Finite Group Databasebeta

← #13319871

Subgroups — #13319871

Order 1792.

970 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6452C211yesmaximal, characteristic
2#11736064C212yesmaximal
3C7×#1463C211yesmaximal, characteristic
4#11729306C211yesmaximal, characteristic
5#11727725C212yesmaximal
6#11725882C218yesmaximal, complemented
7C22×C7×C4⋊C4C2211yescharacteristic
8C22×C4⋊Dic7C411yescharacteristic
9C22×Dic7⋊C4C411yescharacteristic
10C22×Dic7⋊C4C411yescharacteristic
11C22×Dic7⋊C4C411yescharacteristic
12C22×Dic7⋊C4C2211yescharacteristic
13C22×C4×Dic7C2211yescharacteristic
14C2×#6452—22no—
15C2×#6452—22no—
16C2×#6452—22no—
17C2×#6452—22no—
18C2×#6452C414yes—
19C2×#6452C2214yes—
20C7×#407C2212yes—
21#89979C2212yes—
22#89890—22no—
23M16×Dic7—22no—
24#89491—22no—
25#89491C2214yes—
26#89490—22no—
27#89490C2214yes—
28C7×#324C2214yes—
29#89323C2214yes—
30#89264C2218yes—
31#17394—71nomaximal, Sylow
32C23×C4×C7C2×C411yescharacteristic
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C23×Dic7C2×C411yescharacteristic
36C23×Dic7C2×C411yescharacteristic
37C23×Dic7C2×C411yescharacteristic
38C23×Dic7C2×C411yescharacteristic
39C2×C7×M16—22no—
40C2×C7×C4⋊C4—22no—
41C2×C7×C4⋊C4—22no—
42C2×C7×C4⋊C4C2×C412yes—
43C2×C7×C4⋊C4C2312yes—
44C2×C4⋊Dic7—21no—
45C2×C4⋊Dic7—21no—
46C2×C4⋊Dic7—22no—
47C2×C4⋊Dic7C2×C414yes—
48C2×Dic7⋊C4—22no—
49C2×Dic7⋊C4—22no—
50C2×Dic7⋊C4—21no—
51C2×Dic7⋊C4—22no—
52C2×Dic7⋊C4—22no—
53C2×Dic7⋊C4—21no—
54C2×Dic7⋊C4—21no—
55C2×Dic7⋊C4—21no—
56C2×Dic7⋊C4—21no—
57C2×Dic7⋊C4—22no—
58C2×Dic7⋊C4—21no—
59C2×Dic7⋊C4C2×C412yes—
60C2×Dic7⋊C4C2×C414yes—
61C2×Dic7⋊C4C2×C414yes—
62C2×Dic7⋊C4C2×C414yes—
63C2×Dic7⋊C4C2312yes—
64C2×C4×Dic7C2×C412yes—
65C2×C4×Dic7D412yes—
66C2×C4×Dic7D412yes—
67C2×C4×Dic7D412yes—
68C2×C4×Dic7D412yes—
69C2×C4×Dic7C2312yes—
70C2×#1136—22no—
71#6452—24no—
72#6452—24no—
73#6452—24no—
74#6452—24no—
75#6452—24no—
76#6452—24no—
77#6452—24no—
78#6452—24no—
79C7×C22⋊C8—22no—
80C7×C22⋊C8C2314yes—
81C22⋊(C7⋊C8)—22no—
82C22⋊(C7⋊C8)C2314yes—
83#6389—24no—
84Dic7⋊C8—24no—
85Dic7⋊C8—24no—
86C8×Dic7—24no—
87C22×#125—71no—
88#2920—72no—
89#1470—78no—
90#1463—72no—
91#1463—71no—
92#1463D711yescharacteristic
93C24×C7C22×C411yescharacteristic
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—42no—
99C22×C4×C7—42no—
100C22×C4×C7—42no—
101C22×C4×C7C22×C412yes—
102C22×C4×C7C22×C412yes—
103C22×C4×C7C22×C411yescharacteristic
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2×D412yes—
106C22×C4×C7C2411yescharacteristic
107C22×Dic7—24no—
108C22×Dic7—21no—
109C22×Dic7—21no—
110C22×Dic7—21no—
111C22×Dic7—21no—
112C22×Dic7—42no—
113C22×Dic7—42no—
114C22×Dic7—42no—
115C22×Dic7C22⋊C411yescharacteristic
116C22×Dic7C22⋊C412yes—
117C22×Dic7C22⋊C411yescharacteristic
118C22×Dic7C22⋊C411yescharacteristic
119C22×Dic7C22⋊C412yes—
120C22×Dic7C22⋊C411yescharacteristic
