Finite Group Databasebeta

← #13252942

Subgroups — #13252942

Order 1792.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740788C211yesmaximal, characteristic
2C7×#1503C211yesmaximal, characteristic, complemented
3#11729595C211yesmaximal, characteristic, complemented
4#11728543C211yesmaximal, characteristic
5#11728493C211yesmaximal, characteristic, complemented
6#11727246C211yesmaximal, characteristic
7C4×#6382C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9#90168C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C4×C7×C4⋊C4C2211yescharacteristic
12#89860C411yescharacteristic, complemented
13#89857C411yescharacteristic, complemented
14#89853C411yescharacteristic, complemented
15#89851C411yescharacteristic, complemented
16#89820—21no—
17#89820—21no—
18C4×#1143—21no—
19C4×#1143—21no—
20C2×#6382—21no—
21C4×C4⋊Dic7C2211yescharacteristic
22C2×#6379—22no—
23C4×#1136C2211yescharacteristic
24C2×C4×C7⋊C8C2211yescharacteristic
25C7×#328—22no—
26#89327—22no—
27#89324—21no—
28#89265—22no—
29#89249—21no—
30C8×C7⋊C8—21no—
31#9717—71nomaximal, Sylow
32C2×#1157—21no—
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×C4⋊Q8C2×C411yescharacteristic
35C7×#125—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C7×#117—21no—
41C2×C7×C4⋊C4—21no—
42C2×C42×C7—21no—
43C2×C42×C7C2311yescharacteristic
44C2×#1143—21no—
45C2×#1143—21no—
46#6493—21no—
47#6493—21no—
48C2×C4⋊Dic7—21no—
49C2×C4×Dic7—21no—
50C2×#1136D412yes—
51C22×C7⋊C8D411yescharacteristic
52C22×C7⋊C8D411yescharacteristic
53#6438—21no—
54#6437C2×C411yescharacteristic
55C4×D28—21no—
56C4×D28—21no—
57#6432C2×C411yescharacteristic
58C7×C4⋊C8—21no—
59C7×C22⋊C8D412yes—
60C4×C7×C8—21no—
61C7×#95—21no—
62#6409—21no—
63#6409—21no—
64C7⋊#95—21no—
65#6383—21no—
66#6383—21no—
67#6382—24no—
68#6382—21no—
69#6382—21no—
70#6379—44no—
71#6377—21no—
72C4×C7⋊C8—21no—
73C4×C7⋊C8C2×C411yescharacteristic
74C4×C7⋊C8C2×C411yescharacteristic
75C4×C7⋊C8D411yescharacteristic
76C4×C7⋊C8D411yescharacteristic
77#3167—71no—
78#1986—71no—
79#1980—71no—
80C4×#106—71no—
81#1503D711yescharacteristic, complemented
82#1502—71no—
83C8×M16—71no—
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C22×C4×C7—41no—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C2×C22⋊D7—41no—
91C22×Dic7—41no—
92#1157—42no—
93#1157—42no—
94C2×D28—21no—
95C2×D28—21no—
96C2×D28—21no—
97C2×C4×D7—41no—
98C2×Dic14—21no—
99C2×C7×C8—22no—
100C7×C4⋊C4—22no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C22⋊C4—41no—
109C42×C7—41no—
110C42×C7—42no—
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115#1143—42no—
116#1143—42no—
117(C2×D7)⋊C4—41no—
118(C2×D7)⋊C4—41no—
119C4⋊Dic7—22no—
120C4⋊Dic7—21no—
121C4⋊Dic7—21no—
122C4⋊Dic7—41no—
123C4×Dic7—42no—
124#1136—22no—
125#1136SD1614yescomplemented
126C2×C7⋊C8—22no—
127C2×C7⋊C8—22no—
128C2×C7⋊C8—41no—
129C2×C7⋊C8C22⋊C411yescharacteristic
130C2×C7⋊C8C22⋊C411yescharacteristic
131C2×C7⋊C8C22⋊C411yescharacteristic
132C2×C7⋊C8C22⋊C411yescharacteristic
133#533C2×D711yescharacteristic
134#532—71no—
135#498—71no—
136#492—71no—
137#487—71no—
138#486—71no—
139#465—141no—
140#465—141no—
141C4×SD16—141no—
142C4×SD16—141no—
143#431C2×D711yescharacteristic
144C2×#106—141no—
145C2×#104—142no—
146C4×M16—71no—
147C2×C4×C8—71no—
148C4×C4⋊C4—71no—
149C4×C4⋊C4C2×D711yescharacteristic
150#340—141no—
151#339—142no—
152#328—22no—
153#327—142no—
154#322—141no—
155C82—141no—
156C23×C7#12711yescharacteristic
157C22×D7—41no—
158C7×Q8—41no—
159C7×Q8—41no—
160C7×Q8—81no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—81no—
168C2×C4×C7—81no—
169C2×C4×C7C2×C22⋊C411yescharacteristic
170C2×C4×C7C4×D411yescharacteristic
171C2×C4×C7C4×D411yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12111yescharacteristic
174C2×C4×C7#12411yescharacteristic
175C2×C4×C7C2×SD1612yes—
176C2×C4×C7#13512yes—
177C22⋊D7—81no—
178C22⋊D7—81no—
179C2×Dic7—42no—
180C2×Dic7—41no—
181C2×Dic7—81no—
182C2×Dic7—81no—
183D28—41no—
184D28—41no—
185D28—42no—
186D28—81no—
187C4×D7—82no—
188Dic14—42no—
189C7×C8—42no—
190C7⋊C8—21no—
191C7⋊C8—21no—
192C7⋊C8—21no—
193C7⋊C8—24no—
