Finite Group Databasebeta

← #13251505

Subgroups — #13251505

Order 1792.

690 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2973C211yesmaximal, characteristic, complemented
2#11739823C211yesmaximal, characteristic
3#11729357C211yesmaximal, characteristic, complemented
4#11728572C212yesmaximal, complemented
5#11725556C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6503C412yes—
8C2×#6493—21no—
9C2×#6493—21no—
10C2×#6437C2211yescharacteristic
11C7×#435—24no—
12C7×#432C2212yes—
13C7×#407—21no—
14C2×C7×C22⋊C8—21no—
15#89760C412yes—
16#89732—21no—
17#89732—21no—
18C4×C4⋊Dic7C2211yescharacteristic
19#89698C2212yes—
20#89320—21no—
21#89320—21no—
22#89280—24no—
23#9457—71nomaximal, Sylow
24C22×C7×D4C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×C22⋊D7—22no—
28C22×C22⋊D7—22no—
29C22×D28—21no—
30C22×D28—21no—
31C22×D28C2×C412yes—
32C2×C7×M16—22no—
33C22×C7×C8—22no—
34C4×C7×D4—22no—
35C4×C7×D4—22no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C42×C7C2311yescharacteristic
40#6503—44no—
41#6493—44no—
42#6493—44no—
43C2×(C2×D7)⋊C4—22no—
44C2×(C2×D7)⋊C4—22no—
45C2×C4⋊Dic7—21no—
46C2×C4×Dic7C2×C412yes—
47#6437—22no—
48#6437—22no—
49C7×C4⋊C8D414yescomplemented
50C7×C22⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×C22⋊C8—22no—
53C7×#97—22no—
54C7⋊#95—21no—
55C22⋊(C7⋊C8)—22no—
56C4⋊(C7⋊C8)—22no—
57#6377—22no—
58#2973D711yescharacteristic, complemented
59#2541—71no—
60#1555—72no—
61#1466—71no—
62#1285—72no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C22⋊C411yescharacteristic
65C23×D7—22no—
66C2×C7×D4—22no—
67C2×C7×D4—22no—
68C2×C7×D4—41no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C2×C22⋊D7—44no—
81C2×C22⋊D7—44no—
82C2×C22⋊D7—44no—
83C2×C22⋊D7—44no—
84C2×C22⋊D7—82no—
85C2×C22⋊D7—82no—
86C22×Dic7—22no—
87C2×D28—24no—
88C2×D28—42no—
89C2×D28—42no—
90C2×D28—42no—
91C2×D28—42no—
92C2×D28—41no—
93C2×D28—42no—
94C2×D28—41no—
95C2×D28—82no—
96C7×M16—82no—
97C2×C7×C8—24no—
98C2×C7×C8—82no—
99C7×C4⋊C4—22no—
100C7×C22⋊C4—22no—
101C7×C22⋊C4—22no—
102C42×C7C2×D412yes—
103C42×C7#4112yes—
104(C2×D7)⋊C4—44no—
105(C2×D7)⋊C4—44no—
106C4⋊Dic7—24no—
107C4×Dic7—42no—
108C2×C7⋊C8—42no—
109C2×#127—72no—
110C2×#127—71no—
111C2×#121—141no—
112C2×#121—141no—
113C2×C4×D4C2×D711yescharacteristic
114#435—24no—
115#432—72no—
116#432C2×D712yes—
117#407—21no—
118C2×C22⋊C8—21no—
119#392—141no—
120#392—141no—
121#390—72no—
122C4×C4⋊C4—71no—
123#331—144no—
124#323—141no—
125#323—141no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7#10012yes—
136C23×C7C2×C22⋊C411yescharacteristic
137C22×D7—42no—
138C22×D7—42no—
139C22×D7—42no—
140C22×D7—42no—
141C22×D7—42no—
142C22×D7—42no—
143C22×D7—42no—
144C7×D4—44no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—22no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—41no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—81no—
156C2×C4×C7—81no—
157C2×C4×C7C4×D412yes—
158C2×C4×C7#12011yescharacteristic
159C2×C4×C7#12112yes—
160C2×C4×C7#12311yescharacteristic
161C22⋊D7—88no—
162C22⋊D7—88no—
163C2×Dic7—24no—
164C2×Dic7—42no—
165C2×Dic7—42no—
166D28—84no—
167D28—82no—
168D28—84no—
169D28—82no—
170C7×C8—44no—
171C7⋊C8—82no—
172C22×D4—72no—
173C22×D4—142no—
174C22×D4—142no—
175C22×D4—141no—
176C22×D4—141no—
177C22×D4C4×D711yescharacteristic
178C23×C4C22⋊D711yescharacteristic
179C23×C4C22⋊D711yescharacteristic
180C2×M16—22no—
181C22×C8—22no—
182#127—142no—
183#127—142no—
184#127—284no—
185#121—284no—
186#121—284no—
187C4×D4—22no—
188C4×D4—22no—
189C2×C4⋊C4—141no—
