Finite Group Databasebeta

← #13252792

Subgroups — #13252792

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739521C211yesmaximal, characteristic
2C7×#1749C211yesmaximal, characteristic
3#11729288C211yesmaximal, characteristic
4C2×#89262C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6487C2212yes—
7C2×#6432C411yescharacteristic, complemented
8C2×C7×C22⋊C8C2212yes—
9C2×C7×#97—21no—
10#89984—21no—
11#89984—21no—
12C2×C7⋊#95—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2212yes—
15#89707C411yescharacteristic, complemented
16#89701C412yescomplemented
17#89700—21no—
18#89700—21no—
19C7×#336—21no—
20#89262—28no—
21#9683—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C23×Dic7—22no—
26C22×Dic14C2×C412yes—
27C22×C7×C8—22no—
28C2×C42×C7—42no—
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7C2×C411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C22⋊Dic7—22no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C412yes—
39#6487—24no—
40#6487—24no—
41C2×Dic7⋊C4—22no—
42#6432—42no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8D414yes—
45C7×#97—42no—
46C7⋊#95—24no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—24no—
50C7⋊#95—42no—
51#2857—71no—
52#1749D711yescharacteristic
53C2×#328—74no—
54#1434—71no—
55C24×C7C2×D411yescharacteristic
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22×C411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×C4×C7C2×D412yes—
71C22×C4×C7#4112yes—
72C22×Dic7—24no—
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—42no—
76C22×Dic7—42no—
77C22×Dic7—42no—
78C22×Dic7—42no—
79C22×Dic7—42no—
80C22×Dic7—42no—
81C2×Dic14—24no—
82C2×Dic14—42no—
83C2×Dic14—42no—
84C2×C7×C8—24no—
85C2×C7×C8—42no—
86C2×C7×C8—42no—
87C42×C7—22no—
88C42×C7—22no—
89C42×C7—41no—
90C42×C7—41no—
91C22⋊Dic7—42no—
92C22⋊Dic7—42no—
93C4⋊Dic7—24no—
94C4⋊Dic7—42no—
95C4⋊Dic7—42no—
96C4⋊Dic7—42no—
97C4⋊Dic7—42no—
98Dic7⋊C4—44no—
99C2×C4⋊Q8—71no—
100C2×C22⋊Q8—72no—
101C22×C42C2×D711yescharacteristic
102C2×C22⋊C8C2×D712yes—
103C2×#97—21no—
104#391—72no—
105#389—141no—
106#389—141no—
107#384—71no—
108#380—141no—
109#380—141no—
110C2×#95—72no—
111C2×#95—141no—
112C2×#95—141no—
113#336—21no—
114#328—148no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×C22⋊C411yescharacteristic
120C23×C7#12411yescharacteristic
121C23×C7#12711yescharacteristic
122C23×C7#13712yes—
123C23×C7#13712yes—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—44no—
141C2×C4×C7—44no—
142C2×C4×C7#9912yes—
143C2×C4×C7#10412yes—
144C2×C4×C7C4×D412yes—
145C2×C4×C7#12112yes—
146C2×Dic7—44no—
147C2×Dic7—44no—
148C2×Dic7—44no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155C2×Dic7—82no—
156C2×Dic7—82no—
157C2×Dic7—82no—
158C2×Dic7—82no—
159Dic14—84no—
160C7×C8—44no—
161C22×Q8—72no—
162C23×C4—142no—
163C23×C4D2811yescharacteristic
164C23×C4D2811yescharacteristic
165C23×C4C22×D711yescharacteristic
166C22×C8—22no—
167C4⋊Q8—282no—
168C22⋊Q8—144no—
169C22⋊Q8—144no—
170C2×C4⋊C4—72no—
171C2×C4⋊C4—141no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4—141no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4—142no—
176C2×C22⋊C4—142no—
177C2×C42—42no—
178C2×C42C4×D711yescharacteristic
179C2×C42C4×D711yescharacteristic
180C2×C42C22⋊D711yescharacteristic
181C2×C42C22⋊D711yescharacteristic
