Finite Group Databasebeta

← #13276192

Subgroups — #13276192

Order 1792.

2166 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic7⋊C4C211yesmaximal, characteristic
2C2×C7×#380C211yesmaximal, characteristic
3C2×C4×C22⋊Dic7C211yesmaximal, characteristic
4#11729320C214yesmaximal, complemented
5#11729298C212yesmaximal
6#11729298C212yesmaximal
7#11727159C214yesmaximal, complemented
8C24×C4×C7C2211yescharacteristic
9C24×Dic7C2211yescharacteristic
10C24×Dic7C2211yescharacteristic
11C22×C7×C4⋊C4C2212yes—
12C22×C7×C22⋊C4C2212yes—
13C22×C22⋊Dic7C2212yes—
14C22×Dic7⋊C4—24no—
15C22×Dic7⋊C4—24no—
16C22×Dic7⋊C4C418yescomplemented
17C22×Dic7⋊C4C2212yes—
18C22×Dic7⋊C4C2212yes—
19C22×C4×Dic7C2212yes—
20C7×#380—24no—
21C2×C7×#95C2212yes—
22C4×C22⋊Dic7—24no—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2218yes—
25C2×C7⋊#95C2218yes—
26#13923—71nomaximal, Sylow
27C25×C7C2311yescharacteristic
28C23×C4×C7—44no—
29C23×C4×C7C2×C414yes—
30C23×C4×C7D414yes—
31C23×C4×C7D414yes—
32C23×C4×C7C2314yes—
33C23×C4×C7C2311yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C23×Dic7—22no—
36C23×Dic7—22no—
37C23×Dic7—28no—
38C23×Dic7—22no—
39C23×Dic7—22no—
40C23×Dic7—44no—
41C23×Dic7C2×C414yes—
42C23×Dic7C2×C414yes—
43C23×Dic7C2312yes—
44C23×Dic7C2314yes—
45C23×Dic7C2312yes—
46C2×C7×C4⋊C4—24no—
47C2×C7×C4⋊C4—44no—
48C2×C7×C22⋊C4—24no—
49C2×C7×C22⋊C4—28no—
50C2×C22⋊Dic7—24no—
51C2×C22⋊Dic7—24no—
52C2×C22⋊Dic7Q818yes—
53C2×Dic7⋊C4—232no—
54C2×Dic7⋊C4—24no—
55C2×Dic7⋊C4—24no—
56C2×Dic7⋊C4—24no—
57C2×Dic7⋊C4—24no—
58C2×Dic7⋊C4C2×C418yes—
59C2×Dic7⋊C4C2×C418yes—
60C2×C4×Dic7—24no—
61C2×C4×Dic7—44no—
62C7×#95—24no—
63C7⋊#95—24no—
64C7⋊#95—216no—
65C7⋊#95—48no—
66C23×C4⋊C4—71no—
67C2×#380D711yescharacteristic
68C2×C4×C22⋊C4—71no—
69#1441—74no—
70#1436—74no—
71C22⋊#95—72no—
72C22⋊#95—72no—
73C24×C7—24no—
74C24×C7—24no—
75C24×C7C22×C412yes—
76C24×C7C2×D412yes—
77C24×C7C2×D412yes—
78C24×C7C2×Q814yes—
79C24×C7#4114yes—
80C24×C7C2411yescharacteristic
81C22×C4×C7—24no—
82C22×C4×C7—24no—
83C22×C4×C7—24no—
84C22×C4×C7—28no—
85C22×C4×C7—28no—
86C22×C4×C7—28no—
87C22×C4×C7—48no—
88C22×C4×C7—48no—
89C22×C4×C7—48no—
90C22×C4×C7C22⋊C4116yes—
91C22×C4×C7C22×C412yes—
92C22×C4×C7C22×C412yes—
93C22×C4×C7C2×D412yes—
94C22×C4×C7C2×D412yes—
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×D412yes—
97C22×Dic7—216no—
98C22×Dic7—216no—
99C22×Dic7—28no—
100C22×Dic7—22no—
101C22×Dic7—22no—
102C22×Dic7—24no—
103C22×Dic7—216no—
104C22×Dic7—28no—
105C22×Dic7—24no—
106C22×Dic7—22no—
107C22×Dic7—22no—
108C22×Dic7—28no—
109C22×Dic7—24no—
110C22×Dic7—28no—
111C22×Dic7—28no—
112C22×Dic7—48no—
113C22×Dic7—48no—
114C22×Dic7—48no—
115C22×Dic7C22×C414yes—
116C22×Dic7C22×C414yes—
117C22×Dic7C2×Q818yes—
118C22×Dic7#4118yes—
119C7×C4⋊C4—48no—
120C7×C22⋊C4—48no—
121C22⋊Dic7—216no—
122Dic7⋊C4—216no—
123Dic7⋊C4—216no—
124C4×Dic7—48no—
