Finite Group Databasebeta

← #13249567

Subgroups — #13249567

Order 1792.

902 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 143 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×#2848C211yesmaximal, characteristic
2C2×#89983C211yesmaximal, characteristic
3#11729880C211yesmaximal, characteristic
4C2×#89333C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C24×C4×C7C2211yescharacteristic
7C2×C7×C22⋊Q8C412yescomplemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C22⋊Dic7—21no—
10C7×#385—21no—
11C7×#385—22no—
12C7×#380C412yescomplemented
13C7×#380C2211yescharacteristic
14C7×#379—21no—
15#89983—24no—
16C2×C7⋊#95—21no—
17C2×C7⋊#95C2212yes—
18C2×C22⋊(C7⋊C8)—22no—
19C2×C22⋊(C7⋊C8)C2212yes—
20#89333—24no—
21#9154—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C22×C7×Q8C2×C411yescharacteristic
24C23×C4×C7—22no—
25C23×C4×C7—22no—
26C23×C4×C7—44no—
27C23×C4×C7C2×C412yes—
28C23×C4×C7D412yes—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C23×Dic7—22no—
32C7×C22⋊Q8—22no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—44no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C412yes—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C22⋊Dic7—24no—
44C2×C22⋊Dic7—44no—
45C22×C7⋊C8—22no—
46C7×#95—22no—
47C7×#95—21no—
48C7⋊#95—24no—
49C7⋊#95—22no—
50C7⋊#95D414yes—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53C22⋊(C7⋊C8)—44no—
54#2848D711yescharacteristic
55C2×#381—71no—
56#1779—71no—
57C2×#328—72no—
58#1294—72no—
59C24×C7—24no—
60C24×C7—21no—
61C24×C7—21no—
62C24×C7—22no—
63C24×C7—22no—
64C24×C7—42no—
65C24×C7C22⋊C412yes—
66C24×C7C2×D411yescharacteristic
67C2×C7×Q8—22no—
68C2×C7×Q8—42no—
69C22×C4×C7—24no—
70C22×C4×C7—24no—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
73C22×C4×C7—24no—
74C22×C4×C7—24no—
75C22×C4×C7—21no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—42no—
79C22×C4×C7—42no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7—48no—
83C22×C4×C7—48no—
84C22×C4×C7C22⋊C412yes—
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D412yes—
88C22×C4×C7#4112yes—
89C22×Dic7—24no—
90C22×Dic7—44no—
91C22×Dic7—44no—
92C22×Dic7—42no—
93C22×Dic7—42no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—44no—
97C7×C4⋊C4—42no—
98C7×C22⋊C4—44no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—42no—
103C22⋊Dic7—48no—
104C2×C7⋊C8—44no—
105C2×C7⋊C8—42no—
106C24×C4C2×D711yescharacteristic
107C2×C22⋊Q8Dic712yescomplemented
108C2×C22⋊Q8C2×D711yescharacteristic
109C22×C22⋊C4—141no—
110C2×C22⋊C8—72no—
111C2×C22⋊C8—142no—
112#385—21no—
113#385—22no—
114#381—144no—
115#380Dic712yescomplemented
116#380C2×D711yescharacteristic
117#379—21no—
118C2×#95—72no—
119C2×#95—141no—
120#328—144no—
121C23×C7—22no—
122C23×C7—22no—
123C23×C7—24no—
124C23×C7—24no—
125C23×C7—22no—
126C23×C7—22no—
127C23×C7—22no—
128C23×C7—22no—
129C23×C7—22no—
130C23×C7—22no—
131C23×C7—48no—
132C23×C7—48no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7#9914yes—
138C23×C7C2×C22⋊C411yescharacteristic
139C23×C7#12011yescharacteristic
140C23×C7#12311yescharacteristic
141C23×C7#13512yes—
142C23×C7#13712yes—
143C7×Q8—82no—
144C2×C4×C7—22no—
145C2×C4×C7—22no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—22no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—48no—
152C2×C4×C7—48no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—42no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7C4×D412yes—
163C2×C4×C7#12112yes—
164C2×Dic7—44no—
165C2×Dic7—44no—
166C2×Dic7—44no—
167C2×Dic7—84no—
168C2×Dic7—84no—
169C7⋊C8—82no—
170C25C22⋊D711yescharacteristic
