Finite Group Databasebeta

← #13249477

Subgroups — #13249477

Order 1792.

568 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2857C211yesmaximal, characteristic
2C2×#89698C211yesmaximal, characteristic
3C2×#89341C212yesmaximal, complemented
4#11729284C211yesmaximal, characteristic
5#11725440C212yesmaximal, complemented
6C2×C7×C4⋊Q8C2211yescharacteristic
7C2×C7×C22⋊Q8C412yes—
8C22×C42×C7C2211yescharacteristic
9C7×#391C412yes—
10C7×#389—21no—
11C7×#389—21no—
12C7×#384C2211yescharacteristic
13C7×#380—21no—
14C7×#380—21no—
15C2×C22⋊(C7⋊C8)—21no—
16C2×C22⋊(C7⋊C8)—21no—
17#89698—24no—
18C2×C4⋊(C7⋊C8)C2212yes—
19C2×C4⋊(C7⋊C8)C2212yes—
20C2×#6377—21no—
21#89341—24no—
22#89319—21no—
23#9136—71nomaximal, Sylow
24C22×C7×Q8C2×C412yes—
25C23×C4×C7C2×C411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C7×C4⋊Q8—22no—
29C7×C4⋊Q8—22no—
30C7×C22⋊Q8—44no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C412yes—
37C2×C7×C22⋊C4—22no—
38C2×C42×C7—22no—
39C2×C42×C7—22no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7D411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7C2311yescharacteristic
44C22×C7⋊C8—22no—
45C22×C7⋊C8—22no—
46C7×#95—22no—
47C7×#95—22no—
48C22⋊(C7⋊C8)—24no—
49C22⋊(C7⋊C8)—42no—
50C4⋊(C7⋊C8)—22no—
51C4⋊(C7⋊C8)—22no—
52C4⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)D414yes—
54#6377—24no—
55C2×#432—71no—
56#2857D711yescharacteristic
57#1797—71no—
58C2×#333—72no—
59#1296—72no—
60C24×C7C22⋊C411yescharacteristic
61C2×C7×Q8—24no—
62C2×C7×Q8—42no—
63C2×C7×Q8—42no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—22no—
68C22×C4×C7—21no—
69C22×C4×C7—22no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7—41no—
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C7×C4⋊C4—42no—
84C7×C4⋊C4—44no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—42no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C22⋊C4—44no—
92C42×C7—22no—
93C42×C7—21no—
94C42×C7—22no—
95C42×C7—21no—
96C42×C7C2×D412yes—
97C42×C7#4112yes—
98C2×C7⋊C8—24no—
99C2×C7⋊C8—42no—
100C2×C7⋊C8—42no—
101C2×C7⋊C8—42no—
102C2×C7⋊C8—42no—
103C2×C7⋊C8—42no—
104C2×C4⋊Q8C2×D711yescharacteristic
105C2×C22⋊Q8Dic712yes—
106C22×C42C2×D711yescharacteristic
107#432—144no—
108C2×C4⋊C8—72no—
109C2×C4⋊C8—72no—
110C2×C22⋊C8—141no—
111C2×C22⋊C8—141no—
112C2×#97—141no—
113#391Dic712yes—
114#389—21no—
115#389—21no—
116#384C2×D711yescharacteristic
117#380—21no—
118#380—21no—
119#336—141no—
120#333—144no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7#10012yes—
128C23×C7C2×C22⋊C411yescharacteristic
129C7×Q8—84no—
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—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—44no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7C4×D412yes—
153C2×C4×C7#12011yescharacteristic
154C2×C4×C7#12112yes—
155C2×C4×C7#12311yescharacteristic
156C2×C4×C7#13512yes—
157C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C7#13712yes—
159C2×C4×C7#13712yes—
160C2×C4×C7#13711yescharacteristic
161C7⋊C8—44no—
162C7⋊C8—82no—
163C22×Q8C2×Dic712yes—
164C23×C4C2×Dic711yescharacteristic
165C23×C4C22⋊D711yescharacteristic
166C23×C4C22⋊D711yescharacteristic
167C22×C8—142no—
168C22×C8—142no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C22⋊Q8—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C2×Dic712yes—
178C2×C22⋊C4—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42C2×Dic711yescharacteristic
182C2×C42C22⋊D711yescharacteristic
183C2×C42C22⋊D711yescharacteristic
184C2×C42C22×D711yescharacteristic
185C4⋊C8—74no—
186C4⋊C8—142no—
187C4⋊C8—142no—
188C4⋊C8—142no—
189C22⋊C8—144no—
190C22⋊C8—282no—
191#97—144no—
192#95—22no—
193#95—22no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—41no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10011yescharacteristic
203C22×C7#41311yescharacteristic
204C22×C7#41611yescharacteristic
205C22×C7#41711yescharacteristic
206C22×C7#41911yescharacteristic
207C22×C7#41911yescharacteristic
208C4×C7—22no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7—82no—
213C4×C7—82no—
214C4×C7#45212yes—
215C4×C7#46812yes—
216C4×C7#47012yes—
217C4×C7#48512yes—
218C24C22⋊Dic711yescharacteristic
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—42no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4C22⋊Dic711yescharacteristic
236C22×C4C22⋊Dic711yescharacteristic
237C22×C4C22⋊Dic711yescharacteristic
238C22×C4C22×Dic711yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240C22×C4C2×C22⋊D711yescharacteristic
241C2×C8—144no—
242C2×C8—282no—
243C2×C8—282no—
244C2×C8—282no—
245C2×C8—282no—
246C2×C8—282no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C22⋊C4—44no—
256C42—22no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42D4×D712yes—
261C42#115912yes—
262C2×C7—42no—
263C2×C7#155111yescharacteristic
264C2×C7#155411yescharacteristic
265C2×C7#155811yescharacteristic
266C2×C7#155911yescharacteristic
267C2×C7#187311yescharacteristic
268C2×C7#189411yescharacteristic
269C2×C7#190011yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23#640712yes—
277C23C2×C22⋊Dic711yescharacteristic
278Q8—84no—
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—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4D4×Dic712yes—
302C2×C4#649811yescharacteristic
303C2×C4#650011yescharacteristic
304C2×C4#650212yes—
305C2×C4#650512yes—
306C2×C4#650512yes—
307C2×C4#651211yescharacteristic
308C2×C4#651312yes—
309C2×C4#651411yescharacteristic
310C8—284no—
311C8—562no—
312C7#913611yescharacteristic, complemented, Sylow
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#640711yescharacteristic
321C22#8999211yescharacteristic
322C22#8999711yescharacteristic
323C22#8999711yescharacteristic
324C22#9000411yescharacteristic
325C22#9000511yescharacteristic
326C22#9000911yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4#8981712yes—
334C4#8981912yes—
335C4#8982112yes—
336C4#8982412yes—
337C2—42no—
338C2#1172943711yescharacteristic
339C2#1172944711yescharacteristic
340C2#1172945111yescharacteristic
341C2#1172953111yescharacteristic
342C2#1172956111yescharacteristic
343C2#1172958611yescharacteristic
344C2#1172958711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2857C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2857D7
hypocenterC7#9136
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. C2×#89341C2semidirect
5. #11725440C2semidirect
312. C7#9136semidirect

Maximal subgroups

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

subgroupindex
C7×#28572
C2×#896982
C2×#893412
#117292842
#117254402
#91367