Finite Group Databasebeta

← #13248795

Subgroups — #13248795

Order 1792.

800 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 147 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×#2316C211yesmaximal, characteristic
2C2×#89751C211yesmaximal, characteristic
3C2×#89323C211yesmaximal, characteristic
4#11727279C212yesmaximal
5#11725430C212yesmaximal
6C23×Dic14C2211yescharacteristic
7C2×C7×#117—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C22×Dic7⋊C4—21no—
10C7×#389—24no—
11C7×#380—21no—
12C7×#380C2212yes—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2211yescharacteristic
15C2×C22⋊(C7⋊C8)C2212yes—
16#89751—22no—
17#89751C414yescomplemented
18C2×#6384—24no—
19#89323—22no—
20#9015—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C22×Dic14—22no—
27C22×Dic14—22no—
28C22×Dic14—21no—
29C22×Dic14—44no—
30C22×Dic14C2×C412yes—
31C7×#117—42no—
32C7×#117—42no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4—22no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4C2×C412yes—
38C2×C7×C4⋊C4D412yes—
39C2×C7×C4⋊C4D414yes—
40C2×C7×C22⋊C4—22no—
41C2×C42×C7—22no—
42C2×Dic7⋊C4—24no—
43C2×Dic7⋊C4—44no—
44C22×C7⋊C8—22no—
45C7×#95—22no—
46C7×#95—22no—
47C7⋊#95—24no—
48C7⋊#95—21no—
49C7⋊#95C2×C412yes—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52#6384—48no—
53C2×#395—71no—
54#2316D711yescharacteristic
55#1787—72no—
56C2×#324—71no—
57#1294—72no—
58C24×C7C2×D411yescharacteristic
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—24no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7C22⋊C412yes—
69C22×C4×C7C22⋊C412yes—
70C22×C4×C7SD1612yes—
71C22×C4×C7Q1612yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D412yes—
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7#4112yes—
76C22×Dic7—24no—
77C22×Dic7—24no—
78C22×Dic7—22no—
79C22×Dic7—22no—
80C22×Dic7—42no—
81C22×Dic7—42no—
82C22×Dic7—41no—
83C22×Dic7—41no—
84C22×Dic7—41no—
85C22×Dic7—41no—
86C2×Dic14—28no—
87C2×Dic14—24no—
88C2×Dic14—44no—
89C2×Dic14—48no—
90C2×Dic14—44no—
91C2×Dic14—44no—
92C2×Dic14—42no—
93C7×C4⋊C4—24no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—44no—
96C7×C4⋊C4—44no—
97C7×C4⋊C4—82no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C42×C7—44no—
101Dic7⋊C4—48no—
102C2×C7⋊C8—44no—
103C2×C7⋊C8—42no—
104C23×Q8—71no—
105C2×#117—21no—
106C22×C4⋊C4—141no—
107C22×C4⋊C4C2×D711yescharacteristic
108C2×#103—144no—
109C2×C22⋊C8—72no—
110#395—74no—
111#395—142no—
112#389—24no—
113#380—21no—
114#380C2×D712yes—
115C2×#95—71no—
116C2×#95—141no—
117#324—142no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#12011yescharacteristic
122C23×C7#12311yescharacteristic
123C23×C7C2×SD1611yescharacteristic
124C23×C7C2×Q1611yescharacteristic
125C23×C7AGL(1,ℤ/8)11yescharacteristic
126C23×C7#13711yescharacteristic
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—24no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—42no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—44no—
141C2×C4×C7—44no—
142C2×C4×C7—44no—
143C2×C4×C7—81no—
144C2×C4×C7#10314yescomplemented
145C2×C4×C7C4×D412yes—
146C2×C4×C7#12112yes—
147C2×Dic7—24no—
148C2×Dic7—24no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—44no—
152C2×Dic7—44no—
153C2×Dic7—44no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—44no—
157Dic14—28no—
158Dic14—48no—
159Dic14—44no—
160C7⋊C8—82no—
161C22×Q8—72no—
162C22×Q8—141no—
163C22×Q8—142no—
164C22×Q8—142no—
165C22×Q8—284no—
166C23×C4—141no—
167C23×C4—141no—
168C23×C4C22⋊D711yescharacteristic
