Finite Group Databasebeta

← #13242516

Subgroups — #13242516

Order 1792.

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

normal subgroups only

This group has 604 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#11733825C211yesmaximal, characteristic, complemented
2C7×#2011C211yesmaximal, characteristic
3#11729516C212yesmaximal, complemented
4#11725445C211yesmaximal, characteristic, complemented
5#11725283C212yesmaximal, complemented
6C22×C7×D8—21no—
7C2×C7×#127C2211yescharacteristic
8C2×#6441—22no—
9C2×C7×#102—21no—
10C7×#390C2211yescharacteristic
11#89992—21no—
12C2×#6406—22no—
13C2×#6406—22no—
14#89751—21no—
15#89720—22no—
16#89700C2212yes—
17C7×#336C2211yescharacteristic
18#89319C2212yes—
19#7738—71nomaximal, Sylow
20C22×C7×D4—22no—
21C22×C7×D4—21no—
22C22×C7×D4D411yescharacteristic
23C23×Dic7—21no—
24C22×Dic14—21no—
25C2×C7×D8—42no—
26C2×C7×D8—42no—
27C2×C7×D8—42no—
28C22×C7×C8—21no—
29C7×#127—42no—
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C7×C22⋊C4—22no—
32C2×C42×C7C2311yescharacteristic
33C2×C22⋊Dic7—22no—
34C2×C22⋊Dic7—21no—
35C2×C4⋊Dic7D412yes—
36C2×C4⋊Dic7D412yes—
37C2×Dic7⋊C4—22no—
38C2×Dic7⋊C4—22no—
39C22×C7⋊C8—22no—
40#6441—44no—
41#6441—44no—
42C7×#102—42no—
43#6406—44no—
44#6406—44no—
45C7⋊#95—21no—
46C7⋊#95—22no—
47#2683—71no—
48#2041—72no—
49#2011D711yescharacteristic
50#1271—72no—
51#1270—71no—
52C24×C7—21no—
53C24×C7—41no—
54C2×C7×D4—24no—
55C2×C7×D4—22no—
56C2×C7×D4—21no—
57C2×C7×D4—41no—
58C2×C7×D4—41no—
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—42no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—41no—
69C2×C7×D4SD1612yes—
70C22×C4×C7—21no—
71C22×C4×C7C2×D412yes—
72C22×C4×C7C2×D411yescharacteristic
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—44no—
76C22×Dic7—81no—
77C2×Dic14—44no—
78C2×Dic14—81no—
79C7×D8—48no—
80C2×C7×C8—22no—
81C2×C7×C8—41no—
82C2×C7×C8—41no—
83C7×C4⋊C4—41no—
84C7×C22⋊C4—42no—
85C7×C22⋊C4—42no—
86C42×C7—41no—
87C22⋊Dic7—44no—
88C22⋊Dic7—81no—
89C4⋊Dic7—22no—
90C4⋊Dic7—42no—
91C4⋊Dic7D814yescomplemented
92Dic7⋊C4—44no—
93Dic7⋊C4—42no—
94Dic7⋊C4—42no—
95C2×C7⋊C8—42no—
96C2×C7⋊C8—42no—
97C2×C7⋊C8—42no—
98C22×D8—21no—
99C2×#127C2×D711yescharacteristic
100C2×C22⋊Q8—142no—
101C2×#102—21no—
102C2×#102—142no—
103C2×#102—142no—
104#396—142no—
105#395—141no—
106#390C2×D711yescharacteristic
107#389—72no—
108#386—141no—
109#336—72no—
110#336C2×D711yescharacteristic
111C23×C7—22no—
112C23×C7—42no—
113C23×C7—41no—
114C23×C7—42no—
115C23×C7—41no—
116C23×C7—42no—
117C23×C7—82no—
118C23×C7—81no—
119C23×C7—81no—
120C23×C7—82no—
121C23×C7#12011yescharacteristic
122C7×D4—42no—
123C7×D4—42no—
124C7×D4—42no—
125C7×D4—44no—
126C7×D4—44no—
127C7×D4—81no—
128C7×D4—81no—
129C7×D4—81no—
130C7×D4—82no—
131C7×D4—82no—
132C2×C4×C7—21no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—42no—
137C2×C4×C7—41no—
138C2×C4×C7C2×D812yes—
139C2×C4×C7C2×SD1611yescharacteristic
140C2×C4×C7AGL(1,ℤ/8)12yes—
141C2×C4×C7#13711yescharacteristic
142C2×Dic7—24no—
143C2×Dic7—42no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—84no—
149Dic14—84no—
150C7×C8—42no—
151C7⋊C8—82no—
152C22×Q8—141no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4C22⋊D711yescharacteristic
156C23×C4—141no—
157C2×D8—42no—
158C2×D8—42no—
159C2×D8—42no—
