Finite Group Databasebeta

← #13252639

Subgroups — #13252639

Order 1792.

920 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#390C211yesmaximal, characteristic
2C2×#89320C214yesmaximal, complemented
3#11729284C211yesmaximal, characteristic, complemented
4#11729283C211yesmaximal, characteristic, complemented
5C23×C7×D4C411yescharacteristic
6C22×C7×C4⋊C4C411yescharacteristic
7C22×C7×C22⋊C4C412yes—
8C22×C7×C22⋊C4C2212yes—
9C22×C42×C7C2211yescharacteristic
10C7×#390—21no—
11C7×#390—21no—
12C7×#390—21no—
13C7×#390—21no—
14C2×C22⋊(C7⋊C8)C2212yes—
15C2×C22⋊(C7⋊C8)C2212yes—
16C2×C4⋊(C7⋊C8)—21no—
17C2×#6377—21no—
18#89320—28no—
19#89319—21no—
20#89319—21no—
21#9653—71nomaximal, Sylow
22C25×C7C2×C412yes—
23C22×C7×D4—24no—
24C22×C7×D4—21no—
25C22×C7×D4—21no—
26C22×C7×D4—42no—
27C22×C7×D4—42no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7C2×C412yes—
31C23×C4×C7C2311yescharacteristic
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—42no—
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—24no—
40C2×C7×C22⋊C4—24no—
41C2×C7×C22⋊C4—44no—
42C2×C42×C7—42no—
43C2×C42×C7D411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C22×C7⋊C8—22no—
48C22×C7⋊C8—22no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—24no—
51C22⋊(C7⋊C8)D414yescomplemented
52C4⋊(C7⋊C8)—42no—
53#6377—42no—
54C2×#390D711yescharacteristic
55#1797—71no—
56#1749—71no—
57C2×#323—74no—
58C24×C7—22no—
59C24×C7—22no—
60C24×C7—24no—
61C24×C7—24no—
62C24×C7—42no—
63C24×C7—42no—
64C24×C7—42no—
65C24×C7—42no—
66C24×C7M1614yes—
67C24×C7C22×C411yescharacteristic
68C2×C7×D4—24no—
69C2×C7×D4—24no—
70C2×C7×D4—416no—
71C2×C7×D4—42no—
72C2×C7×D4—42no—
73C2×C7×D4—42no—
74C2×C7×D4—42no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—22no—
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—22no—
81C22×C4×C7—24no—
82C22×C4×C7—41no—
83C22×C4×C7—41no—
84C22×C4×C7—41no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D412yes—
94C22×C4×C7#4112yes—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—44no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C42×C7—22no—
103C42×C7—22no—
104C42×C7—41no—
105C42×C7—41no—
106C2×C7⋊C8—24no—
107C2×C7⋊C8—42no—
108C2×C7⋊C8—42no—
109C2×C7⋊C8—42no—
110C2×C7⋊C8—42no—
111C2×C7⋊C8—42no—
112C23×D4Dic711yescharacteristic
113C22×C4⋊C4Dic711yescharacteristic
114C22×C22⋊C4Dic712yes—
115C22×C22⋊C4C2×D712yes—
116C22×C42C2×D711yescharacteristic
117C2×C4⋊C8—141no—
118C2×C22⋊C8—72no—
119C2×C22⋊C8—72no—
120C2×#97—141no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#336—141no—
126#336—141no—
127#323—148no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—24no—
133C23×C7—28no—
134C23×C7—44no—
135C23×C7—44no—
136C23×C7—44no—
137C23×C7—44no—
138C23×C7—42no—
139C23×C7—42no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—42no—
144C23×C7—44no—
145C23×C7—84no—
146C23×C7—84no—
147C23×C7—84no—
148C23×C7—84no—
149C23×C7C2×C22⋊C411yescharacteristic
150C23×C7C4×D412yes—
151C23×C7C2×M1612yes—
152C23×C7#13112yes—
153C7×D4—48no—
154C7×D4—48no—
155C2×C4×C7—28no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—44no—
172C2×C4×C7—42no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—84no—
176C2×C4×C7#9912yes—
177C2×C4×C7#10012yes—
178C2×C4×C7#10012yes—
179C2×C4×C7#12112yes—
180C2×C4×C7#12411yescharacteristic
181C2×C4×C7#12711yescharacteristic
182C7⋊C8—44no—
183C7⋊C8—82no—
184C25C2×Dic712yes—
185C22×D4—24no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—42no—
189C22×D4—42no—
190C23×C4C2×Dic711yescharacteristic
191C23×C4C2×Dic711yescharacteristic
192C23×C4C2×Dic712yes—
193C23×C4C22×D711yescharacteristic
194C22×C8—142no—