194C7⋊C8#10412yescomplemented
195C2×#41—141no—
196C2×SD16—141no—
197C2×SD16—141no—
198C2×M16—72no—
199C22×C8—71no—
200C22×C8—71no—
201C4⋊Q8—71no—
202C4⋊Q8C4×D711yescharacteristic
203C4⋊Q8C4×D711yescharacteristic
204#127—71no—
205#125—21no—
206#124—141no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209#121—141no—
210#121—141no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×D4—141no—
214C4×D4—141no—
215#117—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—141no—
218C2×C42—21no—
219C2×C42—141no—
220C2×C42C22×D711yescharacteristic
221#106—141no—
222#106—141no—
223#106—144no—
224C4⋊C8—21no—
225#104—284no—
226#103—141no—
227#103—141no—
228#102—141no—
229#102—141no—
230C22⋊C8C22⋊D712yes—
231#97—141no—
232C4×C8—21no—
233C4×C8—71no—
234C4×C8—71no—
235C4×C8—71no—
236C4×C8—71no—
237C4×C8—141no—
238#95—21no—
239#95—141no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7#39011yescharacteristic
243C22×C7#49211yescharacteristic
244C22×C7#49511yescharacteristic
245C2×D7—81no—
246C2×D7—81no—
247C2×D7—81no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—82no—
252C4×C7—81no—
253C4×C7—81no—
254C4×C7C2×#10411yescharacteristic
255C4×C7#42111yescharacteristic
256C4×C7C4×SD1612yes—
257C4×C7#46512yes—
258Dic7—82no—
259Dic7—81no—
260#41—282no—
261#41—282no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—141no—
265C2×D4—141no—
266C2×D4—141no—
267C2×D4—141no—
268C2×D4—281no—
269C22×C4—41no—
270C22×C4—281no—
271C22×C4—281no—
272C22×C4D4×D711yescharacteristic
273C22×C4D4×D711yescharacteristic
274C22×C4C2×C22⋊D711yescharacteristic
275SD16—282no—
276SD16—282no—
277M16—74no—
278M16—142no—
279C2×C8—22no—
280C2×C8—71no—
281C2×C8—71no—
282C2×C8—71no—
283C2×C8—71no—
284C2×C8—142no—
285C2×C8—142no—
286C2×C8—281no—
287C4⋊C4—22no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—142no—
296C4⋊C4—141no—
297C4⋊C4—141no—
298C4⋊C4—281no—
299C22⋊C4—41no—
300C22⋊C4—281no—
301C22⋊C4—281no—
302C42—41no—
303C42—42no—
304C42—282no—
305C42C2×C4×D711yescharacteristic
306C42#115711yescharacteristic
307C42D4×D711yescharacteristic
308C42#115911yescharacteristic
309C2×C7—21no—
310C2×C7—21no—
311C2×C7#196811yescharacteristic
312C2×C7#197611yescharacteristic
313C2×C7#198011yescharacteristic
314D7—161no—
315C23—281no—
316C23#650311yescharacteristic
317Q8—41no—
318Q8—41no—
319Q8—81no—
320Q8—282no—
321D4—282no—
322D4—281no—
323D4—281no—
324D4—561no—
325D4—561no—
326D4—561no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—282no—
336C2×C4—281no—
337C2×C4—561no—
338C2×C4—561no—
339C2×C4—562no—
340C2×C4D7×C22⋊C411yescharacteristic
341C2×C4(C22⋊D7)⋊C411yescharacteristic
342C2×C4#644711yescharacteristic
343C2×C4#644811yescharacteristic
344C2×C4D7×SD1612yes—
345C2×C4#648212yes—
346C2×C4C4×C22⋊D711yescharacteristic
347C2×C4#650211yescharacteristic
348C8—42no—
349C8—72no—
350C8—144no—
351C8—141no—
352C8—141no—
353C8—141no—
354C7#971711yescharacteristic, complemented, Sylow
355C22—21no—
356C22—21no—
357C22—561no—
358C22—561no—
359C22—561no—
360C22#8972911yescharacteristic
361C22#8994711yescharacteristic
362C22#8996311yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—562no—
370C4—561no—
371C4#8956012yes—
372C4#8957912yes—
373C4D7×#10411yescharacteristic
374C4#8959311yescharacteristic
375C2—21no—
376C2—21no—
377C2—1121no—
378C2#1172779611yescharacteristic
379C2#1172801111yescharacteristic
380C2#1172811211yescharacteristic
subgroupG/H
centerC2×C4#6447
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1503C2
socleC22×C7#390
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#1503D7
hypocenterC7#9717
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. C7×#1503C2semidirect
3. #11729595C2semidirect
5. #11728493C2semidirect
7. C4×#6382C2semidirect
12. #89860C4semidirect
13. #89857C4semidirect
14. #89853C4semidirect
15. #89851C4semidirect
81. #1503D7semidirect
125. #1136SD16semidirect
194. C7⋊C8#104semidirect
354. C7#9717semidirect

Maximal subgroups

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

subgroupindex
#117407882
C7×#15032
#117295952
#117285432
#117284932
#117272462
C4×#63822
#97177