171C22×Q8C2×Dic711yescharacteristic
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—44no—
175C23×C4—142no—
176C23×C4C2×Dic712yes—
177C23×C4C22⋊D712yes—
178C23×C4C22⋊D711yescharacteristic
179C23×C4C22×D711yescharacteristic
180C22×C8—142no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—44no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C2×Dic711yescharacteristic
187C2×C4⋊C4C2×Dic712yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—144no—
193C2×C22⋊C4—284no—
194C22⋊C8—142no—
195C22⋊C8—142no—
196C22⋊C8—284no—
197#95—22no—
198#95—21no—
199#95—74no—
200#95—144no—
201#95—142no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—24no—
207C22×C7—24no—
208C22×C7—24no—
209C22×C7—24no—
210C22×C7—24no—
211C22×C7—48no—
212C22×C7—48no—
213C22×C7—42no—
214C22×C7—42no—
215C22×C7—42no—
216C22×C7—42no—
217C22×C7#38611yescharacteristic
218C22×C7C2×#9912yes—
219C22×C7#41611yescharacteristic
220C22×C7#41911yescharacteristic
221C22×C7#42112yes—
222C22×C7#45212yes—
223C22×C7#48512yes—
224C4×C7—22no—
225C4×C7—22no—
226C4×C7—42no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—82no—
230Dic7—84no—
231C24—24no—
232C24—21no—
233C24—21no—
234C24—22no—
235C24—22no—
236C24—42no—
237C24C22⋊Dic712yes—
238C24C2×C22⋊D711yescharacteristic
239C2×Q8—22no—
240C2×Q8—42no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—144no—
257C22×C4—284no—
258C22×C4—284no—
259C22×C4—282no—
260C22×C4—282no—
261C22×C4C22⋊Dic712yes—
262C22×C4D4×D712yes—
263C22×C4#115912yes—
264C22×C4C22×Dic711yescharacteristic
265C22×C4C2×C22⋊D711yescharacteristic
266C2×C8—284no—
267C2×C8—282no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—288no—
278C2×C7—22no—
279C2×C7—22no—
280C2×C7—44no—
281C2×C7#151112yes—
282C2×C7#151612yes—
283C2×C7#187311yescharacteristic
284C2×C7#189711yescharacteristic
285C2×C7#189811yescharacteristic
286C23—22no—
287C23—22no—
288C23—24no—
289C23—24no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—48no—
297C23—48no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23#640814yes—
303C23#649811yescharacteristic
304C23#650011yescharacteristic
305C23#650512yes—
306C23#651312yes—
307C23C2×C22⋊Dic711yescharacteristic
308Q8—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—284no—
328C2×C4—284no—
329C2×C4—284no—
330C2×C4—564no—
331C2×C4—564no—
332C2×C4D4×Dic712yes—
333C2×C4#650212yes—
334C8—562no—
335C7#915411yescharacteristic, complemented, Sylow
336C22—21no—
337C22—21no—
338C22—24no—
339C22—24no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—48no—
346C22—48no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22#8981712yes—
352C22#8981912yes—
353C22C2×#640812yes—
354C22#8999211yescharacteristic
355C22#8999711yescharacteristic
356C22#9000511yescharacteristic
357C22#9000812yes—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—81no—
362C4—82no—
363C4—81no—
364C4—564no—
365C2—22no—
366C2—22no—
367C2—44no—
368C2#1172942411yescharacteristic
369C2#1172944312yes—
370C2#1172944711yescharacteristic
371C2#1172946412yes—
372C2#1172946611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2848C2
socleC23×C7C2×C22⋊C4
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2848D7
hypocenterC7#9154
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
4. C2×#89333C2semidirect
5. #11725430C2semidirect
7. C2×C7×C22⋊Q8C4semidirect
12. C7×#380C4semidirect
107. C2×C22⋊Q8Dic7semidirect
115. #380Dic7semidirect
335. C7#9154semidirect

Maximal subgroups

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

subgroupindex
C7×#28482
C2×#899832
#117298802
C2×#893332
#117254302
#91547