Finite Group Databasebeta

← #13249539

Subgroups — #13249539

Order 1792.

756 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#11739525C211yesmaximal, characteristic
2C2×#89704C211yesmaximal, characteristic, complemented
3C7×#1817C211yesmaximal, characteristic, complemented
4C2×#89265C212yesmaximal
5#11725481C212yesmaximal, complemented
6C22×C42×C7C2211yescharacteristic
7C2×#6493C2211yescharacteristic
8C2×#6492—21no—
9C22×C4×Dic7—21no—
10C2×C7×C22⋊C8C2212yes—
11#89985—21no—
12#89984C2211yescharacteristic
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2212yes—
15#89712—22no—
16#89711C412yescomplemented
17#89704—24no—
18#89700C412yescomplemented
19C7×#336—22no—
20#89265—24no—
21#9149—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C22×C22⋊D7—21no—
26C23×Dic7—22no—
27C22×D28C2×C411yescharacteristic
28C22×C7×C8—22no—
29C2×C42×C7—22no—
30C2×C42×C7—22no—
31C2×C42×C7C2×C412yes—
32C2×C42×C7D412yes—
33C2×C22⋊Dic7—21no—
34#6493—22no—
35#6493—22no—
36#6492—44no—
37C2×(C2×D7)⋊C4—22no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C412yes—
42C2×Dic7⋊C4—22no—
43C2×C4×Dic7—24no—
44C2×C4×Dic7—28no—
45C7×C22⋊C8—22no—
46C7×C22⋊C8—22no—
47C7⋊#95—24no—
48C7⋊#95—22no—
49C7⋊#95—22no—
50C7⋊#95—21no—
51C7⋊#95D414yes—
52#2854—71no—
53C2×#376—71no—
54#1817D711yescharacteristic, complemented
55C2×#327—72no—
56#1294—72no—
57C24×C7C2×D411yescharacteristic
58C23×D7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—24no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—42no—
67C22×C4×C7C22⋊C412yes—
68C22×C4×C7C22⋊C412yes—
69C22×C4×C7C22×C411yescharacteristic
70C22×C4×C7C2×D411yescharacteristic
71C22×C4×C7C2×D412yes—
72C22×C4×C7#4112yes—
73C2×C22⋊D7—41no—
74C2×C22⋊D7—41no—
75C2×C22⋊D7—42no—
76C2×C22⋊D7—42no—
77C22×Dic7—28no—
78C22×Dic7—24no—
79C22×Dic7—21no—
80C22×Dic7—22no—
81C22×Dic7—44no—
82C22×Dic7—42no—
83C22×Dic7—42no—
84C2×D28—22no—
85C2×D28—42no—
86C2×C7×C8—44no—
87C2×C7×C8—42no—
88C42×C7—24no—
89C42×C7—24no—
90C22⋊Dic7—42no—
91(C2×D7)⋊C4—44no—
92(C2×D7)⋊C4—44no—
93(C2×D7)⋊C4—42no—
94C4⋊Dic7—22no—
95C4⋊Dic7—42no—
96Dic7⋊C4—44no—
97C4×Dic7—216no—
98C2×#123—141no—
99C2×#121—71no—
100C22×C42—141no—
101C22×C42C2×D711yescharacteristic
102C2×C22⋊C8C2×D712yes—
103#390—72no—
104#389—72no—
105#388—142no—
106#386—141no—
107#380—71no—
108#376—144no—
109C2×#95—72no—
110C2×#95—141no—
111#336—22no—
112#327—144no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7#12011yescharacteristic
117C23×C7#12311yescharacteristic
118C23×C7#13512yes—
119C23×C7AGL(1,ℤ/8)12yes—
120C22×D7—41no—
121C22×D7—42no—
122C22×D7—41no—
123C22×D7—41no—
124C22×D7—42no—
125C2×C4×C7—24no—
126C2×C4×C7—24no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7#10414yes—
139C2×C4×C7C4×D412yes—
140C2×C4×C7#12112yes—
141C22⋊D7—82no—
142C22⋊D7—82no—
143C2×Dic7—28no—
144C2×Dic7—44no—
145C2×Dic7—48no—
146C2×Dic7—48no—
147C2×Dic7—44no—
148C2×Dic7—41no—
149C2×Dic7—42no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154D28—82no—
155C7×C8—82no—
156C22×D4—71no—
157C22×D4—141no—
158C23×C4—142no—
159C23×C4D2811yescharacteristic
160C23×C4D2811yescharacteristic
161C23×C4C22×D711yescharacteristic
162C22×C8—22no—
163#123—284no—
164#121—142no—
165#121—142no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—72no—
168C2×C4⋊C4—142no—
169C2×C22⋊C4—141no—
170C2×C22⋊C4—142no—
