Finite Group Databasebeta

← #13247796

Subgroups — #13247796

Order 1792.

612 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739601C211yesmaximal, characteristic
2C2×C7×#324C211yesmaximal, characteristic
3#11729319C211yesmaximal, characteristic
4#11727682C212yesmaximal
5#11725481C212yesmaximal
6C22×C7×C4⋊C4C2211yescharacteristic
7C2×#6489—21no—
8C22×C4⋊Dic7C2211yescharacteristic
9C2×C7×C22⋊C8C2212yes—
10#89984—21no—
11#89984C2212yes—
12C2×C7⋊#95—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2211yescharacteristic
15C2×#6391—22no—
16#89757—22no—
17#89757—22no—
18C7×#324C414yescomplemented
19#89346—22no—
20#8857—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—21no—
25C23×Dic7—21no—
26C22×C7×C8—22no—
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4C2×C412yes—
30C2×C7×C4⋊C4Q812yes—
31C2×C22⋊Dic7—22no—
32#6489—42no—
33#6489—42no—
34C2×C4⋊Dic7—22no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7—42no—
40C2×C4⋊Dic7D412yes—
41C2×C4⋊Dic7D412yes—
42C2×C4×Dic7—22no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8C2×C414yes—
45C7⋊#95—22no—
46C7⋊#95—22no—
47C7⋊#95—22no—
48C7⋊#95—42no—
49C7⋊#95—42no—
50#6391—44no—
51#2316—71no—
52#1853—72no—
53C2×#324D711yescharacteristic
54#1441—71no—
55#1294—72no—
56C24×C7C2×D411yescharacteristic
57C22×C4×C7—21no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7—41no—
66C22×C4×C7—41no—
67C22×C4×C7C4⋊C412yes—
68C22×C4×C7C4⋊C412yes—
69C22×C4×C7SD1612yes—
70C22×C4×C7SD1612yes—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D412yes—
73C22×C4×C7C2×Q811yescharacteristic
74C22×C4×C7#4112yes—
75C22×Dic7—22no—
76C22×Dic7—22no—
77C22×Dic7—22no—
78C22×Dic7—22no—
79C22×Dic7—41no—
80C22×Dic7—41no—
81C22×Dic7—41no—
82C22×Dic7—41no—
83C22×Dic7—41no—
84C22×Dic7—41no—
85C22×Dic7—41no—
86C22×Dic7—41no—
87C22×Dic7—41no—
88C22×Dic7—41no—
89C22×Dic7—41no—
90C22×Dic7—41no—
91C2×C7×C8—24no—
92C2×C7×C8—42no—
93C2×C7×C8—42no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—24no—
96C22⋊Dic7—42no—
97C22⋊Dic7—42no—
98C4⋊Dic7—24no—
99C4⋊Dic7—42no—
100C4⋊Dic7—42no—
101C4⋊Dic7—42no—
102C4⋊Dic7—42no—
103C4⋊Dic7—42no—
104C4⋊Dic7—81no—
105C4⋊Dic7—81no—
106C4×Dic7—44no—
107C2×#117—141no—
108C22×C4⋊C4—71no—
109C22×C4⋊C4C2×D711yescharacteristic
110C2×#106—142no—
111C2×C22⋊C8C2×D712yes—
112#389—142no—
113#389—142no—
114#380—72no—
115#380—141no—
116C2×#95—71no—
117C2×#95—141no—
118C2×#95—141no—
119#340—142no—
120#324Dic714yescomplemented
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7C2×C4⋊C411yescharacteristic
126C23×C7#12011yescharacteristic
127C23×C7#12311yescharacteristic
128C23×C7C2×SD1611yescharacteristic
129C23×C7C2×SD1611yescharacteristic
130C23×C7AGL(1,ℤ/8)11yescharacteristic
131C23×C7#13711yescharacteristic
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—22no—
138C2×C4×C7—22no—
139C2×C4×C7—22no—
140C2×C4×C7—22no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7—41no—
145C2×C4×C7—41no—
146C2×C4×C7—42no—
147C2×C4×C7—81no—
148C2×C4×C7#10614yescomplemented
149C2×C4×C7C4×D412yes—
150C2×C4×C7C22⋊Q812yes—
151C2×Dic7—42no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159C2×Dic7—42no—
160C2×Dic7—42no—
161C2×Dic7—42no—
162C2×Dic7—42no—
163C2×Dic7—81no—
164C2×Dic7—81no—
165C2×Dic7—81no—
166C2×Dic7—81no—
167C2×Dic7—81no—
168C2×Dic7—81no—
169C2×Dic7—81no—
170C2×Dic7—81no—
171C7×C8—44no—
172C23×C4—141no—
173C23×C4—141no—
174C23×C4D2811yescharacteristic
175C23×C4D2811yescharacteristic
176C23×C4C22×D711yescharacteristic
177C22×C8—22no—
178#117—282no—
179#117—282no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—72no—
183C2×C4⋊C4—72no—
184C2×C4⋊C4—142no—
185C2×C4⋊C4—141no—
186C2×C4⋊C4—141no—
187C2×C4⋊C4—141no—
188C2×C4⋊C4—141no—
189C2×C4⋊C4—282no—
190C2×C4⋊C4Dic1412yes—
191C2×C4⋊C4C2×Dic712yes—
192C2×C22⋊C4—142no—
193C2×C42—142no—
