Finite Group Databasebeta

← #13248972

Subgroups — #13248972

Order 1792.

492 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739887C211yesmaximal, characteristic
2#11736487C211yesmaximal, characteristic
3C7×#1487C211yesmaximal, characteristic, complemented
4#11729389C211yesmaximal, characteristic, complemented
5#11728526C211yesmaximal, characteristic
6#11725547C211yesmaximal, characteristic, complemented
7#11725520C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×#6470—21no—
10#90168C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C7×#423C2211yescharacteristic
13#89999C412yes—
14#89901—21no—
15#89799—21no—
16C4×#1143—22no—
17#89792—21no—
18#89740C412yes—
19#89697C2211yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21C2×#6377C2211yescharacteristic
22#89608—22no—
23#89608—22no—
24C7×Q8⋊C8—22no—
25#89319—21no—
26#89319—21no—
27#89281—21no—
28#89280—21no—
29#9047—71nomaximal, Sylow
30C22×C7×Q8C2×C411yescharacteristic
31C2×#1157—21no—
32C2×C7×M16—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C4×C7×Q8—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C42×C7—21no—
40C2×C42×C7C2311yescharacteristic
41C2×#1143—22no—
42#6493—21no—
43#6493C2×C412yes—
44C22×C7⋊C8—21no—
45C22×C7⋊C8D411yescharacteristic
46C22×C7⋊C8D411yescharacteristic
47#6470—42no—
48#6470—42no—
49C2×#1131—22no—
50C2×C8×D7—22no—
51#6438—21no—
52#6437D411yescharacteristic
53C4×D28—21no—
54C4×D28D412yes—
55#6432D411yescharacteristic
56C7×C4⋊C8—21no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8D412yes—
59C7×C22⋊C8—21no—
60C4×C7×C8—21no—
61#6409—22no—
62C22⋊(C7⋊C8)C2×C412yes—
63(C2×D7)⋊C8—24no—
64#6383—22no—
65#6382—22no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)—21no—
69#6377—21no—
70#6377—21no—
71C4×C7⋊C8—21no—
72#3103—71no—
73#2932—71no—
74#1985—71no—
75#1540—71no—
76#1487D711yescharacteristic, complemented
77#1301—71no—
78#1281—71no—
79C2×C7×Q8—21no—
80C2×C7×Q8—21no—
81C2×C7×Q8—42no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7C22×C411yescharacteristic
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7#4111yescharacteristic
87C2×C22⋊D7—22no—
88#1157—41no—
89#1157—41no—
90#1157—41no—
91#1157—41no—
92C2×D28—21no—
93C2×D28—21no—
94C2×D28C22⋊C412yes—
95C2×C4×D7—22no—
96C2×Dic14—21no—
97C7×M16—41no—
98C7×M16—41no—
99C2×C7×C8—22no—
100C2×C7×C8—41no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C42×C7—41no—
106C42×C7—41no—
107C42×C7C2×D411yescharacteristic
108C42×C7C2×D411yescharacteristic
109C42×C7C2×D411yescharacteristic
110C42×C7#4111yescharacteristic
111#1143—44no—
112(C2×D7)⋊C4—22no—
113(C2×D7)⋊C4—22no—
114C4⋊Dic7—21no—
115C4⋊Dic7C22⋊C412yes—
116C2×C7⋊C8—21no—
117C2×C7⋊C8—21no—
118C2×C7⋊C8—22no—
119C2×C7⋊C8—21no—
120C2×C7⋊C8—21no—
121C2×C7⋊C8—21no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8SD1612yescomplemented
126#1131—44no—
127C8×D7—44no—
128C2×#131—141no—
129#532—71no—
130C2×C4×Q8C2×D711yescharacteristic
131#481—72no—
132#475—141no—
133#460—141no—
134#448—72no—
135C4×SD16—142no—
136#435—142no—
137#435—142no—
138#431—71no—
139#431C2×D711yescharacteristic
140#423C2×D711yescharacteristic
141C2×C4⋊C8—71no—
142#408—141no—
143C2×#97—71no—
144#336—141no—
145#336—141no—
146#332—141no—
147#331—141no—
148Q8⋊C8—22no—
149C23×C7C4×D411yescharacteristic
150C22×D7—22no—
151C7×Q8—22no—
152C7×Q8—22no—
153C7×Q8—41no—
154C7×Q8—41no—
155C2×C4×C7—22no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—41no—
162C2×C4×C7C2×C22⋊C411yescharacteristic
163C2×C4×C7C4×D411yescharacteristic
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7#12311yescharacteristic
168C2×C4×C7C2×SD1611yescharacteristic
169C2×C4×C7#13511yescharacteristic
170C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C7#13711yescharacteristic
172C22⋊D7—42no—
173C22⋊D7—42no—
174C2×Dic7—22no—
175D28—42no—
176D28—41no—
177D28—41no—
178D28—81no—
179C4×D7—42no—
180C4×D7—42no—
181Dic14—41no—
182Dic14—41no—
183C7×C8—42no—
184C7×C8—81no—
185C7⋊C8—22no—
186C7⋊C8—42no—
187C7⋊C8—41no—
188C7⋊C8—81no—
189C2×#41—141no—