171C2×C22⋊C4—142no—
172C2×C22⋊C4—141no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—144no—
176C2×C42—148no—
177C2×C42C4×D712yes—
178C2×C42C22⋊D712yes—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#95—74no—
182#95—144no—
183#95—142no—
184#95—142no—
185#95—141no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—24no—
191C22×C7—24no—
192C22×C7#38611yescharacteristic
193C22×C7#41012yes—
194C22×C7#41611yescharacteristic
195C22×C7#41811yescharacteristic
196C22×C7C2×#10412yes—
197C22×C7#44912yes—
198C22×C7#48212yes—
199C2×D7—81no—
200C2×D7—81no—
201C2×D7—81no—
202C2×D7—82no—
203C2×D7—82no—
204C4×C7—24no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—42no—
208Dic7—48no—
209Dic7—81no—
210Dic7—82no—
211C24—141no—
212C24C2×D2811yescharacteristic
213C2×D4—142no—
214C2×D4—282no—
215C2×D4—281no—
216C2×D4—281no—
217C2×D4—282no—
218C2×D4—282no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—148no—
228C22×C4—144no—
229C22×C4—142no—
230C22×C4—141no—
231C22×C4—284no—
232C22×C4—282no—
233C22×C4—282no—
234C22×C4(C2×D7)⋊C412yes—
235C22×C4(C2×D7)⋊C412yes—
236C22×C4C2×C4×D711yescharacteristic
237C22×C4D4×D712yes—
238C22×C4#115912yes—
239C22×C4C2×C22⋊D711yescharacteristic
240C2×C8—44no—
241C2×C8—42no—
242C4⋊C4—142no—
243C4⋊C4—282no—
244C4⋊C4—284no—
245C22⋊C4—284no—
246C22⋊C4—282no—
247C22⋊C4—282no—
248C22⋊C4—284no—
249C42—24no—
250C42—24no—
251C42—1416no—
252C2×C7—22no—
253C2×C7—22no—
254C2×C7#150012yes—
255C2×C7#150512yes—
256C2×C7#187511yescharacteristic
257C2×C7#189611yescharacteristic
258C2×C7#189911yescharacteristic
259D7—161no—
260C23—22no—
261C23—22no—
262C23—281no—
263C23—281no—
264C23—282no—
265C23—281no—
266C23—282no—
267C23#644511yescharacteristic
268C23#644911yescharacteristic
269C23#646712yes—
270C23#647112yes—
271C23C2×(C2×D7)⋊C411yescharacteristic
272D4—562no—
273D4—562no—
274D4—562no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—148no—
289C2×C4—288no—
290C2×C4—284no—
291C2×C4—288no—
292C2×C4—284no—
293C2×C4—281no—
294C2×C4—282no—
295C2×C4—281no—
296C2×C4—282no—
297C2×C4—281no—
298C2×C4—282no—
299C2×C4#639914yes—
300C2×C4(C22⋊D7)⋊C412yes—
301C2×C4#650212yes—
302C8—82no—
303C7#914911yescharacteristic, complemented, Sylow
304C22—21no—
305C22—21no—
306C22—24no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—561no—
311C22—561no—
312C22—562no—
313C22—561no—
314C22—562no—
315C22#8950612yes—
316C22#8950912yes—
317C22#8973011yescharacteristic
318C22#8977812yes—
319C22#8988511yescharacteristic
320C22#8990311yescharacteristic
321C22C2×#639912yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—288no—
327C4—561no—
328C4—562no—
329C2—22no—
330C2—22no—
331C2—1121no—
332C2#1172728911yescharacteristic
333C2#1172768711yescharacteristic
334C2#1172771911yescharacteristic
335C2#1172854212yes—
336C2#1172858612yes—
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1817C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#1817D7
hypocenterC7#9149
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#89704C2semidirect
3. C7×#1817C2semidirect
5. #11725481C2semidirect
16. #89711C4semidirect
18. #89700C4semidirect
54. #1817D7semidirect
303. C7#9149semidirect

Maximal subgroups

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

subgroupindex
#117395252
C2×#897042
C7×#18172
C2×#892652
#117254812
#91497