182C22⋊C8—22no—
183C22⋊C8D2814yes—
184#97—42no—
185#95—144no—
186#95—142no—
187#95—142no—
188#95—144no—
189#95—282no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—22no—
198C22×C7#39011yescharacteristic
199C22×C7C2×#9911yescharacteristic
200C22×C7#41011yescharacteristic
201C22×C7#41911yescharacteristic
202C22×C7#41911yescharacteristic
203C22×C7C2×#10411yescharacteristic
204C22×C7#42111yescharacteristic
205C22×C7#49112yes—
206C22×C7#49712yes—
207C4×C7—22no—
208C4×C7—22no—
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—41no—
212C4×C7—41no—
213Dic7—84no—
214Dic7—82no—
215Dic7—82no—
216C24C2×D2811yescharacteristic
217C2×Q8—144no—
218C2×Q8—282no—
219C2×Q8—282no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—144no—
229C22×C4—142no—
230C22×C4—142no—
231C22×C4—282no—
232C22×C4—282no—
233C22×C4—282no—
234C22×C4—282no—
235C22×C4—282no—
236C22×C4—282no—
237C22×C4(C2×D7)⋊C411yescharacteristic
238C22×C4(C2×D7)⋊C411yescharacteristic
239C22×C4(C2×D7)⋊C411yescharacteristic
240C22×C4(C2×D7)⋊C411yescharacteristic
241C22×C4C2×C4×D711yescharacteristic
242C22×C4C2×D2812yes—
243C22×C4#115712yes—
244C22×C4C2×C22⋊D711yescharacteristic
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—144no—
249C4⋊C4—282no—
250C4⋊C4—282no—
251C4⋊C4—284no—
252C4⋊C4—282no—
253C4⋊C4—282no—
254C22⋊C4—282no—
255C22⋊C4—282no—
256C42—22no—
257C42—22no—
258C42—41no—
259C42—41no—
260C2×C7—22no—
261C2×C7—22no—
262C2×C7#150611yescharacteristic
263C2×C7#150811yescharacteristic
264C2×C7#151111yescharacteristic
265C2×C7#151511yescharacteristic
266C2×C7#196011yescharacteristic
267C2×C7#196411yescharacteristic
268C2×C7#198011yescharacteristic
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23#643711yescharacteristic
274C23#643811yescharacteristic
275C23#647212yes—
276C23#647212yes—
277C23C2×(C2×D7)⋊C411yescharacteristic
278Q8—564no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—284no—
298C2×C4—284no—
299C2×C4—284no—
300C2×C4—282no—
301C2×C4—282no—
302C2×C4—282no—
303C2×C4—282no—
304C2×C4—282no—
305C2×C4—282no—
306C2×C4—562no—
307C2×C4—562no—
308C2×C4—562no—
309C2×C4—562no—
310C2×C4#638012yes—
311C2×C4#639912yes—
312C2×C4C4×D2812yes—
313C2×C4#649312yes—
314C8—44no—
315C7#968311yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22#8947512yes—
325C22#8948712yes—
326C22#8971111yescharacteristic
327C22C2×#638011yescharacteristic
328C22#8977811yescharacteristic
329C22#8989211yescharacteristic
330C22#8989211yescharacteristic
331C22C2×#639911yescharacteristic
332C22#8991111yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—564no—
340C4—562no—
341C4—562no—
342C2—22no—
343C2—22no—
344C2#1172721011yescharacteristic
345C2#1172757611yescharacteristic
346C2#1172762111yescharacteristic
347C2#1172773411yescharacteristic
348C2#1172773511yescharacteristic
349C2#1172859011yescharacteristic
350C2#1172859111yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1749C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1749D7
hypocenterC7#9683
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
7. C2×#6432C4semidirect
15. #89707C4semidirect
16. #89701C4semidirect
315. C7#9683semidirect

Maximal subgroups

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

subgroupindex
#117395212
C7×#17492
#117292882
C2×#892622
#96837