190C2×C4⋊C4C4×D711yescharacteristic
191C2×C22⋊C4—142no—
192C2×C22⋊C4—142no—
193C2×C22⋊C4D2811yescharacteristic
194C2×C22⋊C4D2811yescharacteristic
195C2×C42—72no—
196C2×C42C22×D711yescharacteristic
197C4⋊C8—142no—
198C4⋊C8C22⋊D714yescomplemented
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—142no—
203#97—22no—
204#97—142no—
205#95—141no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—22no—
209C22×C7—42no—
210C22×C7—42no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—81no—
217C22×C7—81no—
218C22×C7#38611yescharacteristic
219C22×C7C2×#10011yescharacteristic
220C22×C7#41311yescharacteristic
221C2×D7—82no—
222C2×D7—82no—
223C2×D7—82no—
224C2×D7—82no—
225C2×D7—82no—
226C2×D7—82no—
227C2×D7—82no—
228C4×C7—21no—
229C4×C7—22no—
230C4×C7—22no—
231C4×C7—41no—
232C4×C7—41no—
233C4×C7#45312yes—
234Dic7—44no—
235C24—142no—
236C24(C2×D7)⋊C411yescharacteristic
237C24(C2×D7)⋊C411yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—144no—
242C2×D4—282no—
243C2×D4—282no—
244C2×D4—282no—
245C2×D4—282no—
246C2×D4—284no—
247C2×D4—284no—
248C2×D4—284no—
249C2×D4—284no—
250C2×D4—281no—
251C2×D4—282no—
252C2×D4—281no—
253C2×D4—562no—
254C2×D4—562no—
255C2×D4—562no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—142no—
263C22×C4(C2×D7)⋊C411yescharacteristic
264C22×C4(C2×D7)⋊C411yescharacteristic
265C22×C4C2×C4×D711yescharacteristic
266C22×C4C2×D2811yescharacteristic
267C22×C4C2×C22⋊D711yescharacteristic
268M16—82no—
269C2×C8—24no—
270C2×C8—82no—
271C2×C8—282no—
272C4⋊C4—22no—
273C4⋊C4—144no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—284no—
277C22⋊C4—284no—
278C42—282no—
279C42#115712yes—
280C42C2×C22⋊D712yes—
281C2×C7—21no—
282C2×C7—21no—
283C2×C7—42no—
284C2×C7—81no—
285C2×C7#188711yescharacteristic
286C2×C7#190011yescharacteristic
287C2×C7#191011yescharacteristic
288D7—162no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—282no—
299C23—282no—
300C23—282no—
301C23—282no—
302C23—282no—
303C23—282no—
304C23—282no—
305C23#639712yes—
306C23C2×(C2×D7)⋊C411yescharacteristic
307D4—44no—
308D4—564no—
309D4—562no—
310D4—564no—
311D4—568no—
312D4—568no—
313D4—562no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—144no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4C4×C22⋊D712yes—
330C2×C4#649211yescharacteristic
331C2×C4#649312yes—
332C2×C4#651611yescharacteristic
333C8—44no—
334C8—562no—
335C7#945711yescharacteristic, complemented, Sylow
336C22—21no—
337C22—21no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—562no—
349C22—562no—
350C22—562no—
351C22—562no—
352C22—562no—
353C22—562no—
354C22—562no—
355C22C2×#639711yescharacteristic
356C22#8990411yescharacteristic
357C22#8998511yescharacteristic
358C4—22no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—284no—
364C4#9001312yes—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—1122no—
370C2#1172940411yescharacteristic
371C2#1172975611yescharacteristic
372C2#1172977911yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2973C2
socleC22×C7#386
prime radical#11739823C2
cosocleC2×C42×C7C23
hypercenter#2973D7
hypocenterC7#9457
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
1. C7×#2973C2semidirect
3. #11729357C2semidirect
4. #11728572C2semidirect
5. #11725556C2semidirect
49. C7×C4⋊C8D4semidirect
58. #2973D7semidirect
198. C4⋊C8C22⋊D7semidirect
335. C7#9457semidirect

Maximal subgroups

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

subgroupindex
C7×#29732
#117398232
#117293572
#117285722
#117255562
#94577