Finite Group Databasebeta

← #13256105

Subgroups — #13256105

Order 1792.

676 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C7×#2582C211yesmaximal, characteristic, complemented
2C2×#89799C212yesmaximal
3C2×Q8⋊(C7⋊C8)C212yesmaximal, complemented
4#11727334C211yesmaximal, characteristic
5#11725440C211yesmaximal, characteristic, complemented
6C2×C4×C7×Q8C2212yes—
7C22×#1143—22no—
8C2×#6437C2211yescharacteristic
9C7×#398—22no—
10C7×#391—22no—
11C7×#389—22no—
12C7×#384C2211yescharacteristic
13#89799—24no—
14#89799—24no—
15C2×#6383—22no—
16C2×#6383—22no—
17C2×C4⋊(C7⋊C8)C2212yes—
18C2×C4×C7⋊C8C2211yescharacteristic
19Q8⋊(C7⋊C8)—24no—
20#10230—71nomaximal, Sylow
21C22×C7×Q8D412yes—
22C22×D28—22no—
23C22×D28—21no—
24C4×C7×Q8—22no—
25C4×C7×Q8—22no—
26C4×C7×Q8—22no—
27C4×C7×Q8—22no—
28C2×C7×C4⋊C4—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4D412yes—
32C2×C7×C4⋊C4D412yes—
33C2×C42×C7—22no—
34C2×C42×C7C2311yescharacteristic
35C2×#1143—28no—
36C2×#1143—44no—
37C2×#1143—44no—
38C22×C7⋊C8—21no—
39C22×C7⋊C8—22no—
40#6437—22no—
41#6437—21no—
42#6437D412yes—
43C7×#95—24no—
44C7×#95—22no—
45#6383—28no—
46#6383—44no—
47C4⋊(C7⋊C8)—22no—
48C4⋊(C7⋊C8)D414yes—
49C4×C7⋊C8—22no—
50C2×#460—72no—
51#2582D711yescharacteristic, complemented
52#1957—71no—
53C2×Q8⋊C8—72no—
54#1296—71no—
55C23×D7—41no—
56C2×C7×Q8—22no—
57C2×C7×Q8—22no—
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8SD1614yescomplemented
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7C2×D412yes—
67C2×D28—24no—
68C2×D28—22no—
69C2×D28—42no—
70C2×D28—42no—
71C2×D28—42no—
72C2×D28—42no—
73C2×D28—42no—
74C2×D28—42no—
75C2×D28—42no—
76C2×D28—41no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—42no—
82C7×C4⋊C4—42no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—82no—
85C42×C7—42no—
86C42×C7—42no—
87C42×C7C2×D411yescharacteristic
88C42×C7C2×D412yes—
89C42×C7#4111yescharacteristic
90#1143—416no—
91C2×C7⋊C8—22no—
92C2×C7⋊C8—24no—
93C2×C7⋊C8—41no—
94C2×C7⋊C8—41no—
95C2×C7⋊C8—42no—
96C2×C7⋊C8—42no—
97C22×SD16—142no—
98C2×#127—71no—
99C2×C4×Q8C2×D712yes—
100#460—144no—
101#460—144no—
102C2×C4⋊C8—72no—
103C2×#102—142no—
104C2×#102—142no—
105C2×C4×C8—71no—
106#398—22no—
107#391—22no—
108#389—22no—
109#384C2×D711yescharacteristic
110Q8⋊C8—144no—
111C23×C7#12011yescharacteristic
112C22×D7—42no—
113C22×D7—82no—
114C22×D7—82no—
115C22×D7—81no—
116C22×D7—81no—
117C7×Q8—24no—
118C7×Q8—24no—
119C7×Q8—42no—
120C7×Q8—42no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—24no—
124C2×C4×C7—42no—
125C2×C4×C7—42no—
126C2×C4×C7—42no—
127C2×C4×C7—44no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7—42no—
131C2×C4×C7#12012yes—
132C2×C4×C7#12111yescharacteristic
133C2×C4×C7#12112yes—
134C2×C4×C7#12711yescharacteristic
135C2×C4×C7C2×SD1612yes—
136C2×C4×C7#13512yes—
137C2×C4×C7AGL(1,ℤ/8)12yes—
138C2×C4×C7#13712yes—
139D28—44no—
140D28—44no—
141D28—44no—
142D28—82no—
143D28—82no—
144D28—82no—
145C7⋊C8—42no—
146C7⋊C8—44no—
147C22×Q8C22⋊D712yes—
148C22×D4—142no—
149C22×D4—141no—
150C2×SD16—148no—
151C2×SD16—284no—
152C2×SD16—284no—
153C22×C8—142no—
154C22×C8—141no—
155#127—72no—
