Finite Group Databasebeta

← #13249516

Subgroups — #13249516

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×C4⋊C4C211yesmaximal, characteristic
2#11739520C211yesmaximal, characteristic
3#11729285C211yesmaximal, characteristic
4C2×#89247C212yesmaximal
5#11725418C212yesmaximal
6C22×C7×C4⋊C4—21no—
7C22×C42×C7—21no—
8C22×C42×C7C2211yescharacteristic
9C2×#6487C2211yescharacteristic
10C4×C7×C4⋊C4—24no—
11C2×C7×#95C2212yes—
12#89984C2211yescharacteristic
13#89983—21no—
14#89982—21no—
15C2×C22⋊(C7⋊C8)C2212yes—
16#89705—22no—
17#89701C412yescomplemented
18#89700C412yescomplemented
19#89319—22no—
20#89247—24no—
21#9144—71nomaximal, Sylow
22C23×C4×C7—22no—
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×Dic14C2×C411yescharacteristic
27C2×C7×C4⋊C4—28no—
28C2×C7×C4⋊C4—24no—
29C2×C42×C7—22no—
30C2×C42×C7—22no—
31C2×C42×C7—24no—
32C2×C42×C7—42no—
33C2×C42×C7—42no—
34C2×C42×C7C2×C412yes—
35C2×C42×C7D412yes—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7—21no—
38C2×C4⋊Dic7C2×C411yescharacteristic
39C2×C4⋊Dic7C2×C412yes—
40#6487—22no—
41#6487—22no—
42C2×Dic7⋊C4—22no—
43C2×Dic7⋊C4—21no—
44C22×C7⋊C8—22no—
45C7×#95—22no—
46C7×#95D414yes—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—22no—
50C7⋊#95—21no—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53#2855—71no—
54C2×C4×C4⋊C4D711yescharacteristic
55#1817—71no—
56C2×#328—72no—
57#1294—72no—
58C24×C7C2×D411yescharacteristic
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—24no—
66C22×C4×C7—28no—
67C22×C4×C7—24no—
68C22×C4×C7—42no—
69C22×C4×C7—44no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22⋊C412yes—
74C22×C4×C7SD1612yes—
75C22×C4×C7Q1612yes—
76C22×C4×C7C22×C411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7C2×D412yes—
79C22×C4×C7#4112yes—
80C22×Dic7—22no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C2×Dic14—22no—
84C2×Dic14—42no—
85C7×C4⋊C4—216no—
86C42×C7—24no—
87C42×C7—24no—
88C42×C7—48no—
89C22⋊Dic7—41no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92C22⋊Dic7—41no—
93C4⋊Dic7—22no—
94C4⋊Dic7—42no—
95Dic7⋊C4—44no—
96Dic7⋊C4—42no—
97C2×C7⋊C8—44no—
98C2×C7⋊C8—42no—
99C2×C22⋊Q8—71no—
100C22×C4⋊C4—21no—
101C22×C42—21no—
102C22×C42C2×D711yescharacteristic
103C2×C22⋊C8—72no—
104#391—72no—
105#389—72no—
106#387—142no—
107#385—141no—
108#381—141no—
109#380—71no—
110C4×C4⋊C4—24no—
111C2×#95C2×D712yes—
112#336—142no—
113#328—144no—
114C23×C7—22no—
115C23×C7—22no—
116C23×C7C2×C22⋊C411yescharacteristic
117C23×C7#12011yescharacteristic
118C23×C7#12311yescharacteristic
119C23×C7C2×SD1611yescharacteristic
120C23×C7C2×Q1611yescharacteristic
121C23×C7AGL(1,ℤ/8)11yescharacteristic
122C23×C7#13711yescharacteristic
123C2×C4×C7—24no—
124C2×C4×C7—24no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
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—22no—
135C2×C4×C7—28no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—44no—
139C2×C4×C7—48no—
140C2×C4×C7—48no—
141C2×C4×C7—44no—
142C2×C4×C7#10314yes—
143C2×C4×C7#10414yes—
144C2×C4×C7C4×D412yes—
145C2×C4×C7#12112yes—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7—42no—
153Dic14—82no—
154C7⋊C8—82no—
155C22×Q8—71no—
156C23×C4—22no—
157C23×C4C22⋊D711yescharacteristic
158C23×C4C22⋊D711yescharacteristic
159C23×C4C22×D711yescharacteristic
160C22×C8—142no—
161C22⋊Q8—142no—