125C24×C4—71no—
126C24×C4—71no—
127C24×C4C2×D711yescharacteristic
128C22×C4⋊C4—78no—
129C22×C4⋊C4—72no—
130C22×C4⋊C4—72no—
131C22×C4⋊C4—144no—
132C22×C4⋊C4—144no—
133C22×C4⋊C4C2×D712yes—
134C22×C22⋊C4—72no—
135C22×C22⋊C4C2×D712yes—
136C22×C42—72no—
137#380—24no—
138C4×C22⋊C4—144no—
139C2×#95—78no—
140C2×#95—72no—
141C2×#95—78no—
142C2×#95C2×D712yes—
143C23×C7—24no—
144C23×C7—24no—
145C23×C7—24no—
146C23×C7—24no—
147C23×C7—24no—
148C23×C7—24no—
149C23×C7—216no—
150C23×C7—216no—
151C23×C7C2×C22⋊C414yes—
152C23×C7C4×Q818yes—
153C23×C7C22⋊Q818yes—
154C23×C7#12414yes—
155C23×C7C4⋊Q814yes—
156C23×C7C23×C411yescharacteristic
157C23×C7C22×D411yescharacteristic
158C23×C7C22×D411yescharacteristic
159C23×C7C22×Q812yes—
160C23×C7C2×#4112yes—
161C23×C7C2×#4114yes—
162C23×C7H(5,𝔽2)14yes—
163C2×C4×C7—24no—
164C2×C4×C7—24no—
165C2×C4×C7—24no—
166C2×C4×C7—24no—
167C2×C4×C7—24no—
168C2×C4×C7—24no—
169C2×C4×C7—416no—
170C2×C4×C7—44no—
171C2×C4×C7—44no—
172C2×C4×C7—44no—
173C2×C4×C7—416no—
174C2×C4×C7C2×C22⋊C414yes—
175C2×C4×C7C2×C22⋊C414yes—
176C2×Dic7—232no—
177C2×Dic7—24no—
178C2×Dic7—24no—
179C2×Dic7—28no—
180C2×Dic7—28no—
181C2×Dic7—24no—
182C2×Dic7—28no—
183C2×Dic7—28no—
184C2×Dic7—216no—
185C2×Dic7—416no—
186C2×Dic7—44no—
187C2×Dic7—44no—
188C2×Dic7—44no—
189C2×Dic7—44no—
190C2×Dic7—48no—
191C2×Dic7#11718yes—
192C25C22×D711yescharacteristic
193C23×C4—44no—
194C23×C4—72no—
195C23×C4—74no—
196C23×C4—74no—
197C23×C4—72no—
198C23×C4—74no—
199C23×C4—148no—
200C23×C4—142no—
201C23×C4—142no—
202C23×C4—142no—
203C23×C4—142no—
204C23×C4—284no—
205C23×C4C4×D714yes—
206C23×C4D2814yes—
207C23×C4C22⋊D714yes—
208C23×C4C22×D714yes—
209C23×C4C22×D711yescharacteristic
210C23×C4C22×D711yescharacteristic
211C2×C4⋊C4—24no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—78no—
214C2×C4⋊C4—78no—
215C2×C4⋊C4—1432no—
216C2×C4⋊C4—144no—
217C2×C4⋊C4—144no—
218C2×C4⋊C4—144no—
219C2×C4⋊C4—144no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—78no—
223C2×C22⋊C4—144no—
224C2×C22⋊C4—144no—
225C2×C42—144no—
226C2×C42—284no—
227#95—24no—
228#95—1416no—
229#95—144no—
230#95—288no—
231C22×C7—24no—
232C22×C7—216no—
233C22×C7—28no—
234C22×C7—28no—
235C22×C7—216no—
236C22×C7—24no—
237C22×C7#39118yes—
238C22×C7C22×C22⋊C411yescharacteristic
239C22×C7C2×#11712yes—
240C22×C7C2×C4×Q812yes—
241C22×C7#51812yes—
242C22×C7C2×C22⋊Q812yes—
243C22×C7#52512yes—
244C22×C7#52512yes—
245C22×C7C2×#12411yescharacteristic
246C22×C7C2×C4⋊Q811yescharacteristic
247C22×C7#53412yes—
248C22×C7#53412yes—
249C22×C7#55318yes—
250C22×C7#55418yes—
251C4×C7—24no—
252C4×C7—24no—
253C4×C7—84no—
254Dic7—28no—
255Dic7—44no—
256Dic7—48no—
257C24—24no—
258C24—24no—
259C24C2×C4×D712yes—
260C24C2×D2812yes—
261C24#115914yes—
262C24Q8×D714yes—
263C24C2×C22⋊D712yes—
264C24C23×D711yescharacteristic
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—74no—
