Finite Group Databasebeta

← #13242534

Subgroups — #13242534

Order 1792.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#11734019C211yesmaximal, characteristic, complemented
2C7×#2014C211yesmaximal, characteristic
3#11729516C212yesmaximal
4#11725448C211yesmaximal, characteristic, complemented
5#11725284C212yesmaximal, complemented
6C22×C7×SD16—21no—
7C2×C7×#127C2211yescharacteristic
8C2×#6507—21no—
9C2×C7×#102—21no—
10C7×#391C2211yescharacteristic
11C2×#6406—22no—
12C2×#6406—22no—
13#89755—21no—
14#89700C2212yes—
15#89428—22no—
16#89425—22no—
17C7×#336C2211yescharacteristic
18#89319C2212yes—
19#7741—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C22×C7×D4—22no—
22C22×C7×D4—21no—
23C22×Dic14—21no—
24C2×C7×SD16—42no—
25C2×C7×SD16—42no—
26C2×C7×SD16—42no—
27C22×C7×C8—21no—
28C7×#127—42no—
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C42×C7C2311yescharacteristic
31#6507—42no—
32#6507—42no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7D412yes—
35C2×C4⋊Dic7D412yes—
36C2×Dic7⋊C4—22no—
37C2×Dic7⋊C4—22no—
38C2×C4×Dic7—21no—
39C22×C7⋊C8—22no—
40C7×#102—42no—
41C7×#95—22no—
42#6406—44no—
43#6406—44no—
44C7⋊#95—21no—
45C7⋊#95—22no—
46C7⋊#95—22no—
47#2734—71no—
48#2041—72no—
49#2014D711yescharacteristic
50#1272—71no—
51#1271—72no—
52C24×C7—41no—
53C2×C7×Q8—21no—
54C2×C7×Q8—41no—
55C2×C7×Q8—41no—
56C2×C7×Q8D812yes—
57C2×C7×D4—24no—
58C2×C7×D4—22no—
59C2×C7×D4—41no—
60C2×C7×D4—41no—
61C2×C7×D4—42no—
62C2×C7×D4—42no—
63C2×C7×D4—42no—
64C2×C7×D4—41no—
65C2×C7×D4—41no—
66C2×C7×D4—42no—
67C2×C7×D4—41no—
68C2×C7×D4—42no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7C2×D412yes—
73C22×Dic7—22no—
74C22×Dic7—22no—
75C2×Dic14—42no—
76C2×Dic14—81no—
77C2×Dic14—81no—
78C7×SD16—48no—
79C2×C7×C8—22no—
80C2×C7×C8—41no—
81C2×C7×C8—41no—
82C7×C4⋊C4—41no—
83C42×C7—41no—
84C4⋊Dic7—22no—
85C4⋊Dic7—42no—
86C4⋊Dic7—81no—
87C4⋊Dic7SD1614yescomplemented
88Dic7⋊C4—44no—
89Dic7⋊C4—42no—
90Dic7⋊C4—42no—
91C4×Dic7—42no—
92C2×C7⋊C8—42no—
93C2×C7⋊C8—42no—
94C2×C7⋊C8—42no—
95C22×SD16—21no—
96C2×C4⋊Q8—141no—
97C2×#127C2×D711yescharacteristic
98C2×#102—21no—
99C2×#102—142no—
100C2×#102—142no—
101#398—141no—
102#395—142no—
103#391C2×D711yescharacteristic
104#389—72no—
105#387—142no—
106#336—72no—
107#336C2×D711yescharacteristic
108C23×C7—42no—
109C23×C7—81no—
110C23×C7—82no—
111C23×C7—81no—
112C23×C7—82no—
113C23×C7#12011yescharacteristic
114C7×Q8—42no—
115C7×Q8—81no—
116C7×D4—42no—
117C7×D4—42no—
118C7×D4—44no—
119C7×D4—44no—
120C7×D4—81no—
121C7×D4—81no—
122C7×D4—82no—
123C7×D4—82no—
124C2×C4×C7—21no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—41no—
130C2×C4×C7—41no—
131C2×C4×C7—41no—
132C2×C4×C7—42no—
133C2×C4×C7C2×D811yescharacteristic
134C2×C4×C7C2×SD1612yes—
135C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C7#13712yes—
137C2×Dic7—24no—
138C2×Dic7—42no—
139C2×Dic7—42no—
140C2×Dic7—42no—
141C2×Dic7—42no—
142C2×Dic7—42no—
143Dic14—84no—
144C7×C8—42no—
145C7⋊C8—82no—
146C22×Q8—141no—
147C22×Q8C22⋊D711yescharacteristic
148C22×D4—22no—
149C22×D4—21no—
