Finite Group Databasebeta

← #13252931

Subgroups — #13252931

Order 1792.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739521C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic
3#11729284C211yesmaximal, characteristic
4C2×#89247C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6487C2212yes—
7C2×#6432C411yescharacteristic, complemented
8C2×C7×#95—21no—
9C2×C7×#95—21no—
10C2×C7×#95C2212yes—
11#89984—21no—
12#89984—21no—
13C2×C22⋊(C7⋊C8)C2212yes—
14#89707C411yescharacteristic, complemented
15#89701C412yescomplemented
16#89700—21no—
17#89700—21no—
18C2×C4⋊(C7⋊C8)—21no—
19#89319—21no—
20#89247—28no—
21#9714—71nomaximal, Sylow
22C23×C4×C7—22no—
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×Dic14C2×C412yes—
27C2×C42×C7—42no—
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C22⋊Dic7—22no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7C2×C412yes—
38#6487—24no—
39#6487—24no—
40C2×Dic7⋊C4—22no—
41C22×C7⋊C8—22no—
42#6432—42no—
43C7×#95—22no—
44C7×#95—42no—
45C7×#95—42no—
46C7×#95D414yes—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C22⋊(C7⋊C8)—24no—
50C4⋊(C7⋊C8)—42no—
51#2857—71no—
52#1797—71no—
53C2×#328—74no—
54#1434D711yescharacteristic
55C24×C7C2×D411yescharacteristic
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—24no—
59C22×C4×C7—41no—
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7—42no—
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22⋊C411yescharacteristic
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—22no—
82C2×Dic14—24no—
83C2×Dic14—42no—
84C2×Dic14—42no—
85C42×C7—22no—
86C42×C7—22no—
87C42×C7—41no—
88C42×C7—41no—
89C22⋊Dic7—44no—
90C4⋊Dic7—24no—
91C4⋊Dic7—42no—
92C4⋊Dic7—42no—
93C4⋊Dic7—42no—
94C4⋊Dic7—42no—
95Dic7⋊C4—42no—
96Dic7⋊C4—42no—
97C2×C7⋊C8—42no—
98C2×C7⋊C8—42no—
99C2×C7⋊C8—42no—
100C2×C4⋊Q8—71no—
101C2×C22⋊Q8—72no—
102C22×C42C2×D711yescharacteristic
103C2×C4⋊C8—141no—
104C2×C22⋊C8—72no—
105#391—72no—
106#389—141no—
107#389—141no—
108#384—71no—
109#380—141no—
110#380—141no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95C2×D712yes—
114#336—141no—
115#328—148no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#12112yes—
122C23×C7C2×SD1611yescharacteristic
123C23×C7C2×Q1611yescharacteristic
124C23×C7#13712yes—
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—42no—
141C2×C4×C7—42no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7—44no—
145C2×C4×C7—44no—
146C2×C4×C7—82no—
147C2×C4×C7—82no—
148C2×C4×C7—82no—
149C2×C4×C7—82no—
150C2×C4×C7#9912yes—
151C2×C4×C7#10312yes—
152C2×C4×C7#10312yes—
153C2×C4×C7C4×D412yes—
154C2×C4×C7#12411yescharacteristic
155C2×C4×C7#12711yescharacteristic
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7—42no—
159C2×Dic7—42no—
160C2×Dic7—42no—
161C2×Dic7—42no—
162Dic14—84no—
163C7⋊C8—82no—
164C22×Q8—72no—
165C23×C4—22no—
166C23×C4C22⋊D711yescharacteristic
167C23×C4C22⋊D711yescharacteristic
168C23×C4C22×D711yescharacteristic
169C22×C8—142no—
170C4⋊Q8—282no—
171C22⋊Q8—144no—
172C22⋊Q8—144no—
173C2×C4⋊C4—72no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4—141no—
176C2×C4⋊C4—141no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4—142no—
179C2×C22⋊C4—142no—
180C2×C42—42no—
181C2×C42C4×D711yescharacteristic
182C2×C42C4×D711yescharacteristic
183C2×C42D2811yescharacteristic
184C2×C42D2811yescharacteristic