121C22×Dic7C22×C412yes—
122C22×Dic7C22×C412yes—
123C22×Dic7C22×C412yes—
124C22×Dic7C22×C412yes—
125C7×M16—44no—
126C2×C7×C8—24no—
127C7×C4⋊C4—24no—
128C7×C4⋊C4—24no—
129C4⋊Dic7—24no—
130C4⋊Dic7—24no—
131Dic7⋊C4—24no—
132Dic7⋊C4—24no—
133Dic7⋊C4—24no—
134Dic7⋊C4—24no—
135Dic7⋊C4—24no—
136Dic7⋊C4—24no—
137Dic7⋊C4—24no—
138Dic7⋊C4—24no—
139C4×Dic7C22⋊C412yes—
140C4×Dic7C22⋊C412yes—
141C4×Dic7C22⋊C412yes—
142C4×Dic7C22⋊C412yes—
143C4×Dic7C2×D412yes—
144C4×Dic7C2×D412yes—
145C4×Dic7C2×D412yes—
146C4×Dic7C2×D412yes—
147#1136—44no—
148C2×C7⋊C8—24no—
149C2×#125—74no—
150C2×#125—74no—
151C2×#125—142no—
152C2×#125—142no—
153C2×#125—142no—
154C2×#125—142no—
155C22×C4⋊C4—71no—
156C22×C4⋊C4—71no—
157C22×C4⋊C4—71no—
158C22×C4⋊C4—71no—
159C22×C4⋊C4—71no—
160C22×C4⋊C4C2×D711yescharacteristic
161C22×C42—71no—
162#432—74no—
163#432—142no—
164#431—74no—
165#431—142no—
166#423—142no—
167#407—72no—
168#407C2×D712yes—
169C4×M16—142no—
170#324—74no—
171#324—78no—
172#324C2×D714yes—
173C23×C7—22no—
174C23×C7—22no—
175C23×C7—22no—
176C23×C7C2×C22⋊C411yescharacteristic
177C23×C7C2×C22⋊C411yescharacteristic
178C23×C7C23×C411yescharacteristic
179C2×C4×C7—24no—
180C2×C4×C7—22no—
181C2×C4×C7—22no—
182C2×C4×C7—24no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—41no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7C2×C22⋊C412yes—
192C2×C4×C7C22×D412yes—
193C2×Dic7—24no—
194C2×Dic7—22no—
195C2×Dic7—24no—
196C2×Dic7—24no—
197C2×Dic7—22no—
198C2×Dic7—22no—
199C2×Dic7—24no—
200C2×Dic7—41no—
201C2×Dic7—41no—
202C2×Dic7—42no—
203C2×Dic7—42no—
204C2×Dic7—42no—
205C2×Dic7—42no—
206C2×Dic7—42no—
207C2×Dic7—42no—
208C2×Dic7#10114yes—
209C2×Dic7C2×C22⋊C412yes—
210C2×Dic7C2×C22⋊C412yes—
211C2×Dic7C2×C22⋊C412yes—
212C2×Dic7C2×C22⋊C412yes—
213C7×C8—44no—
214C7⋊C8—44no—
215C23×C4—71no—
216C23×C4—71no—
217C23×C4—71no—
218C23×C4—71no—
219C23×C4C4×D711yescharacteristic
220C23×C4C4×D711yescharacteristic
221C23×C4C22×D711yescharacteristic
222C2×M16—22no—
223C2×M16—142no—
224#125—144no—
225#125—144no—
226#125—144no—
227#125—144no—
228#125—144no—
229#125—144no—
230#125—144no—
231#125—144no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—74no—
235C2×C4⋊C4—74no—
236C2×C4⋊C4—74no—
237C2×C4⋊C4—72no—
238C2×C4⋊C4—74no—
239C2×C4⋊C4—72no—
240C2×C4⋊C4—141no—
241C2×C4⋊C4—141no—
242C2×C4⋊C4—142no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4—141no—
245C2×C4⋊C4—141no—
246C2×C4⋊C4—141no—
247C2×C4⋊C4—141no—
248C2×C4⋊C4—142no—
249C2×C4⋊C4—142no—
250C2×C4⋊C4—141no—
251C2×C4⋊C4—141no—
252C2×C4⋊C4—142no—
253C2×C4⋊C4—142no—
254C2×C4⋊C4C4×D712yes—
255C2×C4⋊C4C22×D712yes—
256C2×C42—72no—
257C2×C42—72no—
258C2×C42—72no—
259C2×C42—72no—
260C2×C42—72no—
261C2×C42—72no—
262C4⋊C8—144no—
263C4⋊C8—144no—
264C22⋊C8—22no—
265C22⋊C8—74no—
266C22⋊C8—142no—
267C22⋊C8C22×D714yes—
268#97—144no—
269C4×C8—144no—