150C2×SD16—42no—
151C2×SD16—42no—
152C2×SD16—42no—
153C22×C8—21no—
154C22×C8—142no—
155C4⋊Q8—282no—
156C4⋊Q8—282no—
157#127—42no—
158C2×C4⋊C4—72no—
159C2×C4⋊C4—72no—
160C2×C4⋊C4—142no—
161C2×C4⋊C4—142no—
162C2×C4⋊C4—141no—
163C2×C4⋊C4C22⋊D711yescharacteristic
164C2×C42—141no—
165C2×C42C22×D711yescharacteristic
166#102—42no—
167#102—284no—
168#102—284no—
169#95—22no—
170#95—141no—
171#95—142no—
172#95—142no—
173C22×C7—82no—
174C22×C7—84no—
175C22×C7—161no—
176C22×C7—162no—
177C22×C7—161no—
178C22×C7#44711yescharacteristic
179C22×C7#44812yes—
180C22×C7#45311yescharacteristic
181C22×C7#45412yes—
182C22×C7B(4,𝔽2)11yescharacteristic
183C4×C7—21no—
184C4×C7—21no—
185C4×C7—22no—
186C4×C7—22no—
187C4×C7—42no—
188C4×C7—81no—
189Dic7—44no—
190Dic7—82no—
191C24—41no—
192C2×Q8—21no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—282no—
196C2×Q8—561no—
197C2×Q8—561no—
198C2×Q8#114112yes—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—142no—
214C22×C4—142no—
215C22×C4D4×D712yes—
216C22×C4C2×C22⋊D711yescharacteristic
217SD16—48no—
218C2×C8—22no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—282no—
222C2×C8—282no—
223C2×C8—282no—
224C4⋊C4—41no—
225C4⋊C4—74no—
226C4⋊C4—142no—
227C4⋊C4—282no—
228C4⋊C4—284no—
229C4⋊C4—282no—
230C4⋊C4—282no—
231C4⋊C4—561no—
232C42—41no—
233C42—282no—
234C2×C7—162no—
235C2×C7#159511yescharacteristic
236C2×C7#159912yes—
237C2×C7#162311yescharacteristic
238C2×C7#162712yes—
239C2×C7#165711yescharacteristic
240C23—42no—
241C23—82no—
242C23—82no—
243C23—81no—
244C23—81no—
245C23#650011yescharacteristic
246Q8—42no—
247Q8—81no—
248Q8—564no—
249D4—42no—
250D4—42no—
251D4—44no—
252D4—44no—
253D4—81no—
254D4—81no—
255D4—82no—
256D4—82no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—144no—
267C2×C4—282no—
268C2×C4—282no—
269C2×C4—282no—
270C2×C4—282no—
271C2×C4—282no—
272C2×C4D7×SD1612yes—
273C2×C4#647812yes—
274C2×C4C2×#114111yescharacteristic
275C2×C4#651211yescharacteristic
276C8—42no—
277C8—562no—
278C7#774111yescharacteristic, complemented, Sylow
279C22—82no—
280C22—84no—
281C22—161no—
282C22—162no—
283C22—161no—
284C22#8980411yescharacteristic
285C22#8980811yescharacteristic
286C22#8984911yescharacteristic
287C22#8994212yes—
288C22#8997312yes—
289C4—21no—
290C4—21no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—81no—
295C4—284no—
296C4—562no—
297C2—162no—
298C2#1172745011yescharacteristic
299C2#1172873312yes—
300C2#1172878912yes—
301C2#1172902311yescharacteristic
302C2#1172908211yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2014C2
socleC23×C7#120
prime radicalC2×C7×#127C22
hypercenter#2014D7
hypocenterC7#7741
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2014

Complements

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

NH ≅ G/Ntype
1. #11734019C2semidirect
4. #11725448C2semidirect
5. #11725284C2semidirect
87. C4⋊Dic7SD16semidirect
278. C7#7741semidirect

Maximal subgroups

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

subgroupindex
#117340192
C7×#20142
#117295162
#117254482
#117252842
#77417