185C4⋊C8—282no—
186C22⋊C8—144no—
187#95—22no—
188#95—42no—
189#95—42no—
190#95—142no—
191#95—142no—
192#95D2814yes—
193C22×C7—22no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7#39011yescharacteristic
202C22×C7C2×#9911yescharacteristic
203C22×C7#41011yescharacteristic
204C22×C7C2×#10311yescharacteristic
205C22×C7#41911yescharacteristic
206C22×C7#42111yescharacteristic
207C22×C7#42111yescharacteristic
208C22×C7#46112yes—
209C22×C7#46212yes—
210C4×C7—21no—
211C4×C7—22no—
212C4×C7—21no—
213C4×C7—22no—
214C4×C7—22no—
215C4×C7—22no—
216C4×C7—41no—
217C4×C7—84no—
218Dic7—82no—
219Dic7—82no—
220C24C2×C22⋊D711yescharacteristic
221C2×Q8—144no—
222C2×Q8—282no—
223C2×Q8—282no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—142no—
239C22×C4—142no—
240C22×C4(C2×D7)⋊C411yescharacteristic
241C22×C4(C2×D7)⋊C411yescharacteristic
242C22×C4(C2×D7)⋊C411yescharacteristic
243C22×C4(C2×D7)⋊C411yescharacteristic
244C22×C4#114212yes—
245C22×C4#114412yes—
246C22×C4C2×C4×D711yescharacteristic
247C22×C4C2×D2811yescharacteristic
248C22×C4C2×D2812yes—
249C22×C4#115712yes—
250C2×C8—282no—
251C2×C8—282no—
252C2×C8—282no—
253C4⋊C4—144no—
254C4⋊C4—282no—
255C4⋊C4—282no—
256C4⋊C4—282no—
257C4⋊C4—282no—
258C4⋊C4—282no—
259C4⋊C4—282no—
260C22⋊C4—284no—
261C42—22no—
262C42—22no—
263C42—41no—
264C42—41no—
265C2×C7—22no—
266C2×C7—22no—
267C2×C7#151111yescharacteristic
268C2×C7#151511yescharacteristic
269C2×C7#151811yescharacteristic
270C2×C7#152011yescharacteristic
271C2×C7#196311yescharacteristic
272C2×C7#196411yescharacteristic
273C2×C7#198111yescharacteristic
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×(C2×D7)⋊C411yescharacteristic
279C23#649312yes—
280C23C2×#114211yescharacteristic
281C23C2×#114411yescharacteristic
282C23#651412yes—
283Q8—564no—
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—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—282no—
310C2×C4—282no—
311C2×C4—282no—
312C2×C4—282no—
313C2×C4—282no—
314C2×C4—282no—
315C2×C4#638012yes—
316C2×C4#638412yes—
317C2×C4#638412yes—
318C2×C4C4×D2812yes—
319C2×C4#643711yescharacteristic
320C2×C4#643811yescharacteristic
321C8—562no—
322C7#971411yescharacteristic, complemented, Sylow
323C22—22no—
324C22—22no—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22#8971111yescharacteristic
332C22C2×#638011yescharacteristic
333C22C2×#638411yescharacteristic
334C22#8977611yescharacteristic
335C22#8977711yescharacteristic
336C22#8977711yescharacteristic
337C22#8977811yescharacteristic
338C22#8979112yes—
339C22#8980312yes—
340C4—21no—
341C4—22no—
342C4—21no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—84no—
348C4—562no—
349C4—562no—
350C2—22no—
351C2—22no—
352C2#1172719411yescharacteristic
353C2#1172721011yescharacteristic
354C2#1172724011yescharacteristic
355C2#1172726411yescharacteristic
356C2#1172728711yescharacteristic
357C2#1172730011yescharacteristic
358C2#1172732011yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1434C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1434D7
hypocenterC7#9714
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
7. C2×#6432C4semidirect
14. #89707C4semidirect
15. #89701C4semidirect
322. C7#9714semidirect

Maximal subgroups

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

subgroupindex
#117395212
C7×#14342
#117292842
C2×#892472
#97147