160C22×C8—21no—
161C22×C8—142no—
162#127—42no—
163C22⋊Q8—284no—
164C22⋊Q8—284no—
165C2×C4⋊C4—72no—
166C2×C4⋊C4—72no—
167C2×C4⋊C4—142no—
168C2×C4⋊C4—142no—
169C2×C4⋊C4C22⋊D711yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—142no—
172C2×C22⋊C4—141no—
173C2×C42C22×D711yescharacteristic
174#102—42no—
175#102—284no—
176#102—284no—
177#95—141no—
178#95—142no—
179C22×C7—42no—
180C22×C7—44no—
181C22×C7—82no—
182C22×C7—81no—
183C22×C7—81no—
184C22×C7—82no—
185C22×C7—84no—
186C22×C7—161no—
187C22×C7—162no—
188C22×C7—161no—
189C22×C7#44712yes—
190C22×C7#44811yescharacteristic
191C22×C7#45311yescharacteristic
192C22×C7#45412yes—
193C22×C7B(4,𝔽2)11yescharacteristic
194C4×C7—21no—
195C4×C7—21no—
196C4×C7—22no—
197C4×C7—22no—
198C4×C7—81no—
199Dic7—44no—
200Dic7—82no—
201C24—21no—
202C24—41no—
203C2×Q8—284no—
204C2×Q8—561no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#114212yes—
221C22×C4—21no—
222C22×C4—142no—
223C22×C4—142no—
224C22×C4—284no—
225C22×C4—561no—
226C22×C4D4×D712yes—
227C22×C4C2×C22⋊D711yescharacteristic
228D8—48no—
229C2×C8—22no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—282no—
233C2×C8—282no—
234C2×C8—282no—
235C4⋊C4—41no—
236C4⋊C4—74no—
237C4⋊C4—142no—
238C4⋊C4—282no—
239C4⋊C4—284no—
240C4⋊C4—282no—
241C4⋊C4—282no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—284no—
245C22⋊C4—561no—
246C42—41no—
247C2×C7—82no—
248C2×C7—162no—
249C2×C7#159611yescharacteristic
250C2×C7#159812yes—
251C2×C7#161911yescharacteristic
252C2×C7#162712yes—
253C2×C7#165711yescharacteristic
254C23—22no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—82no—
261C23—82no—
262C23—81no—
263C23—81no—
264C23#650011yescharacteristic
265Q8—564no—
266D4—42no—
267D4—42no—
268D4—44no—
269D4—44no—
270D4—42no—
271D4—81no—
272D4—82no—
273D4—82no—
274D4—81no—
275D4—81no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—144no—
283C2×C4—282no—
284C2×C4—282no—
285C2×C4—282no—
286C2×C4—282no—
287C2×C4—282no—
288C2×C4—564no—
289C2×C4D7×D812yes—
290C2×C4#647412yes—
291C2×C4C2×#114211yescharacteristic
292C2×C4#651411yescharacteristic
293C8—42no—
294C8—562no—
295C7#773811yescharacteristic, complemented, Sylow
296C22—42no—
297C22—44no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22—84no—
302C22—82no—
303C22—162no—
304C22—161no—
305C22—161no—
306C22#8980411yescharacteristic
307C22#8981211yescharacteristic
308C22#8984911yescharacteristic
309C22#8992812yes—
310C22#8997312yes—
311C4—22no—
312C4—22no—
313C4—21no—
314C4—21no—
315C4—81no—
316C4—284no—
317C4—562no—
318C2—82no—
319C2—162no—
320C2#1172745211yescharacteristic
321C2#1172872512yes—
322C2#1172876812yes—
323C2#1172903511yescharacteristic
324C2#1172908211yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
Fitting, prime radicalC7×#2011C2
socleC23×C7#120
hypercenter#2011D7
hypocenterC7#7738
LCSG ▷ C2×C42×C7 ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2011

Complements

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

NH ≅ G/Ntype
1. #11733825C2semidirect
3. #11729516C2semidirect
4. #11725445C2semidirect
5. #11725283C2semidirect
91. C4⋊Dic7D8semidirect
295. C7#7738semidirect

Maximal subgroups

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

subgroupindex
#117338252
C7×#20112
#117295162
#117254452
#117252832
#77387