169C23×C4C22⋊D711yescharacteristic
170C23×C4C22×D711yescharacteristic
171C22×C8—142no—
172#117—42no—
173#117—42no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—144no—
179C2×C4⋊C4—284no—
180C2×C4⋊C4C4×D712yes—
181C2×C4⋊C4D2812yes—
182C2×C4⋊C4C22⋊D714yes—
183C2×C22⋊C4—22no—
184C2×C42—22no—
185#103—288no—
186C22⋊C8—142no—
187C22⋊C8—142no—
188#95—22no—
189#95—22no—
190#95—72no—
191#95—144no—
192#95—141no—
193C22×C7—22no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10311yescharacteristic
203C22×C7#41711yescharacteristic
204C22×C7#45011yescharacteristic
205C22×C7#45111yescharacteristic
206C22×C7#45612yes—
207C22×C7B(4,𝔽2)12yes—
208C22×C7#48311yescharacteristic
209C22×C7#48411yescharacteristic
210C4×C7—24no—
211C4×C7—22no—
212C4×C7—22no—
213C4×C7—42no—
214C4×C7—84no—
215Dic7—44no—
216Dic7—44no—
217C24C2×C22⋊D711yescharacteristic
218C2×Q8—148no—
219C2×Q8—144no—
220C2×Q8—284no—
221C2×Q8—284no—
222C2×Q8—284no—
223C2×Q8—288no—
224C2×Q8—282no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—144no—
235C22×C4—144no—
236C22×C4—142no—
237C22×C4—142no—
238C22×C4—282no—
239C22×C4—282no—
240C22×C4—281no—
241C22×C4—281no—
242C22×C4—281no—
243C22×C4—281no—
244C22×C4(C2×D7)⋊C412yes—
245C22×C4(C2×D7)⋊C412yes—
246C22×C4#114212yes—
247C22×C4#114412yes—
248C22×C4C2×C4×D711yescharacteristic
249C22×C4C2×D2811yescharacteristic
250C22×C4#115712yes—
251C22×C4C2×C22⋊D712yes—
252C2×C8—284no—
253C2×C8—282no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—82no—
259C4⋊C4—288no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—44no—
263C2×C7—21no—
264C2×C7—21no—
265C2×C7—21no—
266C2×C7—21no—
267C2×C7#160412yes—
268C2×C7#160512yes—
269C2×C7#187211yescharacteristic
270C2×C7#188611yescharacteristic
271C2×C7#188911yescharacteristic
272C23—22no—
273C23—22no—
274C23C2×(C2×D7)⋊C411yescharacteristic
275C23#649211yescharacteristic
276C23#649511yescharacteristic
277C23C2×#114211yescharacteristic
278C23#650511yescharacteristic
279C23C2×#114411yescharacteristic
280C23#651611yescharacteristic
281Q8—148no—
282Q8—288no—
283Q8—284no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—81no—
301C2×C4—144no—
302C2×C4—144no—
303C2×C4—282no—
304C2×C4—284no—
305C2×C4—284no—
306C2×C4—282no—
307C2×C4—284no—
308C2×C4—282no—
309C2×C4—282no—
310C2×C4—284no—
311C2×C4#638414yescomplemented
312C2×C4C4×C22⋊D712yes—
313C2×C4#649312yes—
314C8—562no—
315C7#901511yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#8973811yescharacteristic
325C22C2×#638411yescharacteristic
326C22#8974211yescharacteristic
327C22#8974311yescharacteristic
328C22#8998511yescharacteristic
329C22#8999011yescharacteristic
330C22#9000011yescharacteristic
331C22#9001612yes—
332C22#9001812yes—
333C4—22no—
334C4—22no—
335C4—24no—
336C4—42no—
337C4—84no—
338C4—284no—
339C4—284no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#1172931311yescharacteristic
345C2#1172940811yescharacteristic
346C2#1172941411yescharacteristic
347C2#1172963412yes—
348C2#1172965812yes—
subgroupG/H
centerC23#6516
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2316C2
socleC23×C7#120
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2316D7
hypocenterC7#9015
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
17. #89751C4semidirect
144. C2×C4×C7#103semidirect
311. C2×C4#6384semidirect
315. C7#9015semidirect

Maximal subgroups

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

subgroupindex
C7×#23162
C2×#897512
C2×#893232
#117272792
#117254302
#90157