C7Q1612yes—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×Dic7—22no—
75C22×Dic7—22no—
76C2×Dic14—42no—
77C2×Dic14—82no—
78C7×M16—44no—
79C7×M16—42no—
80C7×M16—42no—
81C2×C7×C8—24no—
82C2×C7×C8—44no—
83C7×C4⋊C4—22no—
84C7×C4⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—22no—
87C42×C7—21no—
88C42×C7—21no—
89C42×C7C2×D412yes—
90C42×C7#4112yes—
91C22⋊Dic7—82no—
92C4⋊Dic7—24no—
93C4⋊Dic7—41no—
94C4⋊Dic7—41no—
95C4⋊Dic7—41no—
96C4⋊Dic7—41no—
97Dic7⋊C4—82no—
98C2×C7⋊C8—42no—
99C22×M16—21no—
100C2×C4⋊Q8—71no—
101C2×#117C2×D711yescharacteristic
102#431—72no—
103#424—24no—
104#423—21no—
105#423C2×D712yes—
106#400—72no—
107#399—72no—
108#395—141no—
109#393—141no—
110#389—141no—
111#382—71no—
112#380—141no—
113#332—144no—
114#324—141no—
115#323—141no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—41no—
119C23×C7—41no—
120C23×C7#10312yes—
121C23×C7C2×C22⋊C411yescharacteristic
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7#10312yes—
135C2×C4×C7C4×D412yes—
136C2×C4×C7#12011yescharacteristic
137C2×C4×C7#12112yes—
138C2×C4×C7#12311yescharacteristic
139C2×C4×C7C2×SD1611yescharacteristic
140C2×C4×C7C2×Q1611yescharacteristic
141C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C7#13711yescharacteristic
143C2×Dic7—42no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149Dic14—84no—
150C7×C8—44no—
151C7⋊C8—82no—
152C22×Q8—141no—
153C23×C4C22⋊D711yescharacteristic
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—42no—
157C2×M16—42no—
158C22×C8—22no—
159C4⋊Q8—142no—
160C4⋊Q8—142no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—72no—
164C2×C4⋊C4—141no—
165C2×C4⋊C4—141no—
166C2×C4⋊C4—142no—
167C2×C4⋊C4C4×D711yescharacteristic
168C2×C4⋊C4D2811yescharacteristic
169C2×C22⋊C4—142no—
170C2×C22⋊C4C4×D711yescharacteristic
171C2×C22⋊C4D2811yescharacteristic
172C2×C42C22⋊D711yescharacteristic
173C2×C42C22×D711yescharacteristic
174C4⋊C8—24no—
175C4⋊C8—22no—
176C4⋊C8—142no—
177C4⋊C8C22⋊D714yes—
178C22⋊C8—142no—
179C4×C8—142no—
180#95—72no—
181#95—142no—
182#95—142no—
183#95—141no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—22no—
187C22×C7—42no—
188C22×C7—41no—
189C22×C7—41no—
190C22×C7—81no—
191C22×C7#38611yescharacteristic
192C22×C7C2×#10311yescharacteristic
193C22×C7#41711yescharacteristic
194C4×C7—22no—
195C4×C7—22no—
196C4×C7—22no—
197C4×C7—41no—
198C4×C7—41no—
199C4×C7#45011yescharacteristic
200C4×C7#45111yescharacteristic
201C4×C7#48311yescharacteristic
202C4×C7#48411yescharacteristic
203Dic7—82no—
204Dic7—82no—
205C24(C2×D7)⋊C411yescharacteristic
206C2×Q8—282no—
207C2×Q8—562no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—142no—
213C22×C4—142no—
214C22×C4(C2×D7)⋊C411yescharacteristic
215C22×C4(C2×D7)⋊C411yescharacteristic
216C22×C4(C2×D7)⋊C411yescharacteristic
217C22×C4#114212yes—
218C22×C4#114412yes—
219C22×C4C2×C4×D711yescharacteristic
220C22×C4C2×D2811yescharacteristic
221C22×C4C2×C22⋊D711yescharacteristic
222M16—44no—
223M16—42no—
224M16—42no—
225C2×C8—24no—
226C2×C8—44no—
227C2×C8—282no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—144no—
231C4⋊C4—281no—
232C4⋊C4—281no—
233C4⋊C4—281no—
234C4⋊C4—281no—
235C4⋊C4—562no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—562no—
239C42—21no—
240C42—21no—
241C42#115712yes—
242C42C2×C22⋊D712yes—
243C2×C7—21no—
244C2×C7—21no—
245C2×C7—42no—
246C2×C7#187211yescharacteristic
247C2×C7#191011yescharacteristic
248C2×C7#191511yescharacteristic
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23#638412yes—
254C23C2×(C2×D7)⋊C411yescharacteristic
255Q8—564no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—282no—
269C2×C4—282no—
270C2×C4—282no—
271C2×C4—282no—
272C2×C4—282no—
273C2×C4—282no—
274C2×C4#638412yes—
275C2×C4C4×C22⋊D712yes—
276C2×C4#649211yescharacteristic
277C2×C4#649312yes—
278C2×C4#649511yescharacteristic
279C2×C4C2×#114211yescharacteristic
280C2×C4#650511yescharacteristic
281C2×C4C2×#114411yescharacteristic
282C2×C4#651611yescharacteristic
283C8—44no—
284C8—562no—
285C7#919311yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22—22no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22#8973811yescharacteristic
294C22C2×#638411yescharacteristic
295C22#8998511yescharacteristic
296C4—22no—
297C4—22no—
298C4—22no—
299C4—41no—
300C4—41no—
301C4—562no—
302C4—562no—
303C4#8974211yescharacteristic
304C4#8974311yescharacteristic
305C4#8999011yescharacteristic
306C4#9000011yescharacteristic
307C2—21no—
308C2—21no—
309C2—42no—
310C2#1172931311yescharacteristic
311C2#1172977911yescharacteristic
312C2#1172978511yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2905C2
socleC22×C7#386
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2905D7
hypocenterC7#9193
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2905

Complements

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

NH ≅ G/Ntype
4. #11728262C2semidirect
5. #11725519C2semidirect
285. C7#9193semidirect

Maximal subgroups

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

subgroupindex
C7×#29052
#117397092
#117293252
#117282622
#117255192
#91937