275C22×C4—78no—
276C22×C4—74no—
277C22×C4—78no—
278C22×C4—1416no—
279C22×C4—1416no—
280C22×C4—1416no—
281C22×C4—148no—
282C22×C4—142no—
283C22×C4—144no—
284C22×C4—142no—
285C22×C4—148no—
286C22×C4—144no—
287C22×C4—142no—
288C22×C4—144no—
289C22×C4—148no—
290C22×C4—148no—
291C22×C4—148no—
292C22×C4—142no—
293C22×C4—288no—
294C22×C4—288no—
295C22×C4—288no—
296C22×C4(C2×D7)⋊C4116yes—
297C22×C4C2×C4×D712yes—
298C22×C4C2×C4×D712yes—
299C22×C4C2×D2812yes—
300C22×C4C2×D2812yes—
301C22×C4C2×C22⋊D712yes—
302C22×C4C2×C22⋊D712yes—
303C4⋊C4—48no—
304C4⋊C4—1416no—
305C4⋊C4—1416no—
306C22⋊C4—48no—
307C22⋊C4—1416no—
308C42—288no—
309C2×C7—28no—
310C2×C7#228011yescharacteristic
311C2×C7C2×#39111yescharacteristic
312C2×C7#230811yescharacteristic
313C2×C7#234514yes—
314C2×C7#257914yes—
315C2×C7#258312yes—
316C2×C7#259112yes—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—216no—
324C23—216no—
325C23#645018yes—
326C23#646018yes—
327C23C2×(C2×D7)⋊C414yes—
328C23#649914yes—
329C23#650714yes—
330C23C22×C4×D711yescharacteristic
331C23C22×D2811yescharacteristic
332C23C2×#115912yes—
333C23#654814yes—
334C23C2×Q8×D712yes—
335C23D7×#4114yes—
336C23C22×C22⋊D711yescharacteristic
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—416no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—416no—
348C2×C4—78no—
349C2×C4—1432no—
350C2×C4—144no—
351C2×C4—148no—
352C2×C4—144no—
353C2×C4—148no—
354C2×C4—148no—
355C2×C4—148no—
356C2×C4—144no—
357C2×C4—1416no—
358C2×C4—2816no—
359C2×C4—284no—
360C2×C4—284no—
361C2×C4—284no—
362C2×C4—284no—
363C2×C4—288no—
364C2×C4C2×(C2×D7)⋊C414yes—
365C2×C4C2×(C2×D7)⋊C414yes—
366C7#1392311yescharacteristic, complemented, Sylow
367C22—24no—
368C22—28no—
369C22—216no—
370C22—24no—
371C22—216no—
372C22—28no—
373C22#8974618yes—
374C22#9017712yes—
375C22#9018412yes—
376C22C2×#645012yes—
377C22C2×#646012yes—
378C22D7×#11712yes—
379C22#9021512yes—
380C22#9029218yes—
381C22#9032718yes—
382C22C22×(C2×D7)⋊C411yescharacteristic
383C22C2×#649911yescharacteristic
384C22#9049512yes—
385C22C2×#650711yescharacteristic
386C22#9051912yes—
387C4—24no—
388C4—24no—
389C4—84no—
390C4—148no—
391C4—284no—
392C4—288no—
393C2—28no—
394C2#1173132514yes—
395C2#1173139414yes—
396C2#1173956611yescharacteristic
397C2C2×#8974611yescharacteristic
398C2#1173971211yescharacteristic
399C2#1174011912yes—
400C2#1174023012yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC2×C7×#380C2
socle, cosocleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#380D7
hypocenterC7#13923
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. #11729320C2semidirect
7. #11727159C2semidirect
16. C22×Dic7⋊C4C4semidirect
366. C7#13923semidirect

Maximal subgroups

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

subgroupindex
C23×Dic7⋊C42
C2×C7×#3802
C2×C4×C22⋊Dic72
#117293202
#117292982
#117271592
#139237