195C22×C8—142no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—42no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—44no—
206C2×C42—42no—
207C2×C42C22⋊D711yescharacteristic
208C2×C42C22⋊D711yescharacteristic
209C2×C42C22⋊D711yescharacteristic
210C2×C42C22⋊D711yescharacteristic
211C4⋊C8—282no—
212C22⋊C8—74no—
213C22⋊C8—142no—
214C22⋊C8—144no—
215#97—282no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—44no—
223C22×C7—44no—
224C22×C7—44no—
225C22×C7—88no—
226C22×C7—88no—
227C22×C7—82no—
228C22×C7—82no—
229C22×C7—82no—
230C22×C7—82no—
231C22×C7#39011yescharacteristic
232C22×C7#40711yescharacteristic
233C22×C7#40811yescharacteristic
234C22×C7C2×#9911yescharacteristic
235C22×C7#41011yescharacteristic
236C22×C7C2×#10011yescharacteristic
237C22×C7C2×#10011yescharacteristic
238C22×C7#43514yes—
239C4×C7—22no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—22no—
243C4×C7—22no—
244C4×C7—22no—
245C4×C7—41no—
246C4×C7—82no—
247C24—22no—
248C24—22no—
249C24—24no—
250C24—24no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24#113614yes—
256C24C22×Dic711yescharacteristic
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—416no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C22⋊Dic711yescharacteristic
277C22×C4C22⋊Dic711yescharacteristic
278C22×C4C22⋊Dic711yescharacteristic
279C22×C4C22⋊Dic711yescharacteristic
280C22×C4D4×D712yes—
281C22×C4#115912yes—
282C22×C4C2×C22⋊D711yescharacteristic
283C22×C4C2×C22⋊D711yescharacteristic
284C2×C8—144no—
285C2×C8—282no—
286C2×C8—282no—
287C2×C8—282no—
288C2×C8—282no—
289C2×C8—282no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C42—22no—
298C42—22no—
299C42—41no—
300C42—41no—
301C2×C7—24no—
302C2×C7—84no—
303C2×C7#146211yescharacteristic
304C2×C7#146611yescharacteristic
305C2×C7#146911yescharacteristic
306C2×C7#147111yescharacteristic
307C2×C7#195511yescharacteristic
308C2×C7#196411yescharacteristic
309C2×C7#196511yescharacteristic
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—24no—
315C23—28no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—44no—
327C23—84no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23C2×#113612yes—
332C23D4×Dic712yes—
333C23#651112yes—
334C23C2×C22⋊Dic711yescharacteristic
335D4—48no—
336D4—48no—
337C2×C4—28no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—84no—
358C2×C4#640712yes—
359C2×C4#640712yes—
360C2×C4#640812yes—
361C2×C4#649911yescharacteristic
362C2×C4#650112yes—
363C2×C4#650311yescharacteristic
364C8—284no—
365C8—562no—
366C7#965311yescharacteristic, complemented, Sylow
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22—88no—
377C22—88no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22#8978314yes—
383C22#8997911yescharacteristic
384C22C2×#640711yescharacteristic
385C22C2×#640711yescharacteristic
386C22C2×#640811yescharacteristic
387C22#8999311yescharacteristic
388C22#9000611yescharacteristic
389C22#9001211yescharacteristic
390C4—22no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—41no—
397C4—82no—
398C2—24no—
399C2—84no—
400C2#1172929311yescharacteristic
401C2#1172935711yescharacteristic
402C2#1172936911yescharacteristic
403C2#1172943011yescharacteristic
404C2#1172944011yescharacteristic
405C2#1172947411yescharacteristic
406C2#1172952011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#390C2
socleC23×C7C2×C22⋊C4
prime radicalC23×C7×D4C4
cosocleC23×C4×C7C23
hypercenterC2×#390D7
hypocenterC7#9653
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C2×#89320C2semidirect
3. #11729284C2semidirect
4. #11729283C2semidirect
51. C22⋊(C7⋊C8)D4semidirect
366. C7#9653semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3902
C2×#893202
#117292842
#117292832
#96537