270C22×C7—28no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7#41012yes—
278C22×C7C2×#10112yes—
279C22×C7C22×C22⋊C411yescharacteristic
280C22×C7#56812yes—
281C4×C7—22no—
282C4×C7—22no—
283C4×C7—42no—
284C4×C7—42no—
285Dic7—22no—
286Dic7—22no—
287Dic7—22no—
288Dic7—42no—
289Dic7—42no—
290Dic7C2×#10114yes—
291C24C2×C4×D711yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—71no—
300C22×C4—72no—
301C22×C4—71no—
302C22×C4—72no—
303C22×C4—72no—
304C22×C4—71no—
305C22×C4—71no—
306C22×C4—72no—
307C22×C4—72no—
308C22×C4—72no—
309C22×C4—144no—
310C22×C4—141no—
311C22×C4—141no—
312C22×C4—141no—
313C22×C4—141no—
314C22×C4—282no—
315C22×C4—282no—
316C22×C4—282no—
317C22×C4C2×C4×D712yes—
318C22×C4C2×C4×D712yes—
319C22×C4C2×C4×D711yescharacteristic
320C22×C4D4×D712yes—
321C22×C4D4×D712yes—
322C22×C4C23×D711yescharacteristic
323M16—44no—
324M16—284no—
325C2×C8—24no—
326C2×C8—144no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—144no—
330C4⋊C4—144no—
331C4⋊C4—144no—
332C4⋊C4—144no—
333C4⋊C4—144no—
334C4⋊C4—144no—
335C4⋊C4—144no—
336C4⋊C4—144no—
337C4⋊C4—144no—
338C4⋊C4—144no—
339C42—72no—
340C42—72no—
341C42—72no—
342C42—72no—
343C42—72no—
344C42—72no—
345C42—72no—
346C42—72no—
347C2×C7—22no—
348C2×C7—22no—
349C2×C7—22no—
350C2×C7#287911yescharacteristic
351C2×C7C2×#41011yescharacteristic
352C2×C7C22×#10111yescharacteristic
353C23—22no—
354C23—22no—
355C23—22no—
356C23D7×C22⋊C411yescharacteristic
357C23D7×C22⋊C411yescharacteristic
358C23C22×C4×D711yescharacteristic
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—72no—
372C2×C4—74no—
373C2×C4—72no—
374C2×C4—72no—
375C2×C4—72no—
376C2×C4—144no—
377C2×C4—144no—
378C2×C4—144no—
379C2×C4—142no—
380C2×C4—142no—
381C2×C4—142no—
382C2×C4—144no—
383C2×C4—281no—
384C2×C4—281no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—282no—
388C2×C4—282no—
389C2×C4—282no—
390C2×C4—282no—
391C2×C4D7×C22⋊C412yes—
392C2×C4C2×D4×D712yes—
393C8—44no—
394C8—284no—
395C7#1739411yescharacteristic, complemented, Sylow
396C22—28no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22#8951212yes—
404C22D7×#10112yes—
405C22C2×D7×C22⋊C411yescharacteristic
406C22#9043612yes—
407C4—22no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—74no—
412C4—142no—
413C4—142no—
414C4—142no—
415C4—282no—
416C4—282no—
417C2—22no—
418C2—22no—
419C2—22no—
420C2#1173607511yescharacteristic
421C2C2×#8951211yescharacteristic
422C2C2×D7×#10111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1463C2
socleC22×C7C22×C22⋊C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1463D7
hypocenterC7#17394
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
6. #11725882C2semidirect
395. C7#17394semidirect

Maximal subgroups

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

subgroupindex
C22×#64522
#117360642
C7×#14632
#117293062
#117277252
#117258822
#173947