194#106—284no—
195C22⋊C8—22no—
196C22⋊C8C2×Dic714yes—
197#95—142no—
198#95—142no—
199#95—142no—
200#95—282no—
201#95—282no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7#38511yescharacteristic
215C22×C7C2×#10611yescharacteristic
216C22×C7#42811yescharacteristic
217C22×C7#44811yescharacteristic
218C22×C7#45011yescharacteristic
219C22×C7#45411yescharacteristic
220C22×C7#45611yescharacteristic
221C22×C7B(4,𝔽2)11yescharacteristic
222C22×C7#45811yescharacteristic
223C22×C7#48111yescharacteristic
224C22×C7#48311yescharacteristic
225C4×C7—24no—
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—42no—
229Dic7—82no—
230Dic7—82no—
231Dic7—82no—
232Dic7—82no—
233C24C2×D2811yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—142no—
245C22×C4—142no—
246C22×C4—142no—
247C22×C4—142no—
248C22×C4—281no—
249C22×C4—281no—
250C22×C4—281no—
251C22×C4—281no—
252C22×C4—281no—
253C22×C4—281no—
254C22×C4—281no—
255C22×C4—281no—
256C22×C4—281no—
257C22×C4—281no—
258C22×C4—281no—
259C22×C4—281no—
260C22×C4#113212yes—
261C22×C4#113212yes—
262C22×C4C4⋊Dic712yes—
263C22×C4C4⋊Dic712yes—
264C22×C4C2×Dic1411yescharacteristic
265C22×C4D4×D712yes—
266C22×C4#115912yes—
267C22×C4C22×Dic711yescharacteristic
268C2×C8—24no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—144no—
274C4⋊C4—282no—
275C4⋊C4—282no—
276C4⋊C4—282no—
277C4⋊C4—282no—
278C4⋊C4—282no—
279C4⋊C4—561no—
280C4⋊C4—561no—
281C22⋊C4—282no—
282C22⋊C4—282no—
283C42—284no—
284C2×C7—21no—
285C2×C7—21no—
286C2×C7—21no—
287C2×C7—21no—
288C2×C7#159911yescharacteristic
289C2×C7#160411yescharacteristic
290C2×C7#161111yescharacteristic
291C2×C7#161611yescharacteristic
292C2×C7#185311yescharacteristic
293C2×C7#185711yescharacteristic
294C2×C7#186711yescharacteristic
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#644511yescharacteristic
300C23#644911yescharacteristic
301C23C2×#113211yescharacteristic
302C23C2×#113211yescharacteristic
303C23#647111yescharacteristic
304C23#647211yescharacteristic
305C23C2×C4⋊Dic711yescharacteristic
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—282no—
323C2×C4—282no—
324C2×C4—282no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4—282no—
330C2×C4—282no—
331C2×C4—282no—
332C2×C4—282no—
333C2×C4—282no—
334C2×C4—561no—
335C2×C4—561no—
336C2×C4—561no—
337C2×C4—561no—
338C2×C4—561no—
339C2×C4—561no—
340C2×C4—561no—
341C2×C4—561no—
342C2×C4#639114yescomplemented
343C2×C4#644112yes—
344C2×C4D4×Dic712yes—
345C8—44no—
346C7#885711yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#8949311yescharacteristic
360C22#8950311yescharacteristic
361C22#8950711yescharacteristic
362C22#8951011yescharacteristic
363C22#8951311yescharacteristic
364C22#8951411yescharacteristic
365C22#8952411yescharacteristic
366C22#8952511yescharacteristic
367C22#8972211yescharacteristic
368C22C2×#639111yescharacteristic
369C22#8989311yescharacteristic
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—562no—
375C4—562no—
376C4—562no—
377C4—562no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#1172588711yescharacteristic
383C2#1172589111yescharacteristic
384C2#1172589211yescharacteristic
385C2#1172589411yescharacteristic
386C2#1172768211yescharacteristic
387C2#1172777511yescharacteristic
388C2#1172785611yescharacteristic
subgroupG/H
centerC23#6445
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#324C2
socleC23×C7#120
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenterC2×#324D7
hypocenterC7#8857
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
18. C7×#324C4semidirect
120. #324Dic7semidirect
148. C2×C4×C7#106semidirect
342. C2×C4#6391semidirect
346. C7#8857semidirect

Maximal subgroups

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

subgroupindex
#117396012
C2×C7×#3242
#117293192
#117276822
#117254812
#88577