162C22⋊Q8—142no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—71no—
166C2×C4⋊C4—72no—
167C2×C4⋊C4—142no—
168C2×C4⋊C4—141no—
169C2×C22⋊C4—141no—
170C2×C22⋊C4—141no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—24no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42C4×D712yes—
177C2×C42D2812yes—
178C22⋊C8—142no—
179C22⋊C8—142no—
180#95—22no—
181#95—142no—
182#95—142no—
183#95—142no—
184#95—141no—
185#95C22⋊D714yes—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—22no—
191C22×C7—22no—
192C22×C7—22no—
193C22×C7—22no—
194C22×C7#38611yescharacteristic
195C22×C7#41012yes—
196C22×C7C2×#10311yescharacteristic
197C22×C7#41711yescharacteristic
198C22×C7C2×#10412yes—
199C22×C7#45011yescharacteristic
200C22×C7#45111yescharacteristic
201C22×C7#48311yescharacteristic
202C22×C7#48411yescharacteristic
203C4×C7—24no—
204C4×C7—24no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—48no—
208Dic7—81no—
209Dic7—81no—
210Dic7—82no—
211C24C2×C22⋊D711yescharacteristic
212C2×Q8—142no—
213C2×Q8—282no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—142no—
228C22×C4—141no—
229C22×C4—141no—
230C22×C4(C2×D7)⋊C412yes—
231C22×C4(C2×D7)⋊C412yes—
232C22×C4#114212yes—
233C22×C4#114412yes—
234C22×C4C2×C4×D711yescharacteristic
235C22×C4C2×D2811yescharacteristic
236C22×C4#115712yes—
237C22×C4C2×C22⋊D712yes—
238C2×C8—284no—
239C2×C8—282no—
240C4⋊C4—216no—
241C4⋊C4—142no—
242C4⋊C4—282no—
243C4⋊C4—282no—
244C4⋊C4—284no—
245C22⋊C4—281no—
246C22⋊C4—281no—
247C22⋊C4—281no—
248C22⋊C4—281no—
249C42—24no—
250C42—24no—
251C42—48no—
252C2×C7—21no—
253C2×C7—21no—
254C2×C7—21no—
255C2×C7—21no—
256C2×C7#150312yes—
257C2×C7#150912yes—
258C2×C7#187211yescharacteristic
259C2×C7#189611yescharacteristic
260C2×C7#189911yescharacteristic
261C23—22no—
262C23—22no—
263C23C2×(C2×D7)⋊C411yescharacteristic
264C23#649211yescharacteristic
265C23#649511yescharacteristic
266C23C2×#114211yescharacteristic
267C23#650511yescharacteristic
268C23C2×#114411yescharacteristic
269C23#651611yescharacteristic
270Q8—562no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—28no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4—281no—
293C2×C4—281no—
294C2×C4—282no—
295C2×C4—282no—
296C2×C4—282no—
297C2×C4#637914yes—
298C2×C4#638414yes—
299C2×C4C4×C22⋊D712yes—
300C2×C4#649312yes—
301C8—562no—
302C7#914411yescharacteristic, complemented, Sylow
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22C2×#637912yes—
312C22#8973811yescharacteristic
313C22C2×#638411yescharacteristic
314C22#8974211yescharacteristic
315C22#8974311yescharacteristic
316C22#8977812yes—
317C22#8998511yescharacteristic
318C22#8999011yescharacteristic
319C22#9000011yescharacteristic
320C4—24no—
321C4—24no—
322C4—22no—
323C4—22no—
324C4—48no—
325C4—562no—
326C4—561no—
327C4—561no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#1172722412yes—
333C2#1172722512yes—
334C2#1172928611yescharacteristic
335C2#1172931311yescharacteristic
336C2#1172934911yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C4×C7×C4⋊C4C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenterC2×C4×C4⋊C4D7
hypocenterC7#9144
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
17. #89701C4semidirect
18. #89700C4semidirect
302. C7#9144semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×C4⋊C42
#117395202
#117292852
C2×#892472
#117254182
#91447