190C22×Q8C4×D711yescharacteristic
191C2×SD16—142no—
192#131—282no—
193#131—282no—
194C2×M16—21no—
195C2×M16—142no—
196C22×C8—71no—
197C22×C8—71no—
198C22×C8—142no—
199C22×C8—141no—
200C4⋊Q8—71no—
201#127—71no—
202#124—141no—
203#121—72no—
204#121—141no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×D4—72no—
210C4×D4—141no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C4×D711yescharacteristic
213C2×C42—21no—
214C2×C42C22×D711yescharacteristic
215#106—142no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—141no—
219C4⋊C8—141no—
220C4⋊C8—141no—
221C4⋊C8C22⋊D712yes—
222#103—142no—
223#102—142no—
224C22⋊C8—21no—
225C22⋊C8—72no—
226C22⋊C8—144no—
227#97—141no—
228#97—141no—
229C4×C8—21no—
230C4×C8—141no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7#38611yescharacteristic
234C22×C7C4×SD1611yescharacteristic
235C22×C7#44011yescharacteristic
236C2×D7—42no—
237C2×D7—42no—
238C2×D7—42no—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—42no—
244C4×C7—81no—
245C4×C7#45011yescharacteristic
246C4×C7#45211yescharacteristic
247C4×C7#46111yescharacteristic
248C4×C7#46311yescharacteristic
249Dic7—42no—
250#41—281no—
251#41—281no—
252#41—281no—
253#41—281no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—42no—
257C2×Q8—141no—
258C2×D4—72no—
259C2×D4—142no—
260C2×D4—141no—
261C2×D4—141no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—142no—
265C22×C4C2×C4×D711yescharacteristic
266C22×C4D4×D711yescharacteristic
267C22×C4#115911yescharacteristic
268SD16—284no—
269M16—41no—
270M16—41no—
271M16—284no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—72no—
275C2×C8—141no—
276C2×C8—141no—
277C2×C8—142no—
278C2×C8—141no—
279C2×C8—141no—
280C2×C8—141no—
281C2×C8—284no—
282C2×C8—281no—
283C2×C8—281no—
284C2×C8—281no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—72no—
290C4⋊C4—141no—
291C22⋊C4—142no—
292C22⋊C4—142no—
293C42—41no—
294C42—41no—
295C42#115711yescharacteristic
296C42D4×D711yescharacteristic
297C42D4×D711yescharacteristic
298C42C2×C22⋊D711yescharacteristic
299C2×C7—21no—
300C2×C7—21no—
301C2×C7#187711yescharacteristic
302C2×C7#188111yescharacteristic
303C2×C7#189111yescharacteristic
304D7—82no—
305C23—142no—
306C23(C22⋊D7)⋊C411yescharacteristic
307Q8—22no—
308Q8—22no—
309Q8—41no—
310Q8—41no—
311Q8—281no—
312Q8—281no—
313D4—282no—
314D4—282no—
315D4—282no—
316D4—281no—
317D4—281no—
318D4—561no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—142no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4D7×C22⋊C411yescharacteristic
330C2×C4#644611yescharacteristic
331C2×C4#644711yescharacteristic
332C2×C4D7×SD1611yescharacteristic
333C2×C4#647711yescharacteristic
334C2×C4#648111yescharacteristic
335C2×C4#648211yescharacteristic
336C2×C4C4×C22⋊D711yescharacteristic
337C2×C4#650011yescharacteristic
338C2×C4#650111yescharacteristic
339C8—42no—
340C8—81no—
341C8—142no—
342C8—282no—
343C8—281no—
344C8—561no—
345C7#904711yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—282no—
349C22—282no—
350C22—282no—
351C22#8956011yescharacteristic
352C22#8958911yescharacteristic
353C22#8972811yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—81no—
360C4—282no—
361C4#8959011yescharacteristic
362C4#8959111yescharacteristic
363C4#8994111yescharacteristic
364C4#8995911yescharacteristic
365C2—21no—
366C2—21no—
367C2—562no—
368C2#1172774611yescharacteristic
369C2#1172803011yescharacteristic
370C2#1172817011yescharacteristic
subgroupG/H
centerC2×C4#6500
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1487C2
socleC22×C7#386
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#1487D7
hypocenterC7#9047
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
3. C7×#1487C2semidirect
4. #11729389C2semidirect
6. #11725547C2semidirect
7. #11725520C2semidirect
76. #1487D7semidirect
125. C2×C7⋊C8SD16semidirect
345. C7#9047semidirect

Maximal subgroups

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

subgroupindex
#117398872
#117364872
C7×#14872
#117293892
#117285262
#117255472
#117255202
#90477