156#127—142no—
157#127—141no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊D712yes—
166C2×C4⋊C4C22⋊D712yes—
167C2×C42—22no—
168C2×C42C22×D711yescharacteristic
169C4⋊C8—74no—
170C4⋊C8—142no—
171#102—148no—
172#102—284no—
173C4×C8—142no—
174#95—24no—
175#95—22no—
176C22×C7#39211yescharacteristic
177C22×C7#44812yes—
178C22×C7#44912yes—
179C22×C7#45311yescharacteristic
180C22×C7#45411yescharacteristic
181C2×D7—84no—
182C2×D7—82no—
183C2×D7—161no—
184C2×D7—161no—
185C2×D7—162no—
186C4×C7—21no—
187C4×C7—21no—
188C4×C7—44no—
189C4×C7—82no—
190C4×C7—82no—
191C4×C7#45012yes—
192C4×C7#45212yes—
193C4×C7#46512yes—
194C4×C7#46812yes—
195C24—281no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8#114314yescomplemented
202C2×D4—144no—
203C2×D4—142no—
204C2×D4—282no—
205C2×D4—282no—
206C2×D4—282no—
207C2×D4—281no—
208C2×D4—282no—
209C2×D4—282no—
210C2×D4—282no—
211C2×D4—282no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4C2×C22⋊D711yescharacteristic
216C22×C4C2×C22⋊D712yes—
217SD16—2816no—
218C2×C8—142no—
219C2×C8—144no—
220C2×C8—281no—
221C2×C8—281no—
222C2×C8—282no—
223C2×C8—282no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C42—42no—
233C42—42no—
234C42D4×D711yescharacteristic
235C42D4×D712yes—
236C42#115911yescharacteristic
237C2×C7#162011yescharacteristic
238C2×C7#162311yescharacteristic
239C2×C7#162612yes—
240C2×C7#200012yes—
241C2×C7#200611yescharacteristic
242D7—162no—
243C23—282no—
244C23—562no—
245C23—562no—
246C23—561no—
247C23—561no—
248C23#651611yescharacteristic
249Q8—24no—
250Q8—24no—
251Q8—42no—
252Q8—42no—
253D4—284no—
254D4—284no—
255D4—284no—
256D4—562no—
257D4—562no—
258D4—562no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4#650012yes—
270C2×C4#650111yescharacteristic
271C2×C4#650212yes—
272C2×C4#650311yescharacteristic
273C2×C4C2×#114312yes—
274C2×C4#650512yes—
275C2×C4#651212yes—
276C2×C4#651312yes—
277C8—282no—
278C8—284no—
279C7#1023011yescharacteristic, complemented, Sylow
280C22—564no—
281C22—562no—
282C22—1122no—
283C22—1121no—
284C22—1121no—
285C22#8999511yescharacteristic
286C22#8999912yes—
287C22#9001012yes—
288C22#9001311yescharacteristic
289C22#9001711yescharacteristic
290C4—21no—
291C4—21no—
292C4—44no—
293C4—82no—
294C4—82no—
295C4#8981812yes—
296C4#8981912yes—
297C4#8982012yes—
298C4#8982112yes—
299C2—1122no—
300C2#1172945312yes—
301C2#1172963311yescharacteristic
302C2#1172968912yes—
303C2#1172969311yescharacteristic
304C2#1172969711yescharacteristic
subgroupG/H
centerC23#6516
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#2582C2
socleC23×C7#120
prime radicalC2×#6437C22
hypercenter#2582D7
hypocenterC7#10230
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2582

Complements

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

NH ≅ G/Ntype
1. C7×#2582C2semidirect
3. C2×Q8⋊(C7⋊C8)C2semidirect
5. #11725440C2semidirect
51. #2582D7semidirect
61. C2×C7×Q8SD16semidirect
201. C2×Q8#1143semidirect
279. C7#10230semidirect

Maximal subgroups

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

subgroupindex
C7×#25822
C2×#897992
C2×Q8⋊(C7⋊C8)2
